Muscle Dynamics, Size Principle, and Stability - Research

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-31, NO. 7, JULY 1984
489
Muscle Dynamics, Size Principle, and Stability
YUAN-FANG ZHENG, STUDENT MEMBER, IEEE, HOOSHANG HEMAMI, MEMBER, IEEE,
BRADFORD T. STOKES
Abstract-A mathematical model of skeletal muscle activation during
small or isometric movements is discussed with the following attributes.
A direct relationship between the stimulus rate and the active state is
established. A piecewise linear relation between the length of the contractile element and the isometric force is considered. Hill's characteristic equation is maintained for determining the actual output force
during different shortening velocities. A physical threshold model is
proposed for recruitment which encompasses the size principle and its
manifestations and also exceptions to the size principle. Finally, the
role of spindle feedback in stability of the model is demonstrated by
studying a pair of muscles.
0,
AND
cc
SEC
(a)
PEC
F
I. INTRODUCTION
T HE complex behavior and complicated physiological
structure of skeletal muscle has prompted decades of
study to define these parameters. Much remains to be known
about the actual energy conversion, bioenergetics, and dynamics of the muscle. In spite of this gap in knowledge, models
of the skeletal muscle, that have input-output and behavior
similar to an actual muscle, are desirable. Such models would
be indispensable in any quantitative studies of the behavior of
the skeletal and the nervous systems by mathematical methods
and analog and digital computer simulations. In many locomotion studies [1], ideal torque generators are employed at
the joints to represent the action of groups of muscles. This
idealization ignores muscle characteristics and the system cannot predict actual human locomotion very well. Further, the
inputs to such torque generators do not correspond to valid
physiological inputs to the human muscle from the nervous
system. For prosthetic and orthotic devices, eventually quiet,
efficient, and cosmetically appealing muscle-like force actuators are desirable that could interface with signals from the
nervous systems, and would also provide sensory signals acceptable to the central nervous system. Besides, muscle-like actuators are distributed along the peripheral links, while torquegenerator actuators are concentrated at the joints, and would
make the joints very heavy and cumbersome. All these issues
necessitate postulation of meaningful muscle models. Since
the work of Hill [2] -[41 and Wilkie [5], the mechanical behavior of active muscles has been described by a contractile
component (CC) in series with a noncontractile series elastic
Manuscript received May 21, 1982; revised June 23, 1983. This work
was supported in part by the National Science Foundation under Grant
ECS 820-1240 and in part by the Department of Electrical Engineering,
The Ohio State University, Columbus, OH 43210.
Y.-F. Zheng and H. Hemami are with the Department of Electrical
Engineering, The Ohio State University, Columbus, OH 43210.
B. T. Stokes is with the Department of Physiology, The Ohio State
University, Columbus, OH 43210.
(b)
Fig. 1. Structural model of muscle fibers.
component (SEC) [Fig. 1(a)]. For a long time, the study of
muscles was concentrated on the mechanical properties of the
series elastic element [6] , and the parallel elastic element [Fig.
1(b)] whose nonlinear behavior can be described by reasonably
accurate functions [7]. In some further studies, the force output of the contractile element is assumed to be a function of
an input from the central nervous system [8], [9]. Recently,
Hatze [7], [10] has proposed a control model of the muscle
based on a set of five nonlinear first-order differential equations. This model would produce a simulation system with a
very large dimension.
For the model presented in this paper, a threshold theory
and a corresponding physical implementation are proposed that
encompass the size principle and its physiological manifestations, as well as exceptions to it [11] -[14]. The role of some
of the afferent fibers in stability of this model are also studied,
and some of the muscle input-output attributes are studied by
digital computer simulation. In Section II, a single motor unit
is studied. In Section III, a recruitment and thresholding model are proposed via a physical circuit. In Section IV, the
dynamics of a whole muscle are discussed and some digital
computer simulations are presented. The stability analysis is
carried out in Section V. All signal intensities are represented
by amplitudes in order to simplify the presentation.
