SUPERPOSITION OF MOTOR PROGRAMS-I, RHYTHMIC FORWARD MUV~MENTS IN MAN

A.G. FBLDMAN Institute of Problems of Infarmation Transmission, Academy of Sciences,

Ermaiova L9, 103oSI Moscow, U.SS.R.

Abstract-_lhe so-called invariant ~b~racteristi~s of the elbow muscles are described, An invariant characteristic is a curve on a plot of elbow static. moment vs angle. An invar~a~t ~haracterist~c shows the elbow displacements in response to different amplitudes of toad change when the subjecr is told not to compensate, or not to intervene voluntary to correct the dcfkction of the arm. If an ~n~~~~a~on of the invariant characteristic is known, the degree of central tonic coactivation of antagonistic motoneurones can be estimated. The idea was utilized when analysing rhythmic movements (2-6 Hz) of the forearm that were carried out against a spring resistance (2-10Nm). When the spring was suddenly switched off, the d&e&on of forearm rose. The subject was told not to correct the deflection and not to interrupt the swingings. The mean changes of both muscle torque and joint angle were determined in the experiments so that it was possible to estimate the inclination of the invariant characteristic used at a given fre- quency. Judging by the results, central tonic coactivation of elbow flexors and extensors occurs and increases to maximum as the frequency of movement increases. In addition, the reciprocai activation of antagonistic mu&es is we3 known to occur upon execution of this movement.

ft can be concluded that both the reciprocal and the unjdire~t~o~al central c~~rna~ds to ~tagonjs~i~ motoneuroues occur simultaneously. The fun~tionaS significance~ as well as a scheme of supe~~sition, of these motor commands are discussed. ft is suggested that the commands are universal. i.e. the nervous system operates by using them during the performance of any movement.

TW~E ACTIVITY of antagonistic muscles changes reci- procaify, especially during rhythmic movements. This is known as the law of the reciprocal inhibition dis- covered by SHERRINGTON (1906). An overlapping of e~~~omyo~arns (E?vK&f recorded from the muscles is not a serious objection to the law provided that the contrary change of the EMGs rakes place. But a uni- directional change of activity of the muscles, in par- ticular, their tonic coactivation is sometimes observed, for instance, during jumps in the cat (ZOM- LEWR, X,43& & LEVINE, 1977) as well as during volun- tary efforts in human subjects (AIZERMAN, ANDREEVA, KANDJX & TENNENBAUM, 1974). For the stability of posture and movement to be guaranteed this coacri- vation must often occur (FELDMAN, 1979).

SIGIIRKNGTON (1906) supposed the reciprocal and unid~r~t~ona~ commands that are given to antagonis- tic muscles would exclude eaeb other. This point of view has not been ~hai~eng~d until recently* But it is possible that the nervous system can combine the commands in a proper manner depending upon mntar task. The present study was undertakes to check the hypothesis for rhythmic movements in an elbow joint.

Why does EMG analysis of rhythmic movements make it possible to demonstrate so easily the recipro- ca! activation of antagonistic muscles but reveal very

&de of their co~~~iv~~~~~~ The reason seems to be that the component of coactivation is a tonic one in this case and is easily masked by the rhythmic reciprocal component. A scheme of the masking is represented in Fig. 1. Let us assume that the intensity of tfie central influence on a moto~euro~ pool of a given muscle changes rhythmically from +m to -m (A). The iaff uence with intensity f m is determined as an influence that can recruit m motoneurones. The influence with intensity -m is determined as an influence that can inhibit m working motoneuranes. In addition, the central inflow is capable of switching on tonically n motor units of the muscle (Bj. Consider a case n < m. To obtain a number of the actually recruited motor units, an algebraic sum of these influences must be calculated. If the sum is negative at some intervals of time, one must befieve the number to be zero at that time. As a res,&, a picture of the ‘EMG’ arises in which the tonic component is com- pletely absent (C). Not knowing the genesis of the picture it is impossible to say whether there is a su~r~sit~on of two central programs or not. This example shows that EMG is not a reliable basis to judge the composition of motor program.

