An Evaluation

of Length and Force Feedback to

Soleus Muscles of Decerebrate Cats JAMES

C. HOUK,

Departnzmt

JOSHUA

of Physiology,

J.

SINGER,

Harvtzrd

Medical

AND

IT IS NOW WELL KNOWN that contraction of a muscle is reflexly excited by responses of its spindle receptors to stretch (32, 34) and is reflexly inhibited by responses of its Golgi tendon organs to contraction (10, 31). Many experimental techniques have been used to confirm these observations (7, 12, 23). Nevertheless, the actual importance of each of these reflexes in the gradation of contraction remains obscure because of the lack of an experimental approach which is capable of estimating quantitatively their respective influences (39). Formerly it was believed by many that tendon organs responded and inhibited forces becontraction on1 .y when muscular came excessive. Recent studies (19, 26) have cast doubt on this hypothesis by demonstrating for these receptors a much lower threshold to must ular contract ion than was This find .ing alone does previ ouslv t.h ought. a con tinuo us regulation of not assure from tendon muscular force by signals organs since impulses must be transmitted through one or two interneurons before they may inhibit homonymous motoneurons (7, 31). Studies have shown that these Ib pathways transmit impulses more effectively in spinal than in decerebrate cats (8). It is therefore likely that the gain of this reflex pathway is not constant but, rather, is subject to control by signals from various regions of the nervous system. For example, Ib pathways are facilitated by signals transmitted from the red nucleus (IS). As a result, the relative importance of tendon of contraction organs in the regulation would depend on the particular state of the experimental animal. We have attempted to determine whether or not this regulation of force is significant in decerebrate cats.

publication

April

IO, 1970.

MAKK

IX. GOLDhIAN

School, Boston,

Afassachusetts

02115

Al though the evidence discussed above would suggest that tendon organ inhibition would be less than maximal in this preparation, the decerebrate animal offers the advantage of presenting a reasonably physiological background of extensor contraction about which observations can be made. A criterion for the judgment of the importance of this reflex, loop gain, is borrowed from feedback control theory. The importance of length feedback from muscle spindles also has not been adequately evaluated. In discussing this problem, Matthews (39) has recently stressed the “need to know the ‘gain’ of the servo-loop in order to assess its efficacy.” From the point of view of control theory the gain of length feedback can be expressed as the positional stiffness of the system. The greater the stiffness, the less will the length of the muscle be disturbed by a change in load and the more reliable will be the performance of the system in executing controlled changes in length. Positional stiffness has been measured in soleus muscles of decerebrate cats (1 I, 37) as the slope of the stretch reflex curve, i.e., the increase in muscular force which results from a unit increase in muscle length. The problems with this measurement are twofold. Part of the increase in force would result from the increase in length even if there were no reflex increase in the efferent signal to the muscle. This component of the response results from the well-known length-tension characteristic of skeletal muscle and may completely account for the stretch reflex curve in some preparations (47). Partridge (46) has detailed how the length-tension characteristic and other mechanical properties of the muscle can provide a built-in system for regulating the length of a muscle.

A second probkrn is that an unknown amount of inhibition from tendon organs would tend to decrease the extra force produced by stretch and, hence, decrease the stiffness as measured from a stretch reflex curve. A realistic evaluation of the effectiveness of spindle feedback requires that these effects be separated. In this paper we show how each influence may be independently evaluated. (A preliminary account has been presented (2 I).) METHODS

Prepam cion Intercollicular decerebration was performed under ether anesthesia. The common carotid arteries were ligated in the neck. The forebrain overlying the coIhculi was aspirated, and a transection was made in a plane just caudal t0 both the superior colliculi and the exit of the third nerve. A small portion of the midbrain rostra1 to the transection was aspirated to insure the completeness of the section and to prevent pressure on the brain stem which otherwise might result from hemorrhage or edema. Ether was discontinued after bleeding in the region of the transection had been controlled. We often waited 1 day after decerebration before preparing the animal. for experimental measurements. Crossed extension was stronger in these animals, but “spontaneous” variations seemed more pronounced than in recently decerebrated cats. Lamina of the L, and L, vertebrae were removed, but the dura was not opened until later in the experiment. The following denervation was performed on the left limb. The hamstring, tenuissimus, pudendal, and inferior gluteal nerves were sectioned following a dorsal incision at the hip Just before the decerebration the femoral, the superficial branch of the obturator, and the external femoral cutaneous nerves were sectioned following a ventral incision. The peroneal, sural, tibial, and gastrocnemius nerves were transected in the popliteal. fossa. This extensive denervation eliminated myographic resnonses due to contraction of muscles other th:n soleus and also prevented inhibition of the stretch reflex which seemed to result from noxious stimuli to exposed tissue in the lower part of the limb. The peroneal nerves of both limbs were prepared for stimulation (LO-30/set). Stimulation of the left nerve caused a complete inhibition of the stretch reAex in the left soleus (37).

