Human skeletal muscle may be regarded as an electro-mechanical transducer. Its physiological input is a neural signal originating at the alpha motoneurons in the spinal cord and its output is force and muscle contraction, these both being dependent on the external load. Some experimental data taken during voluntary efforts around the ankle joint and by direct electrical stimulation of the nerve are described. Some of these experiments are simulated by an analog model, the input of which is recorded physiological soleus muscle EMG. The output is simulated foot torque. Limitations of a linear model and effect of some nonlinearities are discussed

Agarwal, G. C.

Cecchini, L. R.

Gottlieb, G. L.

English

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1 P~ece~ing page blank 
13. Further Observations on the Relationship 
of EMG and Muscle Force* 
GYAN C. AGARWAL, LAWRENCE R. CECCHINI, AND GERALD L. GOTTLIEB 
Rush Presbyterian-St. Luke’s Medical Center 
University of Illinois at Chicago Circle 
AND 
Human skeletal muscle may be regarded as an electro-mechanical transducer. Its physiolog- 
ical input is a neural signal originating a t  the alpha motoneurons in the spinal cord and its 
output is force and muscle contraction, these both being dependent on the external load. 
In  this paper some experimental data taken during voluntary efforts around the ankle joint 
and by direct electrical stimulation of the nerve are described, Some of these experiments are 
simulated by an analog model, the input of which is recorded physiological soleus muscle 
EMG. The output is simulated foot torque. Limitations of a linear model and effect of some 
nonlinearities are discussed. 
INTRODUCTION 
Modeling of biological systems requires us to 
face three most challenging problems. First, we 
sometimes do not know and often cannot measure 
the inputs to the real system we are modeling. 
Second, it is usually extremely difficult, if not 
impossible, to measure all state variables, and 
third, such systems almost inevitably contain 
significant nonlinearities which cannot be avoided 
by common analytical devices of piecewise lineari- 
zation or small signal analysis. In  the following 
paper we wish to discuss some models of muscle 
contraction in normal humans, and show how 
we have attempted to resolve or avoid these 
difficulties. 
Human skeletal muscle may be regarded as an 
electro-mechanical transducer, converting the 
electrical signals from the alpha motor-neurons 
into a mechanical force at its origin and insertion. 
For all practical purposes we cannot measure 
this neural input. Associated with muscle activa- 
tion is the electrical activity of the contracting 
muscle itself, which we may record as the electro- 
*This work was partially supported by NSF grant 
GK-17581. 
myogram (EMG). There is a one-to-one relation- 
ship between an action potential in a single motor 
nerve and the EMG of its muscle fibers. Using 
surface electrodes on the belly of the muscle, 
however, we get only a single scalar measure 
from the many motor units which fire in varying 
spacial and temporal patterns. 
Nevertheless, there is some theoretical justifi- 
cation for expecting the average rectified EMG 
(refs. 1 and 2) to be proportional to the level of 
innervation and we have used this signal (ref. 3) 
as our model’s input. 
The only other variables we can measure are 
muscle force and length. For simplicity we have 
fixed muscle length and measured only the iso- 
metric force (actually torque) generated by the 
gastrocnemius-soleus muscles (GSM) at the ankle 
joint. This avoids those nonlinearities in both 
the EMG measurement and in the mechanical 
parameters which are associatediwith changing 
muscle length. 
One of the intrinsic state variables of the muscle 
which we cannot measure is termed the “active 
state” (ref. 4). This rather loosely used term de- 
scribes two conceptually distinguishable changes, 