A method which makes it possible to reveal the coactivating component in the eentrai program of the rhythmic movements ia an elbow joint is based essen- tially on experimental determination of the so-called invariant characteristics (KS) of muscles. Operating by these ~h~a~ter~stj~~ the nervous system brings about shifts of the e~u~~~br~urn point of the muscie-

X2 A. G. FEl.I)MAI\

A

m

0 w t

-m

B

n

O-t

m+pLcLL FIG. I. Central tonic influences on the motoneuronal pool are masked in the electromyogram by the rhythmic ones

(scheme). (A) The intensity of central influence on the

motoneuronal pool changes rhythmically from +m to

--m. The influence with intensity +m is defined as an

influence that can recruit m motoneurones. The influence

with intensity -m is defined as an influence that can in-

hibit m working motoneurones. (B) n Motoneurones are

recruited by the tonic excitatory inflow. (C) The number of

active motoneurones when both types of inflows are com-

bined. A case n < m is shown. Abscissa: time (t).

load system and, as a consequence of it, the move-

ment arises (FELDMAN 1966a,h, 1974; 1979; BIZZI, Po~rr & MORASSCI. 1976). Therefore, the experimental study of ICs utilized during the performance of a movement seems to be an essential stage in the analy-

sis of its construction. To summarize briefly the conception of ICs, note

that there are many controllable parameters of motoneuronal pool activity; for instance, a number of

active motor units, the threshold and the gradient of their recruitment, the frequency of firing and so on. But it seems as if only one can be set by the nervous system independently from the others, as well as from muscle length and tension. This parameter is the threshold, i.e. the muscle length at which the tonic firing of motor units arises (MATTHEWS, 1959a,b; ASA- TRYAN & FELDMAN, 1965; FELDMAN, 1966a; FELDMAN

& ORLOVSKY, 1972; HOUK, 1976; NICHOLS & HOUK, 1976). If the threshold is fixed, the tonic motoneur- onal activity, i.e. a number of the motor units re- cruited as well as the muscle force is a single-valued and increasing function of the length. This function is an IC. With some reservation, ICs are similar to the tension-extension curves characterizing the tonic stretch reflex in the decerebrate cat (MATTHEWS 1959u,b; FELDMAN & ORLOVSKY, 1972). The indepen- dently controllable parameter indicated above is iden- tical with the threshold of the reflex.

Methods of experimental study of KS in normal subjects have been described elsewhere (ASATRYAN &

FELDMAN. 1965; FELDMAN. 196611; 1979) and arc hum-

marized briefly below.

EXPERIMENTAL PROCEDURES

Experimental demonstration of the ICs and analysis of

rhythmic movements in the elbow joint was carried out

using an apparatus shown in Fig. 2. The subject put his

right forearm on the upper plate of the apparatus, clasping

his hand round a vertical handle. The apparatus permitted

flexion and extension of the forearm about the elbow in the

horizontal plane. The elbow angle was measured with a

variable resistance (D). A long steel lever (C) was fastened

to a ball-bearing situated on the axis of rotation (0) of the

platform. The lever was also attached to a rigid steel ring

(A) connected in a diametrical point to the lower plate of

the platform. The ring was the sensitive element of a strain

gauge.

Depending on the type of experiment (demonstration of

the ICs or analysis of rhythmic movements), weights or

springs, respectively. were roped on to electromagnetic

locks (F) fixed to the lever C. The strain gauge was sensi-

tive to the horizontal component of the load only because

the vertical component of the pressure exerted on the lever

C was compensated by a roller (B). For the registration of the ICs it is essential only that, in

statics, the readings of the strain gauge are proportional to the load torque (P) which is equal to muscle torque (M) in

that case. In general, the readings are proportional to

deformation of the ring A. Let Q and R be the torques

applied to the ring in its diametrical points. The torques

arc created by the lever C and the rest of the platform. We consider Q = R because the inertia of the ring is negligible

as compared with the inertia of the elements acting on it.

Hence, the deformation of the ring is proportional to Q = R. The equations of motion of the lever and of the

whole platform are of the form:

Ic$ = P - Q. I;p; = P - M II)

I---’ FIG. 2. The apparatus for the mechanographic analysis of

human forearm movements. A, sensitive element of strain gauge; B, roller compensating a vertical component of

pressure exerted on the lever C to which load is applied; D. goniometer; E. range of the goniometer; F, electromagnetic

locks to fasten a spring or another load to lever C: 0. axis of rotation of the platform.

