A Three-Dimensional Model of Cellular Electrical Activity

Transcription

A Three-Dimensional Model of Cellular
Electrical Activity
by
Yoichiro Mori
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Mathematics
New York University
September 2006
Charles S. Peskin—Advisor
c Yoichiro Mori
All Rights Reserved, 2006
To Toko
iv
Acknowledgements
First of all, I would like to thank my advisor Charlie Peskin for guiding
me throughout my years as a graduate student. I have learned a tremendous
amount from him. Professor Joseph Jerome of Northwestern University has
taught me mathematical analysis, and has given me thoughtful suggestions
on numerous aspects of my thesis work. Professor Glenn Fishman noticed
the connection between my theoretical work and cardiac electrophysiology,
which led to an unexpected physiological application. Boyce Griffith and Sam
Isaacson have given me much advice on computing and numerical analysis.
I have benefited from discussions with many people at Courant, including
Professor Jonathan Goodman, who shared with me his insights on asymptotics, and Professor Dan Tranchina, with whom I discussed on numerous
occasions the possible relevance of my thesis work to neuroscience.
Without the encouragement of Professor Hiroshi Matano, I would not
have decided to switch careers from medicine to mathematical biology. Without the support of Toko, all of this would never have been possible.
v
Abstract
We present a three-dimensional model of cellular electrical activity. This
model takes into account the intricate three dimensional geometry of biological tissue as well as ionic concentration dynamics, both of which are ignored in
conventional models of electrophysiology. Biological tissue is viewed as three
dimensional space being partitioned into the intracellular and extracellular
spaces by the cellular membrane. The concentration of each ionic species
is governed by the drift-diffusion equation and the electrostatic potential is
determined implicitly through the electroneutrality constraint. Ion channel and capacitative currents at the cellular membrane are modeled through
boundary conditions to be satisfied at both faces of the cellular membrane.
This results in a system of nonlinear partial differential equations defined on
domains coupled through the membrane by nonlinear evolutionary boundary
conditions.
We present a detailed asymptotic derivation of the model equations. This
calculation reveals the presence of a hierarchy of mathematical models of
cellular electrical activity, the proposed three-dimensional model being the
most detailed, and traditional models being the simplest in the hierarchy.
This analysis also shows that there are two disparate time scales associated
with the system of equations, one associated with ionic diffusion, and the
other with the electrotonic spread of membrane potential.
vi
Under simplifying assumptions on the model equations, we obtain a global
solvability result using convex analysis and weak compactness arguments.
Our method of proof is constructive, allowing us to simultaneously prove convergence of a time-semidiscretized numerical scheme of the simplified system
of equations.
A numerical method based on a cartesian grid finite volume method is
developed. The presence of two disparate time scales necessitates an implicit
time discretization, and the resulting nonlinear algebraic equations are solved
using an iterative scheme.
The foregoing modeling methodology is applied to cardiac physiology. We
show that simulations using this model can be used to explore the characteristics of a recently observed anomalous mode of cardiac action potential
propagation: cardiac propagation without gap junctions.
vii
Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
1 Introduction
1
1.1 Traditional Cable Model . . . . . . . . . . . . . . . . . . . . .
1
1.2 Limitations of the Cable Model . . . . . . . . . . . . . . . . .
5
1.3 Previous Models . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3.1
The model of Qian and Sejnowski . . . . . . . . . . . .
7
1.3.2
Model of Leonetti et.al. . . . . . . . . . . . . . . . . . . 10
1.4 Outline of Dissertation . . . . . . . . . . . . . . . . . . . . . . 10
2 Model Formulation
14
2.1 Physical Derivation . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1
Drift-Diffusion and Electroneutrality . . . . . . . . . . 14
viii
2.1.2
Electrical Boundary Condition . . . . . . . . . . . . . . 19
2.1.3
Boundary Conditions for Each Ionic Species . . . . . . 21
2.1.4
Transmembrane Ionic Currents . . . . . . . . . . . . . 24
2.1.5
Summary of the Electroneutral Model
. . . . . . . . . 28
2.2 Asymptotic Derivation and Model Hierarchy . . . . . . . . . . 29
2.2.1
Poisson Model . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.2
Non-Dimensionalization . . . . . . . . . . . . . . . . . 33
2.2.3
Multiple Spatiotemporal Scales . . . . . . . . . . . . . 38
√
Inner-Intermediate Matching when lm > β . . . . . . 47
√
Inner-Intermediate Matching when lm ∼ β . . . . . . 64
2.2.4
2.2.5
2.2.6
Electroneutral Model . . . . . . . . . . . . . . . . . . . 74
2.2.7
Equations in the Outer Layer . . . . . . . . . . . . . . 89
2.2.8
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 102
3 Analysis
105
3.1 Simplification and Model Problem . . . . . . . . . . . . . . . . 106
3.2 Weak Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.3 Discretization in Time . . . . . . . . . . . . . . . . . . . . . . 115
3.4 Stability Estimates and Existence . . . . . . . . . . . . . . . . 125
3.5 Uniqueness
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.6 Dirichlet-to-Neumann Map . . . . . . . . . . . . . . . . . . . . 142
4 Numerical Methods
147
4.1 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . 149
ix
4.1.1
Finite Volume Method . . . . . . . . . . . . . . . . . . 149
4.1.2
Embedded Boundary Method . . . . . . . . . . . . . . 153
4.1.3
Cylindrical Geometry and Non-Uniform Meshes . . . . 161
4.2 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . 164
4.3 Solution of Nonlinear Equations . . . . . . . . . . . . . . . . . 166
4.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.4.1
Computational Geometry . . . . . . . . . . . . . . . . 172
4.4.2
Linear Algebra . . . . . . . . . . . . . . . . . . . . . . 173
4.4.3
3D-Cable Model . . . . . . . . . . . . . . . . . . . . . . 173
4.5 Convergence Study: Cylindrical Geometry . . . . . . . . . . . 174
4.5.1
Hodgkin-Huxley Axon . . . . . . . . . . . . . . . . . . 174
4.5.2
Cardiac Geometry . . . . . . . . . . . . . . . . . . . . 185
4.6 General 2D Geometry . . . . . . . . . . . . . . . . . . . . . . 196
4.6.1
Convergence in Space . . . . . . . . . . . . . . . . . . . 203
4.6.2
Convergence in Time . . . . . . . . . . . . . . . . . . . 206
4.7 Numerical Validation of Asymptotics . . . . . . . . . . . . . . 206
4.7.1
Numerical Method . . . . . . . . . . . . . . . . . . . . 210
4.7.2
Comparison . . . . . . . . . . . . . . . . . . . . . . . . 217
5 Application
231
5.1 Physiological Background . . . . . . . . . . . . . . . . . . . . 231
5.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
5.3 Normal Conduction . . . . . . . . . . . . . . . . . . . . . . . . 243
x
5.4 Conduction with Reduced Gap Junctions . . . . . . . . . . . . 248
5.4.1
Phenomenology and Model Comparison . . . . . . . . . 248
5.4.2
Model Comparison . . . . . . . . . . . . . . . . . . . . 255
5.4.3
Varying rNa . . . . . . . . . . . . . . . . . . . . . . . . 258
5.4.4
Varying g gap . . . . . . . . . . . . . . . . . . . . . . . . 265
5.4.5
Varying η . . . . . . . . . . . . . . . . . . . . . . . . . 267
5.5 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 269
Bibliography
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
xi
List of Figures
2.1 A schematic of the relevant spatial scales used in the asymptotic calculations. The solid lines denote the membrane and
the dotted lines are the interfaces between the boundary layers. The inner-most layer has width β, the intermediate layer
√
β. The typical membrane separation is α and the typical
length scale associated with the membrane is lm . . . . . . . . . 43
4.1 Plot of electrostatic potential φ at t = 1, 2, 3, 4msec. Since
the solutions we seek are radially symmetric, the radial crosssection (r > 0) is plotted in the graph. . . . . . . . . . . . . . 177
4.2 Plot of change in ionic concentrations from t = 0 at t = 2msec. 178
4.3 L2 error in space when the axon radius is 1µm. The error is
measured at t = 2ms and 4ms. The reference line indicates
second order convergence. . . . . . . . . . . . . . . . . . . . . 181
4.4 L2 error in time when the axon radius is 1µm. The error is
measured at t = 2ms and 4ms. The reference line indicates
first order convergence.
. . . . . . . . . . . . . . . . . . . . . 184
xii
4.5 The evolution of the electrostatic potential in the cardiac simulation with variable mesh width. Snapshots shown at t =
0.6, 1.2, 1.8, 2.4msec. The mesh size is Nz = 48, Nr = 32 in
this computation.
. . . . . . . . . . . . . . . . . . . . . . . . 189
4.6 The change in ionic concentrations from the initial value, in
the cardiac simulation with variable mesh width. Snapshot at
t = 2msec shown. The mesh size is Nz = 48, Nr = 32 in this
simulation in this computation. . . . . . . . . . . . . . . . . . 190
4.7 L2 error in time. The error is measured at t = 2ms and 4ms.
The reference line indicates second order convergence.
. . . . 194
4.8 L2 error in time. The error is measured at t = 2ms and 4ms.
The reference line indicates first order convergence. . . . . . . 195
4.9 The evolution of the electrostatic potential under Neumann
boundary conditions at the outer boundary of the computational domain. Snapshots shown at t = 0.2, 0.6, 1.0, 1.4msec.
The mesh size is 64 × 64 in this computation. . . . . . . . . . 199
4.10 The change in ionic concentrations from the initial value, computed under Neumann boundary conditions at the outer boundary of the computational domain. Snapshot at t = 1msec
shown. The mesh size is 64 × 64 in this computation.
xiii
. . . . 200
4.11 The evolution of the electrostatic potential under Dirichlet
boundary conditions at the outer boundary of the computational domain. Snapshots shown at t = 0.2, 0.6, 1.0, 1.4msec.
The mesh size is 64 × 64 in this computation. . . . . . . . . . 201
4.12 The change in ionic concentrations from the initial value, computed under Dirichlet boundary conditions at the outer boundary of the computational domain. Snapshot at t = 1msec
shown. The mesh size is 64 × 64 in this computation.
. . . . 202
4.13 L2 error in space, when no-flux boundary conditions are used.
The error is measured at t = 1ms and 2ms. The reference line
indicates second order convergence.
. . . . . . . . . . . . . . 204
4.14 L2 error in space, when Dirichlet boundary conditions are
used. The error is measured at t = 1ms and 2ms. The reference line indicates second order convergence.
. . . . . . . . . 205
4.15 L2 error in time, when no-flux boundary conditions are used.
indicates first order convergence.
. . . . . . . . . . . . . . . . 207
4.16 L2 error in time, when Dirichlet boundary conditions are used.
indicates first order convergence.
xiv
. . . . . . . . . . . . . . . . 208
4.17 Snapshots of simulation when (β, α) = (10−3.5 , 10−1.5 ). Three
curves, the Poisson computation, the raw data and modified
data from the electroneutral models are plotted. The three
curves are virtually indistinguishable.
. . . . . . . . . . . . . 222
4.18 Plot of E2 (Ci ) and E2 (Φ) as a function of time when (β, α) =
(10−3.5 , 10−1.5 ). The abscissa measures dimensionless time τ ,
and the plot ends at τ = Te .
. . . . . . . . . . . . . . . . . . 228
5.1 Relevant voltages in considering ephaptic transmission . . . . 233
5.2 Schematic diagram of the setting of the computational experiments. Each rectangular box represents a cylindrical cardiac
cell.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
5.3 Conduction speed of cardiac action potential when the gap
junction conductance is set to the normal physiological value.
The top graph corresponds to Cin = 0, and the bottom graph
to Cin = 50. The lines on both graphs are conduction velocity
plots for rNa = uniform, 0.5, 0.8, 0.99 from top.
xv
. . . . . . . . 246
5.4 A plot of the electrostatic potential φ. Since we are seeking
radially symmetric solutions, we only plot the radial crosssection, 0 < r < (1 + η)l. Note that the aspect ratio of
the graph is much distorted; the radial direction is stretched
with respect to the axial direction. The four graphs plot φ
at t = 4, 8, 12, 16msec. Note the steep gradient in the electrostatic potential that develops in the gap as the action potential
propagates from cell k to cell k + 1.
. . . . . . . . . . . . . . 251
5.5 The change in ionic concentrations. Note the decrease in Na+
and increase in K+ in the gaps. . . . . . . . . . . . . . . . . . 252
5.6 The sequence of electrostatic potential changes as the action
potential propagates from cell to cell. The above computation
was performed with 3 cells.
. . . . . . . . . . . . . . . . . . . 253
5.7 Snapshots of the electrostatic potential with(below) or without(above) fixed charges in the gap. Note that when fixed
charge is present in the gap, the electrostatic potential is
slightly negative within the gaps. At t = 10msec, the action
potential is approximately one cell ahead with fixed charges
in the gap than without.
. . . . . . . . . . . . . . . . . . . . 254
xvi
5.8 Conduction speed of cardiac action potential computed with
different electrophysiology models. The computation in the
top graph is when Cim = 0 and the lower graph when Cim = 50.
In 1D models, only two meshes, one for r > l and one for r < l
are used. In the φ only models, the ionic concentrations are
assumed not to change in time. . . . . . . . . . . . . . . . . . 256
5.9 Conduction speed of cardiac action potential under reduced
gap junction coupling. The top graph corresponds to Cin = 0,
and the bottom graph to Cin = 50. The lines on both graphs
are conduction velocity plots for rNa = uniform, 0.5, 0.8, 0.9, 0.95, 0.99
from bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
5.10 A plot of the time series of action potential arrival times for
cells 1 through to 15. The bottom trace corresponds to lg =
5nm, the middle to lg = 12nm and the top trace to lg =30nm.
263
5.11 A plot of the conduction velocity for different values of gap
junction coupling, g gap . The graph on top is when Cim = 0
and the one on the bottom is when Cim = 50. For each graph,
the trace corresponds, from the top, to g gap /g0 = 8, 4, 2, 1, 0. . 266
5.12 A plot of the conduction velocity for different values of extracellular space size η. The graph on top is when Cim = 0
and the one on the bottom is when Cim = 50. For each
graph, the trace corresponds, from the top at lg = 5nm,
η = 1, 0.3, 0.1, 0.03, 0.01. . . . . . . . . . . . . . . . . . . . . . 268
xvii
List of Tables
4.1 Parameter values used in the Hodgkin-Huxley simulations of
the axon.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.2 The empirical rates of convergence rps in space for different
values of axonal radii. Values computed at t = 4ms, and
Nr = 16.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.3 The empirical rates of convergence rpt for different values of
axonal radii. Values computed at t = 4ms, and Nr = 8.
4.4 Parameter values used in cardiac simulation.
. . . 183
. . . . . . . . . 187
4.5 The empirical rates of convergence rps for different values of
axonal radii. Values computed at t = 4ms, and Ns = 32. . . . 193
4.6 The empirical rates of convergence rpt . Values computed at
t = 4ms, and Nr = 32. . . . . . . . . . . . . . . . . . . . . . . 196
4.7 Parameter values used in the simulation for general 2D geometries.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
4.8 The empirical rates of convergence rps for both boundary conditions. Values computed at t = 2ms, and Nx = 128. . . . . . 204
xviii
4.9 The empirical rates of convergence rpt for both boundary conditions Values computed at t = 2ms, and NT = 200.
. . . . . 206
4.10 Mp values for spherical geometry for three computational experiments with different values of β and α. Here, cases 1,
2, and 3 correspond respectively to (β, α) values of (β1 , α1 ) =
(10−3 , 10−2 ), (β2 , α2 ) = (10−3.5, 10−1.5 ), and (β3 , α3 ) = (10−4 , 10−1 ).
225
4.11 Mp values for planar geometry for the three computational
experiments with different values of β and α. Here, cases 1,
2, and 3 correspond respectively to (β, α) values of (β1 , α1 ) =
(10−3 , 10−2 ), (β2 , α2 ) = (10−3.5, 10−1.5 ), and (β3 , α3 ) = (10−4 , 10−1 ).
230
5.1 Parameter values used in cardiac simulation.
xix
. . . . . . . . . 237
Chapter 1
Introduction
1.1
Traditional Cable Model
Electrophysiology, because of its importance in many physiological processes
and its quantitative nature, has been a favorite subject in mathematical
physiology. Traditional models of cellular electrical activity are based on the
famous work of Hodgkin and Huxley [13], and may be collectively termed
cable models. These models are based upon an ohmic current continuity
relation on a branched one dimensional electrical cable. We shall here give a
brief derivation of the cable model. For a more complete exposition, we refer
the reader to textbooks such as [23] and [22] or the lecture notes [31]. For a
detailed biophysical description, we refer the reader to [1, 12]
Consider a one dimensional cylinder of radius a and take the z axis to be
the axis of the cylinder. Take the radial coordinate to be r. At r = a, we
1
have the cell membrane, which is composed of a lipid bilayer. All biological cells maintain an electrostatic potential difference across the membrane,
termed the membrane potential. The lipid bilayer acts primarily as a capacitor maintaining this difference in the electrostatic potential. Within the lipid
bilayer are embedded ion channels pumps and transporters that carry current through the membrane. The intracellular milieu (r < a) is filled with an
electrolyte solution, the ions of which act as carriers of the electric current.
With this biophysical picture in mind, we shall derive the cable model, a
system of equations that determines the evolution of the membrane potential.
We shall first make several approximations. We assume that the problem
is effectively one-dimensional so that we can neglect radial electric currents
and radial variations in the electric current or the electrostatic potential. We
also assume that the extracellular space is grounded, i.e., the voltage there is
uniformly equal to 0. We assume further that the electrolyte solution acts as
an ohmic conductor. Under these assumptions, we can apply the Kirchoff’s
law of current conservation to obtain an equation satisfied by the membrane
potential φm
Assume cylindrical symmetry of the system. Let the axial current density
to be i(z, t) where t is time. As assumed, i does not have radial dependence,
and is thus a function of z and t only. Let I(z, t) denote the current density
that goes into membrane at z = z, r = a. By cylindrical symmetry, I is
assumed not to depend on the angular variable.
2
We can immediately write the following current conservation relation:
πa2
∂i
+ 2πaI = 0
∂z
(1.1)
We need to supplement this current continuity equation with some empirical
laws that relate i or I with φm . Let us start with i. As stated above,
we assume that, to a good approximation, the intracellular medium acts as
an ohmic conductor. Let R denote the intracellular cytoplasmic resistivity.
Then, i can be related to φint , the intracellular electrostatic potential by:
i=−
1 ∂φint
R ∂z
(1.2)
Let us now turn to I, the membrane current. We view the membrane
as being a capacitor and a transmembrane current carrier in parallel. We
therefore have:
I = Cm
∂φm
+ g(φm , · · · )
∂t
(1.3)
In the above, Cm is the capacitance per unit membrane area, and g(φm , · · · )
denotes the transmembrane current, carried by ionic channels, transporters
and pumps. The transmembane current g is not only a function of the
membrane potential φm but also a function of other variables, most notably
the ionic concentrations and the gating variables, which characterize the state
of the ionic channel.
3
Substituting (1.2) and (1.3) into (1.1) results in the following:
Cm
a ∂ 2 φint
∂φm
+ g(φm , · · · ) =
∂t
2R ∂z 2
(1.4)
Recall that we assumed that the extracellular potential φext was uniformly
equal to 0. Since φm = φint − φext , we may replace φint with φm in the above.
Cm
∂φm
a ∂ 2 φm
+ g(φm , · · · ) =
∂t
2R ∂z 2
(1.5)
The above is not a closed equation until we can specify the evolution of
the transmembrane current g. The form of this function is model and system dependent. For example, in the work of Hodgkin and Huxley [13], the
transmembrane currents had the following form:
g = gNa (φm − ENa ) + gK (φm − EK ) + gL (φm − EL )
C ext
kB T
log Na
int
q
CNa
C ext
kB T
log Kint
EK =
q
CK
(1.6)
(1.7)
ENa =
(1.8)
gNa = GNa m3 h
(1.9)
gK = GK n4
(1.10)
∂s
= αs (φm )(1 − s) − βs (φm )s,
∂t
s = m, n, h
(1.11)
The first line states that the transmembrane current is a sum of three components, the Na+ current, the K+ current and the leak current, the charge
4
carrier of which is not specified. The three currents all satisfy an ohmic relation, but with an offset given by ENa , EK , EL respectively. These offsets
are given by the equilibrium potentials, given in equations (1.7) and (1.8).
Here, kB is the Boltzmann constant, T is the absolute temperature, q is the
elementary charge, C ext are the extracellular concentrations and C int are the
intracellular concentrations.
It is often assumed that the transmembrane currents are small and that
the ionic concentrations can be approximated as constant in time. Under
this approximation, ENa and EK are given constants. The voltage EL is an
experimentally determined number.
The conductance for the Na+ current and the K+ current are expressed as
the product of the maximal channel conductance GNa and GK respectively,
and of the gating variables m, n, h. The gating variables vary between 0 and 1
and themselves obey an ordinary differential equation, (1.11). The functions
αs and βs are experimentally determined.
1.2
Limitations of the Cable Model
We see that there are several important assumptions that go into the derivation of cable models [23]:
• A one dimensional picture, or more generally, a one dimensional tree
representation of cell geometry is adequate. Geometrical details that
are lost in making this simplified description have negligible effect on
5
electrophysiology.
• The extracellular space can be reduced to a single isopotential electrical
compartment.
• Ionic concentrations are effectively constant in space and time within
each cell separately and in the extracellular space. The diffusive current that may be induced by concentration gradients or the changes in
equilibrium potential are negligible.
Such assumptions are justified in many instances, for example in the isolated
neuronal axon, where the cable model has been extremely successful in explaining the physiology and in making quantitative predictions – a triumph
counted among the greatest successes of mathematics in biology. But there
may be many cases in which any or all of the above assumptions are violated. Neuroscience textbooks [20] show pictures of cells of complex shape
packed together embedded in a tortuous extracellular space. Such pictures
are indicative of the important functional role geometry, ionic concentration
profiles and extra-neuronal space (extracellular space and glia) may play in
the workings of the nervous system.
There have been at least two attempts to tackle the limitations listed
above. The first is the work of Qian and Sejnowski [32], and the other the
work of Leonetti et.al. [26]. We shall first describe the work of Qian and
Sejnowski.
6
1.3
1.3.1
Previous Models
The model of Qian and Sejnowski
Recall the setting of the cable model described above. We now consider
i = 1, · · · , N ionic species as being the current carriers of the electrolyte
medium. These ionic species satisfy the drift-diffusion equation:
∂ci ∂fi
+ 2πaIi
+
0 = qzi
∂t
∂z
∂ci qzi ci ∂φm
fi = −Di
+
∂z
kB T ∂z
(1.12)
(1.13)
where Ii is the transmembrane current carried by the i-th species of ion, and
fi is the ionic flux in the axial direction, which is composed of a diffusive flux
and a drift flux driven by the electrostatic potential gradient. The constant
Di is the diffusion constant for each species of ion and q the elementary
charge, kB the Boltzmann constant and T the absolute temperature. The
constant zi is the valence of each ionic species, to be distinguished from z,
the coordinate direction. Note that we are still assuming a one dimensional
picture.
We need a relation that determines the membrane potential φm . For this,
Qian and Sejnowski propose the following relation:
2
πa (ρ +
N
X
qzi ci ) = 2πaCm φm
i=1
7
(1.14)
where ρ is a fixed immobile charge density (Qian and Sejnowski do not write
their equations in this form in [32], but what they do write there is mathematically equivalent to [32].).
The relationship between this and the cable model can be seen by taking
the time derivative of equation (1.14) and substituting (1.12) whenever
∂ci
∂t
appears.
∂φm
= πa2
2πaCm
∂t
= πa2
N
X
i=1
∂
∂z
∂ci
qzi
∂t
N
X
!
qzi Di
i=1
∂ci qzi ci ∂φm
+
∂z
kB T ∂z
!
− 2πa
N
X
(1.15)
Ii
i=1
Simplification yields:
N
∂φm X
a ∂
Cm
+
Ii =
∂t
2 ∂z
i=1
N
X
i=1
∂ci
qzi Di
∂z
!
a ∂
+
2 ∂z
N
X
(qzi )2 ci
i=1
kB T
Di
!
∂φm
∂z
(1.16)
Therefore, by neglecting that diffusive flux and identifying the sum
P
N (qzi )2 ci
D
as the cytoplasmic resistvity, we recover the cable model.
i
i=1 kB T
Qian and Sejnowski go on to apply their model to the study of the electrical
properties of dendritic spines [33], subcellular structures found on the surface
of dendrites of neurons in the central nervous system on which most excitable
synapses are found [23].
Their formalism seems reasonable in the one dimensional setting that
they considered. It is also attractive in the sense that the connection with
the cable model is quite clear. However, their model only addresses one
8
of the limitations of the cable model listed above, that of neglecting ionic
concentration dynamics.
We would like to generalize their model so that we can include the effects
of the extracellular space as well as of complicated 3D geometry. We note
that it is not immediately clear how one can generalize the model of Qian
and Sejnowski to a three dimensional setting where the presence of the extracellular space is also taken into account. Their model derivation depends
on a convenient feature only available in a one dimensional setting: to each
coordinate location z, there is a corresponding patch of cell membrane as well
as a slab of intracellular space. This allows them to use the simple algebraic
relation (1.14) to determine the membrane potential. In a three dimensional
setting, this is not possible.
One further point we would like to address is computational efficiency.
The model of Qian and Sejnowski, though accepted as being biophysically
more realistic than the cable model [23], seems not to have been widely used
as an alternative of the cable model, probably because of the perception that
simulations with this model are highly computationally intensive. The reason
for the large computational cost that they document in their paper [32] is
most probably due to their explicit time discretization.
One of our goals is to develop a numerical method that is efficient enough
to allow simulations of 3D electrophysiology with modest computational resources. Though the equations we propose are different from the model
of Qian and Sejnowski, an explicit time discretization likewise leads to a
9
formidable computational burden. We shall employ an implicit discretization in time to remove this difficulty, as we shall see in Chapter 4.
1.3.2
Model of Leonetti et.al.
Another study that tackles the above issues is the model by Leonetti et.
al. [26]. They use the Poisson equation and the drift diffusion equations to
describe cellular electrophysiology. Their model is what we call the Poisson
model in Section 2.2. They perform theoretical calculations exploring the
connection between the Poisson model and the cable model, and obtain the
cable model as a long wave approximation to the Poisson model [26]. Their
focus is on exploring diffusive instabilities to the steady state of the Poisson
model. They show that such an instability can lead to bifurcation to a nontrivial steady state, and claim in [27] that such an instability forms one of
the biophysical bases of pattern formation in green algae.
Our primary focus is on performing dynamic simulations of cellular electrical activity, and for this purpose the Poisson model has its difficulties,
among which is the need to resolve an extremely thin boundary layer known
as the Debye layer, along with its associated fast time scale. This issue will
be discussed in Chapter 2.
1.4
Outline of Dissertation
The outline of the dissertation is as follows.
10
In this dissertation, we propose a mathematical model which we call the
electroneutral model that addresses the three limitations of the cable model
listed above. In Chapter 2, we give a detailed description of the model as
well as a systematic derivation of this model using asymptotic analysis starting with the above-mentioned Poisson model of Leonetti et.al. Following this
derivation, we shall see how to reduce systematically either the Poisson model
or the electroneutral model to the traditional cable model. In the course of
this derivation, we shall see that there are electrophysiology models of intermediate complexity which lie in between the Poisson or electroneutral models
and the traditional cable model. We shall derive conditions under which the
cable model may be deemed a reasonable model of electrophysiology. We
believe that this is one of the most careful derivations of the cable model to
date. We shall see in this chapter that the model system possesses two time
scales whose magnitudes are drastically different. One of these time scales
corresponds to ionic diffusion, and the other to the “diffusion” (electrotonic
spread) of the membrane potential. The latter time scale is about four orders of magnitude smaller (i.e., faster) than the former. This discrepancy
has important consequences in developing a numerical scheme.
In Chapter 3, we shall study the behavior of the electroneutral model in
a simplified setting. The full model equations track the evolution of both the
ionic concentrations and the electrostatic potential. This system is difficult
to analyze mathematically. In order to make the system more amenable
to analysis,we shall make (in Chapter 3 only) the simplification that the
11
ionic concentrations are given functions of time. Under this assumption, the
system reduces to a system of elliptic partial differential equations satisfied
by the electrostatic potential, whose evolution is governed by an nonlinear
evolutionary interface condition at the membrane boundaries. We shall prove
an existence-uniqueness result for this system. An important characteristic of
the method of proof is that it is constructive, and gives a proof of convergence
of a temporally semi-discretized numerical scheme. We shall end the chapter
with a brief discussion of the Dirichlet-to-Neumann map and its relation to
our model.
In Chapter 4 we shall develop a numerical scheme that efficiently simulates the electroneutral system. We shall use a finite volume scheme to
discretize the system. As discussed above, the model possesses two disparate
time scales. The spread of the membrane potential is very fast, and thus,
we must use an implicit scheme in time for stability and computational efficiency. We have developed code that implements this algorithm in two
settings: cylindrical geometry and general 2D geometries. We shall perform
convergence studies for both implementations. In the same chapter, we shall
perform a computational comparison of the electroneutral model and the
Poisson model, and show that the electroneutral model approximates the
Poisson model remarkably well.
In the final chapter, we apply our methodology to cardiac physiology.
Recent experimental evidence suggests that genetically engineered mice with
drastically reduced numbers of gap junctions can nonetheless support cardiac
12
action potential propagation [24, 10]. This seems to be due to a high density
of Na+ channels preferentially expressed in the intercalated disc membrane
(membrane facing the narrow space between the ends of two cardiomyocytes).
We perform simulations to show that such propagation is possible for physiologically realistic parameter values. We study the characteristics of this mode
of propagation by seeing how the conduction velocity changes as a function
of several parameters such as the gap junction coupling strength and the gap
width.
13
Chapter 2
Model Formulation
2.1
Physical Derivation
We shall first present a physically motivated derivation of our model equations, and discuss their biophysical significance. This section is largely identical to the presentation given in [30]. There will be certain terms in the
boundary conditions whose determination will be relegated to a more formal
argument using asymptotics in section 2.2.
2.1.1
Drift-Diffusion and Electroneutrality
We begin with equations that hold away from the cell membranes, i.e., in
the cell interior and in the interior of the extracellular space. We assume
that the unknowns and the parameters change continuously within each of
these regions, but we allow for jumps in these quantities as cell membranes
14
are crossed.
Let there be N ion species, where the subscript i denotes each species.
Let ci (x, t) and φ(x, t) denote the ion concentrations and the electrostatic
potential respectively. These quantities are related by the drift-diffusion
equations:
∂ci
= −∇ · fi
∂t
fi = −Di ∇ci +
qzi ci
∇φ
kB T
(ion conservation)
(2.1)
(drift-diffusion flux)
(2.2)
Here, fi denotes the flux of the i-th ion. fi is expressed as a sum of two
terms, the diffusion term and the drift term. Di is the diffusion coefficient
of the i-th ion, qzi is the amount of charge on the i-th ion, where q is the
elementary charge, i.e., the charge on a proton. qDi /(kB T ) is the mobility of
the ion species (Einstein relation) where kB is the Boltzmann constant, and
T the absolute temperature.
Moreover, the electrostatic potential satisfies the Poisson equation:
∆φ = −
1
ǫ
ρ0 +
N
X
i=1
qzi ci
!
(Poisson equation)
(2.3)
where ρ0 is the fixed background charge density (if any), and ǫ is the dielectric constant of the electrolyte solution. Note that we are applying these
equations only in the electrolyte solution, not within the membrane itself.
We shall discuss the treatment of the membrane in the following section. It
15
is interesting to note that these equations are also used in semiconductor
physics, where they are known as the van Roosbroeck equations [36].
Let L0 be a typical length scale, and let c0 be a typical concentration.
We refer to Section 2.2 for a precise prescription of the typical length scale
L0 . By writing the equations in non-dimensional variables, we are led to the
following equation (see Section 2.2 for details):
2
β ∆X Φ = ρ̃ +
N
X
zi Ci
(2.4)
i=1
Here, β = Lrd0 is a non-dimensional parameter in which rd is the Debye length
q
ǫkB T
, x = L0 X and ci = c0 Ci , φ = kBq T Φ. When this parameter
rd ≡
q 2 c0
β is small, we have the limit of electroneutrality, which holds away from
the cell membranes. This is indeed the case in physiological systems, where
rd ≈ 1nm and L0 is typically on the order of microns or more. This limit can
be obtained formally by letting ǫ → 0 in the Poisson equation. This implies
ρ0 +
N
X
qzi ci = 0.
(2.5)
i=1
It should be noted that this does not imply ∆φ = 0, since (2.3) reduces to
∆φ = 0/0 in the limit considered, and thus we have to look to the other equations to find the limiting behavior of the electrostatic potential. Therefore,
the equations to be satisfied away from the membrane are the drift-diffusion
equations (2.1), (2.2) and the electroneutrality condition (2.5).
16
We note that in equation 2.4, the small parameter is multiplying the
highest order derivative, and by formally letting β → 0 we lose one differential
equation. Depending on what boundary conditions we have for the Poisson
equation the smallness of β implies the presence of a boundary layer at the
membrane whose thickness is on the order of the Debye length.
The presence of this boundary layer is one of the main reasons why we
are interested in this electroneutral limit. Our eventual goal is to perform
simulations of electrophysiological systems. It would be computationally
advantageous not to have to resolve these boundary layers. We would like
to develop an effective theory that pertains only to the electroneutral region
in which the effects of the boundary layers have been taken into account. A
systematic derivation of such a model will be relegated to the next section
where this question will be explored in depth. In this section, we shall carry
our physical intuition as far as we can to arrive at this effective theory.
By replacing the Poisson equation (2.3) with the electroneutrality relation
(2.5), we have lost a relation that is satisfied by φ. We may obtain this
relation by considering current continuity. Let j(x, t) be the total electrical
current density (current per unit area) at the position x at time t. Multiply
the flux densities fi by qzi to obtain the current density for each ion, and add
17
to obtain the total current density.
j(x, t) = −(a(x, t)∇φ + ∇b(x, t)
a(x, t) =
b(x, t) =
N
X
(qzi )2 Di
i=1
N
X
kB T
ci (x, t)
qzi Di ci (x, t)
(2.6)
(2.7)
(2.8)
i=1
By charge conservation and electroneutrality, we obtain the equation of current continuity: ∇ · j = 0, and we have the following elliptic equation for
φ.
∇ · (a(x)∇φ + ∇b(x, t)) = 0
(2.9)
Thus, φ satisfies an elliptic constraint (since a(x, t) > 0) instead of the Poisson equation in the limit β = rd /L0 → 0.
We point out that (2.9) may also be derived by differentiating the electroneutrality condition (2.5) with respect to t and using (2.1) and (2.2). We
may understand (2.9) as imposing an elliptic constraint on φ so that electroneutrality is maintained. This procedure of obtaining an equation for
φ from an algebraic relation is employed in index reduction for differential
algebraic systems [34].
18
2.1.2
Electrical Boundary Condition
We shall treat the transmembrane currents and their associated quantities
(as defined in Section 2.1.4) as continuous functions of the spatial coordinates within the membrane. In our model, we are not resolving the individual
membrane protein molecules such as ion channels that give rise to transmembrane currents. All relations that refer to membrane variables should thus
be understood as the result of homogenization with respect to the membrane
coordinates.
Near the membrane, there is an accumulation of electric charge, whose
thickness is on the order of the Debye length rd . The presence of this layer
can also be inferred solely on the basis of the mathematical structure of the
limiting process as β → 0, as we have noted in the previous subsection. In
agreement with this limit, we consider this electric charge to form a layer of
zero thickness concentrated on both sides of the membrane surface.
Let us consider a point x on the membrane. The membrane separates
two regions of space which we call Ω(k) and Ω(l) . We label quantities in Ω(k)
and Ω(l) by the superscripts
k
and l respectively.
Electric current j that hits the membrane will contribute to the change
in surface charge σ or will pass through the membrane as transmembrane
current j. This statement of charge conservation on the Ω(k) face of the
membrane can be expressed as:
∂σ (k)
(x, t) + j (kl) (x, t) = j(k) (x, t) · n(kl) (x)
∂t
19
(2.10)
Here, σ (k) is the surface charge per unit area on face k of the membrane, j (kl) is
the transmembrane current per unit area from Ω(k) to Ω(l) , and j (lk) = −j (kl) .
The functional form of this quantity will be discussed in Section 2.1.4. n(kl)
is the unit normal pointing from Ω(k) to Ω(l) , and thus n(lk) = −n(kl) . The
same relation holds with k and l interchanged on the side of the membrane
facing Ω(l) :
∂σ (l)
(x, t) + j (lk) (x, t) = j(l) (x, t) · n(lk) (x)
∂t
(2.11)
We make two assumptions about the surface charge σ.
• Each patch of the membrane is electroneutral, i.e., σ (k) (x, t)+σ (l) (x, t) =
0. This means that any charge accumulation on one side of the membrane is instantaneously counterbalanced by charge on the other side.
• The membrane and its surface charge layers together behave like a
capacitor. That is to say, the surface charge is proportional to the
transmembrane potential difference.
We shall see that the above assumptions can be derived as a consequence of
the asymptotic calculations to be performed in Section 2.2. We may combine
the above assumptions to deduce:
σ (k) (x, t) = −σ (l) (x, t) = Cm φ(kl) (x, t),
φ(kl) (x, t) ≡ φ(k) (x, t) − φ(l) (x, t)
where Cm is the capacitance per unit area of the membrane. Thus we arrive
20
at the electrical boundary conditions
Cm
∂φ(kl)
(x, t) + j (kl) (x, t) = j(k) (x, t) · n(kl) (x).
∂t
(2.12)
The same boundary condition but with k and l interchanged holds on the
Ω(l) face of the membrane.
We may add (2.10) and (2.11) and use the first assumption to obtain a
relation satisfied by j.
j(k) (x, t) · n(kl) (x) + j(l) (x, t) · n(lk) (x) = 0
This says that any current that comes onto the membrane from one side is
exactly counterbalanced by current coming off the membrane from the other
side, i.e., that there is no overall charge accumulation at the membrane (the
charge accumulation on one face of the membrane is cancelled by the charge
accumulation on the opposite face).
2.1.3
Boundary Conditions for Each Ionic Species
Let us consider the flux of the i-th ion fi at the membrane. The i-th ion con(kl)
tributes a current per unit area of qzi fi · n. Let ji
(x, t) be the contribution
of the i-th species of ion to the transmembrane current per unit area flowing
from Ω(k) into Ω(l) . By considering ion conservation at the membrane we
21
find:
(k)
(kl)
pi (x, t) + ji
(k)
(x, t) = qzi fi (x, t) · n(kl) (x)
(2.13)
(k)
for some flux value pi . In order for the above to be compatible with (2.12),
we must require:
N
X
(k)
pi (x, t) =
i=1
N
X
(kl)
ji
∂φ(kl)
∂σ (k)
(x, t) = Cm
(x, t)
∂t
∂t
(x, t) = j (kl) (x, t)
i=1
(k)
The term pi
should thus be interpreted as the contribution of the i-th
species of ion to the capacitative current.
(k)
To make (2.13) useful, we need an expression for pi
in terms of the ionic
(k)
concentrations and the electrostatic potential. We set pi
to be:
(k)
(k)
pi (x, t) =
∂σi
∂t
(2.14)
where,
(k)
(k)
σi (x, t) = λi (x, t)σ (k)
(k)
(k)
(2.15)
(k)
λ̃ − λi
∂λi
,
= i 2
∂t
rd /D0
(k)
(k)
λ̃i
zi2 ci (x, t)
= PN
2 (k)
i′ =1 zi′ ci′ (x, t)
(2.16)
where D0 is some typical ionic diffusion coefficient and rd is the aforementioned Debye length. There is an arbitrariness to what we should use for
22
the value of D0 . As we shall see in Section (2.2.6), there is a wide range of
admissible values for D0 , and whatever value we choose within this range,
the asymptotic behavior of the above system does not change.
(
Note that λi k) is the fractional contribution of the i-th species of ion to
the surface charge density on face k of the membrane (equation (2.15)). The
(k)
quantity rd2 /D0 is a fast time scale, on which λi
(k)
(k)
relaxes to λ̃i . Thus λ̃i ,
