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Modification of muscle responses by spinal circuitry
t,kwoscience Vol. Il. No. 1, pp. 231-240, Printed in Great Britain
NOTIFICATION
~306-4522/84
1984
$3.00 + 0.00
Pcrgamon Press Ltd C 1984IBRO
OF MUSCLE RESPONSES CIRCUITRY
BY SPINAL
R. B. STEIN and M. N. OC~UZTGRELI Departments of Physiology and Mathematics, University of Alberta, Edmonton, Canada T6G 2H7 Abstract-Recently, we developed a model based on ex~~rncntal data, which includes a pair of antagonistic muscles, a general load against which the muscles act, feedback pathways from muscle sense organs and spinal inhibitory circuits involving IA interneurons and Renshaw cells. Descending inputs can activate the model through combinations of inputs to a-motoneurons, y-motoneurons (via intrafusal muscles and their feedback pathways) or the IA interneurons. The role of each of these connections is analysed here in terms of its effect on the response of the muscles to impulse inputs, with particular interest in the effects on the overall stability of the systems. Increasing muscle stiffness or feedback from muscle receptors tends to produce high frequency oscillations. Coactivation of c1- and ~-motoneurons can lead to cancellation of oscillations, because of delays in the effects of y-motoneurons on contraction. Connections of IA inhibitory interneurons onto antagonist motoneurons accentuate the oscillations, Inhibitory connections from Renshaw cells onto a-motoneurons tend to prevent oscillations, whereas the concentrations onto y-motoneurons may produce them
The motor
system is arranged
in a hierarchical
fash-
ion, Spinal reflexes modify the most peripheral elements (i.e. the muscles). The spinal cord can also produce some of the basic patterns, such as are required for locomotiont~ and scratching,6 which are in turn controlled from brain stem structures. Finally, higher centres such as the motor cortex have executive control of the whole motor system.‘O Although such general ideas are widely accepted, the details and specific roles remain obscure. For example, the role of even the simplest spinal pathway, the monosynaptic reflex remains controversial.” It has been suggested that this reflex is used as part of a follow-up length servo,25 as part of a servo-assisted system for control of length2’,‘” or together with force feedback in the control of stiffness.” The role of force feedback from Golgi tendon organs is even more obscure. its gain is close to zero in decerebrate preparations where it has been measured accurateJy,34 although it might be greater in other preparations. Further experimental work is required to settle some of these controversies, but detailed analysis of simple mathematical models of muscles and reflexes would be useful to examine possible roles of reflexes in modifying the control of muscles. Such a model was proposed in a recent paperZX and some results were derived analytically. However, the number of analytic results which can be obtained are limited, and further results must be derived from computer simulations. Various simulations will be presented in this paper for the whole model and some simplified versions of it, so that the role of each pathway on the stability of the system can be examined. A tendency for instability would be evident behaviorally as a tremor of the limb.
In order to keep this paper relatively short, the full equations and definitions of terms will not be included. Details can be found in Oguztiireli and Stein2* However, the assumptions underlying the model and the terminology will be discussed briefly before proceeding with the results.