Control of rhythmic movements 83

5

E z 0

-5

FIG. 3. Two families of the invariant characteristics of the elbow muscles. Abscissa: elbow angle; ordinate: static muscle torque being positive for flexors and negative for extensors. The registration of each invariant characteristic began from the solid circle. (A) Family of KS of the flexors (above dashed line) and of the extensors (below); dashed line: passive muscle torque vs angle. (B) KS recorded against

the background of maximal tonic coactivation of flexor and extensor muscles.

where ;d is the angle acceleration; I, 1, are the inertia of the whole platform (I = 0.11 kg. m2) and of its lever (I, = 0.03 kg. m2). For comparison, the inertia of the fore- arm is equal to about 0.05 kg.m2. Muscle torque M is considered to be positive if it tends to carry out flexion. Positive direction for a load of torque P is opposite to that for M. From the equation (I), the readings of the strain gauge are proportional to

Q = M + (I - I,)$ = P - I,$. (2)

This formula confirms that muscle torque is displayed on the meter, if 4 = 0. When the load torque P is equal to zero, the meter shows an angle acceleration ;f;. Further- more, if the oscillations of the forearm are close to har- monic ones, the mean strain gauge readings measured for a period are proportional to the mean muscle torque as well as to load torque because the mean value of 0 is equal to zero in that case [see (2)].

The strain gauge was calibrated in terms of moment of forces (Nm) by load application to the lever C while the platform was motionless.

Inoariant characteristics of forearm muscles

In order to describe the ICs of the elbow muscles in their totality, such variables as the muscle moment (M) and the elbow angle (4) will be used.

The shape of the ICs depends essentially on the number of the active muscles. Figure 3(A) shows one family of ICs recorded by the unloading method in man (ASATRYAN & FELDMAN, 1965; FELDMAN, 1979). A dashed line in Fig. 3(A) is the M - 4 characteristic of the passive muscles. The ICs of elbow flexors are situated above the line while those of extensors are below. When an IC of the flexors was recorded, the extensors, judging by the EMG, were not active and vice versa. Each of the ICs shows the elbow displacements in response to different amplitudes of load change when the subject was told not to intervene vohm- tarily to correct the deflection of the arm. For instance, to

determine the upper left-hand IC, a monotonous series of experiments was carried out. In each trial, the elbow flexors were loaded and the subject set a predetermined elbow angle. A solid circle of the IC shows the respective initial combination of static M and 6. After a sudden de- crease of the load the forearm transferred to another equi- librium position, if the instruction indicated above (‘not to correct’) was used. As a result, a new combination of static M and @J was taken up. It gave another point of the IC. The magnitude of the change in load was varied from trial to trial so that it was possible to determine several points of the IC (open circle).

To find out another IC of the family the subject was told to set another initial elbow angle and an analogous series of unloading trials was repeated.

It has been shown that the IC preserves its shape despite rather wide time-space variations in the load change pro- cedure (ASATRYAN & FELDMAN, 1965; FELDMAN, 1979). Therefore, the term invariant characteristic is well founded.

Note that ICs of the family in Fig. 3(A) branch off from different points characteristic of relaxed muscles (dashed line). This means that the transition from one IC of the flexors, for instance, to another is performed by means of the change of the threshold angle at which the motor units begin to recruit. For electromyographic evidence of this point see elsewhere (FELDMAN, 1979). Let us donote the thresholds by &, where k = 1 for the flexors and k = 2 for the extensors. It is worth noting that the 1, and /?* are independently controllable parameters.

Figure 3(B) shows another family of ICs each of which was recorded by the load change method when the subject carried out maximal tonic coactivation of all elbow muscles. The slope of the ICs of the B family is essentially greater than that of the A family. Hence, the stiffness of the muscle-load system increases under the cocontraction.

Using the data on the independently controllable par- ameters /Ii and flZ, let us describe schematically the conver- sion of the A family into the B family. It should be noted

x4

FIG. 4. The change m the number of the active elbow muscles (A and B) as well as of the shape of

invariant characteristics (C) due to regulation of difference between tonic stretch reflex thresholds of the

flexors (/I,) and extensors (pz). Abscissa: elbow angle. Ordinates: active muscle torque (C), index of

activation (A and B) of the flexors (up) and of the extensors (down); this index equal 1 or 0 depending on

whether the respective group of muscles has motor units recruited or not. For further explanation see

text

that the elbow flexors are active, if 4 > pi, while the exten-

sors are active if 4 < /&. Let the index of activation of the

flexors (extensors) equal 1 or 0 depending on whether the

condition of activation of the flexors (extensors) is fulfilled

or not. These indexes are shown in Fig. 4(A) and (B). It can

be seen that if

sz < Pi (3)

the areas of activity of the flexors (F) and of the extensors

(E) do not overlap (Fig. 4A), that is, inequality (3) is a

necessary and sufficient condition for the absence of their

tonic coactivation. If

111 > Pi (4)

the F and E areas overlap (Fig. 4B). The more the differ-

ence between f12 and bi. the greater the area of the overlap-

ping. Thus, the inequality (4) is a necessary condition of the

coactivation.