one the generation of an active force of contrac- 
tion and the other an increase in the passive 
115 
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https://ntrs.nasa.gov/search.jsp?R=19730001390 2020-03-17T07:20:21+00:00Z
116 SEVENTH CONFERENCE ON MANUAL CONTROL 
mechanical impedance of the muscle. Measure- 
ments of both these phenomena have been made 
in animals (ref. 5)  but only the impedance 
changes have been measured in man (refs. 6 
and 7). 
LINEAR MODEL 
At the Sixth Annual NASA-University con- 
ference we presented a linear model for the 
EMG-force relationship ((ref. 8), see also (ref. 9)). 
The topology of this model is shown in figure 1. 
For the linear model, B,  K I ,  and KB are constant, 
passive elements and activation of the muscle 
implies generation of an active force F,. 
The model transfer function is given by 
where 
t i= l ,  2. 
1 
Gi(S)  = ___ 
T,.X+l 
Separation of the transfer function like this 
allows us to tentatively designate G I @ )  as the 
active state component while G & S )  may be con- 
sidered the linearized mechanical component of 
the muscle model shown in figure 1. The input is 
recorded physiological EMG data. For our simu- 
lation the model parameters were chosen as 
Tl=0.105 sec, Tz=0.105 sec, and K=0.25 to 
0.35. Figure 2 shows muscle and model response 
to a rapid contraction. Figure 3 shows the impulse 
response of this linear model. The “impulse 
response” of the muscle was obtained by deliver- 
ing a single maximal shock to its motor nerve. 
FIGURE 1.-A mechanical model of skeletal muscle. The 
series and parallel elastic elements ( K z  and K 1 )  and 
the viscosity B are defined only by the mechanical 
behavior of the muscle and should not imply a mech- 
anism. The force generator F, represents the contractile 
“active state” of the muscle. 
i-mi 0.92, 
at 
-93.u 
4: . I .  .E . . . . . , ib M13R DPERIMENTA? mEm 
FIGURE 2.-Left: subject’s step torque response and 
EMG. Right: model response to subject’s EMG. In- 
stead of replotting EMG on the right, the output of the 
active state stage GI is the fourth curve plotted. 
0.L4t 
EXPERIMENTAL MOOR 
FIQURE 3.-Left: foot twitch in response to a single 
supramaximal shock to the tibial nerve. Right: model 
impulse response and the output of the active state 
stage GI. 
Figure 2 shows that the model is reasonably 
successful in mimicking rapid muscle contraction. 
The output of the first stage of the model is 
plotted as the active state in the lower part of 
the figure. However, it was shown in an earlier 
paper (ref. 8) that these time constants are too 
long to  reproduce a sudden relaxation. Reducing 
one time constant (Tz=O.O1) improves the rate 
of fall of the force (see fig. 8 of ref. 8). It is clear 
that a constant coefficient linear model is not 
adequate to reproduce both contraction and 
relaxation. 
RELATIONSHIP OF THE EMG AND MUSCLE FORCE 117 
The impulse response in figure 3 shows three - 0.50+ 
interesting deficiencies. The negative peak in the t I- 
; z  experimental acceleration curve at 300 msec is not due to  intrinsic muscle activity but is pro- 
duced by the stretch reflex loop causing reactiva- 
tion of the muscle as it relaxes and returns to its 
resting length. 
The second error is the apparent saturation of 
the velocity curve and the third is the double 
hump of the acceleration curve between 100 to 
250 msec. The active state variable does behave 
much like measurements in animal preparations 
indicate it should. It rises abruptly and falls 
slowly but does not remain on a plateau a t  its 
peak value. 
2 
5 
2 0.00- 
4.03.. 
NONLINEAR MODEL 
The nonlinear model incorporates two limiter 
type nonlinearities in order to improve the im- 
pulse response of the model and its performance 
with soleus EMG as the input signal. A block 
diagram of the analog model is shown in figure 4. 
The first limiter N I  is situated after the first 