II. A SINGLE MOTOR UNIT
It is generally accepted that the muscle fiber is composed of
a series arrangement of repeating structure, the sarcomeres,
that extend from Z-disk to Z-disk. Contained in this repeating
structure is a controllable elementary contractile unit. Also,
there are series elastic elements and parallel elastic elements
within each basic unit. Although the structure of these elastic
elements is not exactly known [15], the experimental phenom-
0018-9294/84/0700-0489$0 1.00
C 1984 IEEE
490
9
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-31, NO. 7, JULY 1984
ena strongly suggest their existence. The arrangement of where a1, a2, b1, b2 are appropriate constants. Now for difthese three elements can be seen in Fig. 1. Because the struc- ferent muscles the stimulus rate which excites the maximum
ture is repeated in series in each muscle fiber, this arrangement tension is different, and defining a relative active state is more
is also valid as a lumped model for the entire fiber. Muscle convenient.
fibers normally contract in groups, referred to as muscle units
It is known t-hat in response to the stimulation of the nerves,
[161. The muscle fibers within a single muscle unit are quite muscles need a certain amount of time to contract and differ
homogeneous, even though they may be spread throughout widely in their speed of contraction. For example, the extrathe muscle. Therefore, the physical arrangement of Fig. I ocular muscles that rotate the eyeball require only about 7.5
should be good for a muscle unit. The muscle unit, together ms, but the soleus muscle, a slow antigravity muscle in the
with its associated motorneuron, is known as a motor unit hind limb, requires about 100 ms to reach its maximal tension
[16]. The modeling, based on a single muscle fiber, could [14]. Taking into account the above property of the muscle
also be extended to a motor unit [9], [10].
contraction, the following first-order differential equation is
According to [7], the force developed by the contractile adequate to relate the active state q to Q
component is the product of the active state q (the relative
4q+cq=cQ(r)A 0.Q.1
(4)
amount of Ca++ bound to troponin), the length-tension funcwhere A is the maximum tension a single motor unit can
tion k, and the velocity-tension function g
develop and Q is the "desired" active state of the single motor
f=qkg.
(1) unit while q is the actual active state, and the final values of
In [7], q is a function of the length of the contractile element both in (4) are the same, i.e., the actual active state reaches the
desired one after some time.
and the concentration y of Ca++
Now each stimulation impulse from the nervous system
q = q(y, )
(2) elicits a single twitch lasting for a fraction of a second. SucThe dependence of the active state q in (2) on t could be re- cessive twitches may add up to produce a stronger action, i.e.,
garded as redundant, because in (1), the force is already length a partially or fully fused tetanus. If the stimulus rate is not
dependent through the tension-length function k. This redun- high enough, a single motor unit shows an unfused response
dancy arises from the definition of the active state. As is well (Fig. 2). If the stimulus reaches its critical frequency, the
known, each basic contractile element, the sarcomere, is sur- successive contractions fuse together and cannot be distinrounded by a network called sarcoplasmic reticulum. The guished from one another [21]. But any kind of muscle is
action potential is conducted through the T-system located at composed of a number of motor units and the stimulation
the borders of two adjacent networks. The depolarization of could be assumed to be distributed over motor units, instead
the T-system caused by the action potential depolarizes the of being synchronized. For distributed stimulation, even at
membrane of the reticulum at the triad. This, in turn, triggers a low rate, a smooth contraction can be demonstrated [19].
the release of Ca"+ ions from this area [17]. The action poten- This is the reason why the behavior of a muscle usually shows
tial becomes smaller towards the interior of the membrane of smooth motion. Based on this assumption, (3) and (4) are
the T-system as a function of T-system electrical capacitance. acceptable and simplify the model a great deal.
The length-tension function k could be derived from wellWith the shortening of the muscle, the cross-section area inknown
experimental results by Gordon et al. [22] and by the
creases [9], [10]. The longer the diameter of the muscle
sliding
filament
theory of Huxley and Niedergerke [23]. This
fibers, the less Ca"+ ions could be released; therefore, the relarelation
has
been
approximated by a mathematical expression
tive amount of Ca"+ bound to troponin depends on the muscle
similar
to
a
step
response of a underdamped second-order
length. But according to Hill [4], the active state is defined as
equation
[7]:
the tension developed when the contractile component is
k() = 0.32 + 0.71 exp {-1.1 12( - 1)1
neither lengthening nor shortening. With this definition, the
active state could be measured by the produced isometric
X sin {3 - 722( - 1)} 0.5 8 < t 6 1 .8.
(5)
tension [18]. The relation between free calcium ion concenFig.