One can assume that the IC of the A family is converted

into the IC of the B family as the difference between /I2 and

pi increases. The conversion is illustrated in Fig. 4(C).

Three ICs of the flexors (2-4) with pi = 91 , 77. and 54’ as

well as one IC of the extensors (1) with /Iz = 159’ were

taken from the A family of Fig. 3. The passive muscle

torque was subtracted from the ICs. The dashed lines

denoted by 1 + 2. 1 + 3 and 1 + 4 are algebraic sums of the respective ICs. The slope of the total ICs increases with

increasing B2 - pi. Note that the /I2 - pi = 105” for the

curve indicated by 1 + 4 but it is not a limit. To reach a

limiting value of the slope, which is equal to that of ICs of

the B family, fi, - [<i must be greater than the physiologi-

cal range of angle change ( - 150’ ). Thus, to explain the transformation of the A family into

the B family, there is no need to suppose that the nervous

system independently controls, apart from, or instead of, 8,

and fi2, any other parameters of motoneuronal pool ac- tivity.

The reciprocal activation of flexor and extensor

motoneurones is well known to occur during rhythmical movements. The question is whether there is some joint

central activation of the groups of motoneurones. To

answer this question, the slope of the IC used must be

estimated experimentally. Studies were performed on five

normal subjects aged from 25 to 30yr. Similar qualitative

data were obtained for all of them. In each trial, the subject

set the elbow angle (100-105’) while overcoming the spring

resistance to flexion (6-7 Nm). Then he was carrying out

the rhythmic movements in the elbow joint at a some

stable frequency (1-7 Hz). When the spring was suddenly

switched off, the deflection of the forearm rose. The subject

was told not to correct the deflection as well as not to

interrupt the oscillations. It is worth mentioning that the

instruction ‘not to correct’ is a fine point of the method of

determination of the ICs. The experiments were repeated

three to six times for one and the same frequency of move-

ments. Readings of the goniometer and the strain gauge

were displayed both orthogonally and continuously

(Fig. 5). The subject could not see the display.

RESULTS

Figure 5(A-C) shows orthogonal recordings of the goniometer and strain gauge readings during 2-5 cycles of movements at a low (2.3), moderate (4.1) and maximal (6.2 Hz) frequencies, It should be noted that the maximum (6.2 Hz) is not an absolute one (7-8 Hz, e.g. Fig. 7), as the inertia of the movable part of the apparatus sets the upper limits of the frequency. It means that the ‘central generator’ is not quite autono- mous from the afferent inflow. The trajectory of movement displayed in the coordinates M and 4 is a

Control of rhythmic movements

A B C D Nm 120”

2.3 4.1 6.2

80”

85

IO

F E

5” ? F

.

0 1 ::‘* d d a0 loo’ 00 IOC?

FIG. 5. Effects of unloading during forearm rhythmic movements (A-D) and after their cessation (E and F). (A-C, E and F) Orthogonal recordings of readings of strain gauge (ordinate, Nm) and of goniometer (abscissa, degrees); the numbers show the frequency of oscillations (Hz). (D) Continuous recordings, of muscle moment M and of angle 4. Unloading in (E) and (F) was performed immediately after cessation

of movements with frequency of 2.3 and 6.2 Hz, respectively.

straightforward segment [Fig. 5(A-C)]. After unload- ing of the forearm, it is shifted below and left as both the mean muscle torque and joint angle decrease.

Figure 5(D) shows a continuous recording of M

and 4 at a frequency of 6.2 Hz. Transition to a new position of the forearm is accomplished for 200-300 ms. Rhythmic movement lasts without visible

change of both phase and frequency (P < 0.05). It

shows independence of the generator process from afferent influences created by the unloading.