MODEL
The model we have used consists of two parts: a pair of antagonistic muscles and the spinal circuitry controlling these muscles. The muscles are shown in Fig. 1 within dotted lines. Each consists of an active force-generating element a(t), parallel and internal series elastic elements (with stiffness k, and k,, respectively) and a dashpot with viscosity B. The muscles work against a generalized load consisting of a mass M, an external spring of stiffness k, and a dashpot with viscosity D. The active force generating ability decays exponentially following a neural input, so the muscles behave as linear second order systems (with a viscoelastic time constant and a time constant for the decay of the active state). Despite its simplicity, such a model accounts well for a variety of muscle responses.5 It has been suggested that reflexes are adapted to compensate for some nonlinearities in muscle properties,27 so exclusion of nonlinearities from the present model may seem like a serious omission. However, there are several reasons for this omission. Firstly, a nonlinear system will often behave linearly over a small range, such as would occur during normal, physiological tremor.‘* Secondly, despite recent experimental’9,29,43 and theoretical’5,28 efforts to characterize the nonlinearities, a completely
231
232
R. B. Stein and M. N. Oguztiireli
_. . ..__________________________-_____.___~_~_~._ the connections between r-motoneurons and Ren-_-.-_-_-_-_-_---.-_-_-_-_-_-_-_---_-.-.-.-_-.-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-.-_-_-. ____. . _ shaw cells has been well described’ and is included in I
I
I
Fig. I. A viscoelastic model of antagonistic muscles and a generalized load consisting of a mass M, a dashpot with viscosity D and a spring of stiffness k,. Each muscle consists of an active force-generating element a(t), a dashpot B and parallel k, and internal series k, elastic elements. The y(t) measure positions with respect to a fixed point, whereas the U(I) are deviations from the steady-state position. External forces,f(t) may also be applied. Further details in the text and in Oguztiireli and Stein.‘R
satisfactory nonlinear model is not available. Finally, the methods available for linear systems are much more powerful than those for nonlinear systems, and a linear analysis can serve as the first stage for a later nonlinear analysis. For example, a steady force will change the position of the external load b(t) in Fig. I] and the elements within the muscles yj(t), j = 1. 2. However, the deviations u(r) and u,(t) about the new steady-state positions, in response to a perturbation, will not be affected. Spinal connections to and from the muscles are shown in Fig. 2. Neural inputs n come directly from r-motoneurons and produce muscle responses which interact with the load to produce movement. Length and velocity information g is fed back via muscle spindles to the a-motoneurons and to the 1A interneurons. These interneurons inhibit (-) agonist a-motoneurons and hence provide a natural link between the muscle in addition to the mechanical linkage through the muscles acting on the load. Force feedback from Golgi tendon organs is ignored, because of uncertainties about its efficacy (see Introduction). Renshaw cells R provide negative feedback with a characteristic lag to the large a-motoneurons as well as to the smaller y-motoneurons.‘* The dynamics of
the model.” The y-motoneurons produce negligibly small amounts of force, but do influence contractions by their actions on intrafusal muscles (with properties z), which affect the feedback from muscle receptors. Other connections between spinal neurons could obviously be added to the model. For example. Miller and Scott” considered inhibitory connectiona between IA inhibitory interneurons and from Renshaw cells onto IA inhibitory interneurons. They felt that these connections might contribute to large, low frequency oscillations such as walking (but see Menzies of ul.“). These spinal connections have been omitted in the present paper. since their dynamics have not been determined. and such large oscillations would exceed the linear range considered here. The contractions of the two muscles will also be afPected by descending inputs which may be focussed in different combinations on r-motoneurons. y-motoneurons and IA inhibitory interneurons to one or both muscles. Since the system has been assumed to be linear for present purposes. a brief input (Dirac (3-function or unit impulse) is sutticient to characterize the response to any input. One point of particular interest is the effect of the central connections on the stability of the entire system. The strength of these connections are denoted in Fig. 2 by the 0,. We will begin by setting the strength of these connections to zero to examine responses of the muscles and load in isolation. Then non-zero values of the 0, will be systematically introduced in selected pathways to examine the effect of these pathways on the overall responses. Analytical solutions for the entire system and
Fig. 2. Muscle receptors compare the actual movement within each muscle with the signals from ;‘-motoneurons. which are filtered by the intrafusal muscle properties Z. The feedback signals excite I-motoneurons and IA interneurons in the spinal cord, which inhibit antagonist motoneurons. a-motoneurons excite Renshaw cells which in turn inhibit both a- and y-motoneurons. The 0, represent the strength of the spinal connections. Inputs from other parls of the central nervous system may act on either type of motoneuron or the IA interneurons. Detailed equations and analysis are given in O&rtiireli and Stein.”
