European Journal of

Eur. J. Appl. Physiol.40, 155-164 (1979)

Applied

Physiology

and Occupational Physiology 9 Springer Verlag 1979

Theoretical Analysis of Surface EMG in Voluntary Isometric Contraction H. Miyano and T. Sadoyama Industrial Products Research Institute, Agency of Industrial Scienceand Technology, 21-2, 4-Chome, Shimomaruko,Ota-ku, Tokyo, Japan Summary. A new stochastic model of the surface E M G is suggested and the spectral density of the surface E M G is studied theoretically and experimentally to confirm the validity of this model. Theoretical results show that while the contraction level is not so high, the shape of the spectral density (distribution) does not change and its amplitude is directly proportional to the motor unit firing frequency and recruitment. To illustrate the theoretical results, experiments were carried out for rectus femoris and biceps brachii. The surface E M G was lead off by bipolar surface electrodes. And the spectral density of the surface E M G was calculated using FFT algorithm. From these experimental results, it was confirmed that our theoretical results were almost valid.

Key words: Surface E M G - Stochastic model - Spectral density - Motor unit recruitment - Entropy

The surface electromyogram (EMG) is a summation of action potentials of a great number of motor units, so highly complex in nature. Therefore, frequency analysis of the surface E M G has been considered to be an effective tool for obtaining information about motor functions (Walton, 1952; Chaffin, 1969; Lloyd, 1971; Sato, 1976). But, if an appropriate model of the surface E M G can be obtained, the information about motor function latent in the surface E M G will be revealed easily using the model. The object of this work is to study the model of the surface EMG. Person constructed a model for graphical simulation of E M G pattern, which might be inappropriate for theoretical analysis of the surface E M G (Person and Offprint requests to." H. Miyano, M.D., Human Factors Eng. Division, Industrial Products Research Institute, 21-2, 4-Chome, Shimomaruko, Ota-ku, Zip. 146, Tokyo, Japan

0301--5548/79/0040/0155/$ 02.00

H. Miyano and T. Sadoyama

156

Libkind, 1970). Lindstr6m obtained a model which clarified the ratation between the spectral density of EMG and the action potential conduction velocity (Lindstr6m et aI., 1970). Also, Parker proposed a model, being useful to understand the influence of electrode configuration on observed signals (Parker and Scott, 1973). In this paper, a new stochastic model is suggested, which is not only appropriate for theoretical analysis of the surface EMG, but also includes the models of Lindstr6m and of Parker. The spectral density of the surface EMG was studied theoretically and experimentally to confirm the validity of our model.

Methods Theoretical M e t h o d s Autocorrelation function and spectral density of the surface E M G were obtained theoretically, using a stochastic model which was constructed under the following assumptions: A - i. The number of firings of a motor unit during a given interval has a Poisson distribution. And all the motor units act asynchronously, i.e., the probability P(k, r) of the number k of firings during a given interval r = [ - T, 7] is given by

(1)

P(k, r) = [(alr) ~ exp ( - air)l/k !

where ai is the firing frequency of the unit i, i = 1, 2 , . . . , N. Moreover, for atl i, a~ is nearly equal to a;a~-a. The last part of this assumption may not impose any restrictions on our model (see Appendix). A - 2 . The extracellular action potential el(x, t) of motor unit i, i = 1, 2, ..., N at one dimensional spatial point x and time t e l - 7, 7] is characterized by k

~,(x, 0 =~-~1e(t - t~- I x - x~l/v)d(Ix - x,[)

x el0, L],

(2)

where v is the conduction velocity, xi is motor point of the motor unit i and L is the muscle length. And e(t) is the action potential produced by a motor unit firing at t = 0, d(z) is a monotone decreasing function and ti is the firing time during the interval r. This assumption means that all the action potentials have the same shape. The validity of the assumption may be assured by experimental results obtained by Samejima (1976). Now, the assumption on the output of surface electrodes is introduced, i.e., a more generalized one as the usual assumptiom The output of surface electrodes is arrived at by summing up the action potentials es t), i = 1, 2 , . . . , N o r active motor units. A - 3 . The output G(t) of a unipolar surface electrode can be described by

e,~(t) = I~W(x)E(x, Odx,

(3)

where W ( x ) = (Wl(X), w2(x) . . . . . wN(x)), E(x, t) = (el(x, t), e2(x, t) . . . . . em(X , t))', D ~ (0, min (xi)), wi(x) is the weighting function which represents the influence of ei(x , t) to the output e,(t) and the accented matrix ( . ) ' is the transposed matrix of matrix (.). Also, the output eb(t) of bipolar surface electrodes can be characterized by eb(t) = f D W ( x ) [ E ( x , t) - E(x; O l d x ,

(4)

x' = - x + r~ + 2ro ,

where r0 is the center point of a surface electrode and rc is the distance between centers of surface electrodes (Fig. 1).

Analysis of Surface E M G

~

15 7

Surface Electrodes

eb(t)

+

I

-

Skin

1

wi(x)

I

wi(x)

wi(x) lwi(x)

/ ,.~--~-0

!