which is a function of the bulk concentrations of the various ion species near
face k of the membrane, is a kind of target value to which the fractional
contribution to the surface charge density of the i-th species of ion rapidly
(k)
relaxes. Except during extremely fast transients, λi
(k)
will closely track λ̃i .
(k)
The interpretation of λi
as the fractional contribution of the i-th ion
P (k)
species to the surface charge on face k of the membrane requires that i λi =
1 be satisfied identically, at all membrane locations for all time. To verify
this condition, sum both parts of (2.16) from i = 1 · · · N. The second part
P (k)
P (k)
gives i λ̃i = 1, and the first part therefore shows that i λi = 1 relaxes
to 1, and indeed is identically equal to 1 if it is equal to 1 initially. We assume
in the sequel that this initial condition is indeed satisfied.
In [30], we let:
(k)
pi
=
∂ (k)
λ̃i (x, t)σ (k)
∂t
(2.17)
We subsequently found that this boundary condition renders the system of
equations ill-posed, as we shall show in the Section 2.2.6. This prompted us
to perform an asymptotic study, which led to (2.14)-(2.16).
23
2.1.4
Transmembrane Ionic Currents
We have yet to specify ji , the transmembrane currents. Biophysically, these
are currents that flow through ion channels, transporters, or pumps that
are located within the cell membrane [1, 12, 20]. We use the formalism of
Hodgkin and Huxley for ion channel currents [13, 22, 23], generalized to allow
for nonlinear instantaneous current-voltage relations and ion concentration
effects.
(kl)
ji
(kl)
(x, t) = Ji
x, s(kl) (x, t), φ(kl) (x, t), c(k) (x, t), c(l) (x, t)
(kl)
The transmembrane current Ji
(2.18)
is a function characteristic of the channels
(possibly of more than one type) that carry the i-th species of ion across the
(kl)
membrane separating Ω(k) from Ω(l) . The explicit dependence of Ji
on x
reflects the possible inhomogeneity of the membrane: the density of channels
(kl)
may vary from one location to another. The other arguments of Ji
are as
follows:
(kl)
(kl)
First, there is a vector of gating variables s(kl) (x, t) = (s1 , · · · , sG )
where G is the total number of gating variables in all of the channel types that
arise in our system. (Only some of these influence the channels that conduct
(kl)
ions of species i.) The individual components sg
of s(kl) are dimensionless
variables as introduced by Hodgkin and Huxley [13] that take values in the
24
interval [0, 1] and satisfy ordinary differential equations of the form,
(kl)
∂sg
(x, t) = αg φ(kl) (x, t) 1 − sg(kl) (x, t) − βg φ(kl) (x, t) sg(kl) (x, t)
∂t
(2.19)
for g = 1, · · · , G where αg and βg are positive, empirically defined functions
of the transmembrane potential. In general, the gating variables obey a more
complicated ordinary differential equation:
∂s(kl)
(x, t) = fg s(kl) (x, t), φ(kl)(x, t), c(k) (x, t), c(l) (x, t)
∂t
(kl)
The next argument of Ji
(2.20)
is again the transmembrane potential φ(kl) .
(kl)
Holding the other arguments fixed in Ji
, and letting only φ(kl) vary, we get
the instantaneous current-voltage relationship for current carried by the i-th
ion from Ω(k) to Ω(l) at point x at time t.
(kl)
The last two arguments of Ji
are the vectors of ion concentrations
(k)
(k)
on the two sides of the membrane: c(k) = (c1 , · · · , cN ) and similarly for
c(l) . By including the whole vector of ion concentrations, we allow for the
possibility that the current carried by the i-th species of ion is influenced by
the concentrations of other ionic species on the two sides of the membrane.
As an example of the foregoing, consider the Na+ channel current in the
25
Hodgkin Huxley model, which has the following form.
JNa = gNa (x)m(x, t)3 h(x, t)
ext
cNa (x, t)
kB T
int
ext
log int
(φ (x, t) − φ (x, t)) −
q
cNa (x, t)
(2.21)
In this equation, the regions k and l are identified as the intracellular and
extracellular spaces, denoted by the superscript int and ext. m and h are the
ext
gating variables, and gN a is the ion channel density. Keeping m, h, gNa , cint
Na , cNa
fixed, we happen to have a linear instantaneous current voltage relationship,
but this may not be so in general. The gating variables m and h satisfy a
first order ordinary differential equation of the form (2.19). Functions α and
β are determined experimentally.
Note that the above formalism is more general than the words that we
have used to describe it. The “ions” that we have described do not need to be
charged (one can have zi = 0 for some i) and they can include neurotransmitters (whether charged or not) that have been released at synapses. We may
have ligand-gated channels, by introducing gating variables whose evolution
is governed by the local concentration of chemical species. The only restriction is that the binding of ligands to these channels does not significantly
alter the unbound concentrations of these chemicals, though it would not be
difficult to generalize the formalism to allow for this possibility as well. Other
ion carrying mechanisms such as transporters and pumps may also be easily
incorporated using the above formalism or a slight generalization thereof.
26
A subtle and important question that remains is precisely where to evaluate the electrostatic potential and the various ion concentrations on the two
sides of the membrane. As we have seen in the foregoing sections, there is
a thin space charge layer near each face of the membrane in which the electrostatic potential and the ion concentrations may deviate somewhat from
those in the bulk solution, away form the membranes. Several of the argu(kl)
ments of the functions Ji
that define the transmembrane currents involve
the boundary values of the electrostatic potential or the ion concentrations.
Do these literally refer to the values right on the face of the membrane, or
do they refer to the values in the electroneutral region, near the membrane
but outside the space charge layer?
From a mathematical standpoint, the answer is clear. We are trying to
solve a set of partial differential equations (2.1), (2.2), and (2.5) that are
satisfied away from the membrane under certain boundary conditions with
(k)
(l)
expressions involving ci , ci and φ(kl) . Thus, the boundary conditions will
not be useful unless these quantities are evaluated away from the membrane,
i.e., near the membrane but within the electroneutral region.
(kl)
When Ji
is measured experimentally, the controlled values of voltage
and ion concentrations are always those of the bulk solution. The space
charge layers are, of course, present during these measurements, but they are
experimentally inaccessible, on account of their thinness. This means that the
(kl)
experimentally determined functions Ji
can be used directly in the equa-
tions without corrections. We may therefore claim that the present model
27
in which we have replaced the Poisson equation with the electroneutrality
relation, corresponds more naturally to the nature of electrophysiological experiments. This gives us another reason for our interest in the β = rd /L0 → 0
limit.
2.1.5
Summary of the Electroneutral Model
We can now write down the equations to be solved together with their boundary conditions. Let Γ(kl) denote the membrane separating Ω(k) and Ω(l) . In
Ω(k) and Ω(l) ,
∂ci
+ ∇ · fi
∂t qzi ci
fi = −Di ∇ci +
∇φ
kB T
n
X
0 = ρ0 +
qzi ci
0=
(2.22)
(2.23)
(2.24)
i=1
The boundary conditions at the face of Γ(kl) facing Ω(k) are,
(k)
Cm
∂(λi φ(kl) )
(k)
(kl)
= qzi fi · n(kl) − ji
∂t
(k)
(k)
(k)
(k)
λ̃ − λi
z2c
∂λi
(k)
, λ̃i = PN i i (k)
= i 2
2
∂t
rd /D0
i′ =1 zi′ ci′
(kl)
(kl)
ji = Ji
x, s(kl) , φ(kl) , c(k) , c(l)
(2.25)
(2.26)
(2.27)
(kl)
∂sg
∂t
= fg s(kl) , φ(kl) , c(k) , c(l)
28
(2.28)
The boundary conditions on the Ω(l) face are the same as above with k and
l interchanged. We shall refer to this as the electroneutral model.
One easy extension of the above model is to include chemical reactions
(including the important example of the binding and unbinding of ions to
buffers). We may add reaction terms on the left hand side of (2.22). Since no
chemical reaction destroys or creates charge, reaction terms will always be
consistent with the electroneutrality condition (2.24) as long as we keep track
of all chemical species in the reaction. Inclusion of reaction terms may prove
especially important with respect to calcium, a physiologically important ion
that is heavily buffered in biological cells.
2.2
Asymptotic Derivation and Model Hierarchy
We shall now perform an asymptotic calculation starting from the Poisson
equations to derive the equations of the electroneutral model we just introduced. In particular, this calculation will lead us to the form of the capacitative current term which was postulated but not derived in Section 2.1. By
taking the asymptotics further, we shall find that there is a hierarchy of electrophysiology models of varying complexity. This will in particular illuminate
the relation between the various models and the traditional cable model. In
the course of this derivation, we shall see that the electroneutral model possesses two disparate timescales whose presence has imporatant implications
29
for the later development of a numerical method.
2.2.1
Poisson Model
As before, let Ω(k) denote the subregions partitioned by the cell membranes,
and Γ(kl) denote the membrane between Ω(k) and Ω(l) . In Ω(k) , the equations
to be satisfied are the following.
∂ci
= −∇ · fi
∂t
qzi ci
∇φ
kB T
!
N
X
ρ0 +
qzi ci
fi = −Di ∇ci +
∆φ = −
1
ǫ
(ion conservation)
(2.29)
(drift-diffusion flux)
(2.30)
(Poisson equation)
(2.31)
i=1
These equations are satisfied right up to the face of the boundary, in the
approximately electroneutral region as well as within the space charge layer.
In the Section 2.1, we did not supplement the above with boundary conditions. In fact, we immediately replaced the Poisson equation (2.31) with the
electroneutrality relation and used physical arguments to derive the boundary conditions for the electroneutral model. Here, we shall write down the
boundary conditions for the above system, to be used as a starting point to
perform model reduction.
We first consider the boundary condition for the Poisson equation. The
value of the electrostatic potential and the normal component of the electric
displacement vector D = ǫE, where ǫ is the dielectric constant and E is the
30
electric field, should be continuous at the interface between the cell membrane
and the electrolyte solution. Therefore, at this interface,
φ(mem) = φ(k)
ǫm
∂φ(k)
∂φ(mem)
=
ǫ
∂n(kl)
∂n(kl)
(2.32)
(2.33)
where φ(mem) is the electrostatic potential within the membrane, ǫm the dielectric constant of the cell membrane, and n(kl) the unit normal at the
membrane-electrolyte interface pointing from Ω(k) into the membrane.
It is to be noted that (2.33) is not satisfied at the mouths of ion channels.
If ion channels mouths do not occupy a significant amount of membrane area,
the above boundary condition may be considered reasonable. Fortunately,
ion channels are sparsely distributed even at their peak documented densities
[23]. A more complete analysis, though, may require an averaging procedure
with respect to the membrane coordinate, taking into account the presence
of such microstructural inhomogeneities.
We make the following assumptions:
• The membrane thickness dm (∼ 10nm) is small compared to the curvature radius of the membrane, and is negligible compared to the typical
length scale of the system.
• The bulk of the membrane (away from ion channels, which are sparse
as discussed above) behaves electrically as an insulator with a uniform
31
dielectric constant.
Under these assumptions, φmem varies linearly as one traverses the membrane
from Ω(k) to Ω(l) . Thus,
φ(k) − φ(l)
∂φ(mem)
=
.
∂n
dm
(2.34)
We obtain the following boundary condition,
∗ (kl)
Cm
φ =ǫ
∗
where φ(kl) = φ(k) − φ(l) , Cm
=
ǫm
dm
∂φ(k)
∂n(kl)
(2.35)
and n(kl) is the unit normal on the
∗
membrane pointing from Ω(k) to Ω(l) . Cm
may be considered the intrinsic
capacitance of the membrane. We shall later see that the effective membrane
capacitance Cm as appears in the electroneutral model is slightly different
∗
from Cm
.
The boundary conditions for the drift diffusion equations are simple:
(kl)
qzi fi · n(kl) = ji
(kl)
where ji
(2.36)
are ion channel currents, whose form was discussed in Section
2.1.4.
32
In summary, the boundary conditions we obtain are:
∗ (kl)
Cm
φ =ǫ
∂φ(k)
∂n(kl)
(kl)
qzi fi · n(kl) = ji
(2.37)
.
(2.38)
We shall refer to this model, (2.29)-(2.31) and (2.37)-(2.38), as the Poisson
model.
2.2.2
Non-Dimensionalization
We non-dimensionalize the Poisson model so as to identify dimensionless
parameters whose smallness may be exploited in model reduction. We first
rescale the ionic concentrations ci and the electrostatic potential φ as follows.
φ=
kB T
Φ,
q
ρ0 = qc0 ρ˜0 ,
ci = c0 C i
fi = c0 f˜i ,
(2.39)
ji = γqc0 j˜i
(2.40)
where c0 is the characteristic concentration and γqc0 is the characteristic
magnitude of the transmembrane current per unit area. The constant γ has
units of speed=length/time. Use of the above rescaling leads to the following
33
partial non-dimensionalization.
∂Ci
= −∇ · f˜i
∂t
f˜i = −Di (∇Ci + zi Ci ∇Φ)
!
N
X
rd2 ∆Φ = − ρ˜0 +
zi Ci .
(2.41)
(2.42)
(2.43)
i=1
The boundary conditions are rescaled as,
θ∗ Φ(kl) = rd
∂Φ
∂n(kl)
zi f˜i · n(kl) = γ j˜i
where rd ≡
q
ǫkB T
q 2 c0
(2.44)
(2.45)
is the Debye length introduced in Section 2.1. We have
introduced the dimensionless parameter θ∗ defined as follows.
θ∗ =
∗
Cm
C ∗ kB T /q
= m
ǫ/rd
qc0 rd
(2.46)
Note that kB T /q = 25mV is the natural unit of voltage in electrophysiology,
and that qc0 rd is a natural unit of surface charge density, so qc0 rd /(kB T /q) is
a natural unit of capacitance per unit area. Thus θ∗ expresses the membrane
capacitance per unit area in these natural units. We list here the typical
34
physiological values of the parameters in equations (2.41)-(2.45):
Di ≈ 1µm2 /msec,
rd ≈ 10−3µm
(2.47)
θ∗ ≈ 10−2 ,
γ ≈ 10−5 ∼ 10−3 µm/msec
(2.48)
To complete the non-dimensionalization, we determine a typical length
scale of the system. We take equation (2.45) and integrate over the membrane
surface ∂Ω(k) .
Z
γ j˜i dA =
∂Ω(k)
Z
∂Ω(k)
=
Z
Ω(k)
=−
=−
zi f˜i · n(kl) dA
zi ∇ · f˜i dV
Z
Ω(k)
Z
Ω(k)
(2.49)
∇ · zi Di (∇Ci + zi Ci ∇Φ)dV
∇ · zi Di (Ci ∇µi )dV
where µi = ln Ci + zi Φ is the chemical potential, and dV and dA denote
volume and surface integrals respectively. We have used the divergence theorem in the second equality and the flux expression (2.42) in the third. Let
L0 be the typical length over which the flux and the chemical potential vary.
Balancing the order of magnitude of the surface and volume integrals above,
γ|∂Ω(k) | =
Di |Ω(k) |
.
L20
(2.50)
where |∂Ω(k) | is the surface area of the region Ω(k) and |Ω(k) | is the volume
35
of Ω(k) We therefore set:
L0 =
s
lD0
, where
γ
l=
|Ω(k) |
.
|∂Ω(k) |
(2.51)
The constant D0 is the typical diffusion coefficient for ions. The quantity l is
a measure of the volume per unit surface area, and is a representative length
scale of the distance between membranes. For a cylindrical axon, l corresponds roughly to the diameter of the axon, whereas for an approximately
spherical cell, l corresponds to the diameter of the cell. We note that L0 is
precisely what is termed the electrotonic length in cable theory. Notice that
√
L0 is proportional to l. This is in agreement with the observation in cable
theory that the electrotonic length scales with the square root the diameter
of a cylindrical cable.
Given L0 , we can define a typical time scale T0 as T0 = L20 /D0 = l/γ.
We shall call T0 the diffusion time scale or the slow diffusion time scale. We
note that we can also obtain this time scale by balancing both sides of the
following equality, which can be derived by taking the volume integral of
(2.41).
∂Ci
dV = −
zi
∂t
Ω(k)
Z
=−
=−
Z
Ω(k)
zi ∇ · f˜i dV
Z
∂Ω(k)
Z
36
∂Ω(k)
zi f˜i · n(kl) dA
γ j˜i dA
(2.52)
The T0 time scale thus corresponds to the amount of time it takes for a
membrane-bound region to experience O(1) changes in dimensionless ionic
concentration.
We can now complete our non-dimensionalization using T0 and L0 as
the representative temporal and spatial scales. We introduce the following
dimensionless variables.
x = L0 X,
t = T0 τD
D0
Fi
f˜i =
L0
l
α=
L0
Di = D0 D̃i ,
β=
rd
,
L0
(2.53)
(2.54)
(2.55)
We write (2.41)-(2.45) in dimensionless form:
∂Ci
= −∇X · Fi
∂τD
(2.56)
Fi = −D̃i (∇X Ci + zi Ci ∇X Φ)
β 2 ∆X Φ = −(ρ˜0 +
N
X
(2.57)
zi Ci )
(2.58)
∂Φ
∂n(kl)
(2.59)
i=1
The boundary conditions are,
θ∗ Φ(kl) = β
zi Fi · n(kl) = αj˜i
(2.60)
We have, thus, three constants β, α and θ∗ that characterize the system.
37
These equations are satisfied in a region where the dimensionless volume to
surface ratio is given by α.
Given typical values of l, we can find typical physiological values of the
parameters β and α. Recall that l is the (dimensional) volume to surface
ratio, and thus, roughly measures the separation distance of membranes.
Values typical in the central nervous system can range from 100nm to 10µm.
Combining these numbers with (2.47) and (2.48), we find:
s
lD0
= 10µm ∼ 1mm
γ
r
γ
β = rd
= 10−6 ∼ 10−4
lD0
r
lγ
= 10−3 ∼ 10−1 .
α=
D0
L0 =
(2.61)
(2.62)
(2.63)
Here, ∼ indicates the range of physiologically possible values.
2.2.3
Multiple Spatiotemporal Scales
Before we perform asymptotic calculations on the Poisson model, we would
like to study the system further to see whether there are additional spatiotemporal scales of interest.
38
Multiple Time Scales
Consider the non-dimensionalized Poisson equation (2.58), differentiate in τD
and take the integral over Ω(k) . The left hand side yields:
Z
Ω(k)
∂
2 ∂Φ
β
dA
∂nkl
∂Ω(k) ∂τD
Z
∂Φ(kl)
=
βθ∗
dA.
∂τD
∂Ω(k)
∂
(β 2 ∆Φ)dV =
∂τD
Z
(2.64)
We used the boundary condition (2.59) in the second inequality. The right
hand side yields:
−
Z
Ω(k)
∂
∂τD
ρ˜0 +
N
X
i=1
!
zi Ci dV = −
=
=
=
Z
Z
N
X
zi
Ω(k) i=1
N
X
zi ∇ · Fi dV
Ω(k) i=1
Z
N
X
(2.65)
zi Fi · n(kl) dA
∂Ω(k) i=1
Z
N
X
α
∂Ω(k)
∂Ci
dV
∂τD
(kl)
j˜i dA.
i=1
Balancing the quantities in (2.64) and (2.65), we see that
β
θ∗ ∂Φ(kl)
= O(1)
α ∂τD
(2.66)
We conclude that the membrane potential and hence the electrostatic po∗
tential can vary on the time scale of β θα T0 . We shall refer to this as the
39
membrane potential time scale. It is an interesting coincidence that θ∗ and α
are roughly of the same order of magnitude, as can be seen from (2.48) and
(2.63). Thus, the membrane potential time scale is roughly equal to βT0 .
Given the smallness of β, the membrane potential time scale is considerably
smaller than the slow diffusion time scale T0 . We shall later see that the
membrane potential time scale βT0 corresponds to the “diffusion” time scale
of the membrane potential in the traditional cable model.
There is yet another time scale, which corresponds to charge relaxation:
∂
∂τD
ρ˜0 +
N
X
i=1
zi Ci
!
=
=
N
X
i=1
N
X
i=1
=−
zi ∇ · D̃i (∇Ci + zi Ci ∇Φ)
(zi ∇ · D̃i ∇Ci + zi2 ∇(D̃i Ci ) · ∇Φ + zi2 Ci ∆Φ)
N
X
i=1
zi2 Ci
!
1
β2
ρ˜0 +
N
X
i=1
zi Ci
!
+ other terms
(2.67)
where we have used the Poisson equation (2.58) in the last equality to replace ∆Φ. We see that charge density decays exponentially with a time
constant of β 2 T0 = rd2 /D0 = 1nsec. We can infer that this time scale is
only important where the electrolyte solution may deviate significantly from
electroneutrality, i.e., within the space charge layer. This observation implies
that simulations with the Poisson model not only require a fine spatial mesh
to resolve the space charge layer, but also a fine time step to account for
diffusion within this layer.
40
We thus see that there are three time scales present in the Poisson model,
T0 , βT0 and β 2 T0 . We list the physiological values for these time scales.
T0 = 10−1 ∼ 103 sec
(2.68)
βT0 = 10−2 ∼ 1 msec
(2.69)
β 2 T0 = 1 nsec
(2.70)
The time scale of greatest interest is the βT0 time scale, in which the membrane potential varies. This is also roughly equal to the time scale of channel
gating and chemical neurotransmission. Thus, we shall focus our attention
on this time scale. On the other hand, the β 2 T0 time scale and the space
charge layer within which this time scale is relevant are spatiotemporal details that we do not wish to resolve. We would therefore like to derive a
model that resolves the membrane potential time scale but does not resolve
the charge relaxation time scale or the Debye length scale. The T0 time scale
is important with regard to long term changes in ionic concentrations. We
shall address this time scale, although the treatment will be more limited.
Multiple Spatial Scales and Matched Asymptotics
Based on the above observations, we shall introduce a newly rescaled time
variable τV to capture the rapid fluctuations in membrane potential.
τV =
41
τD
β
(2.71)
Our primary goal is to capture the dynamics in this time variable. We write
Ci , Φ as functions of τV rather than τD . Therefore, (2.56)-(2.58) rescale as a
result of this rescaling of the time variable.
∂Ci
= −β∇X · Fi
∂τV
!
N
X
β 2 ∆X Φ = − ρ˜0 +
zi Ci
(2.72)
(2.73)
(2.74)
i=1
When β ≪ 1, a boundary layer of thickness O(β) develops at the membrane.
This is the space charge layer whose existence we claimed on physical grounds
in Section 2.1. The formal mathematical reason for the presence of this layer
is that Φ satisfies a mixed (Robin) boundary condition (2.59) and this is not
in general compatible with the electroneutrality condition, which is obtained
by formally letting β = 0 in (2.74). The thickness of this layer can be
determined by noting that β 2 multiplies a second order spatial derivative.
We therefore introduce an inner layer of thickness O(β) at the membrane
(Figure 2.1). We shall continue to use the terms space charge layer or Debye
layer to denote this layer.
√
We need to introduce another spatial scale of order O( β) at the mem√
brane. The need for this extra O( β) scale arises because we introduced a
new time scale βT0 , and rescaled time to conform to this time scale. The
√
spatial scale of order O( β) corresponds to the length over which ions can
diffuse in the βT0 time scale. Formally, the necessity for this layer can be
42
lm
β
√
β
α
Figure 2.1: A schematic of the relevant spatial scales used in the asymptotic
calculations. The solid lines denote the membrane and the dotted lines are
the interfaces between the√boundary layers. The inner-most layer has width
β, the intermediate layer β. The typical membrane separation is α and the
typical length scale associated with the membrane is lm .
43
seen by noting that β multiplies a second order spatial derivative in (2.72),
as can be seen by noting that Fi itself is written in terms of spatial derivatives, (2.73). We shall refer to this layer as the intermediate layer or the fast
diffusion layer (Figure 2.1). It is interesting to note that such layers around
the membrane have been documented as the Hodgkin-Frankenhaeuser space
in the case of a cylindrical axon [9]. We thus have three regions to consider
in the asymptotic calculations to follow: the inner and intermediate layers
located at the membrane, and the region away from the membrane, which
we shall call the outer layer.
The above discussion prompts us to expand the physical variables in
√
powers of β instead of β:
Ci (X, τV ) = Ci0 (X, τV ) +
Φ(X, τV ) = Φ0 (X, τV ) +
p
p
βCi1 (X, τV ) + βCi2 (X, τV ) · · ·
(2.75)
βΦ1 (X, τV ) + βΦ2 (X, τV ) · · ·
(2.76)
The other two parameters of the system, α and θ∗ are also small, but we
shall treat them as being O(1) with respect to β. We note that β is typically
a few orders of magnitude smaller than α or θ∗ .
A potentially important question is whether we should treat α or θ∗ as
√
√
being O( β) quantities. The numbers α or θ∗ are usually larger than β but
can be comparable in magnitude especially when l, the (dimensional) volume
√
to surface ratio, is small. We do not let α or θ∗ be O( β) primarily because a
useful mathematical model seems to be difficult to derive under this scaling.
44
One of the consequences of α = O(1) scaling is the following. Recall α is
√
the dimensionless volume to surface ratio. Thus, α = O( β) means that the
intermediate layer scale is comparable to the average membrane separation.
In this case it would not make sense to talk about an outer layer. Since
α = O(1), an outer layer will always exist as β → 0, within the context
of asymptotic calculations. Although α and θ∗ are to be treated as O(1)
quantities relative to β, the smallness of α and θ∗ will be later exploited, in
Sections 2.2.7 and 2.2.4 respectively.
In the ensuing asymptotic calculations, we shall simplify the boundary
conditions by regarding the transmembrane ionic current densities ji as given
functions of position (on the membrane) and time instead of being functions
of Ci , Φ(kl) and the gating variables.
In performing matched asymptotics at the membrane, we introduce a
coordinate system at the membrane ξ = (ξ1 , ξ2 , ξ3), where the ξ1 axis is
taken to be perpendicular to the membrane, while ξ2 and ξ3 are curvilinear
coordinates that run “parallel” to the membrane. The ξ1 axis will be rescaled
√
to yield coordinates in the intermediate layer ξa such that ξ1 = βξ1a and in
the inner layer ξb such that ξ1 = βξ1b .
The question we now ask is how we are going to rescale ξ2 and ξ3 . There
are at least two spatial scales that are relevant: ρκ the dimensionless curvature radius of the membrane and lj the dimensionless length scale on which
one may see O(1) changes in ion channel current density. Let lm be the
smaller of the two spatial scales lj and ρκ . We shall call lm the membrane
45
length scale (Figure 2.1). The question raised at the beginning of this paragraph can be answered by comparing the relative magnitude of this length
√
scale to that of the O( β) length scale.
√
If lm is considerably larger than β, there is no need to rescale ξ2 and ξ3 .
√
√
√
If lm is order O( β), we must scale ξ2 , ξ3 to ξ2 = βξ2a , ξ3 = βξ3a so that the
curvature correction and the ionic fluxes parallel to the membrane are O(1)
quantities when measured in the intermediate layer coordinate ξ a . When
√
lm = O( β), it seems difficult to perform matching between the intermediate
layer and the outer layer, even if a well-defined outer layer may exist. There
would be some form of lateral mixing within the intermediate layer, so that
√
spatial structure O( β) in the physical variables will be smeared out as one
traverses the intermediate layer from the inner-intermediate layer interface to
the intermediate outer layer interface. An analysis of this situation is beyond
the scope of matched asymptotics to be performed here. We therefore have
two cases:
1. lm >
√
β. Perform both inner-intermediate and intermediate-outer
matching. Rescale only the ξ1 coordinate in the intermediate layer.
2. lm ∼
√
β. Perform inner-intermediate matching only. Rescale not only
the ξ1 coordinate but also ξ2 , ξ3 in the intermediate layer.
We point out that there could be situations in which lm is small only along
a certain coordinate direction. For example, if we take a cylindrical axon
√
with diameter O( β), and take ξ2 to be the angular coordinate, and ξ3 to be
46
√
the axial coordinate, the curvature radius along the ξ2 coordinate is O( β)
whereas the curvature radius along the ξ3 coordinate is large (curvature is
negligible). In such cases (and if the cylindrical axon is endowed with near
uniform ion channel density so that lj is large), we need only rescale ξ2 but
not ξ3 in the intermediate layer. An analysis of such a situation can be easily
performed, but we do not pursue this here, since the analysis is essentially
the same as case 2 above. We now address the above cases in turn.
2.2.4
Inner-Intermediate Matching when lm >
We first consider inner-intermediate matching when lm >
√
√
β
β. This corre-
sponds the case 1 listed above. We write down the equations to be satisfied
in the inner and intermediate layers.
Consider the membrane surface facing Ω(k) . We now introduce a coordinate system ξ so that the ξ1 coordinate direction is perpendicular to the
membrane. We let ξ1 = 0 coincide with the membrane face, and let the
positive ξ1 axis point into the region Ω(k) . For simplicity, we shall assume
that the membrane is flat, i.e., that it has no curvature. Therefore, we can
√
take the coordinate system to ξ to be orthonormal. When lm > β, it turns
out that curvature corrections produce only higher order terms that we can
ignore.
In the inner layer, we rescale ξ as:
ξb = (ξ1b , ξ2b , ξ3b),
ξ1 = βξ1b,
47
ξ2 = ξ2b ,
ξ3 = ξ3b .
(2.77)
The equations are:
b
∂Fi1b
∂Fi2 ∂Fi3b
2
+β
+
(2.78)
∂ξ1b
∂ξ2b
∂ξ3b
b
b
∂Ci
b
b ∂Φ
Fip = −D̃i
+ zi Ci b , p = 1, 2, 3
∂ξpb
∂ξp
∂C b
β i =−
∂τV
(2.79)
∂ 2 Φb
2
∂ξ1b
+ β2
∂ 2 Φb
2
∂ξ2b
+
∂ 2 Φb
2
∂ξ3b
!
= − ρ˜0 +
N
X
zi Cib
i=1
!
.
(2.80)
Since the inner layer is adjacent to the membrane, we must supplement the
above with boundary conditions, suitably rescaled:
∂Φb θ ( Φ|ξb =0 − Φ ) = −
1
∂ξ1b ξb =0
1
− zi Fi1b ξb =0 = βαj̃i .
∗
(l)
(2.81)
(2.82)
1
In the intermediate layer we rescale ξ as:
ξ a = (ξ1a , ξ2a, ξ3a ),
ξ1 =
p a
βξ1 ,
48
ξ2 = ξ2a ,
ξ3 = ξ3a .
(2.83)
The equations are:
a
a
∂Fi2 ∂Fi3a
∂Fi1
∂Cia
+β
+
(2.84)
=−
∂τV
∂ξ1a
∂ξ2a
∂ξ3a
a
a
∂Ci
a ∂Φ
Fip = −D̃i
+ zi Ci
, p = 1, 2, 3
∂ξpa
∂ξpa
(2.85)
∂ 2 Φa
β a2 + β 2
∂ξ1
∂ 2 Φa ∂ 2 Φa
+ a2
∂ξ2a 2
∂ξ3
= − ρ˜0 +
N
X
i=1
!
zi Cia .
(2.86)
Substitute (2.75) and (2.76) in the inner layer equations (2.78)-(2.80),
√
and collect like terms in order β. The expansions of Ci and Φ in β induce
√ k
√
expansions of Fi in terms of β. We shall denote the O( β ) term as Fki .
For example,
Fi1b0
Fi1b1
b0
∂Cib0
b0 ∂Φ
= −D̃i
+ zi Ci
∂ξ1b
∂ξ1b
b1
b0
b1
∂Ci
b1 ∂Φ
b0 ∂Φ
= −D̃i
+ zi Ci
+ zi Ci
∂ξ1b
∂ξ1b
∂ξ1b
(2.87)
(2.88)
By applying the same procedure to the equations (2.84)-(2.86), we may obtain analogous expressions in the intermediate layer.
We start with the following considerations. First, note from (2.78) and
49
(2.82) that:
∂Fi1b0
= 0,
∂ξ1b
∂Fi1b1
= 0,
∂ξ1b
Fi1b0 ξb =0 = 0
1
Fi1b1 ξb =0 = 0
1
(2.89)
(2.90)
From this, we find that
Fi1b0 = Fi1b1 ≡ 0
(2.91)
within the inner layer.
Armed with this observation, we now derive matching conditions in terms
of the ionic fluxes. Consider the p = 1 component of (2.79) and (2.85).
Substituting (2.75) and (2.76) into these expressions,
b
p b1
∂Cib
b ∂Φ
b0
βFi1 + βFi1b2 + · · ·
= −D̃i
+
z
C
=
F
+
i
i
i1
∂ξ1b
∂ξ1b
a
a
p
∂Ci
a
a ∂Φ
Fi1 = −D̃i
+ zi Ci
= Fi1a0 + βFi1a1 + βFi1a2 + · · ·
a
a
∂ξ1
∂ξ1
Fi1b
(2.92)
(2.93)
We now introduce a matching coordinate system ξ η in between the inner and
intermediate layers such that,
ξ1a
η
= η(β)ξ ,
lim
β→0
√
β
= lim η = 0
β→0
η
(2.94)
We only rescale the ξ1a coordinate. We now write (2.92) and (2.93) in terms
50
of the matching coordinate ξ η .
p
η η
η η
η η
b1
b2
√ ξ + ηFi1 √ ξ + βηFi1 √ ξ + · · · (2.95)
β
β
β
p
(2.96)
Fi1a = ηFi1a0 (ηξ η ) + βηFi1a1 (ηξ η ) + βηFi1a2 (ηξ η ) + · · ·
Fi1b
η
= √ Fi1b0
β
In the above, we have only written the dependence of the fluxes on ξ η since
the other coordinates are irrelevant here. We now invoke Kaplun’s matching
condition [14] [21], which states that the above two expressions must match
to the same order as β → 0 independent of ξ η . The matching conditions in
the present case are:
η η
√ ξ =0
β
1 b0
η η
η η
b1
a0
η
lim √ Fi1 √ ξ + Fi1 √ ξ − Fi1 (ηξ ) = 0
β→0
β
β
β
1 b1
η η
η η
η η
1 b0
b2
√ ξ + √ Fi1 √ ξ + Fi1 √ ξ
F
lim
β→0
β i1
β
β
β
β
1
− √ Fi1a0 (ηξ η ) − Fi1a1 (ηξ η ) = 0
β
lim Fi1b0
β→0
(2.97)
(2.98)
(2.99)
Condition (2.97) is automatically satisfied by (2.91). Condition (2.98), taken
together with (2.91), yields:
lim Fi1a0 (ηξ η ) = Fi1a0 ξa =0 = 0.
β→0
1
(2.100)
The last matching condition (2.99), combined with (2.91), yields the follow-
51
ing.
lim
β→0
Fi1b2
η
√ ξη
β
1
− √ Fi1a0 (ηξ η ) − Fi1a1 (ηξ η )
β
= 0.
(2.101)
We would now like to evaluate (2.101).
To proceed any further, we need to calculate Ci and Φ to leading order in
the inner layer. We shall come back to expression (2.101) after we perform
this calculation. From (2.91), (2.80) and (2.81) we see that the leading order
terms satisfy the following one dimensional boundary value problem in ξ1a in
the inner layer:
b0
∂Cib0
b0 ∂Φ
+
z
C
i
i
∂ξ1b
∂ξ1b
N
X
= −(ρ˜0 +
zi Cib0 )
0=
∂ 2 Φb0
∂ξ1b
b,(l)
θ∗ Φb0 (ξ1b = 0) − Φ(l),b0 (ξ1
2
i=1
∂Φb0 = 0) = −
∂ξ1b ξb =0
(2.102)
(2.103)
(2.104)
1
Cib0 (ξ1b = ∞) = Cia0 (ξ1a = 0)
(2.105)
Φb0 (ξ1b = ∞) = Φa0 (ξ1a = 0).
(2.106)
The last two conditions come from matching conditions at the inner-interb,(l)
mediate layer interface. Here, ξ1
refers to the inner layer coordinate system
on the Ω(l) side of the membrane (note that we are now working on the Ω(k)
side). We now solve (2.102), (2.103) under the boundary conditions (2.104)(2.106). Since this is a one dimensional boundary value problem, we shall
think of Φb0 and Cib0 as functions only of ξ1b and do not explicitly write their
52
dependence on ξ2b or ξ3b .
Equation (2.102) can be integrated easily to obtain
Cib0 (ξ1b ) = Cib0 (∞) exp −zi (Φb0 (ξ1b ) − Φb0 (∞)) .
(2.107)
This equation can be substituted into (2.103) to yield:
2
−
b0
∂ Φ
2
∂ξ1b
=
ρ̃0 +
N
X
i=1
zi Cib0 (∞) exp −zi (Φb0 (ξ1b ) − Φb0 (∞))
!
.
(2.108)
Here we use an approximation to linearize the above Poisson-Boltzmann
equation. We suppose
zi (Φb0 (ξ b ) − Φb0 (∞)) ≪ 1.
1
(2.109)
This can be justified by the smallness of θ∗ . We shall show this later in this
section. Assuming for now that (2.109) is a valid assumption, we linearize
(2.108) to find:
∂2
2
∂ξ1b
Cib0 (ξ1b ) = Cib0 (∞) 1 − zi (Φb0 (ξ1b ) − Φb0 (∞))
(Φb0 (ξ1b ) − Φb0 (∞)) = Γ2 (Φb0 (ξ1b ) − Φb0 (∞))
(2.110)
(2.111)
where,
2
Γ =
N
X
i=1
zi2 Cib0 (∞)
=
N
X
i=1
53
zi2 Cia0 (0),
Γ > 0.
(2.112)
where Cia0 is shorthand for Cia0 (ξ1a = 0). To derive (2.110) and (2.111), we
have used
ρ̃0 +
N
X
zi Cib0 (∞) = 0
(2.113)
i=1
which follows from
ρ̃0 +
N
X
zi Cia0 = 0
(2.114)
i=1
a consequence of (2.86), and the matching condition (2.105). Solving (2.111)
with (2.103) and (2.106),
Φb0 (ξ1b ) = Φa0 (0) −
σ̃
exp(−Γξ1b )
Γ
(2.115)
where Φa0 (0) is shorthand for Φa0 (ξ1a = 0) and σ̃ is equal to
σ̃ = θ∗ (Φb0 (0) − Φ(l),b0 (0)).
(2.116)
Hence, according to (2.110) and the matching condition (2.105),
Cib0 (ξ1b )
=
Cia0 (0)
zi σ̃
b
1+
exp(−Γξ1 )
Γ
(2.117)
We note that σ̃ is the total excess charge found in the inner layer, as can be
seen as follows. The excess charge contributed by the i-th species of ion σ̃i
54
can be computed as:
σ̃i ≡
Z
∞
Z0 ∞
zi (Cib0 (ξ1b ) − Cia0 (0))dξ1b
zi2 Cia0 (0)σ̃
exp(−Γξ1b )dξ1b
Γ
0
2 a0
z C (0)
= i i 2 σ̃ ≡ λ̃i σ̃
Γ
=
(2.118)
From (2.112), we see that λ̃i is given by:
λ̃i =
zi2 Cia0 (0)
zi2 Cia0 (0)
=
PN
2 a0
Γ2
i′ =1 zi′ Ci′ (0)
We immediately conclude that
by summing σ̃i in i.
N
X
i=1
PN
i=1
σ̃i =
(2.119)
λ̃i = 1. The total excess charge is given
N
X
i=1
!
λ̃i σ̃ = σ̃.
(2.120)
The factor λ̃i thus represents the fraction of excess charge contributed by the
i-th species of ion.
We now have the solutions Cib0 and Φb0 except that σ̃ is expressed in
terms of Φb0 . We shall now express σ̃ in terms of Cia0 (0) and Φa0 (0). First,
we observe by substituting ξ1b = 0 in (2.115) that
σ̃ = Γ(Φa0 (0) − Φb0 (0)).
(2.121)
We next rewrite Φa0 (0) = Φ(k),a0 (0), Φb0 (0) = Φ(k),b0 (0), σ̃ = σ̃ (k) , Γ = Γ(k)
and consider (2.116) and (2.121) as well as the corresponding expressions on
55
the other side of the membrane(the Ω(l) side).
σ̃ (k) = θ∗ (Φ(k),b0 (0) − Φ(l),b0 (0))
(2.122)
σ̃ (k) = Γ(k) (Φ(k),a0 (0) − Φ(k),b0 (0))
(2.123)
σ̃ (l) = θ∗ (Φ(l),b0 (0) − Φ(k),b0 (0))
(2.124)
σ̃ (l) = Γ(l) (Φ(l),a0 (0) − Φ(l),b0 (0))
(2.125)
After some algebra, we find,
σ̃ (k) = −σ̃ (l) = θ(Φ(k),a0 (0) − Φ(l),a0 (0))
1
1
1
1
= ∗ + (k) + (l) .
θ
θ
Γ
Γ
(2.126)
(2.127)
Note that (2.126) is equivalent to the assumptions set forth in Section 2.1.2
to obtain the electrical boundary conditions. The meaning of relation (2.127)
becomes clear once this is written in dimensional terms:
1
1
1
1
= ∗ + (k) + (l) .
Cm
Cm ǫγ
ǫγ
(2.128)
where Γ = γrd . This relation states that the total membrane capacitance
∗
Cm can be computed as the intrinsic membrane capacitance Cm
and the
capacitance of the space charge layers ǫγ (k) , ǫγ (l) in series. We note that in
(2.127), θ∗ is small in magnitude whereas Γ(k) and Γ(l) are order 1. Therefore,
∗
θ ≈ θ∗ , and Cm ≈ Cm
.
56
Now, consider (2.78) and (2.82) at the first non-trivial order:
∂F b2
∂Cib0
= − i1b
∂τV
∂ξ1
− zi Fi1b2 ξb =0 = αj̃i
1
(2.129)
(2.130)
Since our goal is to evaluate (2.101), we would like to obtain an expression
for Fi1b2 . We integrate the above in ξ1b to obtain:
−zi Fi1b2
= αj̃i + zi
Z
0
ξ1b
∂Cib0 b
dξ .
∂τV 1
(2.131)
Since we now have expressions for Cib0 and Φb0 , we can evaluate the above
integral explicitly. Using (2.117), we find:
Z
ξ1b
0
∂
∂Cib0 b
dξ1 =
∂τV
∂τV
=
≡
Z
ξ1b
0
Cia0 (0)
∂Cia0 (0) b
ξ1
∂τV
∂Cia0 (0) b
ξ1
∂τV
+
Z
zi σ̃
b
exp(−Γξ1 ) dξ1b
1+
Γ
ξ1b
0
∂ zi Cia0 (0)σ̃
∂τV
Γ
(2.132)
+ I charge
We therefore have:
−zi Fi1b2 = αj̃i + zi
∂Cia0 (0) b
ξ1 + zi I charge
∂τV
(2.133)
We can finally consider condition (2.101). We would like (2.101) be satisfied regardless of the value of ξ η . For the Fi2b2 term, taking β → 0 in (2.101)
57
amounts to studying the behavior of (2.133) in the limit ξ1b → ∞. Take
ξ1b → ∞ in I charge . By (2.132), we see that
lim zi I charge =
ξ1b →∞
∂ σ̃i
.
∂τV
(2.134)
We thus conclude that:
−zi Fi1b2
=
∂ σ̃i
+ αj̃i
∂τV
+ zi
∂Cia0 (0) b
ξ1 + O(exp(−Γξ1b ))
∂τV
(2.135)
Note that Fi1b2 thus consists of a constant and a linear component in ξ1b as
well as a residual term that decays exponentially. We now expand the intermediate layer expressions of (2.101) at ξ η = 0.
1
1
ηξ η ∂Fi1a0 (0)
√ Fi1a0 (ηξ η ) + Fi1a1 (ηξ η ) = √ Fi1a0 (0) + √
+ Fi1a1 (0) + · · ·
a
∂ξ
β
β
β
1
a0
η
ηξ ∂Fi1 (0)
=√
+ Fi1a1 (0) + · · ·
a
β ∂ξ1
(2.136)
where we have used (2.100) to eliminate Fi1a0 (0). Substituting the above as
well as (2.135) into (2.101),
1 ∂ σ̃i
a1
lim Fi1 (0) +
+ αj̃i
β→0
zi ∂τV
ηξ η ∂Fi1a0 (0) ∂Cia0 (0)
+ O(exp(−Γξ1b )) + · · ·
+
+√
a
∂ξ1
∂τV
β
=0
58
(2.137)
The necessary conditions for the above to be satisfied are:
∂ σ̃i
+ αj̃i
∂τV
∂F a0 (0)
∂Cia0 (0)
= − i1 a
∂τV
∂ξ1
−zi Fi1a1 (0) =
(2.138)
(2.139)
The second expression (2.139) is automatically satisfied as can be seen by
taking (2.85) to leading order. Equation (2.138) is the matching condition
we set out to obtain.
Validity of the linear approximation to the Poisson-Boltzmann equation in the inner layer
We now consider the validity of the linearization of the Poisson-Boltzmann
equation. In this section we shall omit superscripts to avoid cluttered notation: Φ = Φb0 , Ci = Cib0 and ξ = ξ1b . We take the Poisson-Boltzmann
equation (2.108) as our starting point:
∂2Φ
− 2 =
∂ξ
ρ̃0 +
N
X
!
zi Ci (∞) exp (−zi Φ(ξ)) .
i=1
(2.140)
Without loss of generality, we have taken Φ(∞) = 0. We will also assume
in the following that Φ(0) > 0. By the end of the discussion, it will be clear
that the Φ(0) < 0 case can be handled in an identical way. We note that
ρ̃0 +
N
X
zi Ci (∞) = 0.
i=1
59
(2.141)
Our goal is to derive a condition that guarantees the relation
|zi Φ(ξ)| ≪ 1
to hold. We note that the above expression concerns the deviation of the
electrostatic potential within the space charge layer, and not the potential
jump across the cell membrane. Since zi takes integer values not too different
from 1 in absolute value, we shall be content to establish the condition:
|Φ(ξ)| ≪ 1.
(2.142)
Consider the function f (Φ),
f (Φ) = − ρ̃0 +
N
X
!
zi Ci (∞) exp(−zi Φ) .
i=1
Note that (2.140) can be written as
∂2Φ
∂ξ 2
= f (Φ). We see that,
N
X
∂f
=
zi2 Ci (∞) exp (−zi Φ) > 0.
∂Φ
i=1
Therefore, f (Φ) is monotone increasing in Φ. From (2.141), we see that
f (0) = 0. Therefore, f (Φ) > 0 if Φ > 0 and f (Φ) < 0 if Φ < 0.
Suppose that Φ(ξ) > Φ(0) for some values of ξ. Since Φ(∞) = 0, Φ(ξ)
must attain a positive local maximum at some interior point ξM . Since
Φ(ξM ) > 0, f (Φ(ξM )) > 0. But this is impossible since f (Φ(ξM )) > 0 implies
60
that
∂2Φ
(ξM )
∂ξ 2
> 0. Next, suppose that Φ(ξ) is negative for some values of ξ.
Since Φ(∞) = 0, Φ(ξ) must attain a negative local minimum at some interior
point ξm . Since Φ(ξm ) < 0, f (Φ(ξm )) < 0. This is again impossible since this
implies
∂2Φ
(ξm )
∂ξ 2
< 0. This argument may be considered a simple application
of the maximum principle for elliptic partial differential equations.
From the above we conclude that 0 ≤ Φ(ξ) < Φ(0). From Φ ≥ 0, we see
that
∂2Φ
∂ξ 2
≥ 0 for all ξ, and thus,
implies that
∂Φ
(∞)
∂ξ
∂Φ
∂ξ
is non-decreasing. Since Φ(∞) = 0, this
= 0. The two conclusions we have reached so far are:
0 ≤ Φ(ξ) < Φ(0)
(2.143)
∂Φ
(∞) = 0.
∂ξ
(2.144)
From (2.143) we see that (2.142) will be true for all ξ if
|Φ(0)| ≪ 1.
(2.145)
We next integrate (2.140) from ξ = 0 to ξ = ∞.
∂Φ
∂Φ
− (∞) +
(0) =
∂ξ
∂ξ
Z
∞
ρ̃0 +
0
N
X
!
zi Ci(∞) exp (−zi Φ) dξ.
i=1
Noting that the integral on the right hand side is the total charge in the
space charge layer σ̃ and using (2.144), we find,
∂Φ
(0) = σ̃.
∂ξ
61
(2.146)
We may find another relation by multiplying (2.140) by
∂Φ
∂ξ
and again inte-
grating from ξ = 0 to ξ = ∞. Using (2.144) and (2.146) one finds,
1 2
σ̃ =
2
N
X
i=1
!
Ci (∞) (exp(−zi Φ(0)) − 1) − ρ̃0 Φ(0)
≡ F (Φ(0)).
(2.147)
We estimate F (Φ) from below. We note first that
F (0) = 0,
∂F
(0) = f (0) = 0.
∂Φ
(2.148)
We examine the second derivative of F with respect to Φ. For Φ > 0,
∂2F
∂f
=
∂Φ2
∂Φ
N
X
=
zi2 ci (∞) exp (−zi Φ)
i=1
>
X
zi2 ci (∞) exp (−zi Φ)
X
zi2 ci (∞).
zi <0
>
zi <0
Since zi is at least equal to 1 in absolute value,
∂2F
> C −,
∂Φ2
C− ≡
62
X
zi <0
ci (∞).
Using (2.148) and the above, for Φ(0) > 0 we conclude that,
F (Φ(0)) >
C−
Φ(0)2 .
2
Therefore, by (2.147), we find:
σ̃ 2 > C − Φ(0)2 .
For Φ(0) < 0, an identical argument shows that
σ̃ 2 > C + Φ(0)2 .
Since C − and C + are both order 1, (2.145) holds if σ̃ ≪ 1. By (2.126), this
implies:
θΦ(kl),a0 ≪ 1
(2.149)
The nondimensionalized membrane potential Φ(kl),a0 is of order 1. Thus, we
conclude that the linear approximation is valid if
θ≪1
(2.150)
This is the also equivalent to θ∗ ≪ 1 by (2.127). According to (2.46), the
above condition in dimensional form is:
Cm kB T /q
≪ 1.
qc0 rd
63
(2.151)
This means that the linear approximation is valid when the amount of membrane surface charge is small relative to the absolute total charge concentration in a layer of width rd . As noted in (2.48), this ratio is on the order of
10−2 and this justifies the linear approximation.
We shall set the term
∂
∂τV
∂θ
∂τV
to 0 for the following reason. Note:
1
1 ∂θ
∂
1
1
=− 2
=
+
θ
θ ∂τV
∂τV Γ(k) Γ(l)
(2.152)
where we have taken the time derivative of (2.127). Since the rightmost
expression is O(1),
∂θ
∂τV
must be O(θ2 ). We have shown above that linearizing
the Poisson-Boltzmann equation is equivalent to neglecting terms of order
higher than O(θ). In the sequel, we shall treat θ as being constant with
respect to the τV time variable (but see Section 4.7).
2.2.5
Inner-Intermediate Matching when lm ∼
We now consider inner-intermediate matching when lm ∼
√
√
β
β. We first
introduce a coordinate system XV scaled uniformly with respect to X.
X=
p
βXV
64
(2.153)
The relevant equations now become:
∂Ci
= −∇XV · Fi
∂τV
Fi = −D̃i (∇XV Ci + zi Ci ∇XV Φ)
!
N
X
β∆XV Φ = − ρ˜0 +
zi Ci
(2.154)
(2.155)
(2.156)
i=1
We must now introduce a local coordinate system ξ a such that ξ1a is perpendicular to the membrane and ξ2a and ξ3a are curvilinear coordinates on
the membrane. We shall take the ξ1a coordinate vector to be of unit length
with respect to the XV coordinate system, and let ξ1a = 0 correspond to the
membrane surface. The positive ξ1a axis will point into the region of interest,
Ω(k) . We shall take the coordinate system to be right-handed in the sense
that the Jacobian of the coordinate transformation is positive. We introduce
the following:
J≡
∂ξ a
,
∂XV
J = (n, JS ), JS = (p2 , p3 )