233
Spinal modification of muscle various simplifications of it have been published inclined previously,28 and the more mathematically reader should consult the previous paper for these solutions, as well as for mathematical details of the model. This paper will deal only with numerical computations of two types: firstly, from the transfer function for the system, the roots of the equation can first be determined numerically using methods which have been described.” Secondly, the inverse Laplace transforms can be computed to give the response to an impulse input. The roots of the transfer function can take several forms: a real number p, which corresponds to a solution of the form u = uOep’
(1)
where u0 is the initial value of u and the response grows or decays exponentially, depending on whether the rate constant p is positive or negative. Alternatively, the root may be a complex number, which corresponds to a solution u = uOeP’sin (2rrqt + I$)
(2)
where q is the frequency in Hz and C#Jis the phase of an oscillation, which will grow or decay exponentially depending on the sign of p, as above. More complex forms are also possible, if more than one root has the same value. For example, if there are two roots with a value p, the response will be of the form u = u,te”’ rather than the form given by (1) above. examples are given in standard texts.’
(3) Other
RESULTS
Inputs to a-motoneurons. We will begin with the simplest example and systematically add complexity to examine the effects of each element of the system on its overall performance. The simplest example consists of a single impulse (Dirac a-function) to one set of motoneurons (c(, in Fig. 2) in the absence of feedback. Feedback can be eliminated from the model by setting the gain of the pathway from muscle receptors (g) equal to zero, as well as the values of all the 0,. This abolishes central feedback from the Renshaw cells, for example, as well as peripheral feed back. If the load attached to the muscle contains a significant mass as well as a spring, there is some tendency for a damped oscillation to occur in response to a single impulse to the a-motoneurons (compare Fig. 3B, where M = 0.3 kg to Fig. 3A, where the mass is negligible). The parameters of muscle 1 have been set to standard values obtained from experimental data for cat plantaris muscles.28” In Fig. 3, the stiffness of the antagonist muscle 2 is varied from zero (Fig. 3B) to a value five times as large as that of muscle 1 (Fig. 3D). Note that if the
stiffness of muscle 2 is zero, it essentially decouples this muscle from the load (i.e. no force will be transmitted, however much the load moves). The responses of single muscles with and without feedback connections have been considered extensively in the past.40 Increasing the stiffness of the antagonist muscle in Fig. 3 increases the tendency for oscillations to occur at a high frequency (about 25 Hz). Such highfrequency mass-spring oscillations have been observed for example in hand tremor.33 The stiffness of isolated muscles does vary greatly with the level of activity,29 so the effects seen in Fig. 3 will be observed as the steady level of activity in muscle 2 varies. Muscle viscosity may also vary with the level of activity. However, if both the viscosity and stiffness of muscle are increased in parallel, the increased tendency for high frequency oscillations still occurs for a large range of parameters. Feedback ,fiom muscle receptors. In all subsequent calculations the stiffness of muscle 2 will be fixed at a value half that of muscle 1. Although the choice of this particular ratio is somewhat arbitrary, muscle 2 works with gravity while muscle 1 is an antigravity muscle (Fig. 1). In general, antigravity muscles are stronger simply because they must lift the mass of the body against the force of gravity. The form of the feedback used in this paper is
Gl + F,, t,[ - zi,(t- t,) + +I,(t - t,)]
g, 0) = FO,]-u,
(t - t,) + m,(t -
g2(t) = Fo2]uz(t - t2) + m2(r -
+
F0272[SZ(t
-
(4)
fdl f2)
+
h,(t
-
tJ1. (9
This form is somewhat simplified from that considered analytically2* in that we are only including (1) a single time delay r, for each pathway and (2) length and velocity (e.g. ti,) feedback. The delay used in all the simulations is t, = t, = 30 ms, corresponding to a spinal pathway. Longer latency pathway might also be considered.28” However, Eklund et al.” have shown experimentally that many phenomena, previously attributed to longer latency central pathways, can be accounted for by bursts of repetitive activity in muscle receptors acting through shorter latency, spinal pathways. In all the simulations, the velocity sensitivity of the feedback [the parameters T, and z2 in equations (4) and (5)] was set to 0.1 s. This corresponds to the value measured experimentally,23,3’ although greater stability could be achieved with smaller values of this parameter. 39 The effect of higher order derivatives, such as acceleration terms, have also been considered previously. 39 Muscle receptors do show an apparent acceleration sensitivity under some circumstances,35 but its nature is complex’ and its magnitude is small enough that it will not affect the general conclusions reached here. The feedback in equations (4) and (5) is based on
R. B. Stein and M. N. Oguzttireli
234
2
derivatives are zero. This restriction will be removsed in the next section. Figure 4(A) shows the responses of the system to single impulses to the r,-motoneurons with and without feedback. With a feedback gain equal to zero, the response corresponds to the examples shown in Fig. 3. Increasing the feedback gains to both muscles clearly alfects the instability of the system (the tendency for it to oscillate). Initially. the feedback shortens the twitch response to a single impulse and decreases the tendency for mass-spring oscillations at high frequency (upper part of Pig. 4.4). With more feedback lower frequency reflex oscillations are observed, which may decay away or grow depending on the magnitude of the feedback gain (lower p;)rt 01 Fig. 4A). The stability of the system can be quantified from equation (2) using the parameter p. The more negative the value of p. the more rapidly will the oscillation decay. A positive value of p represents an exponentially growing oscillation (i.e. an unstable system). Figure 4(B) shows that the frequency of the reflex oscillation changes very little over a range of feedback gains, where the system goes from being very stable to very unstable. This agrees well with experimental observations on human elbow tremor,?“.?’