Under zero load the strain gauge readings are pro-

portional to acceleration 4 [see (2) for P = 01. The fact that the trajectory of movement displayed in the orthogonal coordinates is a straightforward segment [Fig. 5(A-C)] means that 4 is proportional to I#J. This is a characteristic property of the harmonic oscil- lations. In this case the mean values of M and r$ coincide with the coordinates of the middle of the segment. Using this property let us estimate the in-

clination of the ICs used. The mean deflection of the forearm (A4) caused by

the unloading decreases with increasing frequency and is equal to 23.5, 11.6 and 8.3” respectively for 2.3, 4.1 and 6.2 Hz, while the mean torque being changed (AM) by 5.8, 6.3, 6.7 Nm. AM should be divided by AI$ to measure the slope of the ICs used. Obviously, it increases with increasing frequency (14.2 f 0.7, 31.2 k 5.1, 46.5 k 9.3Nm/radian). This is confirmed also by the fact that the slope of the trajectory of movement [Fig. S(A-C)] behaves in a similar manner.

It is of interest to compare the mean forearm de- flections evoked by the unloading during the rhyth- mic movements and the corresponding stable posture. To do it, the subject was asked to stop the rhythmic movement, preserving the mean position of the fore- arm but not relaxing the muscles. Soon the spring was switched off. The same instruction ‘not to correct the

deflection of forearm’ was given to the subject. Figure 5(E and F) shows the effect of forearm unloading after

stopping the oscillations with frequencies of 2.3 Hz (E) and 6.2 Hz (F). The deflections of the forearm are equal to 24.2” and 5.7”. The inclinations (AMlAb) of the ICs are equal to 13.5 + 0.6 and 57.7 + 9.5 Nm/ radian, being close to those observed during move- ments with a corresponding frequency (P < 0.05).

This comparison shows that in spite of dynamic conditions during rhythmic movements the slope of

the static, i.e. invariant characteristic, can be esti-

mated rather exactly by the unloading method. The data obtained allow one to answer the ques-

tion whether the central tonic coactivation of anta- gonistic motoneurones occurs during rhythmic move- ments or not. To verify this, consider again the A family of ICs for the same subject (Fig. 3A). Let us determine for any IC the decrease of 4 when the muscle torque is changed from 67 Nm to 0. The de- crease proves to be greater than 40” while the greatest decrease in 4 under the same load change during rhythmic movements was equal to 23.5’. Hence it fol- lows that during rhythmic movements (2-6 Hz) those ICs were used the slope of which was dictated by the central tonic coactivation of the antagonistic muscles.

The data described above also show that the coacti- vation increases with increasing frequency up to maximum because the slope (46.6Nm/radian) of the IC used at 6.2 Hz approaches the limiting value (4&102Nm/radian) defined on the basis of the B family in Fig. 3(B).

DISCUSSION

The present data confirm the hypothesis that the central tonic coactivation of fiexor and extensor motoneurones is performed during rhythmic forearm

movement together with their rhythmic reciprocal ac- tivation. Thus, the superposition of two motor pro- grams takes place during such movements.

Rapid active change of forearm posture in man has been shown to be performed by means of transfer of an IC (FELDMAN. 1966a,b). The inclination of the IC was noticeably greater than that of the ICs of the A

family in Fig. 3(A). Hence it follows that the nervous system does employ the coactivation in that case as well. The reciprocal activation of the muscles is welt known to take place during such movements (WAC.H- ~~OLDEK & ALTENRERGER, 1926; WAC.HHOLDER, 1928;

HUFSC‘HMIDT & HI_Y:S~HMIDT. 1954: TANAKA, 1974:

Kors, 1975). Consequently, the superposition of two central programs also takes place during rapid change

of posture. Thus, the data described above do not support

Sherrington’s view that the processes of the reciprocal and the joint activation of antagonistic muscles are

alternatives (SHERRINGTON, 1906). These processes seem to be universal+ that is, the nervous system oper-

ates by them in a proper manner for any movement. Let us consider a scheme of superposition of these

moior commands during rhythmical movements. Figure 6 shows a plane of the controllable parameters

/I, (abscissa) and /12 (ordinate) normalized so that their minimal and maximal values are equal to 0 and 1. respectively. The normalized values are denoted by p, and &. A formula of the normalization is as foi-

lows;

jjrk = (P, - Bk )/(/I: -- B,, ). k = 152 (5)

where & . /?; are minimal and maximal values of Pk. It is worth mentioning that /II and pZ are the elbow angles at which tonic recruitment of motor units of flexors and extensors begins, respectively.