T
1 0 
T
g ≡ JT J = 
 , g S ≡ JS JS
0 gS
(2.157)
(2.158)
The matrix J is the Jacobian matrix of the coordinate transformation and
g the metric associated with the coordinate system ξ a . The vector n is
the ξ1a coordinate vector and coincides with the unit (inward) normal to the
membrane when ξ1a = 0. Vectors p2 , p3 are the coordinate vectors tangent
65
to the membrane at ξ1a = 0, and make up the column vectors of JS . Since
n · n = 1, n · p2 = n · p3 = 0, the metric g is block diagonal with the (1, 1)
element being equal to 1. The 2 ×2 matrix gS is the metric on the coordinate
surfaces corresponding to ξ1a = const. It is important to note that all of the
above are O(1) quantities. This is because we made a change of coordinate
system from XV and not from X.
We rewrite (2.154)-(2.156) in terms of ξa . Although it is possible to calculate this expression by brute force, we choose not to, by seeking expressions
for the gradient and divergence operators in the ξ a coordinate system. What
follows is well-known, but we shall present a derivation nonetheless to clarify
notation.
First, noting that
∂
∂XV
=
∂ξa ∂
,
∂XV ∂ξa
the gradient of an arbitrary function φ
transforms as:
∂φ
∇XV φ = J∂ξa φ = n a + JS ∂S φ,
∂ξ1
∂S =
∂
∂
, a
a
∂ξ2 ∂ξ3
T
(2.159)
We note that JS ∂S is nothing other than the surface gradient operator which
we shall denote as ∇S ≡ JS ∂S .
We now consider divergence. We employ an integration-by-parts argument. The divergence of a vector field v may be defined by the scalar function
ψ such that for arbitrary smooth scalar function φ with compact support,
66
the following holds:
−
Z
(v, ∇XV φ)dXV =
Z
ψφdXV
(2.160)
where (·, ·) is the inner product of vectors in R3 . Using (2.159),
−
Z
(v, ∇XV φ)dXV = −
Z
(v, J∂ξa φ)dXV
Z
= − (J T v, ∂ξa φ)|J|dξa
Z
= (∂ξa · (|J|J T v))φdξa
Z
= (|J|−1∂ξa · (|J|J T v))φdXV
(2.161)
where |J| is the Jacobian (determinant of the Jacobian matrix). Comparing
the above with (2.160), we conclude that:
∇XV · v = |J|−1 ∂ξa · (|J|J T v)
(2.162)
We rewrite this slightly by noting:
|g| = |gS | = |J T J| = |J|2 , thus |J| =
67
p
|gS |
(2.163)
Therefore,
∇XV · v = p
1
p
∂ξa · ( |gS |J T v)
|gS |
p
1
∂ p
1
T
p
|g
|(n,
v)
+
=p
|g
|J
v
∂
·
S
S
S
S
|gS | ∂ξ1a
|gS |
∂
= a (n, v) + κ(n, v) + ∇S · v
∂ξ1
Here, ∇S · v = √1 ∂S ·
|gS |
p
The scalar quantity κ = √1
|gS |
(2.164)
|gS |JST v is the surface divergence operator.
p
∂
|gS | is equal to the sum of the principal
a
∂ξ
1
curvatures (twice the mean curvature) of the surface ξ1a = const. Thus, κ is
equal in particular to twice the mean curvature of the membrane at ξ1a = 0.
We change the coordinate system of (2.154)-(2.156) from XV to ξ a . With
the above expressions,
a
∂Fi1
∂Cia
a
a
=−
+ κFi1 + ∇S · FiS
∂τV
∂ξ1a
∂
∂
a
Fi1 = −D̃i
Ci + zi Ci a Φ
∂ξ1a
∂ξ1
FaiS = −D̃i (∇S Ci + zi Ci ∇S Φ)
!
N
X
∂
∂Φ
zi Ci
β
+κ
+ ∆S Φ = − ρ˜0 +
∂ξ1a
∂ξ1a
i=1
(2.165)
(2.166)
(2.167)
(2.168)
where ∆S ≡ ∇S · ∇S is the Laplace-Beltrami operator on the ξ1a = const
coordinate surfaces.
68
Now, rescale the the coordinate system in the inner layer so that:
ξ b = (ξ1b , ξ2b , ξ3b ),
ξ1a =
p
βξ1b ,
ξ2a = ξ2b ,
ξ3a = ξ3b
(2.169)
The equations in the inner layer are:
∂C b
β i =−
∂τV
p
∂Fi1b
βκFi1b + β∇S · FbiS
+
∂ξ1b
(2.170)
Fi1b = −D̃i
∂ b
b ∂
C + zi Ci b Φ
∂ξ1b i
∂ξ1
FbiS = −D̃i (∇S Cib + zi Cib ∇S Φ)
!
N
X
p
∂Φb
∂
b
b
zi Ci
+ βκ
+ β∆S Φ = − ρ˜0 +
∂ξ1b
∂ξ1b
i=1
(2.171)
(2.172)
(2.173)
which are supplemented with the boundary conditions:
θ
∗
∂Φb −Φ )=−
∂ξ1b ξb =0
1
− zi Fi1b ξb =0 = βαj̃i
( Φb ξb =0
1
(l)
(2.174)
(2.175)
1
Much of the ensuing calculations are the same as the lm >
√
β case, although
we shall find that there is an additional term in the final matching conditions.
We first take (2.170) and (2.175) to leading order to conclude that Fi1b0 ≡ 0
69
√
in the inner layer. Next, take the same equations to O( β).
∂Fi1b1
+ κFi1b0 = 0,
∂ξ1b
Fi1b1 ξb =0 = 0
1
(2.176)
Since Fi1b0 ≡ 0, we conclude that Fi1b1 ≡ 0 in the inner layer. We have reached
√
the same conclusion as (2.91). Similarly to the lm > β case, we find that
(2.100) and (2.101) must be satisfied and that Ci and Φ satisfy (2.102)-(2.106)
√
to leading order. We now turn to the equivalent of (2.130) in the lm ∼ β
case.
b2
∂Fi1
∂Cib0
b0
=−
+ ∇S 0 · FiS 0
∂τV
∂ξ1b
− zi Fi1b2 ξb =0 = αj̃i
1
(2.177)
(2.178)
where:
b0
b0
b0
∇S 0 ≡ JS0 ∂S = JS |ξb =0 ∂S
Fb0
iS 0 = −D̃i (∇S 0 Ci + zi Ci ∇S 0 Φ ),
1
q
1
|gS0 |JS0,T , gS0 ≡ gS |ξb =0
∇S 0 · = p 0 ∂S ·
1
|gS |
(2.179)
(2.180)
We are thus approximating JS and hence gS as being constant along ξ1b coordinate lines. This is true to leading order, since JS has order 1 variation in
√
the ξa coordinate system and we have rescaled so that ξ1a = βξ1b . Thus, the
√
error we incur by approximating JS by JS0 should be order β. We follow
70
the same path as in the lm >
−zi Fi1b2
√
β case.
Z
= αj̃i + zi
ξ1b
0
∂Cib0
b0
+ ∇S 0 · FiS 0 dξ1b .
∂τV
(2.181)
We now evaluate the above integral explicitly. The integral of the first term
√
∂Cib0
is the same as in the lm > β case and is given by (2.132). We consider
∂τV
the integral of the surface divergence term.
Z
ξ1b
0
∇S 0 ·
b
Fb0
iS 0 dξ1
= ∇S 0 ·
= ∇S 0 ·
Z
ξ1b
b
Fb0
iS 0 dξ1
0
b
Fa0
iS 0 ξ a =0 ξ1
1
+ ∇S 0 ·
Z
0
ξ1b
b
a0
Fb0
iS 0 − FiS 0 dξ1
(2.182)
We now evaluate the above integral using (2.115) and (2.117).
Z
ξ1b
0
Fb0
iS 0
−
Fa0
iS 0
dξ1b
=
−D̃i JS0
+ D̃i JS0
Z
Z
0
ξ1b
0
ξ1b
∂S Cib0 + zi Cib0 ∂S Φb0 dξ1b
∂S Cia0 (0) + zi Cia0 (0)∂S Φa0 (0) dξ1b
= D̃i JS0 (Idiff + Idrift1 + Idrift2 + Iresidual )
(2.183)
71
where I = (I2 , I3 )T and,
Ipdiff
∂
=− a
∂ξp
Z
0
Ipdrift2
Ipresidual
b
Cib0 − Cia0 (0) dξ1b
ξ1
∂
zi Cia0 (0)σ̃
=− a
∂ξp 0
Γ
Z ξ1b
∂
=−
zi Cia0 (0) a Φb0 − Φa0 (0) dξ1b
∂ξp
0
Z ξ1b
∂
σ̃
= zi Cia0 (0) a
∂ξp 0 Γ
Z ξ1b
∂Φa0 (0) b
=−
zi (Cib0 − Cia0 (0))
dξ1
∂ξpa
0
Z b
∂Φa0 (0) ξ1 zi Cia0 (0)σ̃
= −zi
∂ξpa
Γ
0
Z ξ1b
∂
=−
zi (Cib0 − Cia0 (0)) a Φb0 − Φa0 (0) dξ1b
∂ξp
0
Z ξ1b
zi Cia0 (0)σ̃
σ̃
b ∂
b
= zi
exp(−Γξ1 ) a
exp(−Γξ1 ) dξ1b.
Γ
∂ξp Γ
0
Z
Ipdrift1
ξ1b
(2.184)
(2.185)
(2.186)
(2.187)
We thus conclude:
−zi Fi1b2
= αj̃i + zi
+ zi I
charge
∂Cia0 (0)
a0 + ∇S 0 · FiS 0 ξa =0 ξ1b
1
∂τV
+ zi ∇S ·
D̃i JS0 (Idiff
72
+I
drift1
+I
drift2
(2.188)
+I
residual
)
We consider the limit of each term as ξ1b → ∞.
lim zi Ipdiff = −
ξ1b →∞
lim
ξ1b →∞
zi Ipdrift1
∂ σ̃i
∂ξpa
(2.189)
∂
= λ̃i Γ
∂ξpa
2
lim zi Ipdrift2 = −zi σ̃i
ξ1b →∞
σ̃
Γ2
∂Φa0 (0)
∂ξpa
(2.190)
(2.191)
The term Ipresidual is a term of order θ2 and may be neglected as long as we are
in the regime in which the linearization of the Poisson-Boltzmann equation
is valid. We thus have,
−zi Fi1b2
∂ σ̃i
+ αj̃i − ∇S 0 · (D̃i σ̃(∇S 0 λ̃i + zi λ̃i ∇S 0 Φa0 (0) − λ̃i Γ2 ∇S 0 Γ−2 ))
=
∂τV
a0
∂Ci (0)
a0 + ∇S 0 · FiS 0 ξa =0 ξ1b + O(exp(−Γξ1b ))
+ zi
1
∂τV
(2.192)
In exactly the same way as in the case lm >
√
β we obtain two matching
conditions:
∂ σ̃i
+ αj̃i − ∇S 0 · (D̃i σ̃(∇S 0 λ̃i + zi λ̃i ∇S 0 Φa0 (0) − λ̃i Γ2 ∇S 0 Γ−2 ))
∂τV
∂ σ̃i
=
(2.193)
+ αj̃i − ∇S 0 · D̃i σ̃i ∇S 0 (ln Cia0 (0) + zi Φa0 (0))
∂τV
a0
∂Fi1 (0)
∂Cia0 (0)
a0
(2.194)
+ ∇S 0 · FiS 0 (0)
=−
∂τV
∂ξ1a
−zi Fi1a1 (0) =
The second expression (2.194) is automatically satisfied as can be seen by
73
taking (2.166) to leading order and noting that Fi1a0 |ξa =0 = 0 by (2.100)
1
(which is valid in the present case as well).
Equation (2.193) is the matching condition that corresponds to (2.138)
√
in the case lm ∼ β. Thus, we have an additional term that describes some
form of drift diffusion along the membrane surface. The second line of (2.193)
may be more illuminating. The surface gradient of the chemical potential
potential µi = ln Cia0 +zi Φa0 , scaled by the diffusion coefficient, gives the drift
velocity of σ̃i along the membrane. This is interesting since drift-diffusion
can be written as the concentration Ci drifting down the chemical potential
gradient. The surface drift term above can be thought of as replacing Ci
with σ̃i .
2.2.6
Electroneutral Model
First of all, we summarize what we have found so far. The major conclusions
of our analysis are the following matching conditions we found at the innerintermediate layer interface. These relations serve as boundary conditions for
equations in the intermediate layer. The relations we have found are (2.100)
√
and (2.138) when lm > β,
Fi0a0 (ξ1a = 0) = 0
zi Fi1a1 (ξ1a = 0) =
(2.195)
∂ σ̃i
+ αj̃i
∂τV
σ̃i = λ̃i σ̃,
(2.196)
σ̃ = θΦ(kl),a0 ,
74
z 2 C a0
λ̃i = PN i i2 a0 .
i′ =1 zi′ Ci′
(2.197)
When lm ∼
√
β, (2.196) above is replaced by:
zi Fia1 (ξ1a = 0) =
∂ σ̃i
+ αj̃i + ∇S · (uµi σ̃i )
∂τV
µi = ln Cia0 + zi Φa0
uµi = −D̃i ∇S µi ,
(2.198)
(2.199)
Our goal now is to find a useful set of equations that is numerically tractable,
and can capture the biophysics as described in the above relations to a suitable degree.
We shall consider the lm >
√
β case for now. We shall address the surface
drift term later. Our attempt at a useful model is the following.
0=
∂Ci
+ β∇X · Fi
∂τV
0 = ρ˜0 +
zi Fi · n(kl) = θ
N
X
zi Ci
(2.200)
(2.201)
(2.202)
i=1
(kl)
∂ λ̃i Φ
∂τV
+ αj̃i
(2.203)
By substituting (2.75) and (2.76) into the above, we obtain a series of
relations which are satisfied by Ci0 , Ci1 , Φ0 , Φ1 , · · · in the outer layer and
Cia0 , Cia1 , Φa0 , Φa1 , · · · in the intermediate layer if Φ and Ci satisfy the above
model rather than the Poisson model. By design, we see that all expressions
that involve Ci0∼3 , Φ0∼3 only in the outer layer and Cia0 , Cia1 , Φa0 , Φa1 only in
the intermediate layer are identical to those obtained for the Poisson model,
75
including the matching conditions (2.195) and (2.196). Therefore the above
system seems to be the model we are looking for; the model is consistent with
√
the Poisson model to expressions of order O( β) in the intermediate layer,
and expressions of higher order in the outer layer. The model does not seem
to possess the fine spatiotemporal scales associated with charge relaxation,
so that we do not have to resolve these scales when performing numerical
simulation.
This model is indeed the original model we proposed in [30]. Unfortunately, this system is ill-posed. We shall now examine this ill-posed behavior
in the hope that this study will lead us to a well-posed model that has desirable approximation properties.
Ill-posed Behavior
For simplicity, we assume we are dealing with two regions, one intracellular
and one extracellular. Consider a situation in which there are no ion channels
in the membrane, and suppose that we have two positive ionic species of
valence 1. We suppose that the positive ionic charges are counterbalanced
completely by immobile charges with valence −1 and a spatially uniform
concentration of 1. Assume, moreover, that the diffusion coefficients for the
two ionic species are the same and are equal to 1. After these simplifications,
76
the electroneutral equations become:
∂Ci
+ ∇XV · Fi
∂τV
(2.204)
Fi = − (∇XV Ci + Ci ∇XV φ)
(2.205)
0=
1 = C1 + C2
(2.206)
√
We have uniformly rescaled the coordinate system to XV = X/ β, and have
chosen to use the time scale τV . In writing the boundary conditions, we shall
take n to be the outward normal pointing from intracellular to extracellular
and the membrane potential [Φ] = Φi − Φe . The boundary conditions on the
intracellular side of the membrane are:
p ∂ σ̃i(i)
β
= Fii · n
∂τV
(i)
(2.207)
σi = Cii θ[Φ]
(2.208)
p ∂ σ̃i(e)
β
= −Fei · n
∂τV
(2.209)
On the extracellular side:
(e)
σi = −Cie θ[Φ]
77
(2.210)
We solve the above with the following initial condition:
[Φ](x, 0) = Φ0 6= 0
(2.211)
Ci (XV , 0) = Ci,0(XV ), C1,0 (XV ) + C2,0 (XV ) = 1
(2.212)
We thus assume that the membrane potential is constant(= Φ0 ) throughout,
whereas the ionic concentration may initially be nonuniform. Intuitively, the
system should relax to an equilibrium state in which the ionic concentration
gradients have disappeared.
We demonstrate now that this initial value problem is ill-posed. First,
differentiating (2.206) with respect to τV , we find,
∇XV · (F1 + F2 ) = 0
(2.213)
∇XV · (−(∇(C1 + C2 ) + (C1 + C2 )∇XV φ)) = 0
(2.214)
By (2.205),
Using C1 + C2 = 1, we find
∆XV Φ = 0
(2.215)
To obtain boundary conditions at the membrane, we may sum (2.207),
78
(2.209) in i and use (2.208) and (2.210) to obtain:
p
βθ
∂[Φ]
∂Φi
∂Φe
=−
=
∂τV
∂n
∂n
(2.216)
The equations (2.215), (2.216), (2.211) together form an initial value problem
for φ and this has a unique solution: Φ does not change, and is constant
within each spatial region.
Now that we know the evolution of Φ, we can go back and substitute this
to obtain equations satisfied by Ci . We obtain:
∂C1
= D∆XV C1
∂τV
(2.217)
C2 = 1 − C1
(2.218)
and the boundary conditions:
∂C1i
∂C1i
βθΦ0
=−
∂τV
∂n
p
∂C1e
∂C1e
∂C1e
=−
=
βθΦ0
∂τV
∂n
∂ne
p
(2.219)
(2.220)
where ne is the normal pointing from the extracellular to intracellular space.
Thus, the evolution equations for the concentrations completely decouple
into two separate diffusion equations in the intracellular and extracellular
spaces.
79
Consider the intracellular concentration C ≡ C1i .
∂C
= ∆XV C
∂τV
p
∂C
∂C
βθΦ0
=−
∂τV
∂n
(2.221)
(2.222)
As we demonstrate below, the above is ill-posed if Φ0 < 0. In light of (2.219)
and (2.220), we must have Φ0 = 0 in order for the diffusion equations on the
two regions to be well-posed simultaneously.
The ill-posedness result for the diffusion equation (2.221), supplemented
with the boundary condition (2.222) kC +
∂C
∂n
= 0 where k is negative, was
formally established recently in [44]. Here, we shall illustrate this by way of
a simple example.
Consider equations (2.221) and (2.222), and let Φ0 = −1, on the upper
half plane XV = (XV , YV ), subject to the condition that C decays to 0 as
YV → ∞. We obtain a family of solutions parametrized by l > 1:
Cl (XV , τV ) = exp
l
l
τV − √ YV
2
βθ
βθ
sin
XV √ 2
√
l −l
βθ
(2.223)
This shows that if the initial data contain any non-zero frequency component
along the membrane, this component will grow exponentially, the exponent
being roughly proportional to the wave number. Thus, the problem is illposed.
Though we have only established this ill-posedness for a highly degenerate
80
situation (no ion channels, same diffusion coefficients, etc.), this instability
is most probably a generic feature of the equations. The term that is causing
this instability is the
the
∂ λ̃i
∂τV
∂C
∂τV
term in the boundary conditions. This came from
term. In general, the boundary conditions are more complicated
functions of the ionic concentrations, but the leading order terms
∂C
∂τV
and
∂C
∂n
will dominate in stability considerations since they involve derivatives. Since
the membrane potential [Φ] is multiplying this term, the diffusion problem
is bound to be ill-posed at least on one side of the membrane.
We note that the above does not invalidate our asymptotic calculations
or their consequences. Equation (2.196) is the boundary condition to be
used to solve for first order correction terms Ci1 and Φ1 given Ci0 and Φ0 .
In (2.196), σ̃i is determined by Ci0 and Φ0 . Therefore, the quantity σ̃i or
its time derivative is an externally given quantity that does not involve Ci1
or Φ1 . Therefore, as long as we seek successive approximations to Ci and
Φ, beginning with the leading order solution, we will not encounter ill-posed
behavior. In (2.203), the
∂ σ̃i
∂τV
term depends on quantities to be solved for,
and is to be determined as part of the solution to the model equations. This
is the crucial difference between the expressions obtained from the asymptotic calculations and the ill-posed model (2.200)-(2.203) inspired by these
calculations. We do not pursue the route of solving for successive approximations to Ci and Φ. Taking this route will inevitably involve solving for
approximations in the intermediate and outer layers separately, and patching these approximations together. This procedure is not very well-suited
81
for numerical computation. It would be numerically more straightforward
if we can instead modify (2.200)-(2.203) to find a system of equations that
is well-posed and asymptotically correct in both the intermediate and outer
layers.
A Well-posed model
We now take a closer look at the above situation in an attempt to obtain a
well-posed system of equations. Equation (2.223) tells us that the time constant associated with exponential growth in the ill-posed solution is at most
βθ2 , since l > 1. The time duration βθ2 measured in the τV time variable
belongs to the charge relaxation regime (actually even faster, by a factor of
θ2 ). The spatial scale that appears is on the order of the Debye length or
shorter (note that length is measured with respect to the XV coordinate,
√
and thus, β is the Debye length scale). Therefore, the instabilities that develop are inconsistent with our ansatz that the evolution of Ci and Φ do not
possess spatiotemporal scales associated with charge relaxation in the space
charge layer. We emphasize, as explained above, that the problem lies not
in the asymptotics we developed, but in the model we proposed, equations
(2.200)-(2.203).
What we would like to do is to smear out the explosive behavior exhibited
by
∂ λ̃i
∂τV
. We propose the following fix. Let λi be a quantity that evolves
82
according to the following differential equation.
z 2 Ci
λ̃i = PN i 2
i′ =1 zi′ Ci′
λ̃i − λi
∂λi
=
,
∂τV
β
(2.224)
This can be thought of as taking into account the charge relaxation time
scale by a simple ODE that involves only λi . This has the effect of filtering
out any temporal structure that exists on a time scale smaller than O(β).
Instead of λ̃i , we shall use λi in (2.203). We note that since we have taken the
relaxation time constant (= β) to be equal for all ionic species, the important
P
relation i λi = 1 holds true as long as this relation is satisfied at the initial
time.
We can find the discrepancy between λi and λ̃i as follows. We can solve
(2.224) so that:
1
λi (τV ) = λi (0) exp (−τV /β) +
β
Z
τV
λ̃i (s) exp
0
s − τV
β
ds
(2.225)
If τV is order 1, expanding λi (s) around τV , one can easily see that:
λi (τV ) =λ̃i (0) exp (−τV /β) + λ̃i (τV ) (1 − exp (−τV /β))
∂ λ̃i
(τV ) (1 − exp (−τV /β)) + · · ·
+β
∂τV
83
(2.226)
We see that λi (τV ) = λ̃i (τV ) + O(β) as long as
∂ λ̃i
∂τV
is O(1). Likewise,
1
∂ λ̃i
∂λi
(τV ) = − λ̃i (0) exp (−τV /β) +
(1 − exp (−τV /β))
∂τV
β
∂τV
∂ 2 λ̃i
+β
(τV ) (1 − exp (−τV /β)) + · · ·
∂τV 2
from which we find that
∂λi
∂τV
(τV ) =
∂ λ̃i
∂τV
(τV ) + O(β) as long as
∂ 2 λ̃i
∂τV 2
(2.227)
is O(1).
It is also possible to show that the error is O(β) when τV = O(β) provided
λi (0) = λ̃i (0). Since λi and
∂λi
∂τV
follow λ̃i and
∂ λ̃i
∂τV
to order O(β), replacing
(2.203) with (2.224)(rescaled so that τD is the time variable) will not alter
the formal approximation properties of the ill-posed model.
The important point is whether this resolves the ill-posedness issue. We
shall now show that this model is stable when applied to the above ill-posed
situation. With the above fix, the equations (2.221) and (2.222) are modified
as:
∂C
= ∆XV C
∂τV
p ∂λ
∂C
βθ
=
∂τV
∂n
C−λ
∂λ
=
∂τV
τλ
(2.228)
(2.229)
(2.230)
where we took Φ0 = −1. We have taken τλ as a parameter for now, and we
shall see what values of τλ are sufficient to remove the instability. As before,
solve the above in the upper half plane and look for solutions of the form
84
(2.223). One finds:
Cl (XV , τV ) = exp (lτV − mY a ) sin(kX a )
√
βθl
2
2
k = m − l, m =
τλ l + 1
(2.231)
(2.232)
Exponential growth corresponds to l > 0. We would therefore like to make
sure that the following equation for l does not have a positive solution for
any real k.
√
βθl
τλ l + 1
2
− l = k2
(2.233)
This is equivalent to showing that the left hand side of the above is non
positive when l ≥ 0. Note that l ≥ 0 implies:
√
βθl
τλ l + 1
2
βθ2 l2
−l
(τλ l + 1)2
2
βθ
≤
−1 l
4τλ
−l =
(2.234)
Therefore, τλ = β is more than adequate to make the above expression non
positive, since θ is a small number much less than 1.
We thus see that, at least for the above situation in which model (2.200)(2.203) fails, the new model is stable. What we have done is to add a stabilizing term to an asymptotically correct but ill-posed system. This is analogous
to the practice of adding numerical diffusion to stabilize an otherwise unstable numerical discretization of a partial differential equation. Therefore, the
85
equations we propose are:
0=
∂Ci
+ β∇X · Fi
∂τV
0 = ρ˜0 +
N
X
zi Ci
(2.236)
(2.237)
i=1
(kl)
∂λi Φ
+ αj̃i
∂τV
∂λi
λ̃i − λi
z 2 Ci
=
, λ̃i = PN i 2
∂τV
β
i′ =1 zi′ Ci′
zi Fi · n(kl) = θ
(2.235)
(2.238)
(2.239)
We shall call this the electroneutral model.
One important difference between the electroneutral model and the Poisson model is what the state variables are. In the Poisson model, specifying
the ionic concentrations at every point in space is enough to describe the
state of the system. The electrostatic potential can be found from the ionic
concentration profile by solving the Poisson equation 2.58 with the boundary conditions 2.59. The difficulty, though, is that we must specify the ionic
concentrations up to the boundary even within the space charge layer. On
the other hand, the state variables for the electroneutral model include the
ionic concentration profile as well as the membrane potential Φ(kl) and the
membrane charge fractions λi . The electroneutral model does not require the
ionic concentrations profiles in the space charge layer, but instead requires
several parameters that specify the concentration profile within this layer.
This means in particular that the initial conditions for the electroneutral
86
model include Ci , Φ(kl) and λi .
An important characteristic of the electroneutral model is that we have
ion conservation in the sense that the following relation holds for each ionic
species.
∂
∂τV
Z
zi Ci dV +
Ω(k)
Z
Γ(kl)
(k)
βθλi Φ(kl) dA
=−
Z
βαj̃i dA.
(2.240)
Γ(kl)
This equation says that the change in the sum of the ionic content of the
region Ωk and of the space charge layer is equal to the transmembrane current
that flows out of this region. This is an important property not only from
a physical point of view, but also from a practical point of view if we are to
perform long-time calculations of ionic concentration dynamics.
If we write the above electroneutral model in dimensional form, we recover
the model equations derived on the basis of physical arguments:
∂ci
+ ∇ · fi
∂t qzi ci
fi = −Di ∇ci +
∇φ
kB T
N
X
0 = ρ0 +
qzi ci
0=
i=1
87
(2.241)
(2.242)
(2.243)
The boundary conditions are:
(k)
(k)
qzi fi
∂λi φ(kl)
(kl)
+ ji
∂t
(k)
(k)
(k)
λ̃ − λi
z2c
∂λi
(k)
, λ̃i = PN i i (k)
= i 2
∂t
rd /D0
z 2′ c ′
′
· n(kl) = Cm
i =1 i
(2.244)
(2.245)
i
In Section 4.7, we shall computationally investigate how well the above system approximates the Poisson model in a one dimensional example.
When lm ∼ β, there is an additional surface drift-diffusion term. The
most natural way to handle this term is probably to replace (2.238) with:
zi Fi · n(kl) = θ
∂λi Φ(kl)
+ β∇S · (uµi σ̃i ) + αj̃i
∂τV
uµi = −D̃i ∇S µi ,
µi = ln Ci + zi Φ
(2.246)
(2.247)
where the membrane differential operator ∇S is now the one induced by the
Euclidean metric associated with distance measured in X coordinate system
rather than in XV coordinate system.
We shall henceforth deal only with (2.235)-(2.239). Including surface
drift-diffusion into model simulations will involve considerably greater computational effort. This is a direction for future investigation.
88
2.2.7
Equations in the Outer Layer
3D Cable Model
We now consider intermediate-outer matching. This, as we have argued, is
a valid consideration since we have taken α = O(1). We consider only the
√
easier case lm > β.
We shall use the following expressions satisfied in the intermediate layer
which follow from (2.84).
∂F a0
∂Cia0
= − i1a
∂τV
∂ξ1
a1
∂Ci
∂F a1
= − i1a
∂τV
∂ξ1
(2.248)
(2.249)
The boundary conditions for the above at ξ1a = 0 are given by (2.100) and
(2.138).
The difficulty we now face is that neither of the above equations can
be solved explicitly. The behavior of their solution as ξ1a → ∞, a piece of
information needed for matching to be successful, is unclear. Fortunately, we
do not need all of this information. To find out what we need, we turn to the
equations satisfied in the outer layer, which can be obtained by substituting
89
(2.76) and (2.75) into (2.72)-(2.74).
∂Ci1
∂Ci0
= 0,
=0
∂τV
∂τV
∂Ci0 ∂Ci2
+
= −∇X · F0i
∂τD
∂τV
N
X
0 = ρ˜0 +
zi Ci0
(2.250)
(2.251)
(2.252)
i=1
0=
N
X
zi Ci2
(2.253)
i=1
Equation (2.250) tells us that Ci to leading order does not change in the τV
time variable. Therefore, we do not need any boundary conditions for Ci0 .
We still need to know the evolution of Φ0 . This can be obtained by summing
(2.251) in i and and using (2.252) and (2.253) to conclude:
N
X
i=1
zi ∇ · F0i = 0
(2.254)
This is the equation satisfied by Φ0 in the outer layer. In order to obtain the
P
0
(kl)
boundary condition for this equation, all we need is N
.
i=1 zi Fi · n
P
Let J = N
i=1 zi Fi . We shall use the usual subscripts and superscripts on
J to denote terms of the expansion of J in β in the different layers, induced
by the expansion of Fi . First note from (2.86) that:
N
X
N
X
zi Cia0 = 0,
i=1
i=1
90
zi Cia1 = 0.
(2.255)
Using this and equation (2.248) and (2.249) above, we see that:
∂J1a1
∂J1a0
=
=0
∂ξ1a
∂ξ1a
(2.256)
Since we know the values of these quantities at ξ1a = 0, we see immediately
that:
N
J1a0
where we have used
= 0,
PN
i=1
−J1a1
∂Φ(kl),a0 X
=θ
+
αj̃i
∂τV
i=1
(2.257)
λ̃i = 1. Similarly to inner-intermediate matching,
we can now establish the matching condition. The result is:
N
X
∂Φ(kl),a0
j̃i = −J 0
+α
θ
∂τV
i=1
(2.258)
We can now use the above as the boundary condition for (2.254) and explicitly
write down the equations satisfied in the outer layer.
∇ · (A∇Φ0 + ∇B) = 0
(2.259)
−(A∇Φ0 + ∇B) · n(kl) = θ
(2.260)
∂Φ(kl),0
+ αIion
∂τV
N
N
N
X
X
X
2 0
0
j̃i
A=
zi Ci , B =
zi Ci , Iion =
i=1
i=1
(2.261)
i=1
Note here that A and B are functions of X only, and do not depend on time,
since Ci0 does not change in the τV time scale.
There is one difficulty here that needs to be pointed out. Equation (2.258)
91
and (2.260) are not the same. In (2.258), Φ(kl) is evaluated just outside the
inner layer, whereas in (2.260), Φ(kl) is evaluated just outside the intermediate
layer. There is a similar concern for the transmembrane current terms j̃i if
they are functions of Ci or Φ(kl) .
From (2.248) and (2.100), and the fact that Cia0 and Φa0 must match to
leading order at ξ1a = ∞ to the outer layer solution, we see that Cia0 and
Φa0 decay to a uniform state after an initial transient (note Cia0 decays to a
constant where as Φa0 decays to a time-varying uniform state, whose value
is equal to Φ0 (ξ1b = 0)). Therefore, after an initial transient, the discrepancy
between Φ(kl),0 , Φ(kl),a0 and Cia0 , Ci0 will be small, and therefore, the difference
between (2.258) and (2.260) will be negligible.
Notice that we have succeeded in obtaining the 3D generalization of the
traditional cable model. This model is valid to leading order outside the
√
intermediate layer of thickness O( β). We shall call this the 3D-cable model.
We note that the 3D-cable model is consistent with the electroneutral
model. This is true by design since the electroneutral model was constructed
√
to be asymptotically equivalent to the Poisson model to several orders in β
in the outer layer. If the 3D-cable model gives the leading order approximation to the Poisson model, it should do so for the electroneutral model as
well.
In fact, the relationship between the 3D-cable model and the electroneutral model is much easier to see than the relationship between the 3D-cable
model and the Poisson model. Consider equations (2.235)-(2.238) of the elec92
troneutral model. We can take the time derivative of the electroneutrality
condition (2.237) and substitute (2.235) to obtain the elliptic equation satisfied by the electrostatic potential, (2.259). Sum (2.238) in i and we find the
boundary condition (2.260). The coefficients A and B in equation (2.259)
are now time dependent, but we can see from (2.235), that to leading order,
the ionic concentrations do not change in the membrane potential time scale.
Thus, A and B are constant to leading order.
Simplified 3D-Cable Model
We reach a further simplification by considering the following situation. Suppose the long time average of the transmembrane currents j̃i is equal to 0.
That is to say, if we average over a sufficient long time, there is no net current
flowing through the membrane. An electrically active cell whose ion channel
currents are quickly counter-balanced by ionic pumps may fit this category.
Then, the ionic concentrations will relax to an equilibrium state in the τD
time scale (we shall discuss the ionic concentration dynamics on the τD time
scale in the next section). If there are no fixed charges ρ̃, or if the fixed
charges are spatially uniform, the resulting ion concentration profile will be
spatially uniform within each region. We apply the above 3-D cable model
93
to this situation. From (2.259)-(2.261), we obtain:
∆Φ0 = 0
−A(k) ∇Φ0 · n(kl) = θ
(2.262)
∂Φ(kl),0
+ αIion
∂τV
(2.263)
The gradient of B vanishes because of the spatial uniformity of Ci0 . Note
that A(k) is a constant that depends only on the region number (k), and
expresses the ohmic conductivity of the electrolyte medium. We shall call
this the simplified 3-D cable model.
Derivation of Standard Cable Model
We may re-derive the traditional cable model, derived in chapter 1, by considering the above simplified 3D-cable model under specialized geometry.
What follows is a “thin-domain” or “lubrication” type asymptotics, where
we seek a simplified system of equations under the assumption that the solution varies rapidly in one direction (the “thin” direction) compared to the
other directions [15].
Consider an infinite cylinder of radius r int . The dimensionless radius
will therefore be η ≡ r int /L0 . This infinite cylinder is surrounded by an
extracellular space which lies between the this cylinder and a concentric
cylinder of radius r ext (> r int ). This extracellular region is insulated at the
outer boundary. We shall let ξ = r ext /r int . Equations of the simplified 3D-
94
cable model (2.262) and (2.263) specialized to this situation are:
∂2Φ
+ ∆D Φ = 0 in Ωint , Ωext
∂Z 2
1 ∂Φ
1 ∂2Φ
∂2Φ
+
+
∆D Φ ≡
∂R2 R ∂R R2 ∂ψ 2
∂Φ
∂[Φ]
−A(k)
+ αIion , [Φ] ≡ Φint − Φext at R = η ±
=θ
∂R
∂τV
∂Φ
−Aext
= 0 at R = ηξ
∂R
(2.264)
(2.265)
(2.266)
(2.267)
To avoid cluttered notation, we have eliminated the superscript 0. In the
above, R is the radial, Z the axial, and ψ the angular coordinate. Equation
(2.266) is satisfied at R = η from both the intracellular (η − ) and extracellular
(η + ) sides. The superscript k denotes either the intra or extracellular region.
We shall now take η to be the small parameter in our system. We rescale
the the radial coordinate to R = ηρ in (2.264)-(2.267) so that the cell membrane corresponds to ρ = 1.
η2
∂2Φ
+ ∆D̃ Φ = 0 in Ωint , Ωext
∂Z 2
1 ∂2Φ
∂ 2 Φ 1 ∂Φ
+
+
∆D̃ Φ ≡
∂ρ2
ρ ∂ρ
ρ2 ∂ψ 2
A(k) ∂Φ
∂[Φ]
−
=θ
+ αIion , [Φ] ≡ Φint − Φext at ρ = 1±
η ∂ρ
∂τV
∂Φ
= 0 at ρ = ξ
−Aext
∂ρ
95
(2.268)
(2.269)
(2.270)
(2.271)
We now expand Φ in powers of η p in the following fashion:
Φ = Φ0 + η p Φ1 + · · ·
(2.272)
We let p = 2 so that we obtain nontrivial expressions when the above substituted into (2.268):
∆D̃ Φ0 = 0
(2.273)
∂ 2 Φ0
+ ∆D̃ Φ1 = 0
∂Z 2
(2.274)
The boundary condition (2.270) requires some care. Upon substitution of
(2.272), we see that a distinguished limit can be obtained by taking α ∼ η.
This is in fact, hardly surprising. We introduced α as the volume to surface
ratio of the domain of interest. The dimensionless radius η is exactly equal
to this ratio (up to a factor of order 1). We shall thus take η = α. Therefore,
∂Φ0,(k)
= 0 at ρ = 1± , ρ = ξ
∂ρ
θ ∂[Φ0 ]
∂Φ1
=
+ Iion
−A(k)
∂ρ
α ∂τV
(2.275)
(2.276)
First of all, (2.273) with (2.275) tells us that Φ0 is constant for fixed Z. In
order to find the Z dependence of Φ0 , we need to look at the next order,
(2.274). The solvability of the this equation with respect to Φ1 requires that
the following identities between an area and a line integral hold for each
96
Z = Z0 .
∂Φ1
ds
ρ=1− ,Z=Z0 ∂ρ
ρ<1,Z=Z0
Z
Z
Z
∂Φ1
∂Φ1
1
∆D̃ Φ dA = −
ds +
ds
ρ=1+ ,Z=Z0 ∂ρ
1<ρ<ξ,Z=Z0
ρ=ξ,Z=Z0 ∂ρ
Z
1
∆D̃ Φ dA =
Z
(2.277)
(2.278)
where dA denotes an area integral and ds denotes a line integral. Applying
the above to (2.274) and (2.276) we find that:
Z 2π
θ ∂[Φ0 ]
Φ0,int
+
= 2π
Iion dψ
πA
∂Z 2
α ∂τV
0
Z 2π
2 0,ext
θ ∂[Φ0 ]
2
ext ∂ Φ
−π(ξ − 1)A
= 2π
+
Iion dψ
∂Z 2
α ∂τV
0
int ∂
2
(2.279)
(2.280)
Dividing by the prefactors and adding the two expressions, we immediately
obtain the cable equations:
G
[Φ0 ]
θ ∂[Φ0 ]
= 2π
+ Iion (Z)
∂Z 2
α ∂τV
Z 2π
1
1
1
1
=
+
, Iion (Z) =
Iion dψ
Geff
πAint π(ξ 2 − 1)Aext
2π 0
eff ∂
2
(2.281)
(2.282)
If 1 ≪ ξ ≪ α−1 (r int ≪ r ext ≪ L0 ), we can take the extracellular space
to be an isopotential compartment and set Geff = πAint without sacrificing
the validity of the above cable equations. In dimensional terms, the above
97
equations take the following familiar form:
∂φm
1 ∂ 2 φm
= pm C m
+ iion ,
R ∂z 2
∂t
pm = 2πr int
R = Rint + Rext
(2.283)
(2.284)
N
X (qzi )2 Di
1
int
=S
cint
,
int
R
kB T i
i=1
S int = π(r int )2
(2.285)
S ext = π((r ext )2 − (r int )2 )
(2.286)
N
X (qzi )2 Di
1
ext
=
S
cext
,
ext
R
kB T i
i=1
Here, φm is the membrane potential and iion is the dimensional transmembrane current, averaged over the angular variable.
We note that the above derivation of the cable model did not assume an
axisymmetric solution to the equations. The axisymmetry, or more strongly,
the constancy of the electrostatic potential for each Z cross-section is a consequence purely of the scaling relations. Related to this is the observation
that the above can be generalized to arbitrary cross-sectional geometry. All
we have used is the divergence theorem as applied to each cross section; we
have made essentially no use of the fact that the cross section was a disc.
Ion Concentration Dynamics in the Slow Diffusion Time Scale
We finally discuss the behavior of the electroneutral model and the Poisson
model in the τD time variable. What follows is a plausibility argument and,
unlike the foregoing discussions, does not provide a systematic treatment of
this important issue.
98
Rescale time in the electroneutral model (2.235)-(2.239) so that the time
variable is now τD instead of τV .
0=
∂Ci
+ ∇X · Fi
∂τD
0 = ρ˜0 +
zi Fi · n(kl) = βθ
N
X
zi Ci
i=1
∂λi Φ(kl)
∂τD
+ αj̃i
(2.287)
(2.288)
(2.289)
(2.290)
We would like to analyze the long time behavior of the above system. More
precisely, we would like to average the above system on a time scale that is
slower than described by the τV time variable but faster than described by
the τD time variable. We shall set our focus on the evolution of the physical
variables in the outer layer.
The only quantity that varies in the τV time variable in the outer layer is,
as we saw in the discussion of the 3D-cable model, the electrostatic potential.
Thus, Φ and Φ(kl) are rapidly varying quantities when observed in the τD time
variable. Suppose there exists an intermediate time scale τM in between
τV and τD , in which the electrostatic potential is varying rapidly but the
ionic concentrations can still be assumed constant. By averaging equations
99
(2.287)-(2.290) in this time scale, one finds:
0=
∂Ci
+ ∇X · F i
∂τD
0 = ρ˜0 +
N
X
zi Ci
(2.291)
(2.292)
(2.293)
i=1
zi Fi · n(kl) = αj̃i
(2.294)
The overlined quantities above are the result of averaging that was performed
over the τM time scale. We have assumed that Φ( kl) remains bounded (as
should be in the physiologically relevant case) so that the capacitative current
term disappears in the averaging process. The ionic concentrations Ci are
not overlined because we have assumed that Ci remains constant in the τM
time scale. We therefore conclude that according to the electroneutral model,
the ionic concentration dynamics in the τD time scale is dictated by (2.291)(2.294).
Compare this to the Poisson model, (2.56)-(2.60) which we reproduce
100
below:
0=
∂Ci
+ ∇X · Fi
∂τD
2
−β ∆X Φ = (ρ˜0 +
θ∗ Φ(kl)
N
X
zi Ci )
(2.295)
(2.296)
(2.297)
i=1
∂Φ
= β (kl)
∂n
(2.298)
(2.299)
We would like to perform a similar averaging for the Poisson model, focusing
on the evolution of physical variables in the outer layer. In the outer layer,
(2.297) reduces to the electroneutrality condition since β is small. Suppose
we can neglect the boundary condition (2.298) which dictates the behavior of
the electrostatic potential in the Debye layer, on grounds that both sides of
(2.298) is subject to rapid fluctuations in membrane potential. This leads to
the same averaged equations (2.291)-(2.294). As long as the average transmembrane current αj˜i is the same in the Poisson and electroneutral models,
we may argue that the ion cocnentration dynamics of the two models in the
outer layer in the slow diffusion time scale are identical.
If the argument above can be fully justified, the electroneutral model
can be viewed as an integration between the averaged slow drift-diffusion
equations (2.291)-(2.294) and the 3D-cable model. The electroneutral model
can be viewed as being consistent with both models, whose validity now
101
encompasses both the membrane potential time scale and the diffusion time
scale. The 3D-cable model and the averaged slow drift-diffusion equations
are valid only in the outer layer, but the spatial validity of the electroneutral
model now extends to within the intermediate fast diffusion layer.
2.2.8
Discussion
We have demonstrated that we can derive the electroneutral model from the
Poisson equations using matched asymptotics. We introduced two spatial
layers at the membrane, the inner space charge layer, and the intermediate
fast diffusion layer. We distinguished two cases depending on the relative
√
magnitude of the thickness of the fast-diffusion layer β and the membrane
√
length scale lm . In the lm ∼ β case, we saw that we have an additional
surface drift diffusion term. The most straightforward model that one can
construct from asymptotic considerations turned out to be ill-posed, which
was fixed by adding a small stabilizing term, chosen so as not to affect the
asymptotic validity of the equations. The resulting electroneutral model was
shown to be valid in the intermediate layer as well as in the outer layer.
We have also succeeded in systematically deriving the standard cable
model from the Poisson equations. We may say that this is an effort to derive the traditional cable model from first physical principles. The above can
be viewed as a significant step toward a full study of the validity of the traditional cable model, an issue of fundamental importance to computational
neuroscience [37]. In the course of this derivation, we have seen that there are
102
models of intermediate complexity in between the Poisson or electroneutral
model and the traditional cable model. The 3D-cable model and the simplified 3D-cable model describe the dynamics of the electrostatic potential
in the fast time scale. We shall call this the model hierarchy. We believe
that each of these models will be suitable in certain situations, the Poisson
or electroneutral models being the most complete.
There are several methodological and asymptotic issues that remain unresolved.
The first point is that we have treated α = O(1) with respect to β.
√
There are certainly cases in which α ∼ β, in which case it may be more
√
√
appropriate to let α = O( β). Calculations with the scaling α = O( β)
do not seem to result in a sensible model. The correct way to handle this
case is not clear, and may involve, for example, a reworking of the nondimensionalization suggested above. Despite this concern, we shall show in
Section (4.7) that the electroneutral model approximates the Poisson very
√
well even when α ∼ β.
The second point is to formulate more precisely the ideas set forth in the
previous section on the long time dynamics of ionic concentrations. It should
also prove beneficial to investigate the deviation between the electroneutral
model and the Poisson model numerically by performing long time calculations.
The third point is that we have not succeeded in performing matched
√
asymptotics between the intermediate and outer layers when lm ∼ β. The
103
expected physical picture is that the intermediate layer acts as a lateral mixing layer. There are ion concentration gradients of high spatial frequency
parallel to the membrane at the inner-intermediate layer interface. The lateral mixing should smear out such gradients by the time one reaches the
intermediate-outer layer interface. Matching between the intermediate and
outer layer will probably require a precise description of this lateral mixing
which takes place in the intermediate layer.
Asymptotics (without an accompanying proof of validity) is subject to
criticism that it only provides a plausibility argument. In the Section 4.7,
we show computational evidence which demonstrates that the electroneutral
model approximates the Poisson model to great accuracy.
104
Chapter 3
Analysis
In this chapter, we perform an analytical study of the system of equations
we proposed in the previous section. As a first step toward an analysis of
the full electroneutral model, we study a simplified version of the model. We
prescribe ion concentration dynamics as well as the gating variable dynamics.
With this simplification, we are left with the electrostatic potential φ as the
only unknown. We shall see that φ satisfies a linear elliptic equation with
nonlinear evolving boundary conditions. We shall prove that this system has
a unique solution. Our method of proof makes use of the Rothe’s method,
or the method of horizontal lines [17] where we first discretize the system
in time and convert the evolution problem into a series of stationary problems. This method of proof parallels the numerical scheme we use to perform
simulations. We obtain an existence-uniqueness result as well as a proof of
convergence of a backward Euler type scheme where the spatial variable is
105
kept continuous. We conclude by noting that, under certain simplifications,
the equation for φ can be seen as satisfying an equation of reaction-diffusion
type, where the laplacian is replaced by −1 times the Dirichlet to Neumann
map.
3.1
Simplification and Model Problem
We first recall the equations for the electroneutral model. We shall slightly
alter notation to avoid cluttered notation. We let Ω be an open connected
and bounded set in R3 . The region Ω is partitioned into two disjoint open
sets, Ωint and Ωext . The interface between the two open sets is denoted Γ.
We denote the boundary of Ω to be Γf . We suppose that Γ ∩ Γf = ∅, and
that Γf ⊂ Ωext . We shall allow Ωint to be composed of multiple disconnected
components, each of which corresponds to the cytoplasm of different cells
contained in the region of interest. We shall for simplicity assume that Ωext
is connected.
The region Ωext would need to be comprised of multiple disconnected
components if we were to model the presence of intracellular organelles. The
interior of the endoplasmic reticulum, or the matrix of a mitochondrion,
would then be considered a component of Ωext (the essentially extracellular
nature of the interior of a membrane-bound organelle is termed ”topological
equivalence” in cell biology).
We shall thus refer to Ωint and Ωext as intracellular and extracellular
106
regions respectively. We let n denote the unit normal on Γ pointing from
Ωint to Ωext . Given this notation, we have,
∂ci
+ ∇ · fi
∂t qzi ci
fi = −Di ∇ci +
∇φ
kB T
n
X
0 = ρ0 +
qzi ci
0=
(3.1)
(3.2)
(3.3)
i=1
The boundary conditions at the face of Γ facing Ωint are,
Cm
∂λint
i [φ]
= qzi fiint · n − ji
∂t
int
∂λint
λ̃int
zi2 cint
i
i − λi
int
, λ̃i = PN i 2 int
=
2
∂t
rd /D0
i′ =1 zi′ ci′
ji = Ji x, s, [φ], cint, cext
∂sg
= fg (s, [φ], cint, cext )
∂t
(3.4)
(3.5)
(3.6)
(3.7)
where φ ≡ φint − φext and ji is the transmembrane current that flows from
Ωint to Ωext . The corresponding boundary condition on the Ωext face of Γ
can be obtained by exchanging the superscripts int and ext in the first two
equations of the above boundary conditions.
We would like to focus our attention on the electrostatic potential. The
107
equations satisfied by the electrostatic potential are:
0 = ∇ · (a(x, t)∇φ + ∇b(x, t))
a(x, t) =
b(x, t) =
N
X
(qzi )2 Di
i=1
N
X
kB T
ci (x, t)
qzi Di ci (x, t)
(3.8)
(3.9)
(3.10)
i=1
supplemented with the boundary conditions:
Cm
∂[φ]
+ iion = −(a∇φint + ∇b) · n
∂t
= −(a∇φext + ∇b) · n
X
iion (x, s, φ, cint, cext ) =
ji
(3.11)
(3.12)
i
If the ionic concentrations above are assumed constant in time, the above
reduces to the governing equations (2.259)-(2.261) of the 3D-cable model of
Section 2.2.7.
We shall assume that the concentrations are given functions of time. We
shall assume further that the gating variables are either functions of the
membrane potential [φ] and time or given functions of time. We denote by
J(x, [φ], t) the functional form of iion after these simplifications are made.
Under this assumption, we are left with a system of equations which is sat-
108
isfied by φ only.
∇ · (a(x, t)∇φ + ∇b(x, t)) = 0
Cm
(3.13)
∂[φ]
+ J([φ], x, t) = −(a∇φint + ∇b) · n
∂t
= −(a∇φext + ∇b) · n
(3.14)
We see first that φ satisfies an elliptic equation in Ωint and Ωext . The boundary conditions are what makes this problem interesting. This is an elliptic