I
0 2
‘;‘ 0
’
ZO El Yz u L +D 5.8 A I
0 2
I
0
0
0.12 Time
0 24 ($1
Fig. 3. (A) A single muscle acting against a negiigibie mass produces a contraction in response to a signal impulse which rises and falls smoothly. (B) Increasing the mass to 0.3 kg produces a rapidly damped oscillation on the failing phase of the contraction. As the stiffness of the antagonist muscle is increased to a value (C) equal to or (D) 5 times as great as that of the agonist, these oscillations become increasingly prominent. All such mass-spring oscillations eventually die away in the absence of feedback. An impulse of unit area (Dirac ~-function) has been used as the input in this and subsequent Figures. Since the system is assumed to be linear larger or smaller responses could be obtained by simple scaling of the input.
the internal length changes of muscles U,(E), rather than the movement of the load u(t). This formulation is used because stimulation of muscles under isometric conditions [u(t) is constant] still produces large changes in feedback from muscle spindle receptors as a result of internal shortening and lengthening. An upward movement represents a shortening of muscle 1 and hence a negative sign is included in these equations and in Fig. 2. In contrast, an upward movement lengthens muscle 2, so the sign is positive. In the present calculations there are no inputs to y-motoneurons, so the values of m,. ml and their
Inputs IO ;‘-motorrrurons. Whether an input is given via the CL- or the y-pathway does not affect the stability of the system.” The stability is determined by the feedback pathway. rather than the inputs. However. the nature of the input will affect the speed and frequency components of the response. This is due to the delays in the feedback pathway, as well as the frequency response of the muscle receptors (equations 4 and 5) and the intrafusal muscle fibers Z. Figure 5 shows responses of the system to impulse inputs via the E- and y-pathways to the two muscles. Note that since the stiffness of the two muscles and hence the visco-elastic time constants are different. the responses to inputs in the x, and X: pathways have quite different time courses. However, in both cases the inputs to the y-pathways produce a slower. smoother response. The parameters of both intrafusal muscles (Z, and ZJ were equal and identical to those which were based on experiused previously,J’ mentally measured values.’ The gains of the y-pathways were adjusted to give a response of similar amplitude to that produced by the n-pathway. Also shovvn in Fig. 5 are the responses to coactivation which often occurs of both X- and y-pathways, naturally.“.‘” Because of the delays in the ;!-pathway. the oscillations it generates tend to be out of phase with those generated by the Y-pathway. Thus, coactivation of both pathways leads to a much less oscillatory response than activation of either alone. The implications of this result will be considered further in the Discussion. IA inhibitor! interncuror7.s. In recent years many of the same pathways which supply inputs to
Spinal
modification
-51
235
of muscle
-201 0
0 I6 Time
032
0
1.2 Feedback
(5)
2.4 goin
Fig, 4. (A) Feedback from muscle receptors can reduce the mass-spring osciliations and shorten a contraction (compare interrupted and solid lines in upper part of (A)). As the feedback gain is increased further, decaying or growing reflex oscillations can be produced (inter~pted and solid lines in lower part of (A)). All values are expressed as multiples of the gain that produces periodic oscillations (which neither decay or grow with time). The feedback gains to the two muscles were assumed to be equal. (B) The frequency of the reflex oscillations varies very little over a range of gains where the oscillation changes from having a rapidly decaying (emzO’) to a rapidly increasing (e2a1) rate constant.