Stimulation of the right nerve elicite ;t crosscdextensor contraction of solerrs which was sometimes used to augment the background contraction upon which the other measurements were made. The gastrocnemius and plantaris muscles were separated from the soleus, and the nerve to soleus was dissected from the surface of the lateral gastrocnemius. The nerve and muscle were bathed in warm mineral oil. The hindlimb was clamped rigidly to a mounting platform using drills through the femur and the tibia. The s&us tendon was left attached to the tip of the calcaneus which was attached directly to a myograph employing semiconductor strain gauges (displacement rate = 0.05 mm/kg). The myograph was rigidly fastened to a carriage which could be clamped in different positions to the platform. Through a threaded-screw arrangement changes in muscle length of 1/3 mm could be consistently delivered manually, aIthough the time course of these inputs could not be accurately regulated. The passive component of muscular force was deducted from the total force to obtain the active component of contraction. Usually the passive force was negligibly small since the FoIeus muscle was well separated from the surrounding tissues. CJnder a dissecting microscope the dura was opened and a small filament of ventral root (from L, or S,) was cut proximally and separated from the remaining root distally. The filament was placed on an electrode pair and stimulated for 0.5-3 set at rates of 15-30/set. Care was taken to prevent the spread of stimulating current. Tile output of the myograph indicated whether or not Q motor fibers were being stimulated. In our initial. experiments we recorded from each filament responses elicited by a singIe shock to the soleus nerve to determine whether or not y motor fibers were present. The filament was subdivided until only a fibers remained, as judged by their conduction velocities. In later experiments we merely adjusted the stimulus voltage to a levelno hiiher than twice a threshold and assumed that no y fibers were stimulated. Our results were not different. Enough filaments were prepared irl this way to produce a total of 50-500 g of contractile force. Thus, Z-ZOpl, ( usually 5%) of the normal efferent output was unavailable to reflex responses. Stimulation of these filaments singly or in combination produced force disturbances within the muscle which could not be controlled by reflex pathways, The magnitude of a disturbance was measured by observing the contraction elicited during a complete inhibi-

786

HOUK,

SINGER,

tion of reflex contraction produced by stimulation of the ipsilateral peroneal nerve. The myograph was connected with an a-c bridge and carrier amplifier (Tektronix 3666), the output of which was disp1ayed on a storage osci lloscope (Tektronix 564). Responses generally reached a steady level in about 200 msec, and readings we re made afte r 0.5 sec. The overall accuracy of the apparatus was tested and found to be better than of the full screen

posed on background contractions (Fi .gs.2, 4, 5), was le ss than this, depending on the magn itude of the background contraction. A greater source of error, however, was the “spontaneous” variation in the background contraction, as illustrated in trace SR of Fig. 5, which sometimes made it difficult to estimate the amplitude of incremental responses. For this reason, several repetitions of each response were taken in the vicinity of each operating point. The experimental arrangement is summarized in Fig. 1. The reflex experiments were successful in nine animals, two of which provided a full set of data suitable for constructing the plots exemplified by Fig. 6, The additional methods used in various control experiments will be described under REsuLTs.