3
shows
a
comparison
of
k
from
(5)
to
others
obtained
tration and contractile response has been studied. Hellam and
Podolsky [19] were the first to use skinned fibers for the from experiments [22]. From Fig. 3, the tension-length relaquantitative study of this relation. They found that the ten- tion can be represented by piecewise linear function
sion was related to the free Ca++ ion concentration via a
0
0<L <0.635
sigmoid curve. This relation is the main argument for support
4.2L - 2.67
0.635 < L < 0.835
of (2). If, however, a definite relation could be established
between the stimulation rate and the contractile force, the
0.97L + 0.022 0.835 < L S 1
(6)
Ca"+ concentration need not be involved in the model and
1 <L < 1.125
-1
consequently the model will be simpler. Many experimental
results [19], [20] suggest that the rate of the nerve impulses
-1.43L + 2.61
1.125 <L <-3.65
determines the rate of the tension output according to a sig- where L is
the relative length and is equal to y/y0, and y0 is
moid curve. Thus, one could directly relate the "desired'
the
at
length
point C in Fig. 3.
active state of a single motor unit Q to the relative input stimuThe
remaining
item which can change the force output of
lus rate r
the contractile element is the velocity of movement between
Q(r)=I-biexp(a,r)-b2exp(a2r), 06Q61 (3) the thick and the thin filaments. The force-velocity relation of
ZHENG etal.: MUSCLE DYNAMICS, SIZE PRINCIPLE, AND STABILITY
100/sec
40/sec
ro
S.
c
15/sec
a)
o
10/sec
Lime
Fig. 2. Contraction response of muscle flbers to stimulus rate.
0.6
1.8
Fig. 3. Comparison of K(L) for Hatze model and experimental results.
Hatze [24] -based
the contractile element-is very similar
on the experiments on
on
to Hill's characteristic equation-based
a
muscle bundle
]
(7)
Vmax
The similarity of the predicted force by these two equations
dictates in favor of Hill's equation if simplicity is the goal.
To summarize, a single motor unit can be described by the
following:
[1 -
Vmax
] [1 +a
fi = q k.1A
amaxJ
Fm
(+
1-
-
Q(ri)
CQ(ri) Ai
=
b1
1 -
exp
(a1 ri) - b2
exp
(a2 ri)
0<L S0.635
0
4.2L
-
2.67
0.97L + 0.022
1
1.43L
0.635 < L < 0.83
0.835 < L < 1
1
+
2.61
<L
1.125
1.125 <L .3.65
where L, as stated before, is the relative length of the contractile element.
RECRUITMENT AND THRESHOLDING
The skeletal muscle is composed of many motor units. Even
within one muscle, motor units vary widely in their properties.
The model derived in the last section is good for a single motor
III.
unit but not valid for the muscle. Motor units are usually
recruited in a sequential order. In large postural muscles for
small forces, only the small, slow motor units may be recruited.
For larger forces, faster and larger motor units cease firing in
the reverse order. This represents the size principle first proposed by Henneman [25], and supported by many experiments [11], [25], [26]. On the other hand, experimental
results of Burke [27] point to functional specialization of
motor unit types. Desmedt's results [28] point still to other
exceptions to the size principle; the ordering appears to depend
on the direction but not the speed of movement [29]. Also,
Grillner [30] points out that other parameters may influence
the recruitment. One feasible hypothesis is that before the
small motor unit reaches its maximum tension, the larger
motor units are not recruited. This is because the large motor
unit is more vulnerable to fatigue. The central nervous system
therefore limits long duration of large motor unit activation.
Here, two physical models of recruitment are discussed. The
first one permanently obeys the size principle. The second one
is programmable from the central nervous system.' Consider
the physical model of Fig. 4 for the first case. The general
signal s, the result of CNS as well as reflex feedback, as a voltage source, is assumed to be the input to this circuit. The
circuit consists of a number of devices in parallel. Each device
pertains to one motor unit, and the current ri in the device is
the signal that the nervous system provides to the motor unit
(8)-(1 1). The ith device consists of three elements in series:
1) a diode conducting only in one direction, 2) a threshold
voltage TG, and 3) a (nonlinear) resistor Ri that determines
the size of the stimulus current ri. A current ri would flow
only when S > Ti. Therefore,
ri
g(V)=-=
Fo
qi + cqi =
491
ri=0
Ti if S > T1
Ri
(12)
if S<T1.