-
2
Q+Y 0
-2 ~
0
0.16
~ 0.32 Time
0
0.16
I .32
(s)
Fig. 5. Activating a-motoneurons (upper traces) to one muscle (A) or its antagonist (B) may produce a lesser or greater tendency for oscillation depending on muscle properties and feedback gain. Activation of y-motoneurons (central traces) to the same muscles produces delayed and somewhat smoother responses because of the properties of the intrafusal muscle fibers and feedback pathway. Note that the oscillations are mainly out of phase with those produced by the a-motoneurons so that coactivation of both c(- and y-motoneurons produces markedly less oscillatory responses (lower traces). NSC
I ,,‘I
M
236
R. B. Stein and M. N. O&tzt?ircii
motoneurons have been found to provide inputs to the IA interneurons.‘* Thus, these interneurons sum both central and peripheral influences. Inputs to the IA, interneurons in Fig. 2 will inhibit the x2-motoneurons to an extent which depends on the parameter 02, which has been zero in earlier sections. Similarly, the parameter 0, linking the IA, interneurons to the ~,-motoneurons has been zero. Figure 6 shows that increasing parameters 0: and 0, in parallel, while maintaining a constant impulse input to the x,-motoneurons, makes the system increasingly unstable. The frequency of the resultant oscillations is considerably less than for the direct pathway to r-motoneurons (compare Fig. 6B to Fig. 4R). Note however that the frequency has not been reduced to half as might occur because the pathway was lengthened by a factor of two to go to an antagonist motoneuron and back. The frequency changes relatively little over a range of values where the system goes from being quite stable to quite unstable (Fig. hC). The strength of the connections was assumed to be a simple constant in Fig. 6, because little is known of the dynamics of this pathway. Thus, descending inputs directly onto IA interneurons will produce responses in the model which are identical in form to those produced by inputs to the r-motoneurons they inhibit. There will merely be a change in polarity (because of the inhibition) and in amplitude (depending on the magnitude of the 0,). Coactivation of x-motoneurons together with their IA inhibitory interneurons will automatically produce reciprocal inhibition, which is a feature of many voluntary and rei3ex movements. On the other hand blocking the 1A inhibitory connections (setting the 0, = 0) would permit coactivation of antagonists, which is useful in dealing with unstable loads.‘,‘” Renshau~ inhibition. The functional role of Renshaw inhibition”-“’ IS less clear than that of the IA inhibition. The dynamic properties of the Renshaw pathway have been studied8 and have been included in the model.” To compare the etTects of the two types of inhibition. we have set the parameters (02 and 0,) of the IA inhibition to zero. and varied the parameters of the Renshaw inhibition (0, and 0,) to the ~-motoneurons in parallel. Figure 7 shows that increasing Renshaw inhibition stabilizes the system for all strengths of connections to x-motoneurons which were studied, cells have inhibitory etfects on Renshaw y-motoneurons as well as on r-motoneurons.” In Fig. 8 the effects of including connections to y-motoneurons (0, and 0,) in addition to those to u-motoneurons are shown. The strength of all connections from Renshaw cells were assumed to be equal. With strong connections a new low frequency oscillation is seen (Fig. 8B), whose rate constant increases monotonically. Thus, although the connections from Renshaw cells to ~-motoneurons improve the stability of the system, connections to
y-motoneurons may eventually compromise stability. However, the gain of the pathways from ;‘-motoneurons was somewhat arbitrarily set to a value that gave a similar amplitude of response to that of the pathway from g-motoneurons, as in Fig. 5. The strengths of the pathways from Renshaw cells to the two types of motoneurons were also assumed equal. Neither assumption is probably true in practice, as will be discussed. so the destabilizing effects of the pathway to ~-motoneurons may be small. DISCXJSSION
A number
of interesting
54
-5+
results have emerged
’
’
’
8
*
t
0
’
‘ 0.16
’
’
’
1
I
*
from
+ 0.32
Time(s) 20
j-----
A ..lAI-1
-t
0
20 Strength
of
connection
Fig. 6. (A) Increasing the sWength of connections (Qz = O,,) to IA inhibitory interneurons from 0 (dotted line) to 0.5 (interrupted line) to I.5 (solid line) increases the tendency for oscillation. (B) The frequency of oscillation is Iower than for the feedback directly into r-motoneurons (cf. Fig. 4) and does not vary greatly over a range where (C) the rate constant goes from quite negative (oscillation decays rapidly) to quite positive values (oscillation grows rapidly).