)~

r0

xi Motor Point

Mosclo -

ro+r c

Region D Motor Unit i

Fig. 1. Schematic diagram of surface E M G detection system

More precisely, for example the output e.(t) m a y be described by

e,(t) = I ~ d Ofgrdr fvW(r, O, x ) e ( x , t)ax/(RZn) ,

(5)

where R is the effective radius of a surface electrode. However, since the same equation as (3) is obtained by exercising integrations on r and 0, so this assumption is used. Representing the equation (4) in s domain (Laplace domain), the following equation can be obtained:

eb(s) = foVCo(x, s)F4x, s)dx ,

(6)

where Wo(x, s) = ((1 - exp ( - his/v) )wi(x) ), i = 1, 2 . . . . . N and hi - - 2xi + 2r0 + re. This equation is used in the next section.

Experimental Methods The surface myoelectric activities of the biceps brachii and the rectus femoris were lead off by bipolar surface electrodes. In all tests, these electrodes were stuck to the skin with the motor point of the muscle between, and the distance between the electrodes was about 30 ram. The experimental situations were the following: (1) Rectus femoris. The subject was fixed on the seat with sitting straight and instructed to extend a wire connected with the load cell in the forward direction. (2) Biceps brachii. The upper arm of the subject was fixed on the horizontal stand. T h e elbow was bended vertically at right angle. The subject was instructed to pull a wire connected with the load cell. The force signal was recorded on magnetic tape, and monitored on an oscilloscope. T h e level of force to be maintained were displayed on the oscilloscope. T h e tests consisted of maintaining a force indicated by the experimenter for 15 s. The subject had to exert a constant force so as to keep the line of the oscilloscope on the reference line displayed on the screen. Different levels of force ranging from 0 kg to the m a x i m u m force of the subject were maintained. The subjects were three normal volunteers, ranging in age from 2 8 - 3 4 years. They were tested three times each.

H. Miyano and T. Sadoyama

158

Results

Theoretical Results Person and Libkind assumed that a stochastic average of the action potential is zero, and all the action potentials have the same shape (1970). Admitting this assumption, the following results on autocorrelation function and spectral density of the surface E M G were obtained theoretically. R - 1. Let denote the stochastic averages of e,(t) and eb(t) by E[e,] and E[eb], respectively. Then,

E[e~] =

EIeb] =

(7)

0.

This is obvious from the above assumption. R - 2 . Let denote the autocorrelation functions of e~(t) and eb(t) by Ru(r) and Rb(r), respectively. Then,

Ru(r) = alp fz)WI(x)W1 (y)Re(v + (y - x ) / v ) d x d y , W l ( X ) =- ( w i ( x ) d ( [ x

-

xil))

i = l,

2 ..... N,

N

R6(r) = aIDfD XlWi(X)W,(y)d(Ix --- R e ( T + ( y -- X -- h i ) / v ) -

x,l)d(ly

(8) (9)

- x,l)[2Re(r + (y - x)/v)

Re(7: + (y - x + hi)/v)]dxdy,

(10)

where Re(p) = E[e(t)e(t + p)]. For the proof of this result, see Appendix. The spectral densities S~(o)) and Sb((o) of the outputs e~ and eb can be obtained by taking the Fourier transform of the autocorrelation functions R,(r) and Rb('c), respectively. Thus, the following result was obtained. R - 3 . Spectral densities S~(o)) and Sb(o)) are given by S.(oJ) = a ] Wffo)/v)12Se(O~),

(11)

Sb(~) = a I W2(o~/v)12Se(O~) ,

(12)

where W1(jo~/v) = fD W1 (x) exp ( - j~ox/v)dx, IWl( /v)12 = W, (l'~O/v) W ; ( - jo~/v), W2(joJ/v) = (fDwi(x)d(lx - xi l)[1 - exp ( - joJh/v)] exp (-jrox/v)dx), i = 1,2,..., N, ]W2(co/v) 12= W:(joJ/v)W~(- joJ/v), j2 = _ 1 and Se(a~) = f~ Re(r) exp ( - jolt)dr. If the assumption that wi(x) = w(x), xi = Xo for any i is ad~nitted, then (11) and (12) reduce to the more explicit forms;

Su(oo) = a N I W(o~/v)12Se(oO , Sb(a)) = aN(2 sin

(o)h/2v))21W(o)/v)12Se(o)) ,

where W(d'a)/v) = fDw(x)d(Ix - x o [ ) exp ( - j~ox/v)dx, ] W(co/v)12 = W ( j c o / v ) W ( - joe~v) and h = - 2x0 + 2re + re

(13) (14)

Analysis of Surface E M G

!59

Su{~)~~)

m

i

0

50

lO0 Frequency [Hz]

150

200

Fig. 2. The normalized spectral density of the surface E M G (contraction level ~ 30%). Calculated results (c = 10 ms, h/v = 5 ms), experimental results (biceps brachii, subject M)

According to Samejima's experimental result (1976), S,(w) may be approximately given by

Su(w) ~ a N f~ f~e~,(t)e.(t') exp ( - jw(t - t'))dtdt' ,

(15)

where

e,(t) = b sin (sr/c)t = 0

0