The multitude of the available meanings of PI and flZ is a square (Fig. 6). A condition of the coactivation is expressed by the inequality (4). For simplicity, let us assume that the minimal values of /I, and & are co- incident (fiIm = b;) and so are the maximal ones (/ji = 8;). Then the condition for coactivation in terms of normalized variables preserves its form:

$2 > PI. (6)

V I 0

7% I

FIG. 6. Superposition of the reciprocal and joint central

commands on the antagonistic muscles during rhythmical movements (scheme). For explanations see text.

This area of values is \ltuated abobe a diagonal pa~b- ing through the point 0 of the square in Fig. 6. ‘~-he more the point (p,. pZ) rises above the diagonal. the stronger is the inequality (6) and. cotlsequcnt~~, the coactivation. Let us suppose now that the point

moves along the diagonal or in parallel, Then /i, and /jL increase or decrease simultaneously. In this c;~sc. the reciprocal change of activity of antagonistic

muscles happens because, if /I, and [iZ, for instance.

increase, the condition of activ~itiol~ of the elbow

flexors (4 > fi,) becomes weaker or is violated and

that of the extensors (4 < /j,) becomes stronger. Figuratively. selecting a line parallel to the dia-

gonal. the nervous system ensures the muscle stiffness required while the movement itself is performuct by

displacement of the point along the line. According to the data obtained. we can believe that as frequency

increases the higher is the line which is selected. This is symbolized by the numbers (Hz) near the linca in

the diagram (Fig. 6). But one should make this con- clusion more precise. From our own experience we know that it is possible to produce rhythmic move-

ments with low frequency both during high and low coactivation of muscles. But it is impossible to pro- duce rhythmic movements with maximal frequency during low coactivation. To take these facts into account, one should consider that. in order to pro- duce the movement with a frequency, for instnncc, of 2 Hz. the nervous system can select either the line marked by 2 in Fig. 6 or an) line situated above

and parallel to it. So. for a given frequency of the movement, the parametric operational area IS a triangle limited by the left and the upper sides of the square and a line marked by the respective number in Fig. 6.

ft is worth empbasizing that displacemenr of the

point (B,. /3,) caused by one of the central programs turns out to be perpendicular to that caused by another. So. the reciprocal and joint commands seem to be, in a mathematical sense, the orthogonal basis for the construction of any motor program. Hence, any combination of I{, and p2 can be provided by the proper combination of the central commands. In par- ticular, one of the parameters can be changed inde-

pendently of another in this way. So, it is possible to control activity of the muscles individually (for in-

stance, to change activity of the flexors, preserving activity of the extensors) by means of superposition of the central commands, each of which evident]! con- trols the groups non-individually. A proof of the proper mathematical theorem is given in Theoretical Appendix.

The superposition of the motor programs seems to be very expedient for different reasons. It should be mentioned that the natural frequency of the muscle load system is dependent on static stiffness of the muscles, i.e. on the slope of the IC (FEI.DMAN. 1979). Therefore, in performing the tonic coactivation. the nervous system seems to bring the natural frequency into accord with that predetermined by the centrat

Control of rhythmic movements 87

I,.3 2.0

75

4.6 ) 0.55 1

5.6

FIG. 7. Decrease of maximal amphtude with increasing frequency of oscillation. Marks: surface EMGs of triceps and biceps brachii muscles (tric. bit), elbow angle (I#$, frequencies of movements (numbers above oscillograms, Hz). On the right: a pIot of maximal amplitude (ordinate) attainable under a given

frequency (abscissa).

generator which accomplishes reciprocal activation of the antagonistic muscles. The fact that the dope of the ICs changed with the rhythm of movements is evidence for the control of the natural frequency (Fig. S[A-Cl).