237
Spinal modification of muscle
muscular contraction. With the mass of 0.3 kg used in Fig. 3 the oscillation was at a high frequency, as found for example in the mechanical oscillations of hand tremor.33 At more proximal joints with larger masses, mechanical oscillations may have a frequency at or even below the frequency of reflexly induced tremors.*’ Feedback from muscle receptors can reduce mechan ical oscillations, but tends to produce other oscillations. This result is in good agreement with experimental
0.16
0
0.32
Time(s)
c ,_
u+21
”
-2
”
”
1
1
1
1
I
1
40 Strength
of
,
t
1+
0.32
0.16
0
i-ii -i 0
’
! 80
connection
Fig. 7. Renshaw inhibition to a-motoneurons, in contrast to 1A inhibition, decreases the tendency for oscillation. Compare trace in (B), where 0, = 0, = 50, to that in (A) where the 0, = 0. The frequency of oscillations (C) is relatively unchanged while the rate constant (D) decreases monotonically (oscillations decay more rapidly).
the simulations presented here, which are listed below. Each will be discussed briefly in terms of available or obtainable experimental data. Increasing muscle stifSness increases the tendency for mechanical oscillations. This was illustrated clearly in Fig. 3, where the stiffness of the antagonist muscle was increased. The damping of a mass-spring system decreases with increasing stiffness, so this result represents an extension from an elementary system to the more complex system considered here. Any muscular contraction will increase muscle stiffness,29 so it will be important experimentally to distinguish mechanical oscillations from those produced by reflex or central mechanisms, which may also increase with
_ 7
”
Oi
’
’
’
’
’
-I I
’
’
D
Strength
of
_
connection
Fig. 8. If Renshsw inhibition to y-motoneurons is included, as well as that to a-motoneurons, slower oscillations are seen, when the connectivity is strong (compare the end of the trace in B to that in A). In fact, three frequencies of oscillation(C) contribute to the responses. The rate constant (D) for one decreases monotonically, while the others increase monotonically, as the strength of connections is increased. The higher frequency oscillation with monotonically decreasing rate constant arises from the a-pathway (cf. Fig. 7), while the lower frequency oscillations with monotonically increasing rate constant arise from the y-pathway.