interface problem where the solutions in the regions Ωint and Ωext are coupled.
The coupling takes the form of a nonlinear evolutionary boundary condition.
There is a time derivative in [φ] corresponding to the capacitative current.
The term J corresponds to the instantaneous current-voltage relationship of
transmembrane currents, and are in general nonlinear functions of [φ].
In the foregoing, we never specified boundary conditions at the outer
boundary Γf . For definiteness, we shall take no-flux boundary conditions on
Γf so that the whole system is isolated:
−(a∇φ + ∇b) · nf = 0 on Γf
(3.15)
where nf is the outward normal on Γf . Other linear boundary conditions on
Γf can be dealt with by standard modifications of the ensuing analysis.
To complete the specification of the problem, we need initial conditions.
109
These are given by specifying φ(kl) at time t = 0:
[φ](x, t = 0) = φ0 (x).
(3.16)
We shall now show that the initial boundary value problem (3.13) supplemented with the boundary conditions (3.14) and (3.15), and initial condition
(3.16) has a suitably defined weak solution.
3.2
Weak Solution
We now define the weak solution of (3.13), (3.14), (3.15), and (3.16). To
this end, we shall first define the relevant function spaces in which we seek a
solution for this system.
We define the function space H as follows:
ψ = (ψ int , ψ ext ) ∈ H = Hint ⊕ Hext
ψ int ∈ Hint = H 1 (Ωint )
Z
ext
1
ext
ψ ∈ Hext = u|u ∈ H (Ω ),
Ωext
(3.17)
(3.18)
udV = 0
(3.19)
where H 1 is the L2 based Sobolev space. It is clear that H is a Hilbert space.
We may take the norm on H to be
2
2
2
kψk2H = ∇ψ ext L2 (Ωext ) + ∇ψ int L2 (Ωint ) + ψ int L2 (Ωint )
110
(3.20)
2
We may omit the term kψ ext kL2 (Ωext ) since the integral of ψ ext over Ωext is 0
and we may thus use the Poincare inequality. We shall often use the L2 inner
product (·, ·) defined on Ωint and Ωext . For any ψ1 , ψ2 ∈ H, we write:
(ψ1 , ψ2 )Ω ≡ (ψ1int , ψ2int )Ωint + (ψ1ext , ψ2ext )Ωext
(3.21)
For any function ψ ∈ H, we let:
[ψ] = ψ int Γ − ψ ext Γ
(3.22)
We note by the trace theorem that [ψ] ∈ H 1/2 (Γ) ⊂ L2 (Γ). We shall frequently make use of this fact.
We solve the initial boundary value problem to time t = T . We define the
weak formulation of the problem as follows. Let w be an arbitrary smooth
test function w ∈ L2 ([0, T ]; H) such that [w] ∈ H 1 ([0, T ]; L2 (Γ)). We let φ
be a function φ ∈ L2 ([0, T ]; H) such that [φ] ∈ H 1 ([0, T ]; L2 (Γ)) that satisfies
the following relation for any such w:
Z
0
T
(∇w, a∇φ + ∇b)Ω dt +
Z
T
0
∂[φ]
+ J([φ], x, t) dt = 0 (3.23)
[w], Cm
∂t
Γ
[φ]|t=0 = φ0
(3.24)
We point out that since [φ] ∈ H 1 ([0, T ]; L2 (Γ)), [φ] ∈ C([0, T ]; L2 (Γ)). Thus
we can make sense of the initial condition (3.24). We shall also place some
111
restrictions on the coefficients of the elliptic problem a and b. We let:
a ∈ W 1,∞ ([0, T ]; L∞ (Ω)),
∇b ∈ W 1,∞ ([0, T ]; L2 (Ω))
(3.25)
We shall impose some conditions on the transmembrane current function
J. The function J(s, x, t), where in s we substitute the transmembrane
potential, is assumed to satisfy the following properties:
|J(0, x, 0)| ≤ C0
(3.26)
uniformly in x and that J is globally Lipschitz continuous.
∂J ≤ Ct ,
∂t ∂J ≤ Cs
∂s (3.27)
both uniformly in s, x, t. We are thus assuming that J is Lipschitz in s and
t. Being Lipschitz in s has the biophysical interpretation that the slope of
the instantaneous current voltage relationship (differential conductance) is
always bounded.
To justify the above definition for the weak solution, we shall first prove
the following.
Proposition 1. Suppose a, b and J are sufficiently smooth functions of x
and t. If the weak solution φ is C 2 , then φ is a classical solution to the
initial boundary value problem. Conversely, if φ is a classical solution, then
φ is a weak solution.
112
Proof. Consider (3.23). Since all quantities involved are continuous, by considering test functions w of the form w1 (t)w2 (x), we find that the integrand
of the time integration in (3.23) must be identically 0. Therefore, for any
smooth function w in H, we have:
∂[φ]
+ J([φ], x, t) = 0
(∇w, a∇φ + ∇b)Ω + [w], Cm
∂t
Γ
(3.28)
We integrate by parts the spatial integrations in (3.28), which we may perform since all functions above are assumed sufficiently smooth. We have:
(∇w, a∇φ + ∇b)Ωint =
=
−
Z
Z
Ωint
∇w(a∇φ + ∇b)dV
w int((a∇φ + ∇b) · n)dA
Γ
Z
Ωint
(3.29)
w(∇ · (a∇φ + ∇b))dV
Likewise, we have:
(∇w, a∇φ + ∇b)Ωext = −
Z
−
Z
Γ
Ωext
w ext ((a∇φ + ∇b) · n)dA
(3.30)
w(∇ · (a∇φ + ∇b))dV
There is a minus sign in front of the boundary term since n is the inward
pointing normal with respect to Ωext . We have also used the no-flux condition
113
at Γf . Therefore, we find:
−
Z
Z
w(∇ · (a∇φ + ∇b))dV −
w(∇ · (a∇φ + ∇b))dV
Ωext
Z
∂[φ]
int
int
+ J([φ], x, t) dA
(a∇φ + ∇b) · n + Cm
+ w
∂t
Γ
Z
∂[φ]
int
ext
(a∇φ + ∇b) · n + Cm
− w
+ J([φ], x, t) dA
∂t
Γ
Ωint
(3.31)
Since w is an arbitrary function in H, we find that:
∇ · (a∇φ + ∇b) = 0
in Ωint
(3.32)
∇ · (a∇φ + ∇b) = const
in Ωext
(3.33)
on Γ
(3.34)
Cm
∂[φ]
+ iion = −(a∇φint + ∇b) · n
∂t
= −(a∇φext + ∇b) · n
Note that we cannot conclude immediately that ∇ · (a∇φ + ∇b) = 0 in Ωext
R
because the test functions w ∈ H, and by (3.19), Ωext wdV = 0. In fact, we
can show that indeed ∇ · (a∇φ + ∇b) = 0 by the following:
Z
Ωext
∇ · (a∇φ + ∇b)dV =
=
Z
ZΓ
Γ
=
Z
−(a∇φext + ∇b) · ndA
−(a∇φint + ∇b) · ndA
Ωint
(3.35)
∇ · (a∇φ + ∇b)dV = 0
Therefore, ∇ · (a∇φ + ∇b) = 0 in Ωext . This shows that a weak solution, if
smooth, is a classical solution. That a classical solution is a weak solution
114
follows easily by tracing back the above argument.
3.3
Discretization in Time
In order to prove that there exists a weak solution to the above system,
we shall discretize in time and consider a series of elliptic problems with
nonlinear boundary conditions. Such an analysis is referred to as the “method
of horizontal lines” or “Rothe’s method” [17]. This analysis was inspired by
the numerical method we use to simulate the system.
We are interested in the solution of the equations from time t = 0 to
t = T > 0. We discretize this time interval into N equal intervals so that
N△t = T . We note that it is possible to consider more general partitions
that are not equispaced. We do not pursue this here since this leads to
cluttered notation.
Suppose
we
were
able
to
find
the
approximation
to
φ
at
tk−1 = (k − 1)△t, (k = 1 · · · N) which we denote by φk−1
∈ H. For a
N
while we shall suppress the dependence of φ on the number of time steps N.
We define φk as the function in H that satisfies the following for all smooth
w∈H
[φk ] − [φk−1 ]
+ J k ([φk ], x)
(∇w, a ∇φ + ∇b )Ω + [w], Cm
△t
k
k
k
ak = a(x, tk ), bk = b(x, tk ), J k ([φ], x) = J([φ], x, tk )
115
= 0 (3.36)
Γ
(3.37)
Note by (3.25) that a and b are smooth enough in t so that point evaluation
in t is allowed. We have also used the assumption that J is smooth enough
in t to allow point evaluation. By a density argument, we can take w to be
any function in H. It is easy to see (following the proof of proposition 1)
that the above is the weak form of system (3.13), (3.14), (3.15) where the
time derivative is discretized in a backward Euler fashion.
In order for φk to be well-defined, we must show that the above indeed
has a unique weak solution. For this purpose we shall use the calculus of
variations.
First, we let:
J˜k (s, x) ≡ J k (s, x) − J k (0, x) ≡ J k (s, x) − J k,0
(3.38)
We thus split J k into two components J˜k and J k,0 so that J˜k ([φ] = 0, x) = 0.
We define the following function K.
K(s, x) =
Note that K(0) = 0. If
∂ J˜k
∂s
=
Z
∂J k
∂s
s
J˜k (σ, x)dσ
(3.39)
0
> 0, then K(s) is a positive convex
116
function. We define the following functional I defined on H.
1
I(ψ) = (∇ψ, ak ∇ψ)Ω + (∇ψ, ∇bk )Ω
2
+ ([ψ], C[ψ])Γ + ([ψ], B)Γ + K([ψ])
Cm
Cm
, B=
+ J k,0
2△t
△t
Z
K([ψ]) ≡ K([ψ], x)dA
C=
(3.40)
(3.41)
(3.42)
Γ
For purposes of the ensuing analysis, we would like K to be a convex functional in [φ]. If
∂J k
∂s
is not always positive, this may not be the case since K
is not convex. To remedy this, we make the following modification when
∂J k
∂s
can be negative.
By assumption (3.27),
∂J k
∂s
> −Cs for some constant Cs . We redefine K
as:
K(s, x) =
Z
s
0
1
J˜k (σ, x)dσ + Cs s2
2
(3.43)
Then, we must redefine the constant C in (3.40) as:
C=
Cm
Cs
−
2△t
2
We would also like C to be positive. We thus take △t <
(3.44)
Cm
Cs
so that C > 0.
Remark 1. The above considerations show that I is not a convex functional
when the transmembrane current exhibits negative differential conductance
unless we take △t to be small enough. As we shall see, the convexity of I
117
is the key ingredient we use to show that φk is well-defined given φk−1 . This
suggests that in the face of negative differential conductance, the backward
Euler scheme may fail if △t is not taken small enough.
We assume that the quadratic forms defined with a is coercive and bounded:
0 < ca ≤ a(x, t) ≤ C a
(3.45)
where ca and C a are constants independent of t. Therefore, ak satisfy the
same bounds. The functional I is well defined as a map from H to R∪{+∞}.
The trace on Γ of a function in H is in L2 (Γ), and thus, [ψ] is in L2 (Γ). The
term K([ψ]) may diverge to +∞ but the rest of the terms remain finite. We
note that I is not identically equal to +∞ since
I(ψ) = 0 if ψ ≡ 0.
(3.46)
If we can show that I has a unique minimum, we will have shown the
existence and uniqueness of a weak solution in H.
Lemma 1. Let ψ ∈ H. The following are equivalent.
1. ψ minimizes I.
2. ψ solves (weakly) the time discrete problem (3.36).
Proof. 1.=⇒ 2.
This can be seen by taking the functional derivative of I. The condition for
118
the weak solution follows since the minimum must be a critical point.
2.=⇒ 1.
If ψ is a weak solution, then it is easy to check by looking at the functional
derivative of I that ψ must be a critical point of I. That ψ must be a
minimizer follows from the fact that I is a convex functional on H. This is
easily seen.
First, we introduce a norm on H that is naturally associated with the
functional I.
Lemma 2.
kψk2A
=
Z
2
Ωint
ai |∇ψ| dV +
Z
2
Ωext
ae |∇ψ| dV +
Z
C[ψ]2 dS
(3.47)
Γ
is a norm on H and is equivalent to k·kH .
Proof. We introduce the norm:
kψk2D = k∇ψk2L2 (Ωint ) + k∇ψk2L2 (Ωext ) + k[ψ]k2L2 (Γ)
(3.48)
We see easily from (3.45) that k·kA and k·kD are equivalent norms. Thus, we
prove the equivalence of k·kH and k·kD .
119
First, we prove that kψkD ≤ c kψkH for some c.
kψk2D = k∇ψk2L2 (Ωint ) + k∇ψk2L2 (Ωext ) + k[ψ]k2L2 (Γ)
≤ k∇ψk2L2 (Ωint ) + k∇ψk2L2 (Ωext ) + 2 kψi k2L2 (Γ) + 2 kψe k2L2 (Γ)
≤ k∇ψk2L2 (Ωint ) + k∇ψk2L2 (Ωext )
+ c1 (k∇ψk2L2 (Ωint ) + kψk2L2 (Ωint ) ) + c2 (k∇ψk2L2 (Ωext ) + kψk2L2 (Ωext ) )
≤ k∇ψk2L2 (Ωint ) + k∇ψk2L2 (Ωext )
+ c1 (k∇ψk2L2 (Ωint ) + kψk2L2 (Ωint ) ) + c3 (k∇ψk2L2 (Ωext ) )
≤ c2 kψk2H
(3.49)
We used the trace theorem in the second inequality, and the Poincaré inequality in the third.
Next we prove that kψkH ≤ c kψkD for some c. We show this by contradiction. Suppose such a c does not exist. Then, we may form a sequence of
functions vk such that
kvk kH = 1
(3.50)
lim kvk kD = 0
(3.51)
k→∞
By Rellich’s compactness theorem, we may extract a subsequence of vk , which
we shall still call vk , such that vk → v in L2 (Ωint ) ⊕ L2 (Ωext ). This implies
120
that in fact, vk is a Cauchy sequence in the k·kH norm since
kvm − vn k2H ≤ kvm − vn k2L2 (Ωint ) + kvm − vn k2L2 (Ωext )
+ k∇vm k2L2 (Ωint ) + k∇vm k2L2 (Ωext )
+ k∇vn k2L2 (Ωint ) + k∇vn k2L2 (Ωext )
≤ kvm − vn k2L2 (Ωint ) + kvm − vn k2L2 (Ωext ) + kvm k2D + kvn k2D
(3.52)
By (3.51), the above can be made arbitrarily small, and thus, vk → v in H.
Using the continuity of the trace operator, we may conclude from (3.50) and
(3.51) that
kvkH = 1,
kvkD = 0.
(3.53)
kvkD = 0 implies that v is identically equal to zero and this contradicts
kvkH = 1.
Now, we proceed to show that I has a minimizer in H. We first show
that I is bounded from below.
Lemma 3. I(ψ) is bounded from below in H. In particular,
I(ψ) ≥ c1 kψk2A − c2
where c1 , c2 are positive constants.
121
(3.54)
Proof.
I(ψ) ≥
1
kψk2A + (∇ψ, ∇bk )Ω + ([ψ], B)Γ
2
(3.55)
We used K([ψ], x) ≥ 0 and hence K([ψ]) ≥ 0.
|(∇ψ, ∇bk )Ω | ≤ k∇ψkΩ ∇bk Ω
1 ∇bk Ω
4ǫ1
ǫ1
1 ∇bk ≤ a kψkA +
Ω
c
4ǫ1
≤ ǫ1 k∇ψkΩ +
(3.56)
where we used the coercivity condition (3.45) in the third inequality.
|([ψ], B)Γ | ≤ kBkL2 (Γ) k[ψ]kL2 (Γ)
≤ ǫ2 k[ψ]k2L2 (Γ) +
≤
1
kBk2L2 (Γ)
4ǫ2
(3.57)
ǫ2
1
kψk2A +
kBk2L2 (Γ)
C
4ǫ2
Therefore, by choosing ǫ1 and ǫ2 sufficiently small,
I(ψ) ≥ c1 kψk2A − c2 > −∞.
(3.58)
We may form a sequence of functions un such that
un ∈ H,
I(un ) ≤ inf I(ψ) +
ψ∈H
122
1
n
(3.59)
Note that the above infimum is not +∞ because of (3.46). We deduce from
the inequality of Lemma 3 that:
c1 kun k2A ≤ inf I(ψ) +
ψ∈H
1
+ c2
n
(3.60)
Thus un is bounded in H. We may extract a subsequence from un , which we
shall still call un , that converges weakly in H to a function u.
Finally, we need a lower semicontinuity property for I(ψ). We shall need
the following lemma.
Lemma 4.
K(v) =
Z
K(v, x)dA
(3.61)
Γ
is lower semicontinuous with respect to weak sequential convergence in L2 (Γ).
Proof. Suppose vk → v in L2 (Γ). Since K(s, x) is a continuous function in
s,
K(v, x) ≤ lim inf K(vk , x)
k→∞
(3.62)
Since K is a positive function, we may use Fatou’s lemma to conclude that:
Z
Γ
K(v)dA ≤ lim inf
k→∞
Z
K(vk , x)dA
(3.63)
Γ
This shows that K is a lower semicontinuous map from L2 (Γ) to R ∪ {+∞}.
The convexity of K shows that K is a convex map. A lower semicontinuous
convex map from a reflexive Banach space to R ∪ {+∞} is lower semicon123
tinuous with respect to weak sequential convergence (p187, Lemma 4.2.2, of
[19]).
We may now prove existence of a unique minimizer.
Theorem 1. The functional I has a unique minimizer in H.
Proof. We first show that weak limit un ⇀ u is a minimizer of I.
I(un ) =
1
kun k2A + (∇un , ∇bk )Ω + ([un ], B)Γ + K([un ])
2
(3.64)
Since the Hilbert norm k·kA is lower semicontinuous, we have:
kuk2A ≤ lim inf kuk2A
n→∞
(3.65)
Since the trace operator is continuous from H to L2 (Γ), un |Γi ⇀ u|Γi and
un |Γe ⇀ u|Γe in L2 (Γ). From Lemma 4, we have:
K([u]) ≤ lim inf K([un ])
n→∞
(3.66)
We also have,
lim (∇un , ∇bk )Ω + ([un ], B)Γ = (∇u, ∇bk )Ω + ([u], B)Γ
n→∞
(3.67)
Collecting (3.65), (3.66), and (3.67), we finally conclude that
I(u) ≤ lim inf I(un )
n→∞
124
(3.68)
This shows that u is a minimizer.
Next, we prove uniqueness. Suppose u and v were both minimizers of I.
Then,
I
u+v
2
2
2
2
1
1
u+v
1
u − v
u + v
u − v
=
+
+K
+ 2
2 A 2 2 A 2 2 A
2
u+v
[u + v]
k
+ ∇
+
, ∇b
,B
2
2
Ω
Γ
kuk2A kvk2A
1
1
+
+ K([u]) + K([v])
4
4
4
4
1
1
+ (∇u, ∇bk )Ω + ([u], B)Γ
2
2
1
1
+ (∇v, ∇bk )Ω + ([v], B)Γ
2
2
1
= (I(u) + I(v))
2
≤
(3.69)
We used the convexity of K to obtain the inequality. If we let m = inf ψ∈H I(ψ),
then I(u) = I(v) = m and I((u + v)/2) ≥ m. Thus we have
This shows that u = v.
3.4
u − v 2 ≤0
A
(3.70)
Stability Estimates and Existence
We have now defined a sequence of approximations at each time step tk =
k△t. We piece together the solutions φkN and would like to argue that φkN
125
converges to a weak solution in an appropriate sense as N → 0. In order
to carry out this program, we need certain stability estimates that do not
depend on the partition number N.
Before we go on any further, we note the following. From (3.27) and
(3.26), we may conclude that:
|J(s, x, t)| ≤ Ct T + Cs |s| + C0
(3.71)
This means that J remains bounded so long as the transmembrane potential
remains bounded. We shall make frequent use of this bound.
Our first stability estimate is the following.
Lemma 5.
where C does not depend on k.
k
[φ ] ≤ C
Γ
(3.72)
Proof. Substitute w = φk in (3.36). Then,
[φk ] − [φk−1 ]
k
k
k
+ J ([φ ], x) = 0 (3.73)
(∇φ , a ∇φ + ∇b )Ω + [φ ], Cm
∆t
Γ
k
k
k
k
Using the Cauchy-Schwartz inequality,
2
2
1 (∇φk , ak ∇φk )Ω − ǫ ∇φk Ω − ∇bk Ω
4ǫ
2
Cm k
k
k
[φ ] + ([φ ], J([φ ], x, tk ))Γ ≤ Cm [φk−1 ]2
+
Γ
Γ
2∆t
2∆t
126
(3.74)
for any positive ǫ. By taking ǫ small enough,
2 Cm k 2
[φ ] +([φk ], J([φk ], x, tk ))Γ ≤ c2 + Cm [φk−1 ]2 (3.75)
c1 ∇φk Ω +
Γ
Γ
2∆t
2∆t
for some constant c1 and c2 . We shall estimate the term ([φk ], J([φk ], x, tk ))Γ .
|([φk ], J([φk ], x, tk ))Γ | ≤ |([φk ], Ct T + Cs |[φ]| + C0 )Γ |
2
≤ Cs [φk ] + |([φk ], Ct T + C0 )Γ |
2 ≤ Cs [φk ] + [φk ]L2 (Γ) kCt T + C0 kL2 (Γ)
2
≤ c3 [φk ] + c4
where we used (3.71) and the Cauchy-Schwartz inequality.
(3.76)
Note that
kCt T + C0 kL2 (Γ) is bounded since the membrane area is bounded. Substituting this back to (3.75), and discarding some positive terms on the left
hand side,
2c3 △t [φk ]2 ≤ [φk−1]2 + 2(c2 + c4 )△t
1−
Γ
Γ
Cm
Cm
We see that [φk ]Γ is bounded by a constant independent of k.
Next, we prove the following bound:
127
(3.77)
Lemma 6. There are constants C1 and C2 such that
k
∇φ ≤ C1
Ω
2
N k
X
[φ ] − [φk−1]Γ
≤ C2
∆t
k=1
(3.78)
(3.79)
Proof. Substitute w = φk − φk−1 in (3.36). We find:
(∇φk − ∇φk−1 , ak ∇φk + ∇bk )Ω
k
+ ([φ ] − [φ
k−1
[φk ] − [φk−1 ]
], Cm
+ J([φk ], x, tk ))Γ = 0
∆t
(3.80)
Use the Cauchy-Schwartz inequality to find that:
1
1
(∇φk , ak ∇φk )Ω − (∇φk−1 , ak ∇φk−1 )Ω
2
2
Cm [φk ] − [φk−1]2
+ (∇φk , ∇bk )Ω − (∇φk−1 , ∇bk )Ω +
Γ
∆t
(3.81)
≤ ([φk−1 ] − [φk ], J([φk ], x, tk ))Γ
Using (3.71) and the Cauchy-Schwartz inequality,
Cm [φk ] − [φk−1 ]2 + (c1 + c2 [φk ]2 )∆t
Γ
Γ
2∆t
(3.82)
We know from lemma 5 that [φk ]Γ is bounded uniformly by a constant.
([φk−1 ] − [φk ], J([φk ], x, tk ))Γ ≤
128
Combining the above with (3.81), we find:
1
1
(∇φk , ak ∇φk )Ω − (∇φk−1, ak ∇φk−1 )Ω
2
2
Cm [φk ] − [φk−1 ]2
+ (∇φk , ∇bk )Ω − (∇φk−1, ∇bk )Ω +
Γ
2∆t
(3.83)
≤ c∆t
We first consider the bound on k∇φkΩ . Summing the above in k = 1 · · · n
for any n ≤ N,
1
1
− (∇φ0 , a1 ∇φ0 ) + (∇φn , an ∇φn ) − (∇b1 , ∇φ0 ) + (∇bn , ∇φn )
2
2
n−1
X1
(∇φk , (ak − ak+1 )∇φk ) + (∇(bk − bk+1 ), ∇φk )
+
2
k=1
n
X
Cm [φk ] − [φk−1]2
+
Γ
2∆t
k=1
(3.84)
≤ cN△t = cT
By repeated use of the Cauchy-Schwartz inequality and the assumption that
the coefficients a and ∇b be Lipschitz in time (3.25), we obtain the following
bound:
k∇φn k2Ω
n−1
n
X
X
2
Cm k
k−1 2
c2 ∇φk Ω △t
[φ ] − [φ ] Γ ≤ c1 +
+
2∆t
k=1
k=1
(3.85)
From this, we conclude that:
k∇φn k2Ω
≤ c1 +
n−1
X
k=1
129
2
c2 ∇φk Ω △t
(3.86)
Application of a discrete Gronwall type inequality to the above immediately
leads to a bound on k∇φn kΩ .
k∇φn kΩ ≤ C1
(3.87)
Substituting this bound into (3.85) and taking n = N, we have:
2
N k
X
[φ ] − [φk−1]
Γ
k=1
∆t
≤ C2
(3.88)
Corollary 1. There is a constant C such that
k
[φ ] 1/2 ≤ C
H
(Γ)
(3.89)
Proof. This follows from the fact that the trace of a function in H is a function
in H 1/2 (Γ). By lemma 5 and lemma 6, φk are contained in a bounded set in
H. Since the mapping:
T : φ ∈ H → [φ] ∈ H 1/2 (Γ)
(3.90)
is continuous, the conclusion follows.
From the solutions at each time step, we construct the following approximate solutions.
130
Define:
φN (x, t) = φkN (x) if tk−1 < t ≤ tk
φ̃N (x, t) = θφkN (x) + (1 − θ)φk−1
N (x), θ =
(3.91)
t − tk−1
, if tk−1 ≤ tk
∆t
(3.92)
Thus, φN is piecewise constant whereas φ̃N is the piecewise linear interpolant
in t.
In terms of these functions, we can recast the bounds just obtained as
follows. There exist constants C such that:
kφN kL∞ ([0,T ];H) ≤ C1
(3.93)
k[φN ]kL∞ ([0,T ];H 1/2 (Γ)) ≤ C2
≤ C3
[φ̃N ] 1
2
(3.94)
H ([0,T ];L (Γ))
(3.95)
We showed in corollary 1 that (3.94) follows from (3.93).
Using the above bound:
Lemma 7. The set {[φN ]} forms a precompact set in L2 ([0, T ]; L2(Γ)).
Proof. Lemma 5 and lemma 6 show that [φÑ ] is bounded in L∞ ([0, T ]; H 1/2(Γ)).
Thus, [φÑ ] is bounded in L2 ([0, T ]; H 1/2 (Γ)) and by (3.95), is also bounded in
H 1 ([0, T ]; L2 (Γ)). Since H 1/2 is embedded compactly in L2 , by Aubin-Lions
lemma (p.106, Proposition 1.3 of [39]), we deduce that {[φÑ ]} form a precompact set in L2 ([0, T ]; L2(Γ)). The bound (3.79) shows that if a subsequence
of {[φÑ ]} is strongly convergent, the corresponding subsequence of {[φN ]} is
131
likewise strongly convergent.
The above lemma allows us to select a subsequence of φN , which we shall
still denote by φNi such that:
[φNi ] → [φ] strongly in L2 ([0, T ]; L2 (Γ))
(3.96)
∂[φ]
∂[φÑi ]
⇀
weakly in L2 ([0, T ]; L2 (Γ))
∂t
∂t
(3.97)
φNi ⇀ φ weakly in L2 ([0, T ]; H)
(3.98)
for some function φ. We shall henceforth refer to this subsequence as φN .
The time step ∆t is no longer T /N. We shall let ∆t = T /MN , where MN
depends on M.
The rest of the proof follows identically to the proof in Jerome [17]. Take
equation (3.36) as our starting point. Let w be a smooth test function in
H 1 ([0, T ]; L2 (Γ)) ∩ L2 ([0, T ]; H). Define w k as the following function:
1
w (x) =
∆t
k
Z
tk
w(x, t)dt
(3.99)
tk−1
Take w k to be the test function in (3.36) and sum in k from k = 1 to n.
MN
X
+
(∇w k , ak ∇φk + ∇bk )Ω ∆t
k=1
MN X
[w k ], Cm
k=1
k
[φ ] − [φ
△t
k−1
]
+ J k ([φk ], x)
(3.100)
Γ
132
∆t = 0
Summing by parts in the difference approximation of the derivative in the
above, we have:
MN
X
−
(∇w k , ak ∇φk + ∇bk )Ω ∆t
k=1
MN X
k=1
+(w
MN
[w k ] − [w k−1]
, Cm [φk−1 ] ∆t
∆t
Γ
, [φ
MN
0
0
])Γ − (w , [φ ])Γ +
MN
X
(3.101)
(w k , J k ([φk ], x))Γ ∆t = 0
k=1
We would now like to take the limit of the above as N tends to infinity. In
order to take this limit, we write the above in integral form:
Z
T
(∇wN , aN ∇φN + ∇bN )Ω dt
0
−
Z
0
+(w
T
(3.102)
(vN , Cm [φN − ])Γ dt
MN
, [φ
MN
0
0
])Γ − (w , [φ ])Γ +
Z
0
133
T
(wN , JN ([φN ]))Γ dt = 0
where:
φN − (x, t) = φk−1
N (x), if tk−1 < t ≤ tk
(3.103)
wN (x, t) = w k (x), if tk−1 < t ≤ tk
(3.104)
vN (x, t) =
(3.105)
[w k ](x) − [w k−1](x)
, if tk−1 < t ≤ tk
∆t
aN (x, t) = ak (x), if tk−1 < t ≤ tk
(3.106)
bN (x, t) = bk (x), if tk−1 < t ≤ tk
(3.107)
JN (·) = J k (·, x), if tk−1 < t ≤ tk
(3.108)
For notational convenience, we rewrite (3.102) as follows:
(∇wN , aN ∇φN )ΩT + (∇wN , ∇bN )ΩT − (vN , Cm [φN − ])ΓT
+(w
MN
, [φ
MN
0
(3.109)
0
])Γ − (w , [φ ])Γ + (wN , JN ([φN ]))ΓT = 0
where ΩT and ΓT denote [0, T ] × Ω and [0, T ] × Γ respectively and (·, ·)ΩT and
(·, ·)ΓT denote the L2 inner products on the respective space time domains.
We shall consider the limit of each term in (3.109) one by one. To do so,
we require the following lemma:
134
Lemma 8.
wN → w strongly in L2 ([0, T ]; H)
(3.110)
[wN ] → [w] strongly in L2 ([0, T ]; L2 (Γ)) = L2 (ΓT )
vN ⇀
(3.111)
∂[w]
weakly in L2 ([0, T ]; L2 (Γ)) = L2 (ΓT )
∂t
(3.112)
Proof. The proof is identical to Lemma 5.2.5 (p.181) of [17]
We first consider the first term in (3.109):
(∇wN , aN ∇φN )ΩT
(3.113)
For aN , we have:
a(t, ·) − a(tk−1 , ·) =
Z
t
tk−1
∂a
dt
∂t
(3.114)
where we have used (3.25), Lipschitz continuity of a with respect to t. By
taking the L∞ norm on both sides of the above,
ka(t, ·) − a(tk−1 , ·)kL∞ (Ω)
t
∂a ≤
dt ≤ C(t − tk−1 )
tk−1 ∂t L∞ (Ω)
Z
(3.115)
where we used the Lipschitz bound (3.25) on a. We therefore have:
ka − aN kL∞ (ΩT ) = max
k
max ka(t, ·) − a(tk−1 , ·)kL∞ (Ω)
tk−1 ≤t≤tk
135
≤ C∆t
(3.116)
Thus,
kaN ∇wN − a∇wkL2 (ΩT ) ≤ kaN kL∞ (ΩT ) k∇wN − ∇wkL2 (ΩT )
(3.117)
+ kaN − akL∞ (ΩT ) k∇wkL2 (ΩT )
Since aN converges to a in L∞ (ΩT ) and ∇wN converges to ∇w in L2 (ΩT ) by
(3.110), aN ∇wN converges strongly in a∇w in L2 (ΩT ). Since by (3.98) ∇φN
converges to ∇φ weakly in L2 (ΩT ),
lim (∇wN , aN ∇φN )ΩT = lim (aN ∇wN , ∇φN )ΩT
N →∞
N →∞
(3.118)
=(a∇w, ∇φ)ΩT = (∇w, a∇φ)ΩT
Next consider:
(∇wN , ∇bN )ΩT
(3.119)
in (3.109).
∇b(t, ·) − ∇b(tk , ·) =
Z
t
tk−1
∂
(∇b(t, ·))dt
∂t
(3.120)
Take the L2 norm on both sides:
k∇b(t, ·) − ∇b(tk−1 , ·)kL2 (Ω)
t
∂
≤
∂t (∇b) 2 dt ≤ C(t − tk−1 ) (3.121)
tk−1
L (Ω)
Z
136
We thus have:
k∇b −
∇bN k2L2 (ΩT )
=
XZ
k
≤
tk
tk−1
XZ
tk
tk−1
k
k∇b(t, ·) − ∇b(tk−1 , ·)k2L2 (Ω) dt
C 2 (t − tk−1 )2 dt
(3.122)
1
1
= C 2 (MN ∆t)(∆t)2 = C 2 T (∆t)2
3
3
Therefore, ∇bN converges strongly in L2 (ΩT ) to ∇b as N → ∞. This, with
(3.110) implies that:
lim (∇wN , ∇bN )ΩT = (∇w, ∇b)ΩT
N →∞
(3.123)
Next we turn to the term:
(vN , Cm [φN − ])ΓT
(3.124)
in (3.109). It follows easily from our estimates that [φN − ] → [φ] in L2 (ΓT ).
Using this fact with (3.112),
lim (vN , Cm [φN − ])ΓT =
N →∞
∂w
, Cm [φ]
∂t
ΓT
(3.125)
We now take a look at
(wN , JN ([φN ]))ΓT
137
(3.126)
in (3.109). Take the difference between JN ([φN ]) and J([φ])
kJ([φ]) − JN (φN )kL2 (ΓT ) ≤ kJ([φ]) − J([φN ])kL2 (ΓT )
(3.127)
+ kJ([φN ]) − JN ([φN ])kL2 (ΓT )
where ΓT denotes [0, T ] × Γ. The first norm in the above goes to 0 since
[φN ] converges strongly [φ] in L2 (ΓT ) by (3.96), and by (3.27) J is uniformly
Lipschitz in [φ].
kJ([φN ]) −
JN ([φ])k2L2 (ΓT )
=
XZ
tk
tk−1
k
J([φk ], x, t) − J([φk ], x, tk )2 2 dt
L (Γ)
(3.128)
Since [φk ] is in L2 (Γ), [φk ] < ∞ pointwise a.e. Therefore, for a.e. x,
k
k
J([φ ], x, t) − J([φ ], x, tk ) =
Z
t
tk
∂J k
([φ ], x, t)dt
∂t
(3.129)
Since J is globally Lipschitz in t by assumption (3.27), we see that:
|J([φk ], x, t) − J([φk ], x, tk )| ≤ C(tk − t)
(3.130)
pointwise a.e. Squaring and integrating over Γ,
J([φk ], x, t) − J([φk ], x, tk )2 2
L (Γ)
138
≤ C 2 (t − tk )2
(3.131)
Integrating this from t = tk−1 to tk , and summing in k we have:
kJ([φN ]) −
JN ([φN ])k2L2 (ΓT )
=
XZ
k
≤
tk
tk−1
X C2
k
3
J([φk ], x, t) − J([φk ], x, tk )2 2
L (Γ)
(∆t)3 =
C2
C2
(MN ∆t)(∆t)2 =
T (∆t)2
3
3
(3.132)
We therefore have:
√
kJ([φN ]) − JN ([φN ])kL2 (ΓT ) ≤ C T ∆t
(3.133)
Thus, we see that JN ([φN ]) → J([φ]) in L2 (ΓT ) as N → ∞. We thus see
that:
lim (wN , JN ([φN ]))ΓT = (w, J([φ]))ΓT
N →∞
(3.134)
where we used (3.111).
Finally, by the fundamental theorem of calculus on Banach spaces, we
have:
[φ
MN
] − φ0 =
Z
0
T
∂[φ̃]
dt
∂t
(3.135)
Thanks to (3.97), we can take the weak limit in the above, and thus,
[φMN ] ⇀ φ(·, T ) in L2 (Γ)
139
(3.136)
We therefore conclude that:
lim (wN , φN )Γ = (w, φ)Γ
N →∞
(3.137)
This concludes the proof of existence of a weak solution.
3.5
Uniqueness
We may prove that this solution is in fact unique by the following standard
argument. Take two weak solutions φ1 , φ2 as defined by (3.23), and take the
difference. For ψ = φ1 − φ2 we have:
Z
T
(∇w, a∇ψ)Ω dt +
0
Z
T
0
∂[ψ]
+ G([φ1 ], [φ2 ], t)[ψ] dt = 0
[w], Cm
∂t
Γ
(3.138)
J([φ1 ], x, t) − J([φ2 ], x, t) = G(φ1 , φ2, t)[ψ]
ψ = 0 at t = 0
(3.139)
(3.140)
Note that since J is Lipschitz in [φ] uniformly in t, G is well defined and
belongs to L∞ ([0, T ]; L∞ ). Since the above is satisfied for arbitrary w, see
that the integrand is 0 for almost all t.
∂[ψ]
+ G([φ1 ], [φ2 ], t)[ψ]
=0
(∇w, a∇ψ)Ω + [w], Cm
∂t
Γ
140
(3.141)
By substituting w = [ψ], we find immediately that:
(∇ψ, a∇ψ)Ω +
d Cm
kψk2Γ + ([ψ], G([φ1 ], [φ2 ], t)[ψ])Γ = 0
dt 2
(3.142)
Since J is Lipschitz, G is bounded from below by some constant G > −CG .
Thus,
d Cm
kψk2Γ ≤ CG k[ψ]k2Γ
dt 2
(3.143)
Since [ψ]|t=0 = 0, we immediately conclude by Gronwall’s inequality that
k[ψ]kΓ = 0 for all t. We can substitute this into (3.142) to find that:
(∇ψ, a∇ψ)Ω = 0
(3.144)
This, together with [ψ] = 0 implies kψkA = 0. This proves uniqueness of the
weak solution.
Now that we know that our solution is unique, we can go back to the
existence proof to see that in fact, not only the subsequence but the whole
sequence itself converges to the true solution, since all subsequences of the
original sequence contains a subsequence that converges to the unique solution. This proves the convergence of the semi-discrete system to the true
solution.
141
3.6
Dirichlet-to-Neumann Map
We end this chapter by introducing a new perspective with which to look at
this system. For simplicity, we shall deal here with the simplified 3D cable
model (2.262) and (2.263) of Chapter 2.
∆φ = 0 in Ωint and Ωext
1 ∂φ ∂[φ]
+ Iion = −
Cm
∂t
R ∂n Γint
1 ∂φ =−
on Γ
R ∂n Γext
1 ∂φ
−
= 0 on Γf
R ∂nf
(3.145)
(3.146)
(3.147)
where [φ] = φint − φext and n is the outward normal on Γ pointing from
Ωint to Ωext . Γext and Γint denote evaluation at the Ωext and Ωint faces of Γ
respectively. Γf is the outer boundary of Ωext and nf is its outward pointing
normal. We have taken the simplification that the electrolyte resistivity R is
the same for both the intracellular and extracellular regions. At the outer rim
of the extracellular space, we may impose Dirichlet or Neumann boundary
conditions.
We can write the above system as an evolution confined to the membrane
by introducing the following map denoted LDN . For any smooth function on
142
[w] defined on Γ, we solve the following boundary value problem:
∆w = 0 in Ωint and Ωext
∂w ∂w =
on Γ
∂n Γint
∂n Γext
w|Γint − w|Γext = [w]
∂w
= 0 on Γf
∂nf
(3.148)
(3.149)
(3.150)
(3.151)
Having solved this boundary value problem, we define:
∂w ∂w LDN [w] =
=
∂n Ωint
∂n Ωext
(3.152)
This is a Dirichlet-to-Neumann map, taking [w] to the normal derivative (on
either side of Γ) of the solution to the boundary problem (3.148)-(3.151).
With the aid of this operator, we can write our evolution equation as:
Cm
1
∂[φ]
+ Iion = − LDN [φ]
∂t
R
(3.153)
We first note:
Lemma 9. LDN can be extended to a non-negative self-adjoint operator on
L2 (Γ).
Proof. Let [v], [φ] be a sufficiently smooth function on L2 (Γ). Let v and φ be
the solution to the boundary value problem (3.148)-(3.151), where we take
143
[w] = [v], [φ]. Then,
([v], LDN [φ])L2 (Γ) =
Z
ZΓ
[v]
∂φint
dA
∂n
∂φint
dA
∂n
Γ
Z
Z
∂φext
∂φint
dA − vext
dA
= vint
∂n
∂n
Γ
Γ
Z
Z
=
vint ∆φint dA +
∇vint · ∇φint dA
Ωint
Ωint
Z
Z
+
vext ∆φext dA +
∇vext · ∇φext dA
Ωext
Ωext
Z
Z
=
∇vint · ∇φint dA +
∇vext · ∇φext dA
=
(vint − vext )
Ωint
(3.154)
Ωext
= (∇v, ∇φ)L2 (Ω)
We have used the Green’s identity in the third equality above, and used the
fact that φ is harmonic (since it satisfies (3.148)) in the fourth equality. The
symmetry of the last line above with respect to v and φ shows that LDN is
symmetric on smooth functions. Substituting v = φ, one immediately sees
that LDN is non-negative. By Friedrich’s extension theorem of semi-bounded
symmetric operators [25], we see that LDN can be extended to a self-adjoint
operator on L2 (Γ).
Note also that:
([v], LDN [φ])L2 (Γ) = (∇v, ∇φ)L2 (Ω)
≤ k∇vkL2 (Ω) k∇φkL2 (Ω)
≤ C k[v]kH 1/2 (Γ) k[φ]kH 1/2 (Γ)
144
(3.155)
where we used the trace theorem in the third line. This tells us that LDN
can also be thought of as mapping H 1/2 (Γ) into its dual H −1/2 (Γ).
Thus, we see that the membrane potential [φ] satisfies an equation of reaction diffusion type, where the laplacian ∆ in a reaction diffusion equation
is replaced by −LDN . In view of the fact that LDN maps H 1/2 (Γ) to H −1/2 (Γ),
it can be shown that −LDN , like the laplacian, is a generator of a smoothing
evolution, if the transmembrane current is simply a time-independent linear
function of [φ] [38]. We see that what we have been calling membrane potential “diffusion” is actually a type of smoothing process generated by −1
times the Dirichlet-to-Neumann map.
In Chapter 2, we derived the traditional cable model from the above
simplified 3D cable model using asymptotic analysis, assuming cylindrical
geometry and taking the radius of the cylinder to be the small parameter.
An independent way to justify the cable approximation is to take the above
Dirichlet-to-Neumann map formulation and expand [φ] in eigen-functions of
LDN , neglecting the nonlinearity of Iion . Note that LDN is self-adjoint and
maps H 1/2 (Γ) to H −1/2 (Γ), thus, (I + LDN )−1 (where I is the identity) is
compact and the eigen-functions of LDN span L2 (Γ). If we can show that the
eigen-functions with radial dependence decay quickly relative to the eigenmodes with axial dependence, we will have shown that the 1D description is
a valid one. This is essentially what Rall achieves in [35], though he does not
phrase his study as one of calculating the spectral properties of the Dirichletto-Neumann map.
145
One future goal of the analysis above will be to study the regularity
properties of the solution, and it should be possible to show that [φ] becomes
progressively regular with time. This will in particular allow us to extend
our well-posedness results to when Iion is not globally but only locally Lipschitz in [φ], a functional form that is often encountered in the biophysical
literature. Such an analysis will, hopefully, also allow us to argue that [φ]
remains bounded in time under physiologically reasonable assumptions on
the transmembrane currents.
146
Chapter 4
Numerical Methods
In this chapter, we shall describe the numerical methods used to simulate
the electroneutral model.
As we have seen in Chapter 2, there are three time scales associated
with the Poisson model: the charge relaxation time scale, the membrane
potential time scale, and the slow diffusion time scale. Associated with the
Poisson model is also a fine spatial structure near the membrane, namely,
the space charge layer. In order to numerically treat the Poisson model, we
need to resolve the space charge layer and the charge relaxation time scale.
Since the Poisson model is computationally demanding, we shall only perform
simulations with the Poisson model in simple one dimensional situations,
where we shall make comparisons with the electroneutral model.
One of the reasons for introducing the electroneutral model was to alleviate this numerical difficulty. We claimed in Chapter 2 on the basis of
147
asymptotic analysis that the electroneutral model correctly captures the effect of having a space charge layer at the membrane without resolving its fine
spatial structure. We were not completely able to dispense with the charge
relaxation time scale, but this time scale now appears only in the simple
ordinary differential equation satisfied by λi , the membrane charge fraction.
This can now be easily handled in numerical computations.
The electroneutral model possesses two time scales of primary importance, the membrane potential time scale and the slow diffusion time scale.
As we have seen, the ratio between these two time scales is about β =
10−4 ∼ 10−6 , and thus, we have two disparate time scales that we need to
resolve. Both time scales are diffusive time scales, the membrane potential
time scale corresponding to “diffusion” of the membrane potential, and the
slow diffusion time scale corresponding to the diffusion of ions. An explicit
time stepping scheme when applied to a diffusive equation requires finer time
stepping as the spatial discretization is refined. We thus develop an implicit
scheme to avoid this time step restriction.
Another issue is spatial discretization. A membrane of complicated geometry may cut through space, and it is a non-trivial issue as to how to
deal with this geometrical complexity. We start with a discussion of spatial
discretization.
148
4.1
4.1.1
Spatial Discretization
Finite Volume Method
We shall use a finite volume discretization in space [28]. The central tool is
the divergence theorem as applied to each computational voxel.
We first recall the equations for the electroneutral model. Let Γ(kl) denote
the membrane separating Ω(k) and Ω(l) . In Ω(k) and Ω(l) ,
∂ci
+ ∇ · fi
∂t qzi ci
∇φ
fi = −Di ∇ci +
kB T
n
X
0 = ρ0 +
qzi ci
0=
(4.1)
(4.2)
(4.3)
i=1
The boundary conditions at the face of Γ(kl) facing Ω(k) are,
(k)
Cm
∂(λi φ(kl) )
(k)
(kl)
= qzi fi · n(kl) − ji
∂t
(k)
(k)
(k)
(k)
∂λi
λ̃ − λi
z2c
(k)
, λ̃i = PN i i (k)
= i 2
2
∂t
rd /D0
i′ =1 zi′ ci′
(kl)
(kl)
ji = Ji
x, s(kl) , φ(kl) , c(k) , c(l)
(4.4)
(4.5)
(4.6)
(kl)
∂sg
∂t
= fg (s(kl) , φ(kl) , c(k) , c(l) )
(4.7)
We shall use the following property of the above system of equations:
Take any region Ωcv contained in Ω(k) and suppose the boundary of this
region is comprised of two components, the Γel component that faces the
149
electrolyte solution, and the Γm component that faces the membrane. It
may be the case that either Γel or Γm is empty. For each ionic species, we
have the following conservation relation in integral form.
Z
∂ci
dV =
qzi
∂t
Ωcv
Z
Ωcv
qzi ∇ · fi dV
Z
qzi ∇ · fi · ndA
Z
qzi ci
=−
qzi Di ∇ci +
∇φ · ndA
kB T
Γel
!
Z
(k)
∂(λi φ(kl) )
(kl)
Cm
+ ji
dA
+
∂t
Γm
=
Γel ∪Γm
(4.8)
The electroneutrality condition is equivalent to saying that:
ρ0 +
N
X
i=1
N
X
qzi ci = 0
at t = 0
(4.9)
∂ci
=0
∂t
for t > 0
(4.10)
qzi
i=1
As long as the electroneutrality condition is satisfied at t = 0, we have only
to consider the time derivative of the electroneutrality condition for time
t > 0. Therefore, we can obtain the electroneutrality condition expressed in
integral form by taking the sum of equation (4.8) in i:
0=−
Z
+
Z
Γel
Γm
!
qzi ci
∇φ · n dA
qzi Di ∇ci +
kB T
i=1
!
N
∂φ(kl) X (kl)
Cm
ji
+
dA
∂t
i=1
N
X
150
(4.11)
We have now derived integral conservation relations (4.8) and (4.11) from
the differential conservation relations (4.1) and the electroneutrality condition (4.3) and the boundary conditions. On the other hand, if the above integral relations hold for arbitrary regions (and assuming that the electroneutrality condition holds at t = 0), we can recover the differential equations
and the boundary conditions. The idea of the finite volume method is to
use these integral conservation relations to discretize the partial differential
equations.
To this end, we partition the spatial region into a finite number of voxels,
and apply (4.8) and (4.11) on each voxel. We then approximate the volume
and surface integrals involved in the integral conservation relations.
Consider the prototypical conservation relation:
∂u
= ∇ · f in Ω
∂t
f · n = g on ∂Ω
(4.12)
(4.13)
For simplicity, we shall consider a 2D situation. We shall discretize space
into polygonal voxels.
We now take one voxel V . For each voxel we designate a representative
location xc where we define the value of the physical variables. This variant
of the finite volume method is termed the cell-centered finite volume method
[8]. Another choice is the vertex-centered finite volume method, in which the
physical variables are defined on the vertices of the polygons [8]. We shall
151
follow the cell-centered approach.
Equations (4.12) and (4.13) can be discretized using a finite volume approach in the following fashion.
Z
∂u
1
∂u dV
≈
∂t x=xc
|V | V ∂t
Z
1
∇ · fdV
=−
|V | V
Z
1
=−
f ·n
|V | ∂V
1 X
Aq F q
≈−
|V | q
(4.14)
where |V | is the volume (in 2D the area) of the voxel, q labels the faces
(polygonal sides), and Aq is the area (in 2D the length) of the face q. F q
is an approximation to the true flux f · n evaluated on face q. The discrete
evolution equation is thus:
∂u
1 X
Aq F q .
=−
∂t
|V | q
(4.15)
′
Suppose we write the flux approximation from voxel Vp to Vp′ to be F p,p . As
′
′
long as F p,p = −F p ,p , we have discrete conservation of the quantity u. Thus,
it is straightforward with the finite volume method construct a conservative
numerical scheme.
This is the framework of the finite volume method. One may construct
a variety of schemes with different choices of mesh geometry (structured /
152
unstructured, triangular / quadrilateral, etc.) and approximation schemes of
fluxes [28, 8].
We now specialize to the specific discretization we use to discretize the
electroneutral model.
4.1.2
Embedded Boundary Method
We have developed finite volume schemes adapted to two types of simulations, one for arbitrary 2D membrane geometry, and the other for cylindrical
geometry. We first discuss the finite volume discretization for arbitrary 2D
membrane geometry. We shall discuss cylindrical geometry in the next section.
We shall use an embedded boundary method, where a uniform cartesian
grid is used over most of the computational domain, except where the grid
is cut by the membrane [18, 6, 29]. One important difference between most
previous work with the embedded boundary method and the present work is
that we are concerned with evolution of physical quantities on both sides of
the membrane. We are dealing here with an interface problem rather than a
conventional boundary value problem.
Henceforth, we shall use the word mesh to denote the uniform cartesian
(or non-uniform rectilinear) grid laid on the computational domain, and the
square (or rectangular) patches that result from this grid. The voxels correspond to the mesh itself if the membrane does not cut through this mesh.
If the membrane cuts through the mesh, we approximate this membrane cut
153
by a straight line, and the resulting two polygons will be our voxels that
correspond to this mesh. We briefly discuss computational geometry issues
further in Section 4.4.1. We shall call a uncut cartesian voxel an ordinary
voxel. A voxel that is cut out of a cartesian mesh by the membrane will
be called a membrane voxel. We shall label our voxels and their associated
quantities with the subscript or superscript p.
When a membrane cuts through a mesh, two voxels will be generated.
These two voxels share a common membrane patch. We shall call such faces
membrane faces, whereas we shall call other faces ordinary faces. We shall
label ordinary faces eq with the subscript or superscript q. We let Ep be the
set of ordinary faces that demarcate the voxel p. We number the membrane
faces γm by m, where m runs through the number of membrane patches,
equal to the number of meshes that are cut through by the membrane. We
let Mp be the set of membrane faces that demarcate the voxel p. Thus, Mp
has one member, or is empty. To each membrane face γm , we assign physical
quantities associated with the membrane. They are the physical variables
associated with transmembrane currents and the membrane charge fraction
λi . In particular, to each membrane patch γm is associated gating variables
sm
g that describe the gating states of ion channels present on that membrane
patch. Since each membrane patch γm has two sides, there are two values of
(l),m
the membrane charge fraction λi
(k),m
and λi
at the membrane Γ(kl) .
In the case of an ordinary voxel, we shall take the representative point
xc , with which the values of the physical variables are associated, to be
154
the center of the voxel. For a membrane voxel, we shall take xc to be the
center of the cartesian mesh from which the voxel was cut. Thus, there
will be cases in which xc geometrically lies outside the voxel. Conceptually,
this involves the smooth extrapolation of a function defined on one side of
the membrane to the other side of the membrane. For each membrane voxel,
there is another membrane voxel that shares the same membrane patch which
was therefore cut out of the same mesh. These two membrane voxels have
representative points xc that coincide. Although these are geometrically the
same points, we shall treat them as different points in terms of computation.
At membrane voxels, we shall thus have two different representative points
which correspond to different voxels, at the same geometrical location.
We first apply the divergence theorem and its approximation to an ordinary voxel. Take an ordinary voxel Vp , with four faces eq , q ∈ Ep . The voxel
is a square with sides of length h. The discrete evolution equation for the
ionic concentration is based on (4.14):
Z
∂ci
1
∂ci dV
≈
∂t x=xc
|Vp | Vp ∂t
Z
1
∇ · fi dV
=−
|Vp | Vp
Z
1
=−
fi · ndA
|Vp | ∂Vp
1 X
≈−
|eq |Fiq
|Vp | q∈E
(4.16)
p
where Fiq is the approximation to the flux of the i-th ionic species going out
155
of voxel Vp through face eq . Therefore, the discrete evolution equation for
ionic concentrations cpi in Vp are:
1 X
∂cpi
|eq |Fiq .
=−
∂t
|Vp | q∈E
(4.17)
p
The first approximation in (4.16) comes from using xc as the representative point for the voxel. Since we have taken xc to be at the center of the
voxel, the error we incur here is O(h2 ).
We now turn to the second approximation in (4.16). We compute the
approximation to the flux, Fiq , at the center of the face eq . If Fiq is exactly
equal to fi · n (n is unit normal on eq pointing from Vp to Vp′ at the midpoint
of eq , the following difference:
1
|eq |
Z
eq
fi · ndA − Fiq
(4.18)
will be O(h2 ). Since |eq | = h and |Vp | = h2 , this will lead to an error O(h)
in the second approximation in (4.16). Therefore, if Fiq is exactly equal to
fi · n at the midpoint of eq , (4.16) yields a O(h) truncation error. We cannot
evaluate Fiq exactly. In order to retain the O(h) truncation error, we must
make sure that Fik is correct to order O(h2 ).
Suppose the face eq is shared by the voxels Vp and Vp′ . Ionic fluxes between
156
voxels are computed as:
l+1
cl+1
i ,φ
cli , φl
Fik
′
′
Fiq = Di
qzi (cpi + cpi ) φp − φp
cpi − cpi
+
h
2kB T
h
′
!
(4.19)
This is just the central difference approximation of the flux. This yields a
O(h2 ) approximation. An important point about the above flux expression is
that Fiq changes sign when the indices p and p′ are exchanged in the above.
This means that the approximation of the flux from Vp′ to Vp is equal to
−1 times the approximation of the flux from Vp to Vp′ . We therefore have
discrete ion conservation with the above flux approximation.
We do not employ up-winding [28] for the advection component in the
flux term in (4.19). Experience shows that this does not cause instability,
presumably because our problem is almost always diffusion dominated.
Because of the symmetry of the uniform cartesian mesh, after we sum Fiq
in q, we find that error terms from opposing faces cancel out to next order
in h, so that the overall truncation error reduces further to O(h2 ). This is
not the case if we use a non-uniform mesh, as we shall see later.
We now consider membrane voxels. Let Vp be a membrane voxel. Our
starting point is the following, which corresponds to (4.16) in the case of an
157
ordinary voxel.
Z
∂ci
1
∂ci dV
≈
∂t x=xc
|Vp | Vp ∂t
Z
1
∇ · fi dV
=−
|Vp | Vp
Z
1
=−
fi · ndA
|Vp | ∂Vp


