Flocking and Synchronization (some connections)

Workshop on
Numerical Aspects of Hyperbolic Balance Laws and Related Problems
27-28 aprile 2012
Università di Ferrara, Dipartimento di Matematica
Flocking and Synchronization (some connections)
G. Aletti, G. Naldi (UniMi)
T. Nieus (IIT, Ge)
OUTLINE
- Flocking (classical Cucker-Smale system)
- Kuramato model
- K model as a C-S model
- MEA recordings experiments
- A computational model
- Theoretical analysis (in progress)
FLOCKING
"Flocking" represents the phenomenon
in which self-propelled
individuals using only limited
environmental information and
simple rules, organize into an ordered
motion
Cucker
⎧
⎪
⎪
⎨
⎪ dVi K
=
⎪
N
⎩ dt
(based on previous work
of Reynolds ‘87, Vicsek et al. ‘95,
G. Flierl, D. Grunbaum, S. Levin, and D. Olson ’99,
……)
dX i
= Vi
dt
N
∑ψ( X
j =11
i
Smale
)
− X j (V j − Vi )
With communication function
ψ (r ) =
κ
(α
2
+r
)
2 β
,
K>0, α>0, β≥0,
Typical flocking Theorem.
On the mathematics of emergence
Felipe Cucker · Steve Smale
Japan. J. Math. 2, 197–227 (2007)
"Synchronization (=syn (same, common) + chronous (time))" is
an adjustment of rhythms of oscillating objects due to their
weak interaction.
Examples:
- Flashing of fireflies in South-East Asia
- Firing of coupled cardiac pacemaker cells (heart’s contraction)
- Synchronous firing of many neurons (Parkinson’s disease)
- Hands clapping in a concert
….
The Kuramoto model
Phase descriptions for synchronization: Winfree (1967),
Kuramoto (1975).
Remark. Kuramoto’ model is considered the main example
For synchronizing agents.
Other people followed the same idea ….
Choi-Ha-Kang-Kang ’11
S-Y Ha, C. Lattanzio, B. Rubino; M. Slemrod ‘11
(Thanks to S.Y. Ha)
(Thanks to S.Y. Ha)
EXAMPLE.
But Kuramoto like models are not the end of the story:
- Communications are not instantaneous;
-  The topoloy of the Network is important
-  the dynamics may be more complicated
-  …..
MEA recordings – Cell culture
Dissociated cells (obtained from pups brain) plated on
MEA recording system develop and make synapses
building up an artificial neural network.
8 hrs
24 hrs
6 DIV
Spontaneous activity along cell culture maturation
div07
div42
After 10 days in vitro the cell culture starts
to fire synchronously.
Main drawbacks of random cell cultures:
1) network topology is not reproducible
2) cell-electrode coupling is rather bad
Bologna LL
et al.
Neuroscienc
e 2009
Cell culture patterning
Principles of designing neural networks
The materials deposited may be:
- Permissive materials to cell growth: polylysine, laminin, or aminosilanes,
- Non-permissive materials: polyethylene glycol, albumin, and chondroitin
Our “receipt” :
- Ink is made by ECM gel (laminin...)
- PDMS supports the feeder culture surrounding the electrode area
Patterned cell cultures
random
patterned
§long-term confinement
(i.e. >20 DIVs)
§long-term culturing
(stable conditions and
survival)
§low density cultures
(i.e. < 100 cells/mm2)
Marconi et al.
Plos One 2012
Network measures
Neural diseases (e.g. Parkinson, Epilepsy) sometime come with abrupt changes in the
properties of the underlying topology (e.g. cell death Mossy cells in the DG).
Watts & Strogatz. Nature 1998. Bullmore & Sporns. Nat. Neurosc. Reviews 2009
Simulazione di Reti Neurali: Fenomeno della sincronizzazione
di M. Bersanelli, M. Tezzele, A. Trivella, G. Zanella
TOPOLOGY <-> SYNCHRONY
Minimal models. Assumptions (all biological):
- Unidirectional connection;
- We consider the probability of firing for each neurons which
depends on the synaptical inputs (how many);
- Background noise;
-  delay in the communications;
- After firing the neuron remain silent.
Links
matrix
Interspike time
Raster plot
- Theoretical analysis
-  sparse control problem (how to produce synchronization by using few neurons)
-  inverse problem
- …
Functional Connectivity Maps
FC measures correlated activities:
Connectivity maps are inferred by:
•
•
•
•
Cross Correlation
Shifted Mutual Information
Joint Entropy
Transfer Entropy
+ multipoint measures that overcome some of
the limitations of pairwise techniques.
Maccione et
al.
submitted