If rhythmic movement is a harmonic (sinusoidal) one, as took place in this study, the muscle torque is known to be proportional to the amplitude of dis- placement as we11 as to the square of frequency. Besides, the increase of the force of the muscles occurs in the phase of their extension [Fig. .5(D)]. So, when the frequency is near to maximum the amplitude of movement must be limited to prevent extreme muscle extension that can cause trauma (RACK & WESTBURY,

1969). The limitation of the amplitude movement is automaticaIly provided by su~rposition of the two motor commands. Actually, Fig. 6 shows that the operational areas in which the point @,, &) can be

shifted by the reciprocal program narrows with in- creasing frequency. This means that maximal ampli- tude of the movement itself must decrease with in- creasing frequency. The reader can learn this from his own experience. This fact is documented in Fig. 7. The subject was performing (out of the apparatus) the rhythmic movements of the forearm trying to reach a maximum amplitude at a frequency which gradually increased up to a limiting value (7.1 Hz). It can be seen in Fig. 7 that the amplitude does decrease (see also A~ZERMAN er al., 1974). Thus, the effect can be easily accounted for by the superposition of motor commands.

As it is seen from Fig. 7, the amplitudes of burst activity of the flexor and extensor increased with in- creasing frequency, This fact is considered to be an electromyographic sign of the increasing tonic coacti- vation of antagonistic muscles.

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(Accepted ?I Muv 1979)

THEORETICAL APPENDIX

A THI:OREM has been mentioned that the nervous system,

by combining the reciprocal and joint activation programs.

can determine the different (if any) combinations of activi-

ties of synergistic and antagonistic muscles controlling a

mechanical degree of freedom and can, in particular, alter

the activity of one muscle group independently of another.

While proving this. we also show that these programs are

not absolutely independent, i.e. changes in one of them can

restrict the variation in the other.

The proof of this theorem is based on the supposition

that the motoneuronal pool of the given muscle group is

controlled monoparametrically by determining the static

threshold [jI, of the stretch reflex (k = I for flexors and

li = 2 for extensors; subscript k is omitted. if the action of

one group only. for clearness. of flexors is considered). It is

worth noting that the alternative hypothesis (that several

independent parameters control the muscle group) does

not reject the theorem completely but somewhat narrows

Its limits. For instance. if the static threshold of each of the

synergistic muscles turns out to be controlled indepen-

dentI>. the theorem must be referred to any antagonistic

muscle pair (flexor ~extensor). Besides. the dynamic sensi-

tivity of the stretch reflex determined by the activity of the

dynamic y-motoneurones could be an additional parameter

controlling the r-motoneuronal pool of the given muscle.

In the case, regulation of the static thresholds allows one to

determine any type of static activity of the pair, but not all

types of its dynamic activity.

To prove the theorem, it is necessary, first of all, to

explain why regulation of the parameter /I allows one effec-

tively to control the activity of one muscle and. ultimately,

its length and tension.

Figure .X(A) shows the determination of the so-called

equilibrium point of the tricomponent system (motoneuro-

nal pool&muscle-load). The solid line beginning from the

threshold point /I is a characteristic of the tonic stretch

reflex. The dashed line is a characteristic of the load. which

need not be external but can be set by the antagonistic

muscles. The point a where the lines intersect is an cquilib-

rium point of the system. Coordinates of this point are

equilibrium meanings of the joint angle and the torque. It

is worth noting that a certain number of motoneurones

recruited corresponds to each point of the solid line so

that. on principle, this number can be found out for the

equilibrium point as well.

Evidently, in order to move the load to another position,

it is necessary to shift the equilibrium point. The change of

the static threshold /I is the most effective method of hring-

ing about the shift: this is shown in Fig. 8(B). There are

considered to be three regimes of muscle work. In the iso-

metric regime, the load characteristic is a vertical line I. In

the Isotonic regime, the load characteristic is a horizontal

P

‘t

(A)

-AL -- --_-O

(B) 9 43

I M 3 I’

P3 --- d I

2 C \O

5--z- ->- B 8' ,'

FIG. 8. Equilibrium ponts of the tricomponent system (load-muscle-motoneurones). (A) Definition of

equilibrium points (EP) (a); solid line: characteristic of the tonic stretch reflex (static muscle torque

versus joint angle); p: its threshold: dashed line: some load characteristic; I’. 6: equilibrium torque and

angle, respectively. (B) Change of the threshold of the reflex (,f?’ -+ /12) as an effective method of the shift

of EP: a: initial EP: b-d: final EPs during isometric (1). isotonic (2) and intermediate (3) regimes of the

muscle contraction: l-3: load characteristics of these regimes; P,, & (k = l-3): equilibrium torques and angles for respective EPs.