X
X
1 
|γm |Fim 
|eq |Fiq +
≈−
|Vp | q∈E
m∈M
(4.20)
p
p
The difference between (4.16) and (4.20) is that there is a contribution from
the membrane face γm . Here, |γm | is the area (length in 2D) of the membrane
face, and the approximation to the transmembrane flux that goes out of voxel
Vp through membrane γm is denoted as Fim . Our discrete evolution equation
for ionic concentrations is thus:
∂cpi
∂t
=−


1 
|γm |Fim  .
|eq |Fiq +
|Vp | q∈E
m∈M
X
X
p
p
(4.21)
As stated earlier, the representative point xc for Vp is the center of the
mesh from which Vp was cut. By taking xc to be the center of the mesh and
not the barycenter of Vp , we incur an error of O(h) at the first approximation
in (4.20).
We first consider fluxes across an ordinary face (i.e. a face that is not a
membrane face). Since the membrane cuts through the mesh, an ordinary
face eq may not have full length h even though it does not correspond to the
158
membrane. Suppose the voxel that shares the ordinary face eq with Vp is Vp′ .
The flux across eq is computed in the same way as (4.19). But since this
flux will in general not correspond to the midpoint of eq , we here incur an
error of O(h) in (4.18), and therefore, this will contribute to an O(1) error in
(4.16). This means that our local truncation error at the membrane can be
O(1). We shall see in computational experiments, however, that this local
truncation error does not spoil the convergence of the numerical scheme, even
at the membrane. This is presumably because the membrane is a structure
of co-dimension 1 with respect to the computational domain, and thus, the
influence of this truncation error diminishes proportionally to h as the mesh
is refined.
Next we consider a membrane face. Let the membrane face γm be shared
by voxels Vp and Vp′ . The membrane ionic flux from Vp to Vp′ is computed as:
l+1
cl+1
i ,φ
qzi Fim = Cm
∂(λpi φpm )
′
+ jip (cp , cp , φpm , sm )
∂t
φpm = φp − φp
′
(4.22)
k
F
l
l i
ci , φ
(4.23)
(l),m
Here, λpi is one of two values λi
(k),m
or λi
depending on whether the voxel
Vp belongs to the region Ω(l) or Ω(k) . We may consider the error we incur in
this expression. We have chosen to evaluate membrane quantities c (vector of
ionic concentrations (c1 , · · · , cN )) and φm using values at the representative
points xc without interpolation or extrapolation to the membrane itself. This
contributes a O(h) error to the computation of the flux, which in turn will
give O(1) error term in (4.16). As for conservation of ionic concentration,
159
the ionic content of voxel Vp :
|Vp |cpi + |γm|
Cm p
λ φm
qzi i
(4.24)
will be conserved as long as the transmembrane current approximation jip
changes sign when p and p′ are exchanged. The expression (4.24) is the total
ionic content in the membrane voxel Vp , taking into account the amount of
ion that resides within the space charge layer (cf. (2.240)).
We finally note that the discretization of the electroneutrality condition
for each voxel can be obtained by multiplying the discrete evolution equations
(4.17) or (4.21) by qzi and summing them over i (under the assumption that
the initial configuration satisfies the electroneutrality condition):
0=
N
X
i=1

qzi 
X
q∈Ep
|eq |Fiq +
X
m∈Mp

|γm |Fim  .
(4.25)
Since Mp is empty when Vp is an ordinary voxel, the above expression is valid
for both ordinary and membrane voxels. This is precisely the discretization
of (4.11). For ordinary voxels, the membrane flux term is absent.
This completes the discussion of the spatial discretization for arbitrary
membrane geometry in 2D discretized with a uniform embedded boundary
mesh. We shall now briefly discuss discretization over cylindrical geometry.
160
4.1.3
Cylindrical Geometry and Non-Uniform Meshes
In addition to the 2D code that can handle arbitrary membrane geometry, we developed a code for cylindrical geometries, where we often utilize
non-uniform meshes in the axial direction. The basic idea behind the discretization is exactly the same as the finite volume discretization we described
above.
We take a cylindrical coordinate system where we shall take the axial
direction to be the z direction. We seek solutions that are axisymmetric so
that we do not have to discretize in the angular direction. We shall take the
radial direction to be the r direction.
We discretize uniformly in r and either uniformly or non-uniformly in z.
We now have a series of voxels whose shape is a torus with a rectangular
cross-section. Take a voxel Vp . Let the width of this voxel in the r direction
be hrp and that in the z direction by hzp . We let hrp < Kh and hzp < Kh for
some constant K uniformly for all l and take h → 0. We apply the divergence
theorem and its approximation to these voxels.
For cylindrical geometry, we require that the membrane conform to the
control volume boundaries. It should be straightforward to write code that
can handle arbitrary axially symmetric geometry, but we have not yet implemented such a scheme.
We take the analogue of (4.16) as our starting point. Suppose our voxel
is characterized by rp0 < r < rp1 and zp0 < z < zp1 , and does not contain
161
membrane faces.
Z
∂ci
1
∂ci dV
≈
∂t x=xc
|Vp | Vp ∂t
Z
1
=−
∇ · fi dV
|Vp | Vp
Z
1
fi · ndA
=−
|Vp | ∂Vp
1 X
≈−
|eq |Fiq
|Vp | q∈E
(4.26)
p
As the representative point xc , we take the point r = (rp0 + rp1 )/2, and z =
(zp0 +zp1 )/2. The error we incur by using this point in the first approximation
above is O(hrp hzp ) = O(h2 ). For discretization of the fluxes in the axial
direction, through face eq from voxel Vp to Vp′ , we take:
′
′
p
p
qzi hzp′ ci + hzp ci φp − φp
cpi − cpi
+
(hzp′ + hzp )/2 kB T hzp′ + hzp (hzp′ + hzp )/2
′
Fiq = Di
!
(4.27)
Since hzp′ and hzp are not the same, calculations using a Taylor expansion
reveals that the leading order error term for the above is O(hzp − hzp′ ).
We shall consider two types of discretizations. In the first case, we have
separate regions within a computational domain where we use a uniform
mesh, but at the boundary of these regions, there is an abrupt change in mesh
width. In this case, we only commit large errors at these region boundaries,
and since these errors are confined to regions of codimension 1, we expect
this error to decrease proportionally to h as h → 0. We can also take hzp to
162
vary smoothly in p. Then, we can conclude that O(hzp − hzp′ ) = O(h2 ) and
we have an approximation of the axial flux that is accurate to order O(h).
For fluxes in the axial direction, through face eq from voxel Vp to Vp′ , we
discretize as follows:
′
′
p
p
qzi hrp′ ci + hrp ci φp − φp
cpi − cpi
+
(hrp′ + hrp )/2 kB T hrp′ + hrp (hrp′ + hrp )/2
′
Fiq = Di
!
(4.28)
We shall sometimes use a nonuniform mesh in the r direction as well, in
which case we are again faced with the above issues.
We therefore, have an O(h) approximation for the fluxes across voxel
faces. This, as we saw in the discussion for the 2D geometry case, leads to
an O(h) consistency error.
For voxels with membrane faces, we discretize in exactly the same fashion
as in the 2D case. Since we let values at xc also be values at the membrane,
we incur a consistency error of O(1), but as we argued in the previous section,
the contribution from the membrane voxels decreases with h, and thus does
not spoil the asymptotic convergence rate.
As for boundary conditions at the outer rim of the computational domain,
we shall usually use a no-flux boundary condition, in which case, there is no
need to discuss implementation of boundary conditions. In the case of general 2D geometry, we shall present convergence study for Dirichlet boundary
conditions at the outer rim. For Dirichlet boundary conditions, we introduce voxels that are just outside the computational domain and adjacent
163
to each of the voxels at the outer rim. We fix the concentrations ci and
electrostatic potential φ at these control volumes, and thereby, impose the
boundary conditions.
4.2
Temporal Discretization
We now discuss temporal discretization.
We label our time step with n, where n is an integer. We let the time
step to be ∆t.
Suppose we know values of sg , ci , φ and λi at time n − 1. We first advance
sg to find values at time n for every membrane patch γm . Suppose γm is
shared by voxels Vp and Vp′ . Then,
sn,m
− sgn−1,m
′
g
= fg (sn,m , φm,n−1
, cp,n−1, cp ,n−1 )
m
∆t
(4.29)
Note that the evolution of the gating variables sg does not involve any spatial
coupling, and thus, can be solved independently for every membrane patch
γm .
Given sn,m
for all m, we advance ci , φ and λi .
g
For any voxel Vp we have the following semi-discrete evolution equation
for concentration:
∂cpi
∂t
=−