AND

GOLDMAN

RESULTS

To evaluate the importance of length feedback from spindles and force feedback from tendon organs we have found it necessary to measure two reflex responses and some mechanical properties of soleus muscles. One reflex which we used was the common stretch reflex, which results primarily from the stretch applied to spindle receptors. The stretch reflex may also be influenced by the mechanical properties of the muscle and by tendon organs which respond to the changes in tension. The other reflex used was the response of the muscle to an internal disturbing force which was produced by stimulation of small filaments of ventral root. Hunt (23) has already used such an input to demonstrate the ability of tendon organ discharges to inhibit the monosynaptic reflex. If the muscle is constrained isometrically, an internal disturbing force would be expected to activate mainly tendon organs leading to a reflex inhibition, but it could also cause small variations in spindle discharge which would influence the reflex. In addition, it was necessary to estimate the lengthtension characteristics and a series elasticity of the muscle. Ideally, one would like to measure all of these responses simultaneously in each preparation. As a practical compromise, we were able to measure stretch reflexes, responses to internal disturbances, and lengthtension characteris tics in rapid succession in each preparation. The series compliance, which was not the same as that normally dealt with in studies of muscle (see below), ~r;ts measured in separate experiments. An important assumption in this anaIysis, as we11 as in previously reported studies, is that the reflex components of responses to stretch and to internal disturbances arise onIy from spindles and tendon organs of the soleus muscle under study. It is quite unlikely that other receptors in the muscle would contribute to these reflexes. The remainder of the left hindlimb was denervated to eliminate any influences from receptors in nearby tissues. Nevertheless, decerebra te preparations are weI1 known for spontaneolrs variations in extensor

LENGTH

AND

rigidity. Usually these variations occurred over a slow enough time course so that all the observations at each operating point could be completed before the state of rigidity changed appreciably. In fact, these slow variations in rigidity were used to advantage to investigate the variation in responses with variations in the magnitude of the background contraction. In adclition, we employed stimulation of the bared peroneal nerves of both limbs to further control the operating point of the stretch reflex. 1. Operating

&mint

The soleus muscles of decerebrate cats were stretched to lengths at which a constant reflex contraction was elicited. The state of the stretch reflex could be characterized by an initial length L, and an initial force F,. These two parameters define an operating point about which small variations of length and force were subsequently delivered. Lo was not allowed to exceed the maximum physiological extension of the muscle. The initial force F, was a function of both the initial length L, and the amount of background reflex excitation to the muscle. The backcould be eliminate has reported a nonlinear response for individual tendon organs when studied over a large range of muscular forces. As the total force supported by the muscle increased the discharge frequency of tendon organs increased at a progressively declining rate (saturation). If the same relationship were- also obey&I by the homonymous population of tendon organs, one would expect that the gain of tendon organ feedback would decrease progressively with operating force. This prediction might help to explain why the reflex stiffness increases with operating force, but it contradicts the progressive increase in the gain of force feedback which we have observed.

It is known that the response characteristics of spindle receptors are modified by their efferent innervation from y motor fibers (25). We therefore considered the possibility that the dependency of our experimental responses on operating force was the result of a dependency on the level of y activity. By this hypothesis the incremental responses during a crossed-extensor reflex, which is known to activate spindles through y fibers (9), should difEer from those elicited during a background stretch reflex, even though the operating forces are equal. Since we consistently found no difference, we are also able to reject this hypothesis. The possibility that the p innervation of intrafusal fibers might be responsible for the dependency on operating force is not so easilv discarded. It is not known whether or not the motoneurons of these fibers receive homonymous inputs from spindles and tendon organs analogous to that of a-motoneurons. If they do, the intrafusal contraction caused by them would probably correlate with the operating force regardless of whether it was elicited by stretch or by a crossed-extensor reflex. Any increase ir 1 th .e static responsiver less of spindies due to the intrafusal con traction (2,

3) would increase the gain of spindle feedback which might account for-the dependency of the reflex responses on operating force. Furthermore, the postulated reflex loop from p-motoneurons would constitute positive feedback which could also augment the gain of spindle and tendon organ pathways. However, the limited and sporadic occurrence of p innervation (particularly in soleus) suggests a minor, and role f’or this system in perhaps vestigial, cats (2, 3). A nonlinearity which appears to explain the experimental results derives from the study of soleus motor units by McPhedran et al. (36) in combination with the studies of recruitment by Henneman et al. (15, 16). The graph in Fig. 12 has been constructed by applying the size principle (15, 16) to the distribution of motor units according to their tetanic tensions (Fig. 6 of ref 36). The number N of recruited motor units is plotted along the abscissa in normalized units. The isometric force P which would result if motor units are recruited in order, smallest to largest, is plotted along the ordinate in normalized units. The points are well fitted by the square relationship. This curve merely formalizes what has already been pointed out (15), that the increase in force due to recruitment of a new motor unit will increase as the number of active motor units i,ncreases. I.0 t

’

Derived from McPhedran, Wuerker and Henneman 2

0.8P

Pm

0.6.