23x
R. B. Stem and M. N. O@~~ttircli
observations. For example. Goodwin ef u/.” trained monkeys to generate constant forces with their jaw muscles. After cutting the sensory supply to these muscles. greater fluctuations in force were found at low frequencies. although oscillations at the presumed reflex frequency were eliminated. Reflex oscillations are quite stable in frequency (Fig. 4), which permits them to be distinguished experimentally from mechanical oscillations.“’ C’ouc~tkation o.wilirrtion.s
of’ r - und ;‘-nmtoneurons
which
would
he producwl
cm
cmc~el
h,~ uctiwting
ritlwr t.r’pe independently. This result may seem surprising initially, since we have shown analytically’“~” that activating one or other set of motoneurons does not affect the stability of the feedback pathways from muscle receptors (except insofar as there are strong connections from Renshaw cells back to ;,-motoneurons). However. coactivation reduces oscillations. because the signals are delayed in the :,-pathway so that they are out of phase with reflex oscillations resulting from activation of the r-pathway. Note that the dramatic cancellation seen in the bottom part of Fig. 5B results from choosing inputs to the two pathways which produced responses of similar amplitudes. Whether this degree of cancellation would occur in natural movements remains to be tested. Experimentally, both type of motoneurons do tend to be excited together.“” rather than having ;‘-motoneurons activated before r-motoneurons. as wax once thought. The role of coactivation in the control of movement has been questioned recently.“.“.” so it is worth considering a number 01 hypotheses. including the cancellation hypothesis discussed in this paragraph. IA
inhihitor~~
interneurons
1rrtion.r (It 11 lower ,freyuency rc+e.\-. This result
tend to produce than the normal
oscilstretch
can be seen clearly by comparing the traces in Figs 4 and 6. The frequency decreases. as the strength of the connections to IA inhibitory neurons is increased, although the change is rather modest (less than l5”;, over the range of values shown in Fig. 6B). One possible role for IA interneurons is that they are involved in regulating the degree of co-contraction. When pathways from IA interneurons are active, alternation of antagonist pairs of muscles will tend to occur. However. subjects cocontract voluntarily, when working against an un-
stable load.’ The co-contraction of antagonists will tend to Increase the stifrness about the joint for a given force level. Increasing stiffness by itself might tend to increase the instability about the joint (see Fig. I). However, if the effect of connections from I A interneurons is reduced by some means in producing co-contraction. stability will be enhanced. T/w
minectiom
fion1
Rrnshu,r~
WII.!
to
As shown in Fig. 7, the action of the inhibitory pathway from Renshaw cells is opposite to that from IA interneurons. Renshaw cells have been suggested to play a role in the generation of physiological tremor.” although there is littlc or no direct experimental evidence for this. In fact, within the linear range investigated here, the effects of Renshaw cells will be to reduce the tendency for high frequency oscillations such as tremor. Adam et cd.’ showed experimentall) that Renshaw cells reduce the response or motoneurons to high frequency noise (50 Hz) and decouple the discharges of motoneurons on a millisecond time scale. Note that the pathway from Renshaw cells to ;*-motoneurons has an opposite effect to that from Renshaw cells to r-motoneurons. In Fig. 8 several different types of oscillations are seen when the connections to both J- and ;t-motoneurons are increased in parallel. The highest frequency oscillation is similar to that seen in Fig. 7 and decays increasingly rapidly as the strength of connections is increased. This oscillation is due to the connection to r-motoneurons. The other. lower frequency oscillations are due to the connection of Renshaw cells to y-motoneurons. This pathway has only recently been studied experimentally.” and its physiological role and strength, relative to the better known pathways to r-motoneurons, requires further investigation. It would be of interest to determine whether the conever become strong nections to y-motoneurons enough to produce oscillations under natural conditions. In conclusion, the model presented here gives a number of interesting results. which agree well with experimental data, to the extent that comparisons can be made. The model also suggests several ideas which should be experimentally testable, and it can be improved as more precise data on various spinal connections become available. x-mttoneuro~l.\
tend to prcr.cwt o.willutions.
REFERENCES
I. Adam D., Windhorst V. and Inbar G. F. (1978) The effects of recurrent inhibitlon on the cross-correlated tiring patterns of motoneurons (and their relation to signal transmission in the spinal cord&muscle channel). Viol C~~h~m. 29, 229-235. 2. Akazawa K.. Milner T. and Stein R. B. (1983) Modulation of reflex EMG and stiffness in response to stretch of human linger muscle. J. Neurophy.sio/. 49, 16-27. 3. Anderson B. F.. Lennerstrdnd G. and Thoden U. (1968) Response characteristics of muscle spindle endings at constant length to variations in fusimotor activation. Actor physiol. .srand. 74, 3Ol- 318. 4. Appenteng K., Prochazka A.. Proske V. and Wand P. (1982) Effect of fusimotor stimulation on la discharge during shortening of the cat soleus muscle at different speeds. J. Physiol.. Land. 329, 509.-526. 5. Bawa P., Mannard A. and Stein R. B. (1976) Predictions and experimental tests of a viaco-elastic muscle model using elastic and inertial loads. Biol C~&v-n. 22, 139.-145.