1 
|γm |Fim 
|eq |Fiq +
|Vp | q∈E
m∈M
X
X
p
164
p
(4.30)
We use a backward Euler scheme to march from time n − 1 to time n.
1 X
cp,n
− cp,n−1
i
i
|eq |Fiq,n(cni , φn )
=−
∆t
|Vp | q∈E
p
p,n p,n
1 X
λi φm − λp,n−1
jip,n
φp,n−1
m
i
|γm | Cm
−
+
|Vp | m∈M
qzi ∆t
qzi
p
(4.31)
Here, the arguments of jip,n are all evaluated at time n. The evolution of λi
is given by:
λ̃p,n
− λp,n
λp,n
− λp,n−1
i
i
i
i
=
2
∆t
rd /D0
(4.32)
where λ̃p,n
is evaluated using cp,n
i
i . We may multiply the above by qzi and
sum in i to get an equation in φp,n .
−
ρp0 +
N
X
qzi cp,n−1
1 X
i
|eq |
qzi Fiq,n
=−
∆t
|Vp | q∈E
i=1
p
!
p,n
X
1 X
φm − φp,n−1
m
−
|γm | Cm
+
jip,n
|Vp | m∈M
∆t
i
PN
i=1
p
(4.33)
This can be viewed as the full discretization of equation (4.25). A subtle point
P
is that we have only imposed the electroneutrality condition ρp0 + i qzi cpi = 0
at time n and not at time n − 1 so that we have retained the term −(ρp0 +
P
p,n−1
)/∆t term in the above. If electroneutrality is strictly satisfied
i qzi ci
at each time step, this term is equal to 0. Since we cannot solve the above
165
system of equations exactly in a numerical computation, electroneutrality is
P
never strictly satisfied. By retaining the term −(ρp0 + i qzi cp,n−1
)/∆t, the
i
above system will tend to redress deviations from electroneutrality.
We now have a rather complicated system of nonlinear equations we must
solve. We solve the above system in the following fashion.
4.3
Solution of Nonlinear Equations
We would now like to solve the nonlinear equations for ci , φ and λi . The major
nonlinearity is in the drift term in the drift-diffusion equation. This couples
the concentration term ci with the gradient of the electrostatic potential. We
solve for φ and λi fixing ci , and subsequently solve for ci fixing φ and λi . We
iterate this procedure.
Let cp,n,r
, φp,n,r , λp,n,r
denote the r-th iterate of the solution procedure,
i
i
where r = 0, 1, 2, · · · . We let:
cp,n,0
= cp,n−1
, φp,n,0 = φp,n−1, λp,n,0
= λp,n−1
i
i
i
i
(4.34)
Thus we set our initial iterate to be equal to the values at time n − 1.
We first solve for φp,n,r . We take (4.33) and take the ionic concentration
166
ci to be constant so that the only unknown is φ.
−
ρp0 +
N
X
1 X
qzi cp,n−1
i
|eq |
qzi Fiq (cn,r−1
, φn,r )
=−
i
∆t
|Vp | q∈E
i=1
p
p,n,r
X
1
φm − φp,n−1
m
−
|γm |Cm
|Vp | m∈M
∆t
PN
i=1
(4.35)
p
−
X p,n,r
1
n−1
|γm |
ji (sn,m , φp,n,r
)
m ,c
|Vp | m∈M
i
X
p
By evaluating ci at cn,r−1
in the flux term Fiq , we avoid dealing with the noni
linearity that arises from the drift term. The only possibility for a nonlinearity in the above is in the transmembrane current term. In many applications,
ji is assumed linear in φm . If not, we linearize as follows:
We first recall the functional form of transmembrane current terms. The
general functional form of ion channel currents is written as:
ji =
X
ji,α
(4.36)
α
(k)
(l)
ji,α = gi,α (s, φm , c(k) , c(l) )Ii,α (φm , ci , ci )
(4.37)
where α labels the types of ion channels present, and ji,α the transmembrane
current through ion channels of this type. gi,α is the density of the such ion
channels per unit area of membrane, and Ii,α is the instantaneous current
voltage relationship. We choose a suitable linearization the instantaneous
167
current voltage relation with respect to φm around φp,n,r−1
.
m
(k)
(l)
(k)
(l)
L
p,n,r−1
Ii,α
(φn,p,r
, ci , ci )(φp,n,r
− φp,n,r−1
)
m , ci , ci ) = DIi,α (φm
m
m
+
(4.38)
(k) (l)
Ii,α (φp,n,r−1
, ci , ci )
m
The term DIi,α will typically be the derivative of Ii,α with respect to φm . We
shall sometimes use simpler linearizations when
∂Ii,α
∂φm
is a complicated expres-
sion. Instead of ji itself, we shall therefore use the following linearization in
its place in (4.35).
jiL,p,n,r =
X
L,p,n,r
ji,α
(4.39)
α
L,n,p,r
n−1
n−1 n−1
L
ji,α
= gi,α(sp,n , φm
, c )Ii,α
(φp,n,r
)
m , ci
(4.40)
Note that sp,n is already a known quantity that we do not have to solve for.
The result is a linear equation in φ, which can now be solved.
Solving for λp,n,r
is simple:
i
λp,n,r
− λp,n−1
λ̃p,n,r−1
− λp,n,r
i
i
i
i
=
∆t
rd2 /D0
where λ̃p,n,r−1
is evaluated using cp,n,r−1
.
i
i
168
(4.41)
Given φp,n,r and λp,n,r
, we solve for cp,n,r
as follows:
i
i
1 X
cp,n,r
− cp,n−1
n,r−1
i
i
|eq |Fiq (cn,r
, φn )
=−
i , ci
∆t
|Vp | q∈E
p
p,n,r p,n,r
X
λi φm − λp,n−1
1
jip,n,r
φp,n−1
m
i
|γm | Cm
−
+
|Vp | m∈M
qzi ∆t
qzi
p
(4.42)
The Fiq term requires explanation. Suppose that the face eq is shared by
voxels Vp and Vp′ . Then we let:
!
p,n,r
p′ ,n,r
c
−
c
n,r−1
i
i
Fiq (cn,r
, φ n ) = Di
i , ci
h
!
′
p′ ,n,r−1
qzi
φp,n,r−1 − φp ,n,r−1
cp,n,r−1
+
c
i
i
+ Di
kB T
2
h
(4.43)
where the above flux expression is to be suitably modified when dealing with
non-uniform meshes (cf. (4.27 and (4.28)). As can be seen from the above
expression, the diffusive flux is treated implicitly, whereas drift flux is left
explicit. This does not seem to lead to stability problems.
With the above discretization of the flux approximation Fiq , (4.42) is a
linear equation in ci . In fact, this is just a familiar discretization of the diffusion equation with a source term and flux boundary conditions. Moreover,
the discrete diffusions in the separate regions demarcated by the membranes
are completely decoupled from one another.
Linear systems for both φ and ci are symmetric positive definite (the
169
equation for φ is only positive semi-definite if we impose no-flux boundary
conditions at the outer rim of the computational domain). We can therefore use the Cholesky decomposition for small systems, and the Conjugate
Gradient Method for larger systems. We shall return to this issue when we
discuss implementation of the above.
We now iterate this procedure in r a suitable number of times, and set
the final iterate to be the values at time n. Note that one iteration is enough
to obtain a first order scheme in time. We also point out that the scheme is
conservative in exact arithmetic: we have ion conservation regardless of how
may iterations we perform.
We iterate so that electroneutrality is more strictly satisfied at time n.
Multiplying (4.42) with qzi and summing in i does not reproduce (4.35)
because the concentrations in the flux approximation Fiq are evaluated using
different values in the two expressions. Therefore, the solution to (4.42) only
satisfies electroneutrality in the limit r → ∞.
Therefore, our termination criterion for the above iteration is to check
whether the electroneutrality condition is satisfied to within a certain tolerance after the r-th iteration. We use the following criterion:
P
k
|Vk ||
P
k,n,r
|
i zi ci
P
k
|Vk |
< ǫtol
(4.44)
In all computations, we take ǫtol = 1 × 10−6 We set this final iterate to be
the value of ci , φ at the next time step, except for the adjustment we discuss
170
below.
When we use no-flux boundary conditions at the outer rim of the computational domain, we perform the following adjustment at the end of each
computational step. We fix the concentrations so that the global amount of
ions is conserved as strictly as possible by:
Qn,r
=
i
X
|Vk |ck,n,r
i
Λn,r
=
i
X
|γ|m
k
ck,n
i
λn,r
i
Cm φn,r
m
qzi
m
n,r
Qn,r
i + Λi − Qinit
ck,n,r
= 1+
n,r
i
Qn,r
+
Λ
i
i
(4.45)
(4.46)
(4.47)
Here, the index r denotes the final iterate, and Qinit is the total amount of
ci at the initial state.
Why do we need to perform this fix when we know that the scheme is in
fact conservative? After all, since the scheme is conservative, Qn,r
+ Λn,r
−
i
i
Qinit = 0 and therefore, the above should have no effect. The unfortunate
reality is that the scheme is conservative only in exact arithmetic. With
floating point arithmetic, errors tend to accumulate and, with time, ion conservation is violated. Computational experiments tells us that this error is
negligible as far as the values of ci are concerned. This has to be corrected
nonetheless because this small violation leads to global charge accumulation.
The scheme presented above has an inherent mechanism to eliminate local
charge accumulation but cannot eliminate global charge accumulation. The
171
above fix is on the order of round-off error at each time step, and therefore, should have little effect on the convergence properties of the numerical
scheme.
We note that this fix is only necessary for the no-flux boundary condition.
When Dirichlet or mixed boundary conditions are imposed at the outer rim
of the computational domain, accumulation in charge in the computational
domain will eventually dissipate through communication with the outer bath.
4.4
4.4.1
Implementation
Computational Geometry
In the 2D code where we handle arbitrary membrane geometry, we must
generate the necessary geometry data at the membrane where the mesh is
cut. We have written a custom mesh generator to perform this routine. It
takes the characteristic function of a region as input to generate the necessary
data. The mesher approximates a cut by the membrane as a straight line, and
cannot handle non-generic cases of degenerate geometry. When the volume
of a membrane voxel is less than 10−5 in volume as an ordinary mesh, this
mesh is ignored.
Though adequate for our purposes at present, we would certainly need a
more sophisticated mesher for more complicated simulations.
172
4.4.2
Linear Algebra
The solution to the nonlinear algebraic equations requires the solution of
a linear system at each iteration. We note that solving for the electrostatic
potential as well as the concentrations involve solving a positive definite symmetric system. We thus either use a direct solver(Cholesky decomposition)
or the conjugate gradient method [43]. The code for cylindrical geometry has
been implemented using Matlab, where we use a direct solver. The code for
general 2D geometry has been written in C++, where we use PETSc for the
linear algebra routines [2]. PETSc is a package that provides sparse linear
solvers and is designed to be suitable for parallel algorithms. Though we do
not yet use parallel machines, having coded in PETSc should facilitate this
transition in the future.
4.4.3
3D-Cable Model
We briefly describe how we perform numerical simulations with the 3D-Cable
model which we introduced in Section 2.2.7 of Chapter 2, We shall make use
of this model, as well as the electroneutral model, in Chapter 5.
For simulations with the 3D-cable model, we need only solve for the
electrostatic potential φ. The ionic concentrations are assumed constant in
time. Therefore, the only variables that need to be updated are φ and the
gating variables defined on the membrane. The algorithm is thus, very simple
to state. Wherever ci occurs in the equations, replace this with the initial
173
concentrations. For the 3D-cable model, there is no need to iterate at each
time step since the equations to be solved are linear and electroneutrality is
not a concern.
4.5
Convergence Study: Cylindrical Geometry
We test the convergence for the cylindrical case. We test the convergence of
the code under two kinds of situations, the standard Hodgkin-Huxley axon
[13, 22, 23], and a cardiac model we shall use in Chapter 5.
4.5.1
Hodgkin-Huxley Axon
We shall use the standard Hodgkin-Huxley parameters. Parameters not required for the Hodgkin-Huxley model as computed with the cable model but
are required with the electroneutral model are the diffusion coefficients for
each ionic species and the initial concentration of the ions. We consider three
ionic species Na+ , Cl− , and K+ . The initial concentrations and the diffusion
coefficients we use are listed in Table 4.1. The membrane charge ratios λi
were initialized so that λi |t=0 = λ̃i . The immobile charge density was
t=0
taken so that electroneutrality is satisfied at each spatial point at t = 0, i.e.,
ρ0 = −
N
X
i=1
qzi ci |t=0
174
(4.48)
T
DN a+
DK +
DCl−
int cN a+ t=0
cext
N a+ t=0
int cK + t=0
cext
K + t=0
int cCl− t=0
cext− Cl
t=0
φm |t=0
Absolute temperature
Diffusion coefficient of K+
Diffusion coefficient of Na+
Diffusion coefficient of Cl−
Initial intracellular concentration of Na+
Initial extracellular concentration of Na+
Initial intracellular concentration of K+
Initial extracellular concentration of K+
Initial intracellular concentration of Cl−
Initial extracellular concentration of Cl−
Initial transmembrane potential, φint − φext
273.15+37 K
1.33µm2 /msec[23]
1.96µm2 /msec[23]
2.03µm2 /msec[23]
10 mmol/l
145 mmol/l
140 mmol/l
5 mmol/l
150 mmol/l
150 mmol/l
-70 mV
Table 4.1: Parameter values used in the Hodgkin-Huxley simulations of the
axon.
The Hodgkin-Huxley model has one free parameter, the value of the equilibrium potential [22]. We take this to be −70mV and the initial value of the
gating variables to be the equilibrium values at −70mV .
We take the axon to be a cylinder of radius lµm and axial length 2lA µm.
Take the z axis along the axis of the cylinder, with the axonal ends at z =
±lA , and the radial axis r from the center of the cylinder. The cylindrical
axon is bathed in an extracellular bath located between the cell membrane
at r = l and r = 2l, where we impose no-flux boundary conditions. We shall
also impose no-flux boundary conditions at z = ±lA . We shall let the total
175
simulation time be Te . We let:
l = 0.1, 1, 10µm
√
lA = 2 l × 103 µm
(4.49)
Te = 5msec
(4.51)
(4.50)
We note that the axonal length is far larger than the radial length. This
length of the axon lA was chosen so that we can see a wave of propagating
action potential. The dependence of lA on l is chosen because L0 , the typical
√
length scale over which the electrostatic potential varies, scales with l (cf.
Section 2.2.2).
At time t = 0, we initiate an action potential by transiently opening Cl−
whose time course is given by:
GCl− =



5t
5 1 + cos 8πz
1
−
cos
2π
− l8A < z <
lA
Te


0
lA
8
and t <
Te
5
otherwise
(4.52)
We thus give a brief change in the membrane chloride conductance at the
center of the axon. An action potential is initiated here and spreads towards
the two ends of the axon. A snapshot from a sample run from this simulation
is shown in Figure 4.1 and 4.2.
We now test convergence in space and time.
176
1ms
2ms
0
mV
mV
0
−50
−100
1
−50
−100
1
2000
0.5
µm
0
−2000
0
0.5
µm
µm
2000
0
0
3ms
µm
4ms
0
mV
0
mV
−2000
−50
−100
1
−50
−100
1
2000
0.5
µm
0
−2000
0
0.5
µm
µm
2000
0
0
−2000
µm
Figure 4.1: Plot of electrostatic potential φ at t = 1, 2, 3, 4msec. Since the
solutions we seek are radially symmetric, the radial cross-section (r > 0) is
plotted in the graph.
177
Na
K
0.1
0.4
mmol/l
mmol/l
0
0.2
0
−0.1
−0.2
−0.3
1
1
2000
0.5
µm
0
−2000
0
0.5
µm
µm
2000
0
0
−2000
µm
Cl
mmol/l
0.06
0.04
0.02
0
−0.02
1
2000
0.5
µm
0
0
−2000
µm
Figure 4.2: Plot of change in ionic concentrations from t = 0 at t = 2msec.
178
Convergence in Space
We take a uniform grid of Nz × Nr over the simulation domain. We take:
Nz = 8 × Nr , Nr = 2n+1
(4.53)
where we let n = 1, · · · 4. We take our time step to be:
∆t = 0.02msec,
NT =
Te
∆t
(4.54)
To measure the convergence rate, we define the discrete p-norm as:
kukLp =
2Nr
X
k=1
|Vk ||uk |p
!1/p
,
1≤p<∞
kukL∞ = max |uk |
(4.55)
(4.56)
k
The convergence rate is measured by comparing the interpolation of the
numerical solution at a finer level to the numerical solution at a coarser
level. For the ionic concentrations ci , let ci computed with an 8Nr × Nr mesh
s
r
be written as cN
i . We define a measure of error ep [ci ; Nr ] as follows.
2Nr →Nr 2Nr
r
esp [ci ; Nr ] = kcN
ci kLp
i −I
(4.57)
Here, I 2Nr →Nr is an interpolation operator from the finer to the coarser grid.
For the electrostatic potential φ, we need to take into account the arbi-
179
radius
0.1
1
10
Lp
L1
L2
L∞
L1
L2
L∞
L1
L2
L∞
rps [C1 , 16] rps [C2 , 16] rps [C3 , 16] rps [Φ, 16]
2.01
2.01
2.00
2.00
2.01
2.01
2.00
2.00
2.01
2.01
1.94
2.01
2.02
2.02
2.00
2.01
2.02
2.02
2.00
2.01
2.01
2.01
1.99
2.01
2.02
2.04
1.95
2.05
2.00
2.03
1.86
2.03
1.41
1.82
1.42
2.03
Table 4.2: The empirical rates of convergence rps in space for different values
of axonal radii. Values computed at t = 4ms, and Nr = 16.
trariness of φ, up to addition of a constant. We thus, measure the error in φ
as:
esp [φ; Nr ] = minkφNr − I 2Nr →Nr φ2Nr − cφ kLp
cφ ∈R
(4.58)
As an empirical measure of convergence rate in space, we use
rps [q; Nr ]
= log2
esp [q; Nr ]
esp [q; 2Nr ]
.
(4.59)
where q can be either ci or φ.
Figure 4.3 plots the rate of convergence of ci and φ measured in the L2
norm, when the axonal radius is taken to be 1µm.
Table 4.2 lists the rate of convergence for both ci and φ at the three radii
with three norms, L1 , L2 and L∞ .
Despite the fact that the local truncation error at the membrane is of order
180
t=2ms
t=4ms
ref.
−6
1
10
L error in c
−4
10
−7
10
2
2
L error in φ
t=2ms
t=4ms
ref.
−5
10
−8
10
1
1
10
N
10
N
r
r
t=2ms
t=4ms
ref.
−6
3
10
L error in c
−7
10
2
L2 error in c
t=2ms
t=4ms
ref.
−7
2
10
−8
10
−8
10
1
1
10
N
10
N
r
r
Figure 4.3: L2 error in space when the axon radius is 1µm. The error is
measured at t = 2ms and 4ms. The reference line indicates second order
convergence.
181
1, we see that the truncation error in the bulk of the computational region
dominates. We see second order convergence for most parameter regions
considered. The deterioration in convergence rate when the axonal radius is
equal to 10µm is attributable to the fact that the concentration gradients
are not fully resolved when Nr = 16. The radial diffusion of ions is rather
slow, and one needs to resolve the slow diffusion layer (Chapter 2) in order
to obtain second order convergence in space.
Convergence in Time
Convergence in time is measured similarly to the spatial case. We vary the
time step so that:
∆t = 0.04 × 2−n , NT ≡
Te
= 125 × 2n
∆t
(4.60)
where n = 1 · · · 4 We take Nr = 8 as our spatial grid to assess convergence
in t.
The convergence rate and error is computed analogously to the spatial
case.
T
T
− I 2NT →NT c2N
kLp
etp [ci ; Nr ] = kcN
i
i
(4.61)
Here, I 2NT →NT is an interpolation operator from the finer to the coarser time
step.
182
radius
0.1
1
10
Lp
L1
L2
L∞
L1
L2
L∞
L1
L2
L∞
rpt [C1 , 500] rpt [C2 , 500] rpt [C3 , 500] rpt [Φ, 500]
0.98
0.98
0.99
1.00
0.99
0.99
1.01
1.00
0.99
0.99
1.04
0.86
0.98
0.98
0.99
0.99
0.99
0.99
1.01
0.98
0.99
0.99
1.01
0.86
0.98
0.98
0.98
0.99
0.99
0.99
1.02
0.98
0.99
0.99
1.10
0.86
Table 4.3: The empirical rates of convergence rpt for different values of axonal
radii. Values computed at t = 4ms, and Nr = 8.
For the electrostatic potential φ, we let
etp [φ; NT ] = minkφNT − I 2NT →NT φ2NT − cφ kLp
cφ ∈R
(4.62)
As an empirical measure of convergence rate in space, we use
rpt [q; NT ]
= log2
etp [q; NT ]
etp [q; 2NT ]
.
(4.63)
where q can be either ci or φ.
Figure 4.4 plots the rate of convergence of ci and φ measured in the L2
norm, when the axonal radius is taken to be 1µm.
Not surprisingly, we see first order convergence in time over all parameter
183
t=2ms
t=4ms
ref.
L error in c
1
10
2
2
L error in φ
−4
t=2ms
t=4ms
ref.
−7
10
−5
10
3
3
10
10
N
N
T
T
−7
10
3
t=2ms
t=4ms
ref.
L error in c
−7
10
2
L2 error in c
2
t=2ms
t=4ms
ref.
−8
10
3
3
10
10
N
N
T
T
Figure 4.4: L2 error in time when the axon radius is 1µm. The error is
measured at t = 2ms and 4ms. The reference line indicates first order
convergence.
184
ranges considered.
4.5.2
Cardiac Geometry
The next test case reflects the geometry we consider in the cardiac application
of our model. For the motivation for this test geometry, we refer to Chapter 5.
We consider Nc = 2 cells of equal length separated by a narrow intercellular
space of width lg . In fact, we consider two “half” cells as we shall see shortly.
The radius of the cell is l and the whole system is bathed in an extracellular
medium of radius 2l.
Similarly to the axonal case, we take z to be the axial direction and r to
be the radial coordinate. We take the origin to be in the middle of the gap.
Formally, the intracellular region can be written as:
−
lg
lA + lg
< z < − and r < l
2
2
lg
lA + lg
<z<
and r < l
2
2
(4.64)
(4.65)
g
). This is what we mean
The intracellular region is open-ended at z = ±( lA +l
2
by “half cell”. We impose no-flux boundary condition at the boundaries of
the intracellular and extracellular regions.
The values for lg , lA and l are given by:
lg = 20nm,
lA = 100µm,
185
l = 11µm
(4.66)
We note that lg is about 4 orders of magnitude smaller than lA , and thus we
use a non-uniform mesh, the details of which we shall describe shortly.
We consider 4 ion types in the calculation, Na+ , K+ , Ca2+ and Cl− . In
the gap we introduce a fixed negative charge density, whose physiological
motivation will be discussed in Chapter 5. We initialize the extracellular
chloride concentration as follows:
cext
Cl− t=0 =



100 − 50 1 −


100mmol/l
r 2
l
mmol/l r < l and − lg /2 < z < lg /2
otherwise
(4.67)
The initial condition for the other ionic species are listed in Table 4.4. We determine ρ so that electroneutrality is satisfied everywhere. The initialization
of the chloride concentration as above introduces an excess fixed negative
charge density in the gaps between cells.
The diffusion coefficients were determined in the following way. The
ohmic cytoplasmic conductance is given by the following as a function of
the ionic concentrations:
N
X
(qzi )2 ci
i=1
kB T
Di
(4.68)
If one computes the cytoplasmic or extracellular conductance according to
values of Di in an aqueous solution, the values used in Table 4.1, we obtain
an overestimate which deviates from the experimentally observed value by a
factor of 2 − 5. We thus scale the diffusion coefficient in aqueous solution
by a uniform factor α so that the cytoplasmic or extracellular conductance
186
T
us
DN
a+
us
DK
+
us
DCa
2+
us
DCl
−
int cN a+ t=0
cext
N a+ t=0
cint
K + t=0
ext cK + t=0
cint
Cl− t=0
cext
Cl− t=0
int cCa2+ t=0
cext
Cl2+ t=0
φm |t=0
Unscaled diffusion coefficient of K+
Unscaled diffusion coefficient of Na+
Unscaled diffusion coefficient of Ca2+
Unscaled diffusion coefficient of Cl−
Initial intracellular concentration of Ca2+
Initial extracellular concentration of Ca2+
273.15+37 K
1.33µm2 /msec
1.96µm2 /msec
0.3µm2 /msec
2.03µm2 /msec
10 mmol/l
145 mmol/l
140 mmol/l
5 mmol/l
10 mmol/l
see text
0.4 µmol/l
2 mmol/l
-90 mV
Table 4.4: Parameter values used in cardiac simulation.
calculated above is approximately within the experimental range. More concretely, we let,
g
observed
=α
N
X
(qzi )2 ci |
t=0
i=1
kB T
Dius
(4.69)
where g observed is the cytoplasmic conductance, which we take to be equal to
the extracellular conductance, Dius is the unscaled diffusion coefficient, and
overline denotes averaging over the computational domain. Following [24],
we let:
1
g observed
= 150Ω · cm
(4.70)
For the ion channel composition for the membrane, we use the model of
Bernus et. al. [3], where the authors model the electrical activity of human
187
ventricular myocytes. The only difference is the localization of the Na+
channels. We concentrate their distribution so that 99% of the total Na+
conductance sits at the membranes facing the gap. This allows an action
potential to propagate across the gap without the two intracellular spaces
being directly connected by a cytoplasmic bridge.
All instantaneous current voltage relations for ionic channels in the model
of Bernus et. al. are linear in the transmembrane voltage. We do not have
to think about linearization of the current voltage relationship to obtain a
linear system. The ionic pump currents are nonlinear in the transmembrane
voltage, but will be treated explicitly. This does not result in numerical
instabilities because ionic pump currents are typically small in magnitude.
We simulate this system for time Te = 4msec. For a while after t = 0,
we add excitation to the system by way of adding Na+ conductance at the
following membrane locations:
Gadd
Na+ =



t
 5 1 + cos π(z+Lz )
z < − l2g and t < τe
1
−
cos
2π
4
lA /2
τe


0
Lz = (lA + lg )/2,
otherwise
(4.71)
τe = 1msec
(4.72)
Thus, we stimulate the system at one end of the cell located in z < 0, and
see the action potential propagate into the next cell. Snapshots from this
simulation are shown in figures 4.5 and 4.6.
188
0.6ms
1.2ms
0
mV
mV
0
−50
−100
−50
−100
20
20
50
10
µm
0
−50
50
0
10
µm
µm
0
0
1.8ms
µm
2.4ms
0
mV
0
mV
−50
−50
−100
−50
−100
20
50
10
µm
0
−50
20
50
0
10
µm
µm
0
0
−50
µm
Figure 4.5: The evolution of the electrostatic potential in the cardiac simulation with variable mesh width. Snapshots shown at t = 0.6, 1.2, 1.8, 2.4msec.
The mesh size is Nz = 48, Nr = 32 in this computation.
189
Na
K
1
−2
mmol/l
mmol/l
0
−4
−6
0.5
0
−8
20
20
50
10
µm
0
−50
50
0
10
µm
µm
0
0
µm
Cl
0.08
0
0.06
−2
mmol/l
mmol/l
Ca
−50
0.04
0.02
−4
−6
0
20
50
10
µm
0
−50
20
50
0
10
µm
µm
0
0
−50
µm
Figure 4.6: The change in ionic concentrations from the initial value, in the
cardiac simulation with variable mesh width. Snapshot at t = 2msec shown.
The mesh size is Nz = 48, Nr = 32 in this simulation in this computation.
190
As we remarked above, the gap width lg and the cell length lA are 4 orders
of magnitude apart. Therefore, we cannot afford to use a uniform mesh,
since this will require considerable computational power. We therefore use a
non-uniform mesh, both in the axial and radial directions.
In the axial direction, we lay a mesh whose width is of order lg when
− l2g < z <
lg
2
and of order lA away from the gap where z ∼ ± l2A . For meshes
in between, we interpolate the two widths with an approximate geometric
sequence. In the radial direction, we lay a mesh of width of order lg near
r = l and of order l where r = l ± l. Again, we interpolate the meshes in
between with an approximate geometric sequence. We give the details of this
construction below.
We first define the following function f (z) on 0 ≤ z ≤ (lA + lg )/2:
f (z) =



2z


lg



2
 lg b
0≤z≤
+1
log 1 + b z − l2g





 nz −1 z −
lA
lA
2
+ nz
lg
2
lg
2
≤ z ≤ zβ
zβ ≤ z ≤
(4.73)
lA
2
where nz is an integer parameter that we specify, and b and zβ are determined
so that f is continuously differentiable at z = zβ . We define the voxel
boundaries zk using f (z) as follows:
zk = f
−1
nz
k ,
Nz /2
191
k = 0, · · · Nz /2
(4.74)
where Nz /2 is a multiple of nz . This construction adjusts the voxel width
depending on whether the location is far away from the intercellular gap. For
z < 0, we take the voxel boundaries to be the reflection of the zk above with
respect to z = 0.
In the radial direction, we shall take the following mesh. We shall first
define the following function g analogous the the f above. For r > l let,
g(r) =




2
lg b
log (1 + b (r − l)) l ≤ r ≤ rβ


 nr (r − l) + nr
2l
(4.75)
rβ ≤ r ≤ 2l
where nr is an integer parameter that we specify, and b and rβ are determined so that g is continuously differentiable at r = rβ . We define the voxel
boundaries rk using g(r) as follows:
rk = g
−1
nr
k ,
Nr /2
k = 0, · · · Nr /2
(4.76)
where Nr /2 is a multiple of nr . For r < l, we take the points 2l − rk as the
voxel boundaries. This construction again has the benefit of concentrating
the meshes toward the membranes and near the gaps.
The coarsest level starts with 2 meshes − l2g < z <
for z < − l2g and z >
lg
,
2
lg
2
and 5 meshes each
a total of Nz = 12 meshes in the axial direction.
This corresponds to nz = 6 in equation (4.73). In the radial direction, we
start with Nr = 8 meshes, which corresponds to nr = 4 in equation (4.75).
192
Lp
L1
L2
L∞
rps [C1 , 32] rps [C2 , 32] rps [C3 , 32] rps [C4 , 32] rps [Φ, 32]
1.90
1.91
1.94
1.94
2.00
1.95
1.87
1.89
1.97
2.00
1.88
1.89
2.10
1.83
2.00
Table 4.5: The empirical rates of convergence rps for different values of axonal
radii. Values computed at t = 4ms, and Ns = 32.
We progressively refine the mesh so that:
Nz = 6 × 2n , Nr = 4 × 2n
(4.77)
where n = 1 · · · 4. We take the time step to be:
∆t = 0.02msec,
NT =
Te
∆t
(4.78)
Spatial convergence is assessed in exactly the same way as in the axonal
case. In Figure 4.7, we plot the L2 error of the numerical scheme.
As we can see, the scheme converges supralinearly despite the local truncation error at the membrane, which our foregoing analysis suggested was
large. The scheme attains almost second order convergence in all of the
variables.
193
−4
1
−6
10
t=2ms
t=4ms
ref.
10
−7
10
2
−6
L error in c
2
L error in φ
10
1
t=2ms
t=4ms
ref.
1
10
10
N
N
r
r
−8
3
L2 error in c
2
L error in c2
10
t=2ms
t=4ms
ref.
−10
10
1
t=2ms
t=4ms
ref.
1
10
10
N
N
r
L2 error in c
4
r
−6
10
−7
10
t=2ms
t=4ms
ref.
1
10
N
r
Figure 4.7: L2 error in time. The error is measured at t = 2ms and 4ms.
The reference line indicates second order convergence.
194
−4
1
L error in c
−5
10
2
t=2ms
t=4ms
ref.
2
L error in φ
10
2
3
10
t=2ms
t=4ms
ref.
−7
10
2
10
3
10
10
N
N
T
T
−8
3
−7
10
L2 error in c
L2 error in c
2
10
t=2ms
t=4ms
ref.
−8
10
2
3
10
t=2ms
t=4ms
ref.
−9
10
2
10
3
10
N
10
N
T
L2 error in c
4
T
−7
t=2ms
t=4ms
ref.
10
2
3
10
10
N
T
Figure 4.8: L2 error in time. The error is measured at t = 2ms and 4ms.
The reference line indicates first order convergence.
Convergence in Time
We vary the time step so that:
∆t = 0.04 × 2−n , NT ≡
Te
= 100 × 2n
∆t
(4.79)
where n = 1 · · · 5. As the spatial mesh, we use Nr = 32, Nz = 48. Convergence in the L2 norm is plotted in Figure 4.8.
195
Lp
L1
L2
L∞
rpt [C1 , 800] rpt [C2 , 800] rpt [C3 , 800] rps [C4 , 800] rps [Φ, 800]
1.01
1.02
1.03
1.01
1.04
1.02
1.02
1.02
1.01
1.05
1.04
1.03
1.06
1.04
1.05
Table 4.6: The empirical rates of convergence rpt . Values computed at t =
4ms, and Nr = 32.
We again see first order convergence for all variables, in accordance with
our analysis in the foregoing.
4.6
General 2D Geometry
For general 2D geometry, we take a circular cell in a two dimensional square
computational domain. Let the computational domain be of size l. Take
the origin of the domain to be at the center of the computational domain,
and take the x and y axes parallel to the sides of the square computational
domain. Then we let the intracellular space be defined by:
2x
l
2
+
2y
l
2
<
105
256
(4.80)
For the value of l, we take:
l = 0.1µm
196
(4.81)
This is a particularly small volume, but is a relevant scale for microstructures
in the central nervous system.
At the rim of the computational domain, we impose either no-flux or
Dirichlet boundary conditions. For Dirichlet boundary conditions, we set
the ci to be equal to their initial values, and φ to be uniformly equal to 0. If
we can demonstrate that the scheme performs well under these two boundary
conditions, we can infer that the scheme will perform well for mixed boundary
conditions.
In order to observe appreciable changes in ionic concentrations over the
time range of the computational study, we scaled the maximal conductances
by a factor of 5 and decreased the diffusion coefficient by a factor of 5 with
respect to the values of Table 4.1. The ionic makeup of the simulation is
therefore Na+ , K+ and Cl− , where the concentrations and the diffusion coefficients used are summarized in Table 4.7. As before, the immobile charge
density is determined so that electroneutrality is strictly satisfied at t = 0
and we initialize λi with λ̃i evaluated using the initial concentrations and
membrane potential.
We add the following the membrane conductances for 0 ≤ t ≤ τe to
197
T
DN a+
DK +
DCl−
int cN a+ t=0
cext
N a+ t=0
int cK + t=0
cext
K + t=0
int cCl− t=0
cext− Cl
t=0
φm |t=0
GNa
GK
GL
Diffusion coefficient of K+
Diffusion coefficient of Na+
Diffusion coefficient of Cl−
Maximal Na+ channel conductance
Maximal K+ channel conductance
Leak conductance (carried by K+ ions)
273.15+37 K
0.266µm2 /msec
0.392µm2 /msec
0.406µm2 /msec
10 mmol/l
145 mmol/l
140 mmol/l
5 mmol/l
20 mmol/l
150 mmol/l
-70 mV
600 mS/cm2
180 mS/cm2
1.5 mS/cm2
Table 4.7: Parameter values used in the simulation for general 2D geometries.
initiate an action potential.
add
add
Gadd
Cl− = GNa+ = GK+