0.2

0.4

0.6

0,8

1.0

N NM FIG. 12. A recruitment nonlinearity. This relation between isometric force of contraction P and number of recruited motor units N was constructed from the di .stribution of motor units according to the size of their contraction assuming that they are recruited in the order smallest to largest. The simple-square relationship fits well the sequence of derived points (e),

In terms of our arr;llysis, the gain factors would both increase with operating force. To determine whether this recruitment nonlinearity could be responsible for the observed variations in inhibition and stiffness, we assumed that the output variable of the motoneuron pool is the number N of motor units recriited. The predictions derived are illustrated by curves 4 in Fig. 10. The calculations are outlined in APPENDIX c. It is apparent that there is at least qualitative agreement with the experimental results of Fig. 6. When the results of an analysis at one operating point in Fig. 6 are used to evaluate the parameters of the model equation 17, the predictions at the other operating points are in good agreement with the data as illustrated by the dashed curves in Fig. 6. These results indicate that the static nonlinearity which is responsible for the dependency of operating force on reflex stiffness and force feedback has two special features. First, it is common to both spindle and tendon organ pathways. This might have been suspected since both curves in Fig. 6 show their greatest changes in the same range of operating forces, i.e., between 0 and 300 g Fo. Second, the nonlinearity must have a shape similar to that shown in Fig. 12, i.e., the gain of the reflex should increase approximately pathways linearly with the number of motor units recruited or with the square root of the operating force. The ability of the proposed recruitment nonlinearity to satisfy both of these features leads us to believe that it is responsible for the major static nonlinearity of this system. It may also be responsible for some of the effects described as “multiplicative” by Kim and Partridge (30). These were observed when neck, vestibular, and stretch reflexes were combined. With this marked nonlinear feature, it may seem inconsistent that the stretch reflex has previously been reported to be linear throughout most of its range (11, 37, 47). Matthews (37) anticipated this problem when he suggested that autogenetic inhibition might be responsible for linearizing the stretch reflex. The dashed curve in Fig. 10 represents the manner in which stiffness would be expected to increase with operating force in the absence

HOUK,

SINGER,

of force feedback. Force feedback, even though its loop gain is small, has a linearizing effect which becomes quite pronounced for large operating forces. Without this linearization, it might not be possible for the stretch reflex to be both effective and free of instability oscillation over a wide range of operating points. SUMMARY

The regulation of muscular contraction by its own proprioceptors has often been likened to that of control systems which, through high-gain feedback loops, provide reliable control of unreliable effecters in the presence of various disturbances. A set of experiments was designed to test whether the static gains of the reflex pathways are sufficiently large to provide the beneficial features ascribed to them. Reflex responses were measured in isometric soleus muscles of decerebrate cats and control experiments were conducted under barbiturate anesthesia. A background contraction supported by a stretch reflex, sometimes augmented by a crossed-extensor reflex, provided an operating point about which incremental responses were measured. Tetanic stimulation of small filaments of ventral root produced contractions which, in the presence of a reflex background, were smaller than the contractions in the absence of reflexes. The diminution was attributed to force feedback whose gain could be evaluated from the responses. Values for the loop gain of force feedback typically fell in the range of 0.2-0.8. We estimate that more than half of this gain is mediated by tendon organ pathways. The remainder is mediated by the reflex pathways of the spindles which experience some internal shortening during the contraction. The stiffness of the stretch reflex was measured from the responses to a small change in muscle length. Values in the range of NO-600 g/mm were observed. Reflex stiffness is determined by three components: the gain of length feedback from spindles, the inherent stiffness of the muscle, and the gain of force feedback. Muscular stiffness was measured while reflexes were inhibited by stimulating the ipsilatera1 peroneal nerve. Its value was directly prapartional to the operating force and