Control of rhyth imic movements

line 2. In the remaining cases, the load characteristic is some line 3 not coinciding with lines I or 2. The initial equilibrium point for all of the regimes was chosen to be identical (a) so that the system is in equilibrium at +r = 42 when the muscle torque equals P,. One can see that when the threshold is attered f@’ + 8’) the system transfers to a new equilibrium point fb, c or d, depending on the regime). The shift of this point leads to a change in the equilibrium torque (PZ - PI) in the isometric regime, of the joint angle t#r -&) in the isotonic regime, of both the torque [Pa --* P3) and the angfe f& -+ 4p,) in the intermcdiatc regime. It is evident that the physiological range of the change of the muscle torque and joint angle can be achieved provided there is a sufficiently wide range of the parameter p.

It is worth noting that the shift of the equilibrium point is nat only a necessary but a sufficient condition of the initiation of active movement. Based upon it, one can de- scribe the dynamics and, as web, the main electromyo- graphic events during an active change of posture (see for details, Feldman, 1974; f 979). To prove the theorem, we consider again the multitude of permissible meanings of the parameters normalized by (5) and denoted by fir and 8,. This multitude presents a square (Fig, 9A). Note that the correlation

where A& (k = 1, 2) is an increase of &. is executed on both a diagonal of the square passing tbrougb point D and any line gorng in paraliel. In accord with the explanation given before, the correlation (7) and direction of the diag- onal symbolize the program of reciprocal activation.

It is worth making a reservation that the coefficients before A& in (5) and in the following correlation (8) are taken to be equal to 1, although their actual meanings are not known. Nevertheless, this reservation does not put the limits on the theorem to be proved.

The correlation

is executed on the other diagonal of the square and on any jine going in parallel. ‘This correlation and the direction of the diagonal symbolize the program of the joint activation. By definition, simultaneous inhibition of both muscle groups is a particular manifestation of the program of joint activation.

Let e, and eZ be the unit vectors directed in parallci to the diagonals of the square (Fig. 9A). Let also B = (B, ~ &) be a vector characterizing a current state of the controlling system and &, indicate this vector when both programs are not used a& ail. The exact meaning of & is not known.

Figure 9fg) shows that

b - B0 = alet + a2e2 (9)

where u1 and aa are scalar variables determinative the con- tribution of the program of the reciprocal (at) and joint (aA) activation. These variables are considered to be the time functions which are chosen by the nervous system in relation on motor tasks. The formula (9) proves to be right for any fr,.

We have shown that any state of the system can be considered as a result of successive or simultaneous apph- cations of two central programs under consideration.

The principle of superposition of motor programs in the form of the vector eq (9) can be generatized for the control

89

(Cl

FSG. 9. Vector representation of the motor programs on the plane of the controlling parameters ?, and Pa which are the normalized threshold of activation of flexors and extensors, respectively. (A) Vector Ip0 shows a state of the system when both motor programs are off; e, and e, : unit vectors symbolizing these programs+ (13) Expansion of the voluntary program B into basis e, and e2 ; a, and aZ : variables showing a degree of utilizing of the motor pro- grams. (C) Multitude of permissible meanings of or and aa

when &, = (l,O)+

of a limb. One of the possible generalizations is described elsewhere (hmsf~~, WY).

Now, let us determine the available meanings of aI and a2. Consider. for simplicity, a case /lo i= (1, 0) which means full relaxation of the muscle under zero action of both programs.

BY (91,

BY - 1 = (a~ - a,).$/2

The parameters Br , fit2 change from 0 to t and therefore, by

(1017

Geometrically, the inequalities (1 I) determine a square with diagonals going along the axis of the coordinates (Fig. 9Cf. There is a single-vaiued corresponding between its points and points of the square on the plane of the parameters Dr. B2, so that the first square transfers into the second by means of turning by n/4.

One may see in Fig. 9(C) that the independence of two programs cannot be absolute because the possibie mean-

90 :I. G. Et-1 I)\I.\\

ings of one of the variables (LI, or u2) arc limited bq an

increase in the other. In particular, when the maximal joint

activation of the muscles occurs (uIz = , 21. the program 01

reclprocitg is reduced to zero ((I, = 0) Therefore the

assumption that the reciprocal and joint activation com-

mands arc aitcrnativcs (SHI.KKtI\G’Tos. 1906) ib not abwl-

utely unjustilicd because they do prove to bc alternative 111

the extreme c;w~s. In this connection nnc ma! suggest that