200 2y 2 1 − cos 2π t
y < 0 and t < τe
l
τe
=


0
otherwise
τe = 1msec
(4.82)
(4.83)
(4.84)
We run the simulation for a total of Te = 2msec. Snapshots from the simulation are given in figures 4.9, 4.10, 4.11, and 4.10.
198
0.6ms
50
50
0
0
mV
mV
0.2ms
−50
−50
1
1
1
0.5
µm
1
0.5
0.5
0
0
µm
µm
0.5
0
µm
1.4ms
50
50
0
0
mV
mV
1.0ms
0
−50
−50
1
1
1
0.5
µm
0
0
1
0.5
0.5
µm
µm
0.5
0
0
µm
Figure 4.9: The evolution of the electrostatic potential under Neumann
boundary conditions at the outer boundary of the computational domain.
Snapshots shown at t = 0.2, 0.6, 1.0, 1.4msec. The mesh size is 64 × 64 in this
computation.
199
K
2
1
1
0
mmol/l
mmol/l
Na
0
−1
1
−1
−2
1
1
0.5
µm
0
0
1
0.5
0.5
µm
µm
0.5
0
µm
1.4ms
0.5
50
0
0
mV
mmol/l
Cl
0
−50
−0.5
1
1
1
0.5
µm
0.5
0
0
µm
µm
1
0.5
0.5
0
0
µm
Figure 4.10: The change in ionic concentrations from the initial value, computed under Neumann boundary conditions at the outer boundary of the
computational domain. Snapshot at t = 1msec shown. The mesh size is
64 × 64 in this computation.
200
0.6ms
50
50
0
0
mV
mV
0.2ms
−50
−50
1
1
1
0.5
µm
1
0.5
0.5
0
0
µm
µm
0.5
0
µm
1.4ms
50
50
0
0
mV
mV
1.0ms
0
−50
−50
1
1
1
0.5
µm
0
0
1
0.5
0.5
µm
µm
0.5
0
0
µm
Figure 4.11: The evolution of the electrostatic potential under Dirichlet
boundary conditions at the outer boundary of the computational domain.
Snapshots shown at t = 0.2, 0.6, 1.0, 1.4msec. The mesh size is 64 × 64 in this
computation.
201
K
2
1
1
0
mmol/l
mmol/l
Na
0
−1
1
−1
−2
1
1
0.5
µm
0
0
1
0.5
0.5
µm
µm
0.5
0
µm
1.4ms
0.5
50
0
0
mV
mmol/l
Cl
0
−50
−0.5
1
1
1
0.5
µm
0.5
0
0
µm
µm
1
0.5
0.5
0
0
µm
Figure 4.12: The change in ionic concentrations from the initial value, computed under Dirichlet boundary conditions at the outer boundary of the
computational domain. Snapshot at t = 1msec shown. The mesh size is
64 × 64 in this computation.
202
4.6.1
We lay a uniform mesh of Nx × Nx over the computational domain, where
the membrane cuts through the uniform mesh as described in the above. We
vary Nx in multiples of 2.
Nx = 24+n
(4.85)
where n = 1, · · · , 5. We let the time step be:
∆t = 0.02msec,
NT =
Te
= 100
∆t
(4.86)
Convergence is measured similarly to the cylindrical cases discussed above.
We give a table of the convergence rates in Table 4.8. We plot the errors of computational results for calculations with both the no-flux (Figure
4.13) and Dirichlet boundary conditions (Figure 4.14). We see that the convergence rate is in between first and second order, and possibly because of
boundary effects, does not reach second order accuracy. The convergence rate
seems to be better when no-flux boundary conditions are imposed. This may
stem from the low order approximation used to enforce dirichlet boundary
conditions.
203
B.C.
No-flux
Dirichlet
Lp
L1
L2
L∞
L1
L2
L∞
rps [C1 , 128] rps [C2 , 128] rps [C3 , 128] rps [Φ, 128]
1.52
1.54
1.51
1.50
1.49
1.52
1.51
1.50
0.92
1.02
1.48
1.50
1.07
1.04
1.15
2.24
1.02
1.01
1.14
2.25
0.94
1.02
1.23
2.21
Table 4.8: The empirical rates of convergence rps for both boundary conditions. Values computed at t = 2ms, and Nx = 128.
−6
1
L error in c
−7
10
−2
10
2
2
L error in φ
10
t=1ms
t=2ms
ref.
−3
10
t=1ms
t=2ms
ref.
2
2
10
10
N
N
x
x
−2
10
3
−3
10
2
L error in c
L2 error in c
2
−2
10
−3
10
t=1ms
t=2ms
ref.
t=1ms
t=2ms
ref.
2
2
10
10
N
N
x
x
Figure 4.13: L2 error in space, when no-flux boundary conditions are used.
The error is measured at t = 1ms and 2ms. The reference line indicates
second order convergence.
204
−1
10
−6
1
−2
10
2
L error in c
2
L error in φ
10
−7
10
t=1ms
t=2ms
ref.
−3
10
t=1ms
t=2ms
ref.
2
2
10
10
N
N
x
x
−2
3
L error in c
−2
10
2
L2 error in c
2
10
−3
10
t=1ms
t=2ms
ref.
−3
10
t=1ms
t=2ms
ref.
2
2
10
10
N
N
x
x
Figure 4.14: L2 error in space, when Dirichlet boundary conditions are used.
The error is measured at t = 1ms and 2ms. The reference line indicates
second order convergence.
205
B.C.
No-flux
Dirichlet
Lp
L1
L2
L∞
L1
L2
L∞
rpt [C1 , 200] rpt [C2 , 200] rpt [C3 , 200] rpt [Φ, 200]
1.00
1.00
0.99
1.00
1.00
1.00
0.99
1.00
1.00
1.01
0.69
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.99
1.00
1.00
1.01
1.00
1.00
Table 4.9: The empirical rates of convergence rpt for both boundary conditions
Values computed at t = 2ms, and NT = 200.
4.6.2
Convergence in Time
For convergence in time, we let:
∆t = 0.08 × 2−n msec,
NT =
Te
= 25 × 2n
∆t
(4.87)
where n = 1, · · · , 5. For the spatial grid, we take Nx = 64.
We give a table of the convergence rates in Table 4.9. We plot the errors of
computational results for calculations with both the no-flux (Figure 4.15) and
Dirichlet boundary conditions (Figure 4.16). We see first order convergence
in all cases.
4.7
Numerical Validation of Asymptotics
In this section we shall test the behavior of the electroneutral model against
that of the Poisson model by way of numerical simulations based on the fi206
−8
10
−4
2
2
L error in c
L error in φ
1
10
−5
10
−9
t=1ms
t=2ms
ref.
10
t=1ms
t=2ms
ref.
2
2
10
10
N
N
T
T
−8
3
2
L error in c
2
L error in c
2
10
−9
10
−10
t=1ms
t=2ms
ref.
10
2
t=1ms
t=2ms
ref.
2
10
10
N
N
T
T
Figure 4.15: L2 error in time, when no-flux boundary conditions are used.
The error is measured at t = 1ms and 2ms. The reference line indicates first
order convergence.
207
−8
10
−4
2
2
L error in c
L error in φ
1
10
−5
10
−9
10
t=1ms
t=2ms
ref.
t=1ms
t=2ms
ref.
2
2
10
10
N
N
2
L error in c
2
L error in c
3
T
2
T
−9
10
t=1ms
t=2ms
ref.
2
−10
10
t=1ms
t=2ms
ref.
2
10
10
N
N
T
T
Figure 4.16: L2 error in time, when Dirichlet boundary conditions are used.
The error is measured at t = 1ms and 2ms. The reference line indicates first
order convergence.
208
nite volume method we described above. We confine numerical validation
to test cases which reduce to one dimensional computations. This is because the Poisson model requires extremely small time steps, which makes
it computationally overwhelming to compare the two models in a full two
or three dimensional setting. We shall consider two geometrical situations,
one spherical, and one planar, the details of which we shall describe below,
in relation to numerical issues particular to this section. In this section, we
shall primarily use dimensionless variables. For reference, we shall reproduce
below the equations of the electroneutral model and the Poisson model in
nondimensional form. The electroneutral model is:
0=
∂Ci
+ β∇X · Fi
∂τV
0 = ρ˜0 +
N
X
zi Ci
(4.89)
(4.90)
i=1
(kl)
∂λi Φ
+ αj̃i
∂τV
∂λi
λ̃i − λi
z 2 Ci
=
, λ̃i = PN i 2
∂τV
β
i′ =1 zi′ Ci′
zi Fi · n(kl) = θ
(4.88)
209
(4.91)
(4.92)
The Poisson model is:
∂Ci
= −β∇X · Fi
∂τV
(4.93)
!
N
X
β 2 ∆X Φ = − ρ˜0 +
zi Ci
(4.94)
(4.95)
i=1
θ∗ Φ(kl)
∂Φ
= β (kl)
∂n
(4.96)
4.7.1
(4.97)
Numerical Method
Spherical Geometry
First, consider spherical geometry. We take a spherical cell of radius l. Denote its center by O, and let r be the radial coordinate. We seek solutions
to the equations (electroneutral of Poisson) which depend only on the radial
coordinate r. What we have is thus a one dimensional problem. The region
characterized by r < l is the intracellular space. We confine our simulation
domain to r < 2l and impose no-flux boundary conditions at r = 2l. Thus,
our extracellular space is the region l < r < 2l. The l we use here as the radius of the cell is to be identified with the l we used to denote the dimensional
volume to surface ratio introduced in Section 2.2.2.
We now rescale length so that L0 , (2.51), is the representative length
scale. The dimensionless cell radius is now α =
210
l
.
L0
We shall continue to use
r as our dimensionless coordinate. Thus, r < α is the intracellular region
and α < r < 2α is the extracellular region.
We discretize the simulation domain into spherical shells whose k-th shell
Vk is the region rk−1 < r < rk where r0 = 0, rNr = α and r2Nr = 2α. We thus
have Nr voxels in both the intracellular and extracellular spaces. In order
to perform computations with the Poisson model, we need to fully resolve
the Debye layers located adjacent to the membrane. For this purpose, we
use a non-uniform mesh with finer meshes near the membrane. We use the
following procedure. Define the following function f (z):
z
z
z
+ a3
+ a2 exp − √
f (z) = a1 exp −
β
α
β
where a1 , a2 , a3 are positive constants, and β =
rd
L0
(4.98)
is the dimensionless Debye
length defined in Section 2.2.2. Take yk , k = 0, · · · , Nr to be:
yk = f (0) +
f (l) − f (0)
k
n
(4.99)
For k = 0, · · · , Nr we let
rn−k = l − f −1 (yk )
(4.100)
rn+k = l + f −1 (yk )
(4.101)
Note that r0 = 0, rNr = α and r2Nr = 2α as it should. The above prescription
has the effect of exponentially clustering mesh points with spatial scales rd
211
and
√
rl L0 near the membrane. The length β is the dimensionless Debye
√
length and β is the dimensionless fast diffusion length scale. The constants
a1 , a2 , a3 give us the relative number of points we would like to place in the
respective spatial regions. We let:
a1 = 2, a2 = 2, a3 = 1
so that approximately
2
n
2+2+1
(4.102)
= 25 n mesh points reside in both the Debye
layer and the fast-diffusion layer.
In what follows, we shall often have
√
β ∼ α. In this case the fast diffusion
length scale and the whole cell length scale are of the same order. This is a
scaling we avoided when we performed asymptotics as explained in Section
2.2.3. We shall see that in fact, even when α is comparable in magnitude
√
to β, the electroneutral model does remarkably well in approximating the
Poisson model.
Planar Geometry
For planar geometry, we consider an infinite series of parallel planes separated
by the dimensionless length α. Take the x coordinate to be perpendicular to
these parallel planes. The parallel planes are located at x = nα where n ∈ Z.
We identify each of these planes as cell membranes. We seek solutions to the
equations that depend only on x and is periodic with period 2α. Thus, we
need only consider the region −α ≤ x ≤ α and impose periodic boundary
212
conditions. We shall call the region −α < x < 0 the intracellular region, and
0 < x < α the extracellular region.
We now discretize the simulation domain into voxels. We let the k-th
voxel Vk be the region defined by xk−1 < x < xk , k = 0, · · · , 4Nx where
x0 = −α, x2Nx = 0 and x4Nx = α. We thus have 2Nx voxels in both the
intracellular and extracellular spaces. We use nonuniform meshes for the
same reason as in the case of spherical geometry. Using (4.98) and (4.99),
we define the positions xk as follows:
xk =
f −1 (yk )
2
x2Nx +k = l + xk
x2Nx −k = l − xk
(4.103)
x4Nx −k = 2l − xk
(4.104)
where k = 0, · · · , Nx . As in the case of spherical geometry, we set the
constants a1 , a2 , a3 as in (4.102).
Computational Scheme
In both the spherical and planar geometries, we use the finite volume method.
In this section, we shall prescribe the transmembrane currents as functions
of time, thereby obviating the necessity to keep track of gating variables and
their kinetics. We shall use τ ≡ τV as our time variable, which is scaled with
the membrane potential time βT0 .
The simulation method for both the electroneutral model and the Poisson
model is a one-dimensional version of the finite volume method. We let the
213
physical variables be defined at
rk−1 +rk
2
for the spherical case, and at
xk−1 +xk
2
for the planar case. The membrane voxels are k = Nr , Nr + 1 in the spherical
case and k = 1, 2Nx , 2Nx + 1, 4Nx in the planar case. We use the values of
ionic concentrations and electrostatic potential at membrane voxels voxels as
the corresponding values on the membrane.
For the electroneutral model, we use a scheme that is a one dimensional
modification of the scheme we described for cylindrical or arbitrary 2D geometries. The scheme is even simpler since all transmembrane currents are
given functions of time.
There is however, one small difference. If we look at the dimensionless
electroneutral and Poisson models, there is one parameter that is different:
θ∗ and θ. There is a functional relation between the two constants which
is given by (2.127). The parameter θ is a function of ionic concentrations
through its dependence on Γ. Even though we argued at the end of Section
(2.2.4) that θ is effectively constant in time, in this section, we shall take θ
as variable in time, whose time dependence is determined by (2.127). We
reintroduce this time dependence to make the comparison between the two
models as true to the asymptotics as possible. This is the only computation
in which we shall treat θ as variable in time. When the electroneutral model
is applied to biophysical situations, no predictive power is lost by taking θ to
be constant, since θ varies so little in time. To the author’s knowledge, time
dependence of membrane capacitance has never been documented.
Our strategy for solving the Poisson model is basically the same as the
214
method we use for the electroneutral model. At each time step, we first fix
concentrations, solve for the electrostatic potential, and then fix the electrostatic potential, and solve for the concentrations and iterate.
Suppose we are to march from time n − 1 to time n. Let Cik,n,r , Φk,n,r
denote the r-th iterate of the solution procedure, where r = 0, 1, 2, · · · . We
let:
Cik,n,0 = Cik,n−1, Φk,n,0 = Φk,n−1
(4.105)
Thus we set our initial iterate to be equal to the values at time n − 1. Let
hk denote the width of voxel Vk , and ek−1 , ek the faces that are shared with
the voxels Vk−1 and Vk respectively. At ordinary voxels, we discretize the
Poisson equation as:
N
X
Φk,n,r − Φk−1,n,r
Φk+1,n,r − Φk,n,r
= −|Vk |
zi Cik,n,r−1
− |ek−1 |
β |ek |
(hk + hk+1 )/2
(hk + hk−1 )/2
i=1
(4.106)
2
At membrane voxels, we discretize as follows. We take voxel VNr in spherical
geometry as our example. Equations for other membrane voxels are similar.
|eNr |βθ
∗
r ,n,r
r ,n−1
ΦN
− ΦN
m
m
∆τ
− |eNr −1 |β
= −|Vk |
2
N
X
ΦNr ,n,r − ΦNr −1,n,r
(hNr + hNr −1 )/2
zi Cik,n,r−1
(4.107)
i=1
r ,n,r
ΦN
= ΦNr ,n,r − ΦNr +1,n,r
m
r ,n−1
ΦN
= ΦNr ,n−1 − ΦNr +1,n−1
m
215
(4.108)
(4.109)
Once we have solved for Φn,r , we solve for Cin,r in the same way as in
the electroneutral model, similarly to (4.42), but with the following modifications. Unlike the electroneutral case, we do not have a capacitative current
term. In the the flux expression Fiq in (4.42), we used (4.43) where the diffusive flux was treated implicitly whereas the drift term was kept explicit. For
the Poisson model, we let,
!
k,n,r
k+1,n,r
C
−
C
i
i
Fiq (Cin,r , Cin,r−1, Φn,r−1 ) = D̃i
(hk + hk+1 )/2
!
hk+1 Cik,n,r + hk Cik+1,n,r
Φk,n,r−1 − Φk+1,n,r−1
+ D̃i zi
hk + hk+1
(hk + hk+1 )/2
(4.110)
Note the ionic concentrations are treated implicitly in the drift term as well.
This seemed to lead to slightly better numerical stability.
As in the electroneutral case, we iterate this procedure a suitable number
of times in r. Here again, one iteration yields a first order scheme in time,
but performing multiple iterations was found to enhance stability.
Despite the above stabilizing measures, stable computations of the Poisson model required very small time steps. Not surprisingly, this time step
turns out to be on the order of the charge relaxation time scale. We note
that the electroneutral model does not require fine time stepping for stable
computations. In fact, time steps that are two or three orders of magnitude
greater lead to stable computations whose simulation results do not differ
appreciably from the Poisson model calculation. This is a significant amount
of computational saving, and serves as a posteriori justification for the model
216
reduction we performed in Chapter 2.
4.7.2
Comparison
Spherical Geometry
In order to perform simulations, we need to set concrete parameter values.
We consider four ionic species with the following diffusion coefficient and
valence.
D1 = 2D0
z1 = 1
(4.111)
D2 = D0 /2
z2 = 1
(4.112)
D3 = D0
z3 = −1
(4.113)
D4 = D0 /2
z4 = 2
(4.114)
where D0 is the characteristic diffusion coefficient set to be:
D0 = 1µm2 /msec
(4.115)
We set the non-dimensionalized capacitance θ∗ and the Debye length to be:
θ∗ = 10−2 ,
rd = 10−3µm
(4.116)
All of these values are set in accordance with physiological values (cf (2.47)
and (2.48)), the relative magnitudes of which were discussed in Section 2.2.2.
217
We consider three values for l, the cell radius. We set
l = 0.01, 0.1, 1µm
(4.117)
At time t = 0 we turn on a constant current of ionic species i = 4
that flows in from the extracellular space(r > l) into the intracellular space
(r < l). The strength of this current is given by γ whose value we set to be:
γ = 10−2 µm/msec
(4.118)
As we stated in equation (2.48) of Section 2.2.2, the physiological value of
γ ranges approximately from γ = 10−5 ∼ 10−3 µm/msec. Thus, the value
γ = 10−2 µm/msec is at the very high end of what is physiologically reasonable. We note that smaller values of γ tend to favor the electroneutral
approximation. Thus, using larger values of γ makes for a more stringent
test of validity of the electroneutral model.
Recall that β, α and θ∗ are the nondimensional parameters that characterize the system of equations. The parameter θ∗ has a fixed value. For the
three values of l which we shall consider, the corresponding values of β and
218
α are:
l = 0.01µm :
(β, α) = (10−3 , 10−2 )
(4.119)
l = 0.1µm :
(β, α) = (10−3.5 , 10−1.5 )
(4.120)
l = 1µm :
(β, α) = (10−4 , 10−1 )
(4.121)
Note that when l = 0.01 or l = 0.1,
√
β and α are comparable in magnitude.
Another nondimensional constant of interest is the ratio between the Debye length and the cell radius:
β
rd
=
α
l
(4.122)
When l = 0.01, this value is 0.1. This means that the Debye layer extends
over one-tenth of the simulated region. If the electroneutral model can be
shown to perform reasonably well in this case, the model would have passed
a fairly stringent test of validity.
We shall start our simulation at time τ = −Tr where Tr is positive. The
reason for this will become clear shortly. For the electroneutral model, we
219
must set the initial conditions at τ = −Tr for Ci (r, τ ).
C1 (r, −Tr ) = 1 + Cg (2|α − r| − α)
C2 (r, −Tr ) = 2 − C1 (r, −Tr )
C3 (r, −Tr ) = 2
(4.124)
C4 (r, −Tr ) = 10−6 for r < α
ρ̃0 (r) = −
(4.123)
4
X
i=1
C4 (r, −Tr ) = 10−3 for r > α
zi Ci (r, −Tr )
(4.125)
(4.126)
We let Cg = 0.9 so that there is a steep initial gradient of the ionic concentrations. At the membrane boundary, we must specify Φm (τ ) and λi (α±, τ )
at τ = −Tr .
Φm (−Tr ) = Φ(α−) − Φ(α+) = −
z 2 Ci (α±, −Tr )
λi (α±, −Tr ) = P4 i 2
k=1 zk Ck (α±, −Tr )
θ∗
θ
(4.127)
(4.128)
where the double signs correspond in the above equation.
For the Poisson model, we need to specify the initial ionic concentrations.
Given initial conditions for the electroneutral model, we set the corresponding
initial conditions for the Poisson model to be:
4πα2 λi (α−, −Tr )θ∗
if r < α (4.129)
zi (4π/3)α3
4πα2 λi (α+, −Tr )θ∗
Ci (r, −Tr ) = Cielectroneutral (r, −Tr ) +
if r > α (4.130)
zi (4π/3)((2α)3 − α3 )
Ci (r, −Tr ) = Cielectroneutral (r, −Tr ) −
The rationale for setting Ci as above is the following. The initial conditions
220
for the electroneutral model says that each ionic species contributes a surface charge amount λi θ∗ times the membrane area 4πα2 . To set the initial
conditions for the Poisson model, we need to take into account this contribution. We spread this surface charge contribution uniformly throughout the
intracellular and extracellular spaces.
The problem with this initialization is that the excess charge should not
be uniformly distributed but should be distributed so that the concentration
profile shows a space charge layer near the membrane. Since we do not know
the exact details of this concentration profile a priori, we let the system
relax between −Tr < τ < 0 so that the Poisson system settles down to a
state where the bulk is approximately electroneutral and the excess charge
accumulates near the membrane. During this period, we set the membrane
current equal to zero and the dimensionless diffusion coefficients to be equal
to D̃i = 1. We let Tr = 10β, 10 times the charge relaxation time.
At t = 0, we turn on a current of strength α that flows in from the
extracellular side into the intracellular side. We let our simulations last until
∗
τ = Te = 2 θα , which is approximately the time it takes to depolarize the
membrane potential from −1 to 1. We let Nr = 200, and the timestep
∆τ =
β
.
5
Using a larger time step led to numerical instabilities with the
Poisson model. A snapshot from a sample run of this simulation is shown in
Figure 4.17.
Raw data produced by the electroneutral model do not capture the ionic
concentration or electrostatic potential profiles in the Debye layer. But it
221
τ=Te/4
τ=3Te/4
1.4
1.2
1.2
1.1
1
C
C1
1
1
0.9
0.8
0.8
0.7
0.6
0.6
0.01
0.02
0.03
r
0.04
0.05
0.06
0.01
0.02
τ=T /4
0.04
0.05
0.06
0.05
0.06
τ=3T /4
e
e
0
0.5
0.4
−0.1
0.3
−0.2
0.2
Φ
Φ
0.03
r
−0.3
0.1
−0.4
0
−0.5
−0.1
0.01
0.02
0.03
r
0.04
0.05
0.06
0.01
0.02
0.03
r
0.04
Figure 4.17: Snapshots of simulation when (β, α) = (10−3.5, 10−1.5 ). Three
curves, the Poisson computation, the raw data and modified data from the
electroneutral models are plotted. The three curves are virtually indistinguishable.
222
is possible to produce an approximate profile in the Debye layer based on
the asymptotic theory developed in Chapter 2. We can see from (2.115) and
(2.117) that the Debye layer has the effect of adding a correction term to
Ci and Φ that decays exponentially with distance from the membrane. The
decay length and the magnitude of the correction term can be approximated
by the values of Φ, Ci and λi evaluated at the membrane. For the Φ, we
modify the raw data as follows:
Φmodified = Φ + δΦ
(4.131)
θΦm
+ |r − α|
if r > α
δΦ = − + exp −Γ
Γ
β
θΦm
− |r − α|
= − − exp −Γ
if r < α
Γ
β
q
±
Γ = zi2 Ci (r = α±)
(4.132)
(4.133)
(4.134)
where the double signs correspond in the last line. For the ionic concentrations Ci ,
Cimodified = Ci + δCi
(4.135)
λi (α+ )θΦm Γ+
|r − α|
if r > α
exp −Γ+
zi
β
λi (α− )θΦm Γ−
− |r − α|
if r < α
exp −Γ
=−
zi
β
δCi = −
(4.136)
(4.137)
We note that δCi and δΦ are expressed completely in terms of raw data
computed with the electroneutral model. When comparing the electroneutral
223
model with the Poisson model, we shall use the above modified profile for
the electroneutral model.
In order to quantify the difference between the electroneutral and Poisson
calculations, we introduce the following norm on the computational domain.
Suppose the quantity u is defined at each voxel. We define the discrete
p-norm as:
kukLp =
P2Nr
p
k=1 |Vk ||uk |
P
2Nr
k=1 |Vk |
!1/p
,
1≤p<∞
kukL∞ = max |uk |
k
(4.138)
(4.139)
In defining the Lp norm in (4.138), we have divided by a normalizing factor
so that kukLp gives an average measure of the “Lp deviation”. In particular,
limp→∞ kukLp = kukL∞ . For ionic concentrations Ci , we shall use the relative
error:
electroneutral
C
− CiPoisson Lp
i
Ep (Ci) =
kCiPoisson kLp
(4.140)
This is a more stringent criteria than using the absolute error (without the
denominator in the above) especially for C4 whose initial concentration is
very small.
For the electrostatic potential Φ, it does not make sense to use the relative
error since an arbitrary constant constant may be added to Φ. We thus,
224
1
2
3
Lp
L1
L2
L∞
L1
L2
L∞
L1
L2
L∞
Mp (C1 )
3.79 × 10−5
4.67 × 10−5
2.87 × 10−4
7.20 × 10−5
8.46 × 10−5
2.32 × 10−4
1.74 × 10−6
2.58 × 10−6
1.19 × 10−5
Mp (C2 )
6.98 × 10−5
8.43 × 10−5
2.35 × 10−4
2.76 × 10−5
4.40 × 10−5
1.57 × 10−4
1.79 × 10−6
2.98 × 10−6
8.59 × 10−5
Mp (C3 )
6.56 × 10−5
7.71 × 10−5
2.86 × 10−4
3.25 × 10−5
5.47 × 10−5
1.89 × 10−4
4.25 × 10−7
1.97 × 10−6
7.53 × 10−5
Mp (C4 )
1.55 × 10−4
1.74 × 10−4
5.85 × 10−4
2.99 × 10−5
5.15 × 10−5
2.06 × 10−4
3.31 × 10−7
1.71 × 10−6
1.57 × 10−4
Mp (Φ)
2.73 × 10−5
5.95 × 10−5
1.40 × 10−4
4.70 × 10−5
1.03 × 10−4
1.71 × 10−4
1.62 × 10−5
4.16 × 10−5
6.95 × 10−5
Table 4.10: Mp values for spherical geometry for three computational experiments with different values of β and α. Here, cases 1, 2, and 3 correspond respectively to (β, α) values of (β1 , α1 ) = (10−3 , 10−2 ), (β2 , α2 ) =
(10−3.5, 10−1.5 ), and (β3 , α3 ) = (10−4 , 10−1).
measure the error in Φ as:
Ep (Φ) = min Φelectroneutral − ΦPoisson + cp Lp
cp ∈R
(4.141)
Note that it makes sense to measure the error in Φ in terms of the absolute
error, since Φ is dimensionless, and its typical magnitude is 1. Though Ep (Φ)
may in general be difficult to compute in closed form, this is possible when
p = 1, 2, ∞, values of p for which we shall compute Ep (Φ) in the following.
In table (4.10), we list the Mp (u), u = Φ or Ci where:
Mp (u) = max Ep (u)
0≤τ ≤Te
(4.142)
We see that for all parameter ranges tested here, the error falls within
225
order 10−4. This translates to a 0.01% error in Ci and an error of about
0.025mV in the dimensional electrostatic potential φ. This degree of correspondence is all the more remarkable because the asymptotic analysis
√
of Chapter (2) assumed that α = O(1) with respect to β. In cases
(β, α) = (10−3 , 10−2 ) or (10−3.5, 10−1.5 ) α is in fact comparable in magni√
tude to β. It is notable that the relative error is order 10−4 even for C4
which has a vanishing small concentration. This tells us that we may include ions of very small concentration into our model framework, which is
significant if we are to include calcium dynamics.
We see that M∞ (Ci ) is significantly larger than M1 (Ci ) or M2 (Ci ). Despite the modification we performed on the raw data for the electroneutral
model, the deviation between the electroneutral and Poisson models are still
concentrated at the Debye layer. Since this layer is very small in volume,
the L1 and L2 errors are not significantly affected. The discrepancy between
the L1,2 and L∞ error is most pronounced in the β = 10−4 case in which
β
,
α
the ratio between the total volume of the simulation domain and the spatial
extent of the Debye layer, is the smallest.
It is interesting to note that M∞ (Ci ) does not seem to show a significant
decrease as β decreases. This may be partly due to the fact that the error
within the Debye layer is controlled by the magnitude of θ∗ which was taken
constant for the three computational runs. This may also explain why there
appears to be little improvement in Mp (Φ) as β is decreased. Since Φ satisfies
an elliptic equation, a discrepancy at the membrane boundary translates into
226
a discrepancy over the whole domain. This is probably the reason why not
only M∞ but also M1,2 show little improvement as β becomes smaller.
We have also plotted in Figure (4.18) the time course of E2 as a function of
time for both the electrostatic potential and the concentrations when (β, α) =
(10−3.5, 10−1.5 ). We see that for Ci there is an initial transient in which
the discrepancy between the electroneutral and Poisson models are large.
This transient corresponds to the time in which the initial steep gradient in
concentrations dissipates.
Planar Geometry
Comparison of the two models in planar geometry follow similar lines. We
use the same parameter physical parameter values and set l to the same three
values used for spherical geometry.
The initial conditions are also similar:
C1 (x, −Tr ) = 1 + Cg (2|α − x| − α) C2 (x, −Tr ) = 2 − C1 (x, −Tr ) (4.143)
C3 (x, −Tr ) = 2
(4.144)
C4 (x, −Tr ) = 10−6 for x < 0
ρ̃0 (x) = −
4
X
i=1
C4 (x, −Tr ) = 10−3 for x > 0
zi Ci (x, −Tr )
(4.145)
(4.146)
We need to consider two pairs of membrane boundary conditions since there
are two membranes, one at x = 0 and the other at x = ±α(note the mem-
227
−4
1.2
x 10
Φ
C
1
C
2
C
3
C
1
4
0.6
2
L error
0.8
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time
Figure 4.18: Plot of E2 (Ci ) and E2 (Φ) as a function of time when (β, α) =
(10−3.5, 10−1.5 ). The abscissa measures dimensionless time τ , and the plot
ends at τ = Te .
228
brane at x = α and x = −α are identified as being the same point).
Φm (0, −Tr ) = Φ(0−) − Φ(0+) = −
θ∗
θ
θ∗
Φm (±α, −Tr ) = Φ(−α) − Φ(α) = −
θ
zi2 Ci (0±, −Tr )
λi (0±, −Tr ) = P4
2
k=1 zk Ck (0±, −Tr )
z 2 Ci (±α, −Tr )
λi (±α, −Tr ) = P4 i 2
k=1 zk Ck (±α, −Tr )
(4.147)
(4.148)
(4.149)
(4.150)
where the double signs correspond in the above. The two independent membrane systems is what makes this example of interest compared to the spherical example. The transmembrane current that passes through one of the
membranes affects not only the membrane potential of itself but also of the
other membrane. This example also serves as justification for the cardiac
calculations we perform later in which similar dimensions are used for the
extracellular space in the intercalated disc.
As in the spherical case, we let the system relax for −Tr < τ < 0 and
turn on a current α at the membrane located at x = 0, so that the current
enters from the x > 0 side into the x < 0 side.
∗
We run the simulation for τ = Te = 4 θα . We turn on a transmembrane
current of strength α composed of the 4-th ionic species, that goes through
the membrane at x = 0 from the x > 0 side to the x < 0 side. The
simulation time is taken twice as long as the spherical case, since we have
two membranes, and thus, we require roughly twice as much time to have
229
1
2
3
Lp
L1
L2
L∞
L1
L2
L∞
L1
L2
L∞
Mp (C1 )
1.06 × 10−4
1.13 × 10−4
1.77 × 10−4
1.56 × 10−4
1.76 × 10−5
2.33 × 10−4
2.29 × 10−5
3.11 × 10−5
7.68 × 10−5
Mp (C2 )
1.37 × 10−4
1.50 × 10−4
2.66 × 10−4
8.67 × 10−5
1.04 × 10−5
2.15 × 10−4
1.92 × 10−5
2.90 × 10−5
8.99 × 10−5
Mp (C3 )
1.38 × 10−4
1.47 × 10−4
2.13 × 10−4
9.10 × 10−5
1.13 × 10−5
2.16 × 10−4
1.63 × 10−5
2.75 × 10−5
7.90 × 10−5
Mp (C4 )
3.20 × 10−4
3.37 × 10−4
4.16 × 10−4
9.12 × 10−5
1.12 × 10−4
1.91 × 10−4
3.01 × 10−5
5.54 × 10−5
1.65 × 10−4
Mp (Φ)
2.90 × 10−5
3.43 × 10−5
6.38 × 10−5
5.55 × 10−5
1.03 × 10−4
1.27 × 10−4
3.85 × 10−5
4.77 × 10−5
6.97 × 10−5
Table 4.11: Mp values for planar geometry for the three computational experiments with different values of β and α. Here, cases 1, 2, and 3 correspond respectively to (β, α) values of (β1 , α1 ) = (10−3 , 10−2 ), (β2 , α2 ) =
(10−3.5, 10−1.5 ), and (β3 , α3 ) = (10−4 , 10−1).
the transmembrane potential Φm change from −1 to 1. We let Nx = 100 so
that there are 200 voxels in both the extracellular and intracellular spaces.
As with the spherical case, we take the ∆τ = β5 .
We measure the error between the electroneutral and Poisson models
similarly to the spherical case. We modify the raw data taking into account
contributions from the Debye layer, and compute the error between the electroneutral and Poisson models. We list computed errors in Table (4.11).
The values of the error are much the same as that for the spherical error,
and again, we see excellent agreement against the two models.
230
Chapter 5
Application
In this section, we apply the foregoing model to a specific biological system:
cardiac action potential propagation with or without gap junctions.
5.1
Physiological Background
The heart makes use of an electrical signal to synchronize its beat. The
electrical pulse originates from the sinoatrial node in the right atrium of
the heart and spreads through the atria, bundle of His, Purkinje fibers and
the ventricles in succession, initiating a coordinated sequence of muscular
contractions [11].
The heart is made of cardiac cells, each of which is demarcated by a
cell membrane. Since the cell membrane is effectively an electric insulator,
electrical activity in one cell will not significantly affect the electrical activity
231
of another, so long as the two cells are not too close. This poses a problem
for action potential propagation in cardiac tissue, since electrical activity in
one cell must be rapidly communicated to the next cell. It is widely accepted
that this communication is achieved through gap junctions [11, 1] which are
pores of low electrical resistance that link two cells. Electrical activity in
one cell can spread to a neighboring cell through these gap junctions thus
initiating an electrical response in the neighboring cell.
These gap junctions are located primarily at the intercalated discs. The
intercalated discs are narrow gaps between two cardiac cells, situated perpendicular the muscular fiber axis [4]. The electrical signal thus primarily
spreads in a direction parallel to the muscular fiber axis.
There has been some controversy as to whether gap junctions are truly
necessary for propagation of the electrical signal [40]. It was first suggested
in [41] that gap junctions were in fact not necessary for electrical coupling of
cardiac cells. In [41], Sperelakis et. al. proposed that the presence of a narrow
intercellular gap between cardiac cells, the intercalated disc, was sufficient to
make propagation possible.
Consider propagation of an electrical signal from cell A to cell B. Cell
A and cell B are separated by an narrow gap (intercalated disc), and suppose that no gap junctions connect the two cells. To explain the cardiac
propagation mechanism proposed by Sperelakis, we consider the electrostatic
232
extracellular space
Vext
cell A
cell B
VB
intercalated disc Vid
Figure 5.1: Relevant voltages in considering ephaptic transmission
potential at three distinct points (Figure 5.1):
VB : voltage in cell B
(5.1)
Vid : voltage in the intercalated disc
(5.2)
Vext : voltage in the extracellular space away from the intercalated disc
(5.3)
Suppose cell A is fully depolarized. Then, there is a strong transmembrane current flowing in from the extracellular space into cell A. The membrane of cell A facing the intercalated disc possesses ionic channels, and
therefore, there must be a transmembrane current flowing in from the intercalated disc into cell A. Current continuity implies that there is a current
flowing into the gap from the extracellular space away from the gap. Ohm’s
law states that there must be a voltage difference that is driving this current:
Vext − Vid > 0. This difference should increase proportionally to the electrical
resistance of the intercellular gap, and thus, a narrow gap implies a greater
value of Vext − Vid . Now, consider the voltage-gated ion channels embedded
233
in the membrane of cell B facing the gap. These channels sense the voltage
difference VB − Vid . Consider:
VB − Vid = (VB − Vext ) + (Vext − Vid)
(5.4)
Even if VB and Vext may not change as a result of electrical activity of cell A,
strong transmembrane currents through the membrane of cell A facing the
gap will generate a positive Vext − Vid . This will make VB − Vid more positive,
which may be enough to open enough voltage-gated ion channels to excite
cell B. We shall refer to the above mechanism as the ephaptic mechanism,
borrowing a term used in the neurosciences [16].
This hypothesis has been given renewed attention recently in light of two
experimental observations. The first is that Na+ channels were found to
be preferentially expressed at the plasma membranes facing the intercalated
discs, [7, 24]. The second is that mice engineered not to express gap junctions
in the heart(connexin 43 knockout mice) are viable, and seem to exhibit nearordinary cardiac conduction velocities [10].
Whether this kind of propagation is physiologically possible is a matter
of debate. Sperelakis and coworkers have employed simple descriptions of
cardiac networks to argue that cardiac propagation is possible without gap
junctions [42]. In [24], the authors address this question using a simple ohmic
current model with realistic cardiac ion channel dynamics.
In this chapter, we shall apply the methodology developed in the foregoing
234
gap width, lg
lA
Sodium channel density, r Na
l
gap junctions (conductance: g gap )
negative fixed charge density Cim
Figure 5.2: Schematic diagram of the setting of the computational experiments. Each rectangular box represents a cylindrical cardiac cell.
chapters to perform a more complete study of this issue. Our methodology
allows us to explore the effect membrane geometry, extracellular space and
ionic diffusion have on ephaptic propagation, aspects which cannot be addressed using conventional models of electrophysiology. We shall use the
mouse as a model system so that we can compare our results to the experiments performed on connexin 43 knockout mice obtained in Dr. Fishman’s
lab at the NYU medical school [45, 10].
5.2
Model
We shall use the mouse heart as our model system, for which experimentally
measured values are available against which we can compare our results [45,
10].
Take a strand of Nc > 3 cardiac cells, all of which are assumed to be
cylindrical in shape (Figure 5.2). We take the radius of the circular crosssection of the cell to be l = 24.7/2µm and the length of the cell to be
235
lA = 157.9µm (experimentally measured values by Fishman et. al. personal
communication). We place the Nc cells so that the cylindrical axes of all
the cells lie along a single line. We shall take this line to be the z axis,
and the radial coordinate to be the r axis. We shall seek radially symmetric
solutions, thus obviating the necessity to deal with the angular coordinate.
The Nc cells have radius l, and we take the computational domain to be a
cylindrical domain of radius (1 + η)l whose axis coincides with the center line
of the strand of cells. The parameter η is a parameter that to be varied but
will ususally taken to be η = 1. Thus, the extracellular space corresponds to
l < r < (1 + η)l as well as the gaps between cells.
Label the cells from k = 1 · · · Nc . The gap between cell k and k + 1 will
be taken uniformly equal to lg in width. The gap width lg will be varied to
see its effects on propagation. Cells k = 1 and k = Nc are “half cells” in the
sense that they only have length lA /2, cut in half at the center. Take the
origin of the coordinate system to be in the middle of the gap between cell 1
and 2. Formally, the intracellular region can be written as:
lg
lA + lg
<z<−
r < l and −
2
2
lg
lg
r < l and k(lA + lg ) +
< z < (k + 1)(lA + lg ) −
2
2
(5.5)
(5.6)
where k = 0, · · · Nc − 3.
lg
lA + lg
r < l and (Nc − 2)(lA + lg ) +
< z < (Nc − 2)(lA + lg ) +
2
2
(5.7)
236
T
us
DN
a+
us
DK
+
us
DCa
2+
us
DCl
−
int cN a+ t=0
cext
N a+ t=0
cint
K + t=0
ext cK + t=0
cint
Cl− t=0
cext
Cl− t=0
int cCa2+ t=0
cext
Cl2+ t=0
φm |t=0
Unscaled diffusion coefficient of K+
Unscaled diffusion coefficient of Na+
Unscaled diffusion coefficient of Ca2+
Unscaled diffusion coefficient of Cl−
Initial intracellular concentration of Ca2+
Initial extracellular concentration of Ca2+
273.15+37 K
1.33µm2 /msec
1.96µm2 /msec
0.3µm2 /msec
2.03µm2 /msec
10 mmol/l
138 mmol/l
140 mmol/l
4 mmol/l
10 mmol/l
see text
0.4 µmol/l
2 mmol/l
-90 mV
Table 5.1: Parameter values used in cardiac simulation.
g
and (Nc − 2)(lA + lg ) +
The intracellular region is open-ended at z = − lA +l
2
lA +lg
.
2
This is what we mean by “half cell”. We impose no-flux boundary
conditions at the outer rim of the computational domain, be it extracellular
or intracellular.
We consider 4 ion types in the calculation, Na+ , K+ , Ca2+ and Cl− . The
value of most parameters are given in Table 5.1.
In the gap we introduce a fixed negative charge density, which we shall
vary. As we shall see, variations in this amount affects the success and
speed of propagation. Though such excess charges are not documented in
the literature, we shall explore the effects this has on propagation speed.
We introduce a fixed charge density in the following way. We initialize the
237
chloride concentration to be non-uniform at the initial time, while keeping
all other ions to have uniform concentration, and use electroneutrality to
prescribe a fixed charge density. Wherever there is non-uniformity in the
chloride concentration, there is a non-uniform fixed charge density.
We initialize the extracellular chloride concentration as follows:
cext
Cl− t=0 =