(13)
This physical system establishes the relationship between the
above-defined general signal S and the two muscle control
parameters, stimulus rate, and the motor unit. If the general
control signal is regarded as from a "voltage source," the responses "current" could be considered as the stimulus rate
which will depend on both the threshold "voltage" Ti and the
nonlinear "resistance" Ri. The values and the arrrangement of
those thresholds should vary widely in different skeletal muscles, but is definitive for each specific muscle. If the voltage
is beyond the threshold of a particular motor unit, this motor
unit will be recruited. -The large motor units have high thresholds, and begin to be active only when the high output tension
is needed. This means if Ti for larger motor units is higher
than Ti for smaller motor units, the size principle is obeyed.
In recent years, different reports of exceptions to the size
principle have appeared [27] -[34]. Even Henneman, who
first proposed the size principle, questioned its validity [25]:
"Does it (central nervous system) select from among the various motor units just those it requires for a certain task? Does
it mobilize just the large, powerful, rapidly contracting units
to supply the speed and the power needed by a high jumper
to clear a 7-foot bar? May it activate only small, slower units
to provide the delicacy and precision a watchmaker needs?
The answers are not obvious." With the physical model proposed here, these questions may be partially answered at least
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-31, NO. 7, JULY 1984
492
S
T
c
R.
Ri
I
Charge
Control
r.
I II1
Fig. 4. Threshold circuit.
for large postural muscles. The motor units are recruited in
the sequential order according to the size principle. When a
large force is needed in a very short time, a large step function,
beyond the threshold of all the motor units, is input. All the
motor units become active and contract. Large motor units
need less time and reach the maximum tension much faster
than the small motor units. This gives the appearance that the
large motor units are recruited first. In reality, the smaller
motor units are not recruited later than the large units but are
slower. The above explanation could very well account for
the contradiction in part between the size principle and its
exceptions.
On the other hand, if the hypothesis is proposed that the
recruitment is controlled by the central nervous system, one
may consider discharge circuits that reduce the threshold T1
and allow a larger motor unit to fire earlier (Fig. 5). As seen
in this figure, two control valves-one for charge and one for
discharge-may allow Ti to be increased or decreased. The
increase in Ti retards firing, and the decrease in Ti expedites
firing of the correponding motor neuron. Such mechanisms
under the control of the central nervous system may provide
many patterns of recruitment for hand and finger muscles
involved in delicate manipulative tasks.
For the purposes of this paper, it is necessary to develop a
relation for a motor unit i involving the input signal s, the
stimulus rate ri, the threshold tension T7, and the maximum
tension (in grams) Ai of the motor unit. Qualitatively, this
relationship can be explained as follows. For fast muscle
fibers, the active state reaches its maximum at about 100 Hz
stimulus rate. The stimulus rate of slow muscle fibers may be
about one-half to one-third of that for fast muscles. The
stimulus rate ri should be selected such that when the input S
reaches threshold Ti, the stimulus rate for the motor unit i
keeps constant. The input signal at this moment should be
equal to the threshold of the next larger motor unit. For a
quantitative relation, two definitions are needed. Let u(x)
be a unit step
u(x)=1 if x>0
u(x)=0 if x.0.
Let Ti <Ai, and let Ti and Ai be expressed in the same units
(grams of force). The stimulus rate ri is related to S, Ti and Ai
as follows:
ri= 1
Ri
(S-
T1)(Ai- Ti)-1
Fig. 5. Threshold circuit with charge and discharge control.
IV. A FIVE MOTOR UNIT HYPOTHETICAL
MUSCLE MODE
Combining (14) with the model of a single motor unit derived
in Section II, the control model of a particular muscle, excluding its series and parallel viscoelastic elements, and composed
of n motor units, can be expressed as follows:
fi = CQk )(
4i + Ci qi = CiAi Q(ri)
(17)
k = same as in (1 1)
(18)
ri = Gi(S - T1) (As - T) -' (U(S - T,)
- U(S - Ai)) + Gi U(S - Ai)
(19)
n
F= ,
i-1
(20)
fi
where a, Vmax, C, bi, a1, b2, a2 , G, A, T, and n are constants
depending on the properties of the motor units:
n = the total number of the motor units in a
muscle
=
1/Ci the contraction time constant for motor
unit i
=
the
maximum tension of motor unit i
Ai
=
the
threshold tension of motor unit i
Ti
b21, ali, and a21 = constants in accordance with the active
state and stimulus relation
=
Gi the stimulus and amplitude ratio
Gi= 1/R.