stage to limit amplitude of the active state thus 
producing a plateau at its peak value. The second 
limiter Nz is situated in the second stage to limit 
the maximum rate of change of foot torque. 
Figure 5 shows the effect of the active state 
limiter and figure 6 shows the effect of the velocity 
limiter. 
TIME (SEC) 
FIQURE 5.-Effect of the active state limiter. 
2 0.50 
1;; 
3 - 
c 
DISCUSSION 0 
0.00 
3.79 The nonlinearities used in this model only show 
nonlinearities (which were used because they were 
actually needed are higher order linear stages 
that the simple first order dynamics of the two 
stages are easily improved upon. However, these 
easy to implement) are not suitable because we 
are still dealing with linear phenomena. What is 
c 
.Y. 
-1.20 
435.01 T 
ELECTROMYOGRAM ACTIVE STATE FOOT TORQUE 
TIME (SEC) 
FIQURE 4.-Analog diagram of the model showing 
the position of the nonlinear elements. FIGURE 6.-Effect of the velocity limiter. 
SEVENTH CONFERENCE ON MANUAL C O ~ T R O L  
0 
0 .2 .4 .6 .a 1.0 1.2 
PEAK TWITCH (FT-KG.M.) 
FIGURE 7.-A plot of the peak velocity of the isometric 
twitch contraction vs the peak twitch in the soleus a t  
different levels of stimulus voltage. 
which have a plateau in their response. That, a t  
least, will improve the impulse response but its 
impact on the response with EMG inputs is not 
clear. 
The impulse response of the physiological sys- 
tem is much more complex than what is shown in 
figure 3. In  a slightly different experiment the 
tibial nerve at the back of the knee in popliteal 
fossa is stimulated with different levels of voltage 
stimulus. The EMG at the soleus at the lower 
levels of stimulus shows the familiar H-wave. 
As the stimulus level is increased, the alpha 
motor fibers are directly stimulated producing 
the M-wave (ref. 9). A plot of the peak velocity 
(with plateau) versus the peak twitch in such 
electrically evoked twitches is shown in figure 7. 
The circles are for H-wave response and the 
crosses are for M-wave response. Note that this 
relationship is nearly linear. The peak velocity 
is a function of the response, i.e. the input level 
itself. Thus the major change that is required in 
this model is that the mechanical parameters 
must be made functions of the input. 
It seems clear, though, that a relatively simple 
model can be used to predict isometric muscle 
activity from the EMG. 
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
REFERENCES 
MOORE, A. D.: Synthesized EMG Waves and Their 
Implication. Amer. J. Phys. Med., vol. 46, 1967, pp. 
LIBKIND, M. S.: I1 Modelling of Interference Bio- 
electrical Activity. Biophysics, vol. 13, 1968, pp. 811- 
821. (English transl.) 
GOTTLIEB, G. L.; AND AGARWAL, G. C.: Filtering of 
Electromyographic Signals. Amer. J. Phys. Med., vol. 
HILL, A. V.: The Abrupt Transition From Rest to 
Activity in Muscle. Proc. Roy. SOC., London, Ser. B, 
BAHLER, A. S.; FALES, J. T.; AND ZIELER, K. L.: The 
Active State of Mammalian Skeletal Muscle. J. Gen. 
Physiol., vol. 50, 1967, pp. 2239-2253. 
WILKIE, D. R.: The Relation Between Force and 
Velocity in Human Muscle. J. Physiol., vol. 110, 1950, 
BERMAN, B. M.; AND STARK, L.: 
Studies in Postural Control System. I. Torque Dis- 
turbance Input. IEEE Trans. Systems Science and 
Cybernetics, vol. SSC-6, 1970, pp. 116-121. 
GOTTLIEB, G. L.; AND AGARWAL, 6. C.: A Model of 
Muscle Activity. Proceedings of the Sixth Annual 
Conference on Manual Control. Wright Patterson 
Air Force Base 1971, pp. 177-193. 
GOTTLIEB, G. L.; AND AGARWAL, G. C.: Dynamic 
Relationship Between Isometric Muscle Tension and 
the Electromyogram in Man. J. App. Physiol., vol. 30, 
1302-1316. 
49, 1970, pp. 142-146. 
V O ~ .  136, 1949, pp. 399-420. 
pp. 249-280. 
1971, pp. 345-351. 


Further observations on the relationship of EMG and muscle force

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