AND

GOLDMAN

also depended on the operating length. Using a value for force feedback measured at a comparable operating point, the gain of spindle feedback was deduced. Our results indicate that length feedback from spindles increases the stiffness of the stretch reflex severalfold (about 5 times) over what it would be if the muscle were regulated without feedback. Consequently, the displacement suffered in response to an external disturbance force would be reduced by about 80%. Also, the gain is sufficiently large to reduce the dependency of movement control on nonlinear and time-varying properties of skeletal muscle. The gain of force feedback is rather small. We estimate that it would reduce the sensitivity of the system to muscular fatigue by about 407&. A disadvantage is that it diminishes the reflex stiffness, Other possible roles for tendon organs are discussed. The gains of both spindle and tendon organ pathways increase approximately as the square root of the operating force. This static nonlinearity may result from the recrui tmen t of progressively larger motor units. Force feedback provides a compensation for this nonlinearity which may extend the range of operating forces over which length feedback is both stable and effective. GLOSSARY

OF

PRINCIPAL

SYMl3OLS

D = disturbance force produced by stimulating ventral root filaments E = hypothetical efferent excitatory signal to muscle F = net force developed by muscle (P + D) L = external length of muscle N = number of active motor units P = force produced reflexly in muscle T= force produced when entire muscle is active X= internal length of muscle Subscripts

to variables

0,1,2 = operating M = maximal Conuen

points value

t ions

Lower case variables designate incremental variations about an operating point; the letter 2 has been italicized to distinguish it from the arabic numeral 1 [ ] designates functional dependency ( ) encloses multiplicative terms “evaluated at”

Purametees

describes

C = compliance common to all muscle fibers G = partial gain of spindle reflex pathways H= partial gain of tendon organ reflex pathways I = loop gain of tendon organ pathways J = stiffness, or net gain, of spindle reflex pathways K = muscular stiffness, slope of L-T curve at operating point k = normalized muscular stiffness, K/F, APPENDIX INCREMENTAL

A:

1.

functions

Transfer

STEADY-STATE ANALYSIS

for

sum (Fig.

the incremental incremental force of the reflex force 9).

reflex

and

changes

in length

are reIated

(5)

T ix*1 P[E, X] = PIEo, X0] + -

(E -

EO) +

K(X

d + (J + K)x

f =

(7)

13-I

where I = HT[X,]/E, is the incremental of force feedback via tendon organs stiffness GTF,l /E&f is the positional feedback via spindles. Making the conversion to external substituting equation 5,

loop gain and J = of length length

by

d -+ (J + K)Z

fr--During I = 0.

responses

During

responses

-=f 2

terms

where

aTF1 1 ax X0

is the slope of the length-tension curve at the operating length weighted by the fraction of the muscle which is activated. Note that K designates the internal stiffness of the muscle. Because of the series compliance this quantity is not immediately available to measurement and must be deduced from external measurements, a procedure which is formalized in section 2 of this APPENDIX. When the higher order terms are dropped and the incremental variables p = P[E, XJ - P[E,, X0], e=E - E,, and x = X - X, are substituted. Kx

to

internal

force

disturbances

f 1 - = _r__-p cl I + I + (J + K)C

(2)

c +

(6)

where G and H represent, respectively, incremental gains of spindle and tendon organ pathways up to the efferent nerve signal. Changes in efferent signal are assumed to resuXt from a linear combination of spindle excitation and tendon organ inhibition. Substituting equation 6 into equation 4,

P>

- X0)

EM

order

9.

1 + I + (J + K)C

the maximal value of efferent excitation occurs when the whoIe muscle is active. Expanding this function as a Taylor series about an operating point denoted by E,, X, one obtains

E 11

external

x-I-Cf

(1)

T[X]

is EM which

vq p = ----

(4)

e = Gx - I-If

E Rl

]K.=---E. E,

e + Kx

E Al

bY

responses

E = -

+ higher

-rFg

f=d+p Internal

behavior of the muscle. which is measured is the p and the disturbance d

where C is the common compliance in Fig. A simple feedback equation is chosen

The force P in Fig. 9 produced by the muscle fibers which are reflexly controlled depends on both the efferent excitation E (a hypothetical variable) and the length of the muscle fibers X. The length dependence is characterized by a function T [X], the length-tension characteristic during tetanic contraction of all the muscle fibers. fn section 4 of RESULTS it was argued that efferent excitation may be treated, at least operationally, as the progressive recruitment of parallel motor units having length-tension characteristics identicaI to that of the overall muscle save for a multiplicative constant. The reflex force can therefore be expressed as the product of two terms, T [X] representing the shape of the dependency on length and E/E, representing the proportion of the muscle which is actively recruited.