Equation (17) may be further simplified. Kanosue et al.
[35] have proposed that the isometric tension (or the active
state) has the following relationship with the stimulus rate:
Q(ri) =
(14)
(15)
(16)
Q(r,) = 1 - bli exp (alir,) - b21 exp (a2iri)
(ri<10)
0.16
[u(S- Tt)
(
0.042ri
1
-
0.26
(10 <
ri < 30).
(30 <
ri).
(21)
ZHENG et al.: MUSCLE DYNAMICS, SIZE PRINCIPLE, AND STABILITY
In order to test this mode, a hypothetical five-motor unit muscle
with no load and no elastic elements is assumed. The constants
of all five motor units are listed in Table I.
From Table I, it is apparent that unit 5 has the largest maximum tension and the fastest contraction time, and that unit I
has the smallest and the slowest.
Two sets of digital computer simulations of this hypothetical
muscle are presented here. The first simulation is concerned
with large tension development. A step function is input to
the muscle whose value is beyond the thresholds of all five
motor units. All motor units become active and contract. The
response is shown in Fig. 6. The rise time of the maximum tension is rather small. It looks as if only the fast and large motor
units are recruited. Actually, all the motor units are recruited
and the tension and the contraction time are mainly dominated
by the large motor units.
In the second simulation, the input signal is a ramp signal,
requiring the muscle to develop maximum tension in 5 s. Let
the input signal be expressed as
S = 960 t(U(t) - U(t - 5)) + 4800 U(t - 5).
The stimulus rate response of the motor units by (19) are
0< t<0.1042
288 t
r = 3
0.1042 < t
30
(144t - 135) 0.093 < t < 0.3021
r2 =
r3
=
r4
=
0.3021 < t
(57.6t - 16.8) 0.297 < t < 0.8125
0.8125 < t
30,
(28.8 t - 22.5) 0.7813 < t < 1.8229
493
TABLE I
Unit 1
Unit. 2
Unit 3
Unit 4
Unit 5
Ci(S-1)
Ai(g)
Ti(g)
Gi(c/s)
25.58
30.70
38.38
51.17
76.75
100
200
500
1000
3000
0
90
280
750
1700
30
30
30
30
30
s
1.2
F
t
sec
(g)
4800
30
t
(sec)
Fig. 6. The response of the hypothetical model to a step input.
0.12
1.8229 < t
1.7708 < t < 4.8958
(9.6t - 17)
r5 =
30
4.8958 < t.
(22) The actual active state of all motor units are shown in Fig.
7(a)-(e). Note that a sigmoid curve is presented for each motor
From (21) the following active states can be obtained:
unit. The total force tension is the summation of all five motor
0.16
0< t 0.1042
unit forces and is shown in Fig. 8. The developing force is not
quite smooth because (21) is employed instead of (17). In the
=
Q(r>) 12.096 t 0.26 0.034. t < 0.1042
second simulation, the force develops in a long time and the
0.1042 < t
motor units are recruited one by one, and the size principle is
more obvious.
0.16
0.093< t<0.163
V. SPINDLE AND GOLGI FEEDBACK
0.163 < t < 0.3021
Q(r2 ) = 6.048t - 0.827
The above model is implemented in a model of the moveI
0.3021 < t
ment of the elbow joint in a plane (Fig. 9) in order to study
0.16
0.2917 < t < 0.4653
the effect of spindle and Golgi feedback (36], [37]. This
is idealized in Fig. 10 with a pair of muscles.
joint
0.4653
<
t
<
0.9656
0.8125
Q(r3)= 2.4192tFor small movement at the joint, the dynamics of muscle
0.8125 < t
could be largely simplified. The lengths of the muscles can be
assumed constant, and (18) is simply k = 1. ft is reasonable to
0.7813 < t < 1.12847
0.16
assume that the stimulus rate will not exceed 30 Hz so that the
Q(r4) = 1.2096t 1.205 1.12847 < t < 1.8229
muscle tension is always less than the maximum value. With
these assumptions (21) is
1.8229 < t
30
1.2
-
Q(rs ) =
0.16
0.4032 t - 0.974
1
1.7708t < t < 2.8125
2.8125 < t < 4.8958
4.8958 < t.