100 − Cim 1 −


100mmol/l
r 2
l
mmol/l r < l and − lg /2 < z < lg /2
otherwise
(5.8)
The initial condition for the other ionic species are listed in Table 4.4. We
determine ρ so that electroneutrality is satisfied everywhere:
ρ = −(CNa + CK + 2CCa − CCl )
(5.9)
The initialization of the chloride concentration as above introduces an excess
fixed negative charge density in the gaps between cells whose magnitude is
controlled by the parameter Cim .
The diffusion coefficients were determined in the same way as in Chapter
4. The ohmic cytoplasmic conductance is given by the following as a function
of the ionic concentrations:
N
X
(qzi )2 ci
i=1
kB T
Di
(5.10)
If one computes the cytoplasmic or extracellular conductance according to
238
values of Di in an aqueous solution, the values used in Table 4.1, we obtain
an overestimate which deviates from the experimentally observed value by a
factor of 2 − 5. We thus scale the diffusion coefficient in aqueous solution
by a uniform factor α so that the cytoplasmic or extracellular conductance
calculated above is approximately within the experimentally observed range.
More concretely, we let:
g
observed
=α
N
X
(qzi )2 ci |
t=0
i=1
kB T
Dius
(5.11)
where g observed is the experimentally observed cytoplasmic conductance, Dius
is the unscaled diffusion coefficient, and overline in the above expression
denotes averaging over the computational domain. Following [24], we let:
1
g observed
= 150Ω · cm
(5.12)
We shall use the model of Bondarendko et. al. [5] as the ion channel
model in our simulations with the following modifications. The model of
Bondarenko et. al. contains a large submodel that handles intracellular calcium dynamics. Including calcium dynamics in our framework would require
a detailed description of the geometry of the sarcoplasmic reticulum as well
as of buffering kinetics. We do not include such details here. Therefore, our
model only uses a subset of the model of Bondarenko et. al. that pertains
to transmembrane ionic currents. The model of Bondarenko et. al. provides
239
two sets of values for certain parameters, one for the cardiac cells in the
septum, and one for cells in the apex of the heart. We shall use parameter values for the ventricular septum. Furthermore, we do not include the
background Na+ conductance and the background Ca2+ conductance. The
background Na+ conductance, as well as the background Ca+ conductance
to a lesser extent, induces an unwanted spontaneous membrane potential oscillation which manifests itself after running the simulation for about 50msec
without external stimulation.
The reason for this oscillation is not clear. One reason may be that we
have neglected intracellular calcium handling. Without calcium buffering and
calcium sequestration into the endoplasmic reticulum, calcium concentration
in the intracellular space tends to increase. This in turn may drive the Na-Ca
pump, which has the effect of increasing Na+ concentration inside the cell,
thereby depolarizing the cell. With a background Na+ conductance, this
effect is probably amplified, leading to cellular automaticity.
The next parameter of interest is the gap width lg . The exact width of
the gap is unknown, and we shall vary this parameter to see its effect on
propagation speed.
As documented in [24, 7], recent evidence suggests the preferential localization of Na+ channels on the membranes facing the intercellular gap. We
shall vary the proportion of Na+ channels expressed in the gap, rNa , while
keeping the total number of Na+ channels on each cell to be constant.
We also introduce gap junctions that straddle the intercellular gap. Take
240
two cells k and k + 1 facing the same gap. We introduce current shunts
between two voxels in cell k and k + 1 which faces the gap at the same radial
coordinate.
Gap junctions are often modelled as simple ohmic resistances that connect two cells. In our model, a simple ohmic description is not enough: we
need to know the permeability of gap junctions to each ionic species. We
view gap junctions as being a cytoplasmic pore, and thus assume that the
permeabilities of each ionic species through gap junctions are proportional to
the diffusion coefficient of each species. We utilize a prototypical ion channel
model in which the channel is modelled as a one dimensional pore, and the
electric field is assumed constant throughout the channel. This yields the
following current voltage relationship for the gap junction currents [22], [31].
Iigap = αgap qzi Di
θigap ≡
θigap
(ck exp(θigap /2) − ck+1
exp(−θigap /2))
i
sinh(θigap /2) i
qzi V gap
,
kB T
V gap = V k − V k+1
(5.13)
(5.14)
Here, Iigap denotes the gap junction current from cell k to cell k + 1. The
scaling factor αgap , which has the units of (length)−1 , is determined so that
the net conductance is in accordance with experimentally measured values.
Linearize the above expression around θgap = 0. We find:
Iigap
≈ αgap qzi Di
cki + ck+1
i
2
θigap
= αgap
241
(qzi )2 Di cki + ck+1
i
kB T
2
V gap (5.15)
Using the above expression we set αgap so that:
gap
gexperimental
N X
(qzi )2 Di cki + ck+1
i
= αgap
k
T
2
B
t=0
i=1
(5.16)
The values of g gap has been experimentally determined in the paper of Yao et.
al. ([45]). We shall vary this parameter within the experimentally observed
range to assess its effect on propagation failure and conduction velocity.
The numerical method we use is as described in Chapter 4. One ingredient
that is new here is the handling of the gap junction current. Suppose we are
marching from time n − 1 to time n. We let:
θin−1,gap
f L (cin−1 , θin,gap , θin−1,gap )
= αgap qzi Di
n−1gap
sinh(θi
/2)
∂f (θn,gap − θin−1,gap ) + f (θin−1,gap )
f L (cin−1 , θigap ) =
∂θigap θgap =θn−1,gap i
Iin,gap
i
f=
cn,k
i
(5.17)
(5.18)
i
exp(θigap /2)
− cn,k+1
exp(−θigap /2)
i
(5.19)
This is a partial linearization of the the nonlinear current voltage relationship
(5.13).
In summary, we shall be concerned with the effect of the following five
242
parameters on the success of propagation and propagation speed:
η : Size of extracellular space
(5.20)
lg : Width of gap
(5.21)
Cim : Immobile charge density in the intercalated disc
(5.22)
rNa : Ratio of Na+ channels expressed facing the gap
(5.23)
g gap : Gap junction conductance
(5.24)
The computational method we shall use was explained in Chapter 4. The
spatial discretization is taken in such a way that the meshes are concentrated
near the gaps between cells. The exact prescription is identical to that explained in Chapter 4. We shall take 16 meshes in the radial direction and
24 × (Nc − 1) meshes in the axial direction. We set the time step to be
0.02msec.
5.3
Normal Conduction
We compute the conduction velocity in the following way. We take the
area-averaged value of the membrane potential for each cell. We record the
time points tk at which cell k reaches a threshold membrane potential of
φth = −30mV and use linear regression over values t3 , · · · , tN c−2 to compute
the velocity of the action potential as (number of cells)/(msec). We multiply
this value by (lA + lg ) to convert this into units of velocity. We let Nc = 10
243
in the calculations below.
We initiate an action potential in cell 1 by adding the following to the
Na+ conductance to the following membrane locations of cell 1:
Gadd
Na+ =



t
 20 1 + cos π(z+Lz )
1
−
cos
2π
4
lA /2
τe


0
Lz = (lA + lg )/2,
otherwise
(5.25)
τe = 1msec
(5.26)
Here, the origin of the z axis is taken to be at the center of the gap between
cell 1 and cell 2.
We first tested the model for normal conduction. According to [45], the
intercellular conductance in the longitudinal direction, which approximately
corresponds to the conductance across gap junctions over the intercalated
discs are:
gap
gtotal
= 5.58 × 10−4 mS
(5.27)
We shall distribute the gap junction conductance uniformly over the membrane facing the intercalated disc. As the gap junction density, we shall
take:
g gap = 5.58 × 10−4 /πl2 mS/µm2
(5.28)
where l, as we defined earlier, is the radius of the cell. For the other four
244
parameters, we let:
η=1
(5.29)
lg = 2, 5, 10, 20, 30, 40, 50nm
(5.30)
Cim = 0, 50
(5.31)
rNa = uniform, 0.5, 0.8, 0.99
(5.32)
where rNa = uniform indicates that we assumed uniform distribution of Na+
channels at the gap. The conduction speed is plotted in Figure 5.3.
When the Na+ channel density is uniform, we see that propagation speed
is insensitive to gap width at approximately 34cm/sec. When we redistribute
Na+ channels so that the density is higher in the gap, propagation velocity
decreases with gap width, and this decrease is greater the higher the Na+
channel density in the gap. When a fixed charge density is introduced, the
conduction speed falls to between 28 − 35cm/sec. It should be noted that
this decrease is not merely because lg is shorter. In fact, lg /lA ∼ 10−4 , and
so, the change in lg in itself has little to do with the fall in conduction speed.
The reason for this decrease in conduction speed with the decrease in lg
needs to be clarified. There can be at least two factors that contribute to
this decrease. The first is that concentration changes can decrease the driving
force for Na+ channels, which are responsible for membrane depolarization.
The narrower the gap, the quicker the Na+ ions will be depleted in the gaps
between the cells, thereby reducing the equilibrium potential for Na+ ions.
245
34
speed, cm/sec
32
30
28
26
24
0
5
10
15
20
25
30
gap width, nm
35
40
45
50
0
5
10
15
20
25
30
gap width, nm
35
40
45
50
36
speed, cm/sec
34
32
30
28
26
Figure 5.3: Conduction speed of cardiac action potential when the gap junction conductance is set to the normal physiological value. The top graph
corresponds to Cin = 0, and the bottom graph to Cin = 50. The lines on
both graphs are conduction velocity plots for rNa = uniform, 0.5, 0.8, 0.99
from top.
246
The second factor may be that the greater resistance of the gap makes it
difficult for current to flow through channels facing the gap. This will make
the channels facing the gap less effective, thereby decreasing propagation
velocity. This drop in propagation velocity as lg decreases is also documented
in [24].
According to [10], the normal conduction velocity is about 40cm/sec in
the transverse direction and 60cm/sec in the longitudinal direction. Thus,
the above value is an underestimate of the physiological conduction velocity,
though not too far off. In fact, given that the ion channel model of Bondarenko et. al. [5] was only calibrated to voltage clamp data, it is reassuring
to see the propagation velocity to be relatively close to the experimentally
observed value.
The source of the discrepancy between the computed and physiological
conduction velocity is not clear. One source of error may be that we are only
considering a single strand of cardiac cells, and we are ignoring the three
dimensional arrangement of cardiac tissue. In a true cardiac preparation,
electric current can go through many pathways to get from one cell to another, thereby reducing the effective resistance between two cells. This is
only one possibility, and it is difficult to pin down the exact source of this
discrepancy. We shall thus not try to adjust any of the parameters to attain
the experimentally observed velocity.
247
5.4
5.4.1
Conduction with Reduced Gap Junctions
Phenomenology and Model Comparison
We take the following parameter values to describe the characteristics of
cardiac conduction under reduced gap junction coupling. According to [45],
the gap junction conductance at the intercalated disc space for connexin 43
(dominant gap junction expressed in cardiac tissue) knockout mice is:
gap
gtotal
= 1.10 × 10−5 mS
(5.33)
We distribute this gap junction conductance uniformly over the membranes
facing the intercalated discs. By comparing this value with (5.27), we see
that the gap junction conductance is reduced to approximately 2% of normal.
The other parameters, Cim , lg and rNa are not known. We take:
η = 1,
lg = 5nm,
Cim = 0, 50,
rNa = 0.95
(5.34)
We initiate an action potential in cell 1 by adding the following external
248
Na+ conductance:
Gadd
Na+ =



t
 5 1 + cos π(z+Lz )
1 − cos 2π τe
4
lA /2


0
Lz = (lA + lg )/2,
otherwise
(5.35)
τe = 1msec
(5.36)
Here, the origin of the z axis is taken to be at the center of the gap between
cell 1 and cell 2. This additional conductance is smaller compared to that
used when the gap junctional coupling was taken to be the normal physiological value. If (5.35) is used when gap junctional coupling is maximal,
we cannot initiate an action potential because much of the injected current
escapes through the gap junctions to neighboring cells. Under reduced gap
junctional coupling, (5.35) is enough to initiate an action potential.
We find successful cardiac action potential propagation, as can be seen,
in Figure 5.4, at about 10cm/sec. The change in concentration is plotted in
5.5. We now describe the sequence of events that underlie the propagation
of the electrical signal.
Consider the propagation of the action potential from cell 1 to cell 2 By
externally turning on additional Na+ conductance, cell 1 depolarizes, which
activates Na+ channels (t = 0.8msec in Figure 5.6). Since these Na+ channels
are preferentially expressed on the membranes facing the intercellular gap,
the opening of these Na+ channels generates a strong current flowing into the
249
gap from the extracellular space outside of the gap. Since the gap is narrow,
a large negative deflection in the extracellular voltage is seen within the gap.
An important point is that there is a voltage gradient within the gap from
r = l to r = 0 . Therefore, the voltage at r = 0 is most negative with respect
to the voltage in the extracellular bulk, r > l (t = 1.6msec in Figure 5.6).
The Na+ channels facing the gap on cell 2 sense a depolarized membrane
potential because the voltage in the gap is negative with respect to the extracellular bulk. This activates the Na+ channels on cell 2, leading to its
depolarization (t = 2.4msec in Figure 5.6). This cycle of events repeats itself between cell k and k + 1, resulting in a propagating action potential
(t = 3.2msec in Figure 5.6).
We have also plotted the change in Na+ ,K+ ,Ca2+ and Cl− concentrations
in Figure 5.5. The increase in K+ concentration in the gap may also facilitate
conduction across the gap by pushing the resting potential of the membrane
to a more positive value.
When there is some negative charge in the gap, we see that a diffusion
potential develops in the gap, as can be seen in Figure 5.7. Without any external stimulus, the electrostatic potential in the gap is negative with respect
to the extracellular bulk, r > l. This tends to accelerate the propagation of
the action potential as can be seen at t = 10msec in Figure 5.7.
We shall repeatedly come back to explore these sequence of events as we
perform further computational experiments with different parameter values.
250
4ms
8ms
0
mV
mV
0
−50
−100
−50
−100
20
µm
20
500
10
0
−500
500
10
0
µm
µm
0
0
12ms
µm
16ms
0
mV
0
mV
−500
−50
−100
−50
−100
20
µm
20
500
10
0
−500
500
10
0
µm
µm
0
0
−500
µm
Figure 5.4: A plot of the electrostatic potential φ. Since we are seeking
radially symmetric solutions, we only plot the radial cross-section, 0 < r <
(1 + η)l. Note that the aspect ratio of the graph is much distorted; the radial
direction is stretched with respect to the axial direction. The four graphs
plot φ at t = 4, 8, 12, 16msec. Note the steep gradient in the electrostatic
potential that develops in the gap as the action potential propagates from
cell k to cell k + 1.
251
Na
K
3
−10
mmol/l
mmol/l
0
−20
2
1
0
20
µm
20
500
10
−500
0
500
10
0
µm
µm
0
−500
0
Ca
µm
Cl
0
mmol/l
mmol/l
0
−0.5
−10
−20
−1
20
µm
20
500
10
0
−500
500
10
0
µm
µm
0
0
−500
µm
Figure 5.5: The change in ionic concentrations. Note the decrease in Na+
and increase in K+ in the gaps.
252
0.8ms
1.6ms
0
mV
mV
0
−50
−100
−50
−100
20
10
µm
0
20
100
10
−100
µm
0
µm
0
2.4ms
100
3.2ms
0
mV
0
mV
0
−100
µm
−50
−100
−50
−100
20
10
µm
0
0
20
100
10
−100
µm
µm
0
0
−100
µm
100
Figure 5.6: The sequence of electrostatic potential changes as the action
potential propagates from cell to cell. The above computation was performed
with 3 cells.
253
5ms
10ms
0
mV
mV
0
−50
−100
−50
−100
20
µm
20
500
10
−500
0
500
10
0
µm
µm
0
0
5ms
µm
10ms
0
mV
0
mV
−500
−50
−100
−50
−100
20
µm
20
500
10
0
−500
500
10
0
µm
µm
0
0
−500
µm
Figure 5.7: Snapshots of the electrostatic potential with(below) or without(above) fixed charges in the gap. Note that when fixed charge is present
in the gap, the electrostatic potential is slightly negative within the gaps. At
t = 10msec, the action potential is approximately one cell ahead with fixed
charges in the gap than without.
254
5.4.2
Model Comparison
Before we proceed further to discuss the effect of the four parameters above on
the conduction velocity when the gap junction conductance is significantly
reduced, we would like to see if any difference will arise if we use simpler
electrophysiology models. We perform computations with the parameters:
η=1
(5.37)
lg = 2, 2.5, 3, 4, 5, 6, 7, 8, 10, 12, 15, 20, 25, 30, 40, 50nm
(5.38)
Cim = 0, 50
(5.39)
g gap = 1.10 × 10−5 /(πl2 )mS/µm2
(5.40)
rNa = 0.95
(5.41)
In Figure 5.8, we plot the propagation speed as a function of gap width, as
calculated with the full electroneutral model, the 3D cable model, the 1D
electroneutral model, and the 1D cable model. The 1D electroneutral model
and the 1D cable model calculations were performed by taking one mesh for
0 < r < l and one mesh for l < r < (1 + η)l in the electroneutral model
calculation and the 3D cable model calculation.
We see that there is a significant difference between the profile of the
traces between the 1D models and the 3D models. The 1D models significantly overestimate the conduction velocity when the gap width is greater,
and yield underestimates when the gap width is smaller.
255
12
full
φ only
1D
1D, φ only
speed, cm/sec
10
8
6
4
2
0
5
10
15
20
25
30
gap width, nm
35
40
45
50
14
full
φ only
1D
1D, φ only
speed, cm/sec
12
10
8
6
4
2
0
5
10
15
20
25
30
gap width, nm
35
40
45
50
Figure 5.8: Conduction speed of cardiac action potential computed with
different electrophysiology models. The computation in the top graph is
when Cim = 0 and the lower graph when Cim = 50. In 1D models, only two
meshes, one for r > l and one for r < l are used. In the φ only models, the
ionic concentrations are assumed not to change in time.
256
When Cim = 0, the difference between models with or without ionic
concentration effects produce almost identical results. We may conclude that
when Cim = 0, ionic concentration changes need not be taken into account
to produce accurate results. It is interesting to note that there is a slight
deviation between the full 3D and 3D φ only calculations when the gap width
is smaller than lg = 5nm. This can be attributed to the rapid accumulation
of Na+ ions in the gap, which has the effect of reducing the driving force
for Na+ channels. Thus, the full 3D model produces a conduction velocity
slightly lower than the φ only model.
When Cim = 50, there is a sizable difference model calculations with or
without ionic concentration effects. When concentration effects are not taken
into account, the conduction velocity is underestimated in this example.
This shows that the use of the full 3D model is meaningful and may
lead to quantitatively and even qualitatively different results compared with
simpler electrophysiology models.
The reason for the discrepancy between the 1D and 3D models is not
clear. One difference between the two simulations is that in the 3D model
calculations, an electrostatic potential gradient develops within the gap. In a
1D calculation, the gap is treated as single compartment, and thus, there is no
radial gradient within the gap. In 1D computations, therefore, the membrane
facing the gap near r = l will feel an electrostatic potential drop greater
than in 3D computations whereas the opposite is true for the membrane
near r = 0.
257
The peak conduction velocity is reached at about 5nm in the 3D model
calculations and at around 30nm in the 1D model calculations. It is interesting that in [24], in which a 1D model is used, the authors report peak
conduction velocities at around lg = 30 − 100nm, a value that comes close
to the value we obtain using a 1D model. We note that it is not possible
to make a straightforward comparison between our results and the results in
[24] however, since many of the parameters used are different.
5.4.3
Varying rNa
We first study the effect of Cim , rNa and lg on conduction velocity, while fixing
g gap to the value used above:
g gap = 1.10 × 10−5 /(πl2 )mS/µm2
(5.42)
We let:
η=1
(5.43)
lg = 2, 2.5, 3, 4, 5, 6, 7, 8, 10, 12, 15, 20, 30, 40, 50nm
(5.44)
Cim = 0, 50mmol/l
(5.45)
rNa = uniform, 0.5, 0.8, 0.9, 0.95, 0.99
(5.46)
The plots for conduction velocity is given in Figure 5.9.
We first note that even when the Na+ channel conductance is close to
258
0, we do see action potential propagation, albeit at a very small speed (3 ∼
4cm/sec). This is because cardiac action potentials have a long plateau phase
during which the cell is depolarized, and a relatively small gap junctional
conductance is enough to inject sufficient current to the next cell to induce
depolarization.
The propagation speed increases with an increase in rNa . For each fixed
value of rNa , it is interesting to note that the propagation velocity does not
increase (or decrease) monotonically with lg . The speed attains its maximum
at around lg = 5nm, and this value is at most about 12cm/sec.
A narrow gap has two opposing effects on conduction velocity. Suppose
cell A and cell B are separated by a narrow gap and an excited action potential is to propagate from cell A into cell B. As discussed earlier, a narrower gap
implies a greater drop in the electrostatic potential in the gap with respect
to the extracellular bulk(r > l). This facilitates action potential propagation
by making it easier for ion channels on cell B facing the gap to be activated.
On the other hand, a narrower gap means that it is more difficult for electric current to flow from the extracellular bulk into cell B through the ion
channels facing the gap. Cell B will not fully depolarize unless a sufficient
amount of current is injected into it. Thus a gap too narrow will tend to
slow down action potential propagation.
This probably explains why the action potential propagation speed does
not vary monotonically with gap width. For reasons that are yet to be
clarified, the propagation speed slowing effect dominates for smaller (< 5nm)
259
gap width and the accelerating effect dominates for larger gap width. At lg =
5nm, these effects balance and the propagation velocity reaches a maximum.
It is interesting to note that lg = 5nm is approximately equal to the distance
between two membranes when gap junctions straddle two membranes [24, 4].
We finally note that this non-monotonic behavior can also be seen in the
1D computations plotted in Figure 5.8. In [24], where the authors employ a
1D model, they also observe this non-monotonic behavior of the conduction
velocity as a function of gap width. The above explanation is essentially the
same as that given in [24] to explain this behavior.
In addition to the two effects above, narrower gaps imply greater changes
in ionic concentrations. A fast rise in Na+ concentration should slow down
conduction since this will decrease the driving force for Na+ channel currents.
On the other hand, fast activating K+ channels will raise the extracellular
K+ concentration, raising the K+ equilibrium potential and facilitating depolarization of the next cell to be excited. Our computational results plotted
in Figure 5.8 suggest that these concentration effects, though quantitatively
significant (when Cim = 50), do not qualitatively change the dependence of
propagation speed on gap width. Even if we neglect concentration changes,
the graph still retains the same character.
A propagation speed of 12cm/sec, the maximum simulated conduction
speed, is unfortunately an underestimate of the experimentally observed
propagation velocity of 20 − 30cm/sec [10]. The value at which the conduction velocity assumes its maximum in the computations performed here
260
is at Cin = 50, rNa = 0.99, and lg = 5nm. It is difficult to raise the conduction velocity any further without changing the gap junction conductance.
We note, however, that the ratio between the normal and reduced conduction
velocities is comparable (∼ 12/30 in the simulated case and ∼ 25/50 in the
experimental case).
In addition to the general features of conduction velocity plots described
above, one notices in Figure 5.9 that there is a small kink in the graph when
the conduction velocity is at about 5 − 6cm/sec. In Figure 5.10, we plot the
time points at which the action potential reaches cell k when lg = 5, 12 and
30nm while taking Cim = 0 and rNa = 0.95. What is unexpected is that at
lg = 12nm, we see a biphasic time series. Conduction is rapid between two
cells k and k + 1 when k is even, but is slow when k is odd.
As explained in the foregoing, at lg = 5nm, the propagation can be attributed primarily to the ephaptic mechanism. At lg = 30nm, as can be
seen from Figure 5.9, there is no longer any difference in conduction velocity
between the cases rNa = uniform and rNa = 0.95. We can therefore infer that
when lg = 30nm, propagation can be attributed primarily to the slow accumulation of membrane charge due to electric currents through gap junctions.
At lg = 12nm, these two mechanisms play an equal role. Fast conduction
between cells k and k + 1 when k is even is driven primarily by the ephaptic
mechanism, and the slow conduction between cell k and k + 1 when k is odd
is driven by the gap junction mediated currents.
To clarify the situation, let us walk through the sequence of events when
261
12
speed, cm/sec
10
8
6
4
2
0
0
5
10
15
20
25
30
gap width, nm
35
40
45
50
0
5
10
15
20
25
30
gap width, nm
35
40
45
50
12
speed, cm/sec
10
8
6
4
2
0
Figure 5.9: Conduction speed of cardiac action potential under reduced gap
junction coupling. The top graph corresponds to Cin = 0, and the bottom
graph to Cin = 50. The lines on both graphs are conduction velocity plots
for rNa = uniform, 0.5, 0.8, 0.9, 0.95, 0.99 from bottom.
262
70
60
50
msec
40
30
20
10
0
0
5
10
15
cell number
Figure 5.10: A plot of the time series of action potential arrival times for
cells 1 through to 15. The bottom trace corresponds to lg = 5nm, the middle
to lg = 12nm and the top trace to lg =30nm.
263
the propagation is biphasic. Let us start with propagation from cell 1 to cell
2. Cell 1 depolarizes and a strong electric current rushes in through the gap,
generating a negative electrostatic potential in the gap with respect to the
extracellular bulk. This drop, is unfortunately, not sufficient to depolarize cell
2. This drop in the electrostatic potential in the gap soon dissipates without
initiating a depolarization in cell 2. The action potential of cell 1, meanwhile,
reaches a plateau phase. Since the electrostatic potential inside cell 1 is higher
than that of cell 2, electric current flows to cell 2, slowly depolarizing the cell.
An important point is that as cell 2 is more depolarized, cell 3 also receives
current from cell 2, thereby slowly depolarizing, but at a slower rate than
cell 2. Cell 2 eventually reaches threshold, and depolarizes. Depolarization
of cell 2 induces a negative drop in the electrostatic potential in the gap
between cell 2 and 3. This time, the ephaptic mechanism is successful, and
cell 3 rapidly depolarizes, unlike what happened between cell 1 and 2. The
key difference is that cell 3 was primed to be more excitable during the slow
depolarization phase of cell 2. This cycle repeats itself because cell 3 must
now activate an unprimed cell 4.
What we have found is that under reduced gap junction coupling, when lg
is very small, conduction is primarily due to the ephaptic mechanism, while
the gap junction mediated mechanism dominates when lg is larger. There is
a transition zone between the two regimes, in which the ephaptic and gap
junction mediated mechanisms alternate in propagating the action potential
to the neighboring cell. This biphasic mode of propagation is a most likely
264
a new finding.
A natural question to ask is: are there finer structures in the transition
zone? Could there be triphasic or other complicated patterns of transmission
between strands of cells? These questions remain to be explored.
5.4.4
Varying g gap
We finally consider varying the gap junction conductance between two cells.
We let:
g gap = 0, g0 × 2n , n = 0, · · · 3
g0 = 1.10 × 10−5 /πl2
(5.47)
(5.48)
and let:
η=1
(5.49)
lg = 2, 2.5, 3, 4, 5, 6, 7, 8, 10, 12, 15, 20, 30, 40, 50nm
(5.50)
Cin = 0, 50mmol/l
(5.51)
rNa = uniform, 0.95
(5.52)
We plot the propagation speeds in Figure 5.11.
The most notable point here is that propagation fails completely when
lg ≥ 7nm and lg ≥ 12nm when Cim = 0 and Cim = 50 respectively, if
there are no gap junctions connecting two adjacent cells. The propagation
265
speed, cm/sec
20
15
10
5
0
0
5
10
15
20
25
30
gap width, nm
35
40
45
50
0
5
10
15
20
25
30
gap width, nm
35
40
45
50
speed, cm/sec
20
15
10
5
0
Figure 5.11: A plot of the conduction velocity for different values of gap
junction coupling, g gap . The graph on top is when Cim = 0 and the one on
the bottom is when Cim = 50. For each graph, the trace corresponds, from
the top, to g gap /g0 = 8, 4, 2, 1, 0.
266
speed is seen to increase with greater gap junction density at the gap, and
at 8 times the experimentally observed density, the propagation velocity can
reach approximately 18cm/sec.
5.4.5
Varying η
The final variable whose effect on conduction velocity we shall address is the
effect of η, the size of the extracellular space on conduction velocity. We let:
g gap = 1.10 × 10−5/(πl2 )mS/µm2
(5.53)
η = 0.01, 0.03, 0.3, 0.1, 0.3, 1
(5.54)
lg = 2, 2.5, 3, 4, 5, 6, 7, 8, 10, 12, 15, 20, 30, 40, 50nm
(5.55)
Cim = 0, 50mmol/l
(5.56)
rNa = 0.95
(5.57)
We plot the propagation velocity corresponding to the parameters above in
Figure 5.12.
The most interesting feature of this plot is that the conduction velocity
tends to decrease uniformly with a decrease in η, when the gap width is
around lg = 5nm. This is probably due to the fact that the ephaptic conduction mechanism requires a large current flux into the cell for rapid conduction. When the extracellular space is limited, it becomes more difficult
to draw enough current into the gap due to greater extracellular resistance.
267
12
speed, cm/sec
10
8
6
4
0
5
10
15
20
25
30
gap width, nm
35
40
45
50
0
5
10
15
20
25
30
gap width, nm
35
40
45
50
12
speed, cm/sec
10
8
6
4
Figure 5.12: A plot of the conduction velocity for different values of extracellular space size η. The graph on top is when Cim = 0 and the one on the
bottom is when Cim = 50. For each graph, the trace corresponds, from the
top at lg = 5nm, η = 1, 0.3, 0.1, 0.03, 0.01.
268
One experimental way to test whether the ephaptic mechanism is in play
may be to limit the extent of the extracellular space and see whether this
has a decreasing effect on conduction velocity.
5.5
Future Directions
We first enumerate some other parametric studies we would like to perform
in the future to better understand this unusual mode of conduction. One
parameter we would like to explore is how the conduction velocity in the
ephaptic mode is influenced by the parameters that dictate Na+ channel
kinetics. Preliminary computations suggest that the speed of the upstroke
of the activation of Na+ channels largely determines the maximum possible
conduction speed. Another parameter whose effect on propagation should
be relatively straightforward to explore is the electrolyte conductance of the
extracellular space.
A fundamental limitation of the calculations performed above is that
the model only considers a single strand of cells. It would be interesting
to perform a similar study where the cells are arranged in a complicated
network.
If this ephaptic mechanism is indeed an important component of cardiac
propagation in healthy or diseased hearts, how should we scale up our computations to the whole-heart level? The most widely used model for wholeheart electrophysiology is the bidomain model [22]. One direction would be
269
to modify the bidomain framework so that it can account for ephaptic effects.
This would be an interesting research question to address, worthy of another
chapter.
270
Bibliography
[1] D.J. Aidley. The Physiology of Excitable Cells. Cambridge University
Press, New York, 4th edition, 1998.
[2] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley,
L.C. McInnes, B.F. Smith, and H. Zhang. PETSc Web page, 2001.
http://www.mcs.anl.gov/petsc.
[3] O. Bernus, R. Wilders, C.W. Zemlin, H. Verscelde, and A.V. Panfilov. A
computationally efficient electrophysiological model of human ventricular cells. Am. J. Physiol. Heart Circ. Physiol., 282:2296–2308, 2002.
[4] D.M. Bers. Excitation-Contraction Coupling and Cardiac Contractile
Force. Kluwer Academic Publishers, Dordrecht, Netherlands, 2001.
[5] V.E. Bondarenko, G.P. Szigeti, G.C.L. Bett, S.J. Kim, and R.L. Rasmusson. Computer modle of action potential of mouse ventricular myocytes.
Am. J. Physiol. Heart Circ. Physiol., 287:1378–1403, 2004.
271
[6] D. Calhoun and R.J. Leveque. A cartesian grid finite volume method
for the advection diffusion equation on irregular domains. Journal of
Computational Physics, 157:143–180, 2000.
[7] S.A. Cohen. Immunocytochemical localization of the rh1 sodium chanel
in adult rat heart atria and ventricle. presence in terminal intercalated
discs. Circulation, 94:3083–3086, 1994.
[8] J.H. Ferziger and M. Peric. Computational Methods for Fluid Dynamics.
Springer-Verlag, New York, 2nd edition, 2001.
[9] B. Frankenhaeuser and A.L. Hodgkin. The after-effects of impulses in
the giant nerve fibers of loligo. J. Physiol., 131:341–376, 1956.
[10] D.E. Gutstein, G.E. Morley, H. Tamaddon, D. Vaidya, M.D. Schneider,
J. Chen, K.R. Chien, H. Stuhlmann, and G.I. Fishman. Conduction
slowing and sudden arrhythmic death in mice with cardiac-restricted
inactivation of connexin 43. Circulation Research, 88(8):333–339, 2001.
[11] A.C. Guyton and J.E. Hall. Medical Physiology. W.B. Saunders Company, Philadelphia, 10th edition, 2000.
[12] B. Hille. Ion Channels of Excitable Membranes. Sinauer Associates, 3rd
edition, 2001.
[13] A.L. Hodgkin and A.F. Huxley. A quantitative description of the membrane current and its application to conduction and excitation in nerve.
Journal of Physiology, 117:500–544, 1952.
272
[14] M.H. Holmes. Introduction to Perturbation Methods. Springer-Verlag,
New York, 1995.
[15] S. Howison.
Practical Applied Mathematics.
Cambridge University
Press, Cambridge, UK, 2005.
[16] J.G.R. Jeffreys. Nonsynaptic modulation of neuronal activity in the
brain: electric currents and extracellular ions. Physiological Reviews,
75:689–723, 1995.
[17] J.W. Jerome. Approximation of Nonlinear Evolution Systems. Academic
Press, New York, 1983.
[18] H. Johansen and P. Colella.
A cartesian grid embedded boundary
method for Poisson’s equation on irregular domains. Journal of Computational Physics, 147:60–85, 1998.
[19] J. Jost and X. Li-Jost. Calculus of Variations. Cambridge University
Press, 1998.
[20] E.R. Kandel, J.H. Schwartz, and T.M. Jessel. Principles of Neural Science. McGraw-Hill/Appleton & Lange, New York, 4th edition, 2000.
[21] J.P. Keener. Principles of Applied Mathematics. Perseus Books, New
York, 1998.
[22] J.P. Keener and J. Sneyd. Mathematical Physiology. Springer-Verlag,
New York, 1998.
273
[23] C. Koch. Biophysics of Computation. Oxford University Press, New
York, 1999.
[24] J.P. Kucera, S. Rohr, and Y. Rudy. Localization of sodium channels in
intercalated disks modulates cardiac conduction. Circulation Research,
91(12):1176–82, 2002.
[25] P.D. Lax. Functional Analysis. Wiley Interscience, New York, 2002.
[26] M. Léonetti. On biomembrane electrodiffusive models. European Physical Journal B, 2:325–340, 1998.
[27] M. Léonetti, E. Dubois-Violette, and F. Homblé. Pattern formation of
stationaly transcellular ionic currents in fucus. Proc. Natl. Acad. Sci.
USA, 101(28):10243–10248, 2004.
[28] R.J. Leveque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, New York, 2002.
[29] P. McCorquodale, P. Colella, and H. Johansen. A cartesian grid embedded boundary method for the heat equation on irregular domains.
Journal of Computational Physics, 173:620–635, 2001.
[30] Y. Mori, J.W. Jerome, and C.S. Peskin. A three-dimensional model
of cellular electrical activity. Transport Theory and Statistical Physics,
2004. to appear.
274
[31] C.S. Peskin. Mathematical Aspects of Neurophysiology. Courant Institute of Mathematical Sciences, Lecture Notes, New York, 2000.
[32] N. Qian and T.J. Sejnowski. An electro-diffusion model for computing
membrane potentials and ionic concentrations in branching dendrites,
spines and axons. Biol. Cybern., 62:1–15, 1989.
[33] N. Qian and T.J. Sejnowski. When is an inhibitory synapse effective?
Proc. Natl. Acad. Sci. USA, 87:8145–8149, 1990.
[34] P.J. Rabier and W.C. Rheinbolt. Theoretical and numerical analysis of
differential-algebraic equations, volume VIII of Handbook of numerical
analysis, pages 183–540. North-Holland, 2002.
[35] W. Rall. Distribution of potential in cylindrical coordinates and time
constants for a membrane cylinder. Biophys. J., 9:1509–1541, 1969.
[36] W. Van Roosbroeck. Theory of flow of electrons and holes in germanium
and other semiconductors. Bell System Tech. J., 29:560–607, 1950.
[37] A. Scott. Neuroscience, a mathematical primer. Springer-Verlag, New
York, 2002.
[38] G.R. Sell and Y. You. Dynamics of Evolutionary Equations. SpringerVerlag, New York, 2000.
275
[39] R.E. Showalter. Monotone Operators in Banach Space and nonlinear
partial differential equations. American Mathematical Society, Providence, Rhode Island, 1997.
[40] N. Sperelakis. An electric field mechanism for transmission of excitation
between myocardial cells. Circulation Research, 91:985–987, 2002.
[41] N. Sperelakis and J.E. Mann Jr. Evaluation of electric field changes in
the cleft between excitable cells. J. Theor. Biol., 64:71–96, 1977.
[42] N. Sperelakis and Ramasamy L. Propagation in cardiac muscle and
smooth muscle based on electric field transmission at cell junctions: an
analysis by pspice. IEEE Eng. Med. Biol. Mag., 21:177–190, 2002.
[43] Henk A. van der Vorst. Iterative Krylov methods for large linear systems, volume 13 of Cambridge monographs on applied and computational
mathematics. Cambridge University Press, 2003.
[44] J.L. Vázquez and E. Vitillaro. Heat equation with dynamical boundary conditions of reactive type. Communications in Partial Differential
Equations, to appear.
[45] J.A. Yao, D.E. Gutstein, F. Liu, G.I. Fishman, and A.L. Wit. Cell
coupling between ventricular myocyte pairs from connexin43-deficient
murine hearts. Circulation Research, 93(8):736–743, 2003.
276

A Three-Dimensional Model of Cellular Electrical Activity

Transcription

Similar documents

Diffusion and Osmosis Worksheet

Structure of Cell Membranes (Insane in the membrane)

Entertainment/Leisure Only

- Membrane vacuum press

POLYGLASS® U.S.A., Inc. Corporate Office 1111 West Newport

MusselBound Install

BIOCHEMISTRY IN PERSPECTIVE Organelles and Human Disease

OutDry technology achieves a fully waterproof, windproof and

Diffusion and Osmosis Worksheet Answers

Interview with an amoeba - ACCESS