P[E, X]

The

to incremental

shortening

J+K --I

+ I + (J + K)C

d = 8.

(10)

Equation 8 represents the view that external length is the regulated variable, whereas equation 7 represents the view that internal length is the regulated variable. The difference is the appearance in equation S of a component of force feedback mediated by spindIes and the stiffness of the muscle, the term (J + K)C in the denominator. If it is agreed that the purpose of the stretch reflex is to maintain an angIe of the joint, the view that external Iength is the regulated variable is more appropriate. 2. Relation length-tension

between curves

The internal tion 2, can be ments using the cxtcrnal length

external

and

internal

stiffness of the muscle, K in equaevaluated from external measurerelationship between internal and in equation 5. If T[X] and T’[L]

808

HOUK,

represent, respectively, the length-tension characterisl:ics, T[X]

and

AN13

external

GOLDMAN

stimulation

during

of two

musck

portions

the stimulation

be written

if I, = X + CF

stiffness when the whole

The internal active is

Evaluation

= T’CL]

internal

SINGER,

from

of the ventral

root

of both simukaneously

equation

can

2.

pCE,~Xl= PCE,,X,] + K,(X - Xl) I'[E,,X] = P[E,,X,] + K,(X - Xl)

is

and

(13)

(14) i'[E 1+2' X] = P[E,, XI] + I'[$, X1] + (K1 + K&(X - X1) (15) and

substitution

El, E,, and El +-2 represent

of JL/ 8X yields aT[

the operating points the three conditions. during each condition

of the efierent signals under Since these remain constant

L,]

of stimulation (no reflexes), the term in (E - Eo) in equation 2 becomes zero. Higher order terms

Next, a11 partial derivatives operating point (Eo, X,, L,,) equation

arcs

multiplied

by

cvall~i~tcdat

are and

the sictes of the

both

the

excitation

ratio

E,JE,lm

E.

E31 ax The

term

on

the

x, left

&~‘[I,]

1 -c-, is

equal

of the muscle K (see eqwztion

dF dL f Fo, Lo

K’ 1

stiffrwss

2). The numerator

on the right can be defined as the external K’ and the term aF/aL when evaluated operating point is also equal. to K’, The K=-

stiffness at the result is (II)

- CK’

In practioc K’ is evaluated in the abscncc of Wflexes using the ventral root filament which happens to be available to provide an operating force F, at the operating length L,,. The methods of HremL,Ts, section 4 yield direct measures of K’z--

X - X,

= -CP[E,X]

(16)

since the change in external length, 2 5, is zero under the isometric conditions rcprescnts tXle change in force applied stitution of equation 16, using the operating points, into equations 13-H the exphcit dependency on X.

[

to internal

are neglected. The internal operating length X, is the same for all three responses. I& corresponds to zero active force at a fixed internal length. The relation between internal length and contractile force can be written

(1 + K,C)P[QX]

= P[E,,X,]

Pn

(I + K,C)P[E,,X]

= &,X1]

(IS)

1+2’q = PLE We define

JTp q

P[E,,

decrement

X] + P[E,, P[E,,X]

Substituting result into

32;

+ wy -~

x11

1 + (K1 + K,)C

a fractional

a =----

in equation and P[E,X] to C. Subappropriate eliminates

as

X] - qq+,,

Xl

+ P[E,,X] 17 and

equations

(19)

18 into

( 2 0)

19, and

the

20, J$CP[E,,

A=--

X] + K,CP[E,,

X]

r3r.J F, ,I+, or its normalized

version

P(E,,X)

kf = - -___- i F, aL IF,, I,,,

(1 21

and P(E,,X)

sulting from the tion of ventral

muscular

If the operating force during a ruflcx is l$, the muscular stiffness will be K’ = k’F,,. Implicit in this proccdurc is the use of the ratio of operating forces, e.g., FO/F,, to estimate the ratio of effcrcnt operating points, e.g., EJE,.

evaluated of external

of each porthe internal stiffnesses at each operating point, arc using equation 11 and measured values stiffness. Then, assuming a value for C,

the predicted

‘i’hc c’o~~m01~series compiiance, C

in Fig. $i, is for the decrement which is seen in the of two portions of the muscle together

as contrasted to the sum of their separate contract ions (Kl~:sIII.‘rS, sertinn 5). Expressions me1ric force of contraclion during

for the

the isoscpar:W

decrement

can be calculated

21, This technique theoretical points in Fig.

equation

the

APPENDIX responsibIe contraction

are the measured forces rc’-

separate stimulation root. K, and K,,

?‘IIP I *I-

gain, force

rlncd-lnnp VIVY1U

was 8.