Q(ri) = 0.042ri 0.26.
-
(23)
For small movements, (15) could be linearized. Suppose at
the operating point the stimulus rate ri = ri0, Q(r) is expanded
in Tqwlor --rip-
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-31, NO. 7, JULY 1984
494
Qi
°2
100
(sec)
0.08
0.03
(a)
(b)
Triceps brachii
Q4
Fig. 9. The human elbow.
1000
(sec)
0.8
0.3
0.8
1.8
(c)
(d)
Fig. 10. A simple planar model of the flow.
Motor Unit 1
t
(sec)
Motor Unit i
Motor Unit 2
(e)
Fig. 7. Active state responses to a ramp input.
F
S
s
4800
0
Fig. 11. Linearization around the operating point So.
Equation (24) can be diagrammatically represented in Fig. 11
for each motor unit. For any specified operating point, only
one of the Ari is active. And it could be expressed as
3200
Ar-= -GiIAS.
1600
ITi +n
All other Ari's are zero. With (24), (17) is now
.
ec
4.5
3
1.5
A4j +C1Aqi=CiAQA
t
or
Fig. 8. The total force output for the hypothetical model with a ramp
input.
q1 + Ciqi = 1.26CiAS
A4i +
Q(riO + Ari) Q(r1o) +
=
dQ(rio) Ari + d2Q(r'i) A2ri + *
dr1
-
d2r
Ignoring higher order terms
Q
dQ(r,0) Ar,
dr,
=
0.042 Ar1.
(24)
Ci qi = 1.26CiAS.
(25)
Equation (24), only, denotes one motor unit of the muscle
depending on the operating point chosen. The Hill characteristic equation has been linearized before [9]
V= (qifi +a
f+) b
-
ZHENG et al: MUSCLE DYNAMICS, SIZE PRINCIPLE, AND STABILITY
.495
Aqi=xs, Aq x6, AS=u1, AS=u2.
(32)
These equations may be put in a convenient state space form
Let a spring, in series with the contractile element, represent the
x =Ax + Bu
elastic element. Let x denote the muscle length-the same for
or
all motor units. Let yi denote the length of the contractile
component (Fig. 1), and let VI = j. It follows that
0
0
1
0
0
0
xl
xl
=
fi KI(x - yi)
mgl/l 0
RI/ -RI 0
X2
X2
0
f=K1x- Ki V
O -KR -K/C 0 K/C
X3
X3
Afi = K1Ax - Ki Vi
O
KR 0 -K/C 0 K/C
X4
X4
EAfi= KiA - AKiAVi
O
O
0
'0
-C. 0
XS
XS
AF= Kx - KLA Vi.
(27)
o
o
0
0
0
X6
X6
I.Replacing A VI in (27) with (26) one obtains
0
0
AF=KA* - IK(Xa Aqi + ^Afi).
0
0
(28)
0 ' 0
At the operating point 3 V/lq1 and a V/3f1 are the same for
+
(33)
0
0
every motor unit.
Since for only one of the motor units qi is not equal to zero
CiA, 0
av
av
.0 CiA.
AF= KAx - K-AF- K-Aqi
afi
.aq
The numerical values of the parameters are selected as follows:
v I and a f..
let a
(29)
R = 0.045 m
1 = 0.2 m
C1 = 25.58
E afi(qiC
B
1=0.12kg m2
C2 = 30-70
Equation (29) becomes
K = (50 *10-3)-l
m = 1 kg
C3 = 38.38
AF=KA*-KB AF--Aqi.
(30)
.C
K= 17 N/m
C4 = 51.17
Equations (25) and (30) describe the dynamics of the muscle
C5 = 76.75
when it is subject to a small disturbance.
Gi
Suppose there are a pair of flexor and extensor muscles workAi=Dj
ing on the forearm. Initially, there is a small disturbance so
Di = 0.042 30.
that the forearm leaves the perpendicular position a small
angle AE (Fig. 9). The changing of the length of each muscle If the value of Ci is chosen to be 25.58, then the numerical
is relatively Ax, = -A91 and Ax2 = -A1l. The Newtonian values of A and B are
equation for expressing this model should be
0
1
0
0
0
0
IA, -=AF1 di - AF2d2 + mgl sin AE.