BB: SENSITIVITY *x-Y

tmncfpr I*n+*Y-1+

TO fllnrtinn L-“-s’~Y”,

for a system such as that can be written as

used

front

to obtain

FATIGUE nr Y-.

regulating

rhrd-lnnp YLYYIU

-...-

muscular

LENGTH

AND

FORCE

where A is the gain of the forward portiorl of the loop, including the muscle, B is the gain of the feedback path, and the muscle is constrained isometrically. The product AB is the loop gain. Muscular fatigue can be thought of as a variation in the gain of the muscle, thus a variation of A. The sensitivity of the system to this variation is evaluated by differentiating the feedback equation, assuming R is constant.

FEET~BilCIi

Its expansion (E,,N,,) yields

so9 as a ‘1’ayIor

series

around

the

point

E=E,+ Using dEO/E,,

incremental variables and N,, provides the incrementa

letting N, relation

=

(eqelation

6)

dA dW= The yield

result

(I + AB)a

is divided

dW -W

by

= -

the

original

1

dA -A

I+AB

equation

to

EXPLAINING INHIBITION WITH

In DIscuSSIoN, section 3 we suggest that a nonIinearity in relation between efferent signal and muscular force might explain the variations with operating force of inhibition and stiffness which appear in Fig. 6. In this appendix we show how curves D in Fig. 10 may be derived to test the hypothesis. The analysis departs from that in APPENDIX A by assuming the relation between efferent signal and muscle activation depicted in Fig. 12. Let N, the number of active motor units, represent the efferent variable rather than E, and let E now represent the degree of excitation or activation of the muscle such that E -=-, E 31 The subscript M equation representing can then be written

Y =51A lx=x, denotes maxima1 the recruitment

feedback by

equation

n = Gx - Hf

where dW/W represents the fractional change of the gain of the closed loop system which results from a fractional change, dA/A, of the muscle gain. In the absence of feedback, a 10% change in A due to fatigue wil1 result in a 10% decrease in muscular force. In the presence of force feedback this sensitivity to fatigue is reduced by the factor I/(1 + AB). Substituting values for the loop gain of force feedback from Table I, this factor becomes 0.82 and 0.55, respectively, for the 200-g and 400-g operating points. The postulated fatigue would now result in a 8.2 and a 5.5% decrease in muscular force, which represents some improvement. Force feedback appears to diminish the effects of fatigue by 18-45x. APPENDIX C: A MODEL VARIATION IN REFLEX AND REFLEX STIFFNESS OPERATING FORCE

The incremental must be replaced

since n now represents the incremental efferent signal from the motoneuron pool. The remainder of the model is described, as before, by equations I-j. The simultaneous solution of these equations yieIds

d + (J,d$+kFo) f=----

I

-~

(22) where Jfif = 2GT[X,1]/N,, and I, = 2HT[X”]/N,, are, respectively, the maxima1 values of spindle stiffness and tendon organ loop gain which occur when F, = FII; k is the normalized muscular stiffness (equatiob 12). Equation 22 is like equation 8 except that the dcpcndence on the operating force is made explicit. and k depend on the operating length X, predicting a dependency on L, as well. The latter was not grossly apparent in our experimental results, probably because T[X,,] varies by only 20% over the range of operating lengths used (Fig. 7). The following values were used in constructi&g the dashed curves superimposed on the data points in Fig. 6.

Jw L,

Jnr = 500 g/mm I 31 = 0.45 k

= 0.16 g/mm

C

= 0.0005

per

g F,

mm/g

ACKNOWLEDGMENTS

F, FM values. The nonlinearity

We thank Dr. E. Henncman for many suggestions and Dr. P. B. C, Matthews for reviewing the manuscript. This research was supported, in part, by Institutes of Health Grants NS-08762 to and NB-07805 to E. Henneman. J. Singer ported by a National Institutes of Health toral fellowship and M. Goldman by a assistantship provided by Harvard Medical

helpful critically National J. Houk was suppredocsummer School,

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