146.35
0
0.375 -0.375
0
0
If AOeis small, d, and d2 remain constant. Let
0 -0.765 -20
0
20
0
d1 =d2 =R
0
0.765 0
-20
0
20
= Ae.
sin
0
0
0
0
-25.58
0
Now the system equations for small motion aTe
0
0
O
0
0
-25.58
IA6= AF1R - AF2R + mgl AO
0
0
K
K
AF1 =-KAeR- AF1 + Aqi
0
0
C
C
0
0
~~K
A =2KAOR - - AF2 + /qi
Vi= a-K Aq1+
-
Af1.
(26)
_
_
[U21
AF2
-
A4i = -CiAqi + CiAi ASi
A4Q = -C;'Aq; + CAiAS!
let
X1 =AE, X2 =AO,
AF1=X3, AF2 =X4,
(31)
32.23 03 1
0 32.23
The eigenvalues of matrix A are -25.58, -25.58, -20, -19.97,
-4.0616, and 4.0316. Since X6 iS positive, this system is un-
496
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL.
BME-31,NOJ. 7,
JULY 1984
stable for small amplitudes. The muscle spindle and Golgi
tendon organs provide this negative feedback which could
stabilize the system.
u =Kx
of this model is that all the parameters should be measured
experimentally. This should be possible with more experiments, although the identif'ication problem is difficult. The
simulation results have shown that the muscle tension can be
developed either very quickly or gradually. Both obey the size
principle. As demonstrated, this model allows inclusion of
or
spindle length and velocity feedback and also Golgi force
feedback.
This model is a piecewise linear model. Although the model
x2
is nonlinear, the parameters of the model are those arrived at
[UI
kll k12 k13 k14 001
X3
(
experimentally and verified by small signal conditions and
analysis. Thus, there may appear an inconsistency in the logic
21 k22 k23 k24
J X4
of using large-size motor units in small movements. Large-size
x5
motor units, however, could be recruited in small movements
when it is desired to oppose excessive external forces, or to
To study the effect of Golgi feedback state, variables X3 and provide stable platforms for delicate and accurate movements.
X4 are fed back by selection of different gains for kl3, k14,
REFERENCES
k23, and k24, and leaving the other elements of c to be zero.
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2 =
-25.58
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X4 = -3.2 - 11.2i
X5 = -3.2 + lli
6 = -2.854.
Based on this analysis, it appears that length and velocity
feedback are essential for small motion stability while the
Golgi feedback must have other functions.
VI. DIsCuSSION AND CONCLUSIONS
Mathematical models of the skeletal muscle are very hard to
establish. The main reason is that the structure and properties
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Yuan-Fang Zheng (S'82) received the B.S. degree in electronics from Tsinghua University,
Peking, China, and the M.S. degree in electrical
engineering from The Ohio State University,
Columbus, in 1970 and 1980, respectively.
He worked for the Ningxia Electronics Company, China, from 1970 to 1978. He was a
graduate student in the The Chinese Academy of
Science from 1978 to 1979. He is currently a
Teacher and on the Research Staff of the Department of Electrical Engineering at The Ohio
State University, and he is pursuing a Ph.D. program. His research interests are robotics, locomotion, and digital and control systems.
W
X
Hooshang Hemami (S'66-M'67) received the
B. S. degree in electrical engineering from the
~* University of Teheran, Teheran, Iran, in 1958.
He further studied with Y. W. Lee and A. G.
Bose at the Massachusetts Institute of Technology, Cambridge, and with R. L. Cosgriff at
The Ohio State University, Columbus.
He is currently a Professor of Electrical Engineering at The Ohio State University where
he teaches courses in nonlinear, digital, and
control systems. His research activities are in
locomotion, robotics, mechanics, dynamics, and control of movement
and pattern recognition.
Bradford T. Stokes received the Ph.D. degree from the University of
Rochester, Rochester, NY, in 1973.
He is an Associate Professor of Physiology at The Ohio State University College of Medicine, Columbus. He is also the Scientific Director
and Co-Principle Investigator of the Spinal Cord Injury Research Center
at Ohio State, sponsored by the National Institute of Neurological Communicable Diseases and Stroke. His research interests lie in the areas of
development of vertebrate motor systems and the influence of the extracellular environment on neural function.