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modification
of muscle
239
during fictitious 6. Berkenbilt M. B., Deliagina T. G., Orlovsky G. N. and Feldman A. G. (1980) Activity of motoneurons scratch reflex in the cat. Brain Rex 193, 427-438. on the response 7. Brown M. C., Goodwin G. M. and Matthews P. B. C. (1969) After-effects of fusimotor stimulation of muscle spindle primary efferent endings. J. Physiol.. Lond. 205, 677-694. Viol. 8. Cleveland S. and Ross H.-G. (1977) Dynamic properties of Renshaw cells: frequency response characteristics. Cyhern. 27, 175-184. New York. 9. D’Azzo J. J. and Houpis C. H. (1966) Feedback Control System Analysis and Synthesis. McGraw-Hill, K.-E., Hagglund J. V. and Wallin E. V. (1982) The ‘late’ reflex responses to muscle stretch: the 10. Eklund G., Hagbarth ‘resonance hypothesis’ vs the ‘long-loop hypothesis’. J. Physiol., Lond. 326, 79-90. J. E. (1976) Motor-unit activity responsible for the 8 to 12Hz component of human II. Elble R. J. and Randall physiological finger tremor. J. Neurophysiol. 39, 370-383. of the recurrent inhibition of z- and y-motoneurones in the 12. Ellaway P. H. and Murphy P. R. (1981) A comparison cat. J. Physiol., Lond. 315, 43-58. 13. Goodwin G. M., Hoffman D. and Luschei E. S. (1978) The strength of the reflex response to sinusoidal stretch of monkey jaw closing muscles during voluntary contraction. J. Physiol., Lond. 279, 81-l I I. 14. Grillner S. (1981) Control of locomotion in bipeds, tetrapods and fish. In Motor Control (ed. Brooks V. B.) Am. Physiol. Sot. Handbook of Physiology Sect. I, Vol. 2, pp. 117991236. Williams and Wilkins, Baltimore. model of skeletal muscle. Biol. Cybern. 25, 103-I 19. 15. Hatze H. (1977) A myocybernetic of antagonist muscles. IEEE Trans. Auto. 16. Hogan N. (1983) Adaptive control of mechanical impedance by coactivation Conlrol. 17. Houk J. C. and Rymer W. Z. (1981) Neural control of muscle length and tension. In Motor Control (ed. Brooks V. B.) Am. Plzysiol. So<. Handbook of Physiology Sect. I, Vol. 2, pp. 2577233. Williams and Wilkins, Baltimore. 18. Hultborn H. (1977) Transmission in the pathway of reciprocal Ia inhibition to motoneurones and its control during the tonic stretch reflex. Prog. Brain Res. 44, 235-254. 19. Huxley A. F. (1980) Reflections on Muscle: Sherrington Lectures XIV. Liverpool University Press. Liverpool. 20. Joyce G. C. and Rack P. M. H. (1974) The effects of load and force on tremor at the normal human elbow joint. J. Physiol., Lond. 240, 375-396. 21. Matthews P. B. C. (1972) Mammalian Muscle Receptors and Their Cen/ral Actions. Arnold, London. 22. Matthews P. B. C. and Muir R. B. (1980) Comparison of electromyogram spectra with force spectra during human elbow tremor. J. Physiol., Lond. 302, 427441. 23. Matthews P. B. C. and Stein R. B. (1969) The sensitivity of muscle spindle afferents to small sinusoidal changes in length. J. Physiol., Lond. 200, 723-743. 24. Menzies J. E., Albert C. P. and Jordan L. M. (1978) Testing a model for the spinal locomotor generator. Sot. Nrurosci. Abs. 4, 1219. 25. Merton P. A. (1953) Speculations on the servo-control of movement. In CIEA Found. Symp., The Spinul Cord (ed. Wolstenholme G. E. W.) pp. 2477255. Churchill, London. 26. Miller S. and Scott P. D. (1977) The spinal locomotor generator. Expl. Brain Res. 30, 387403. 27. Nichols T. R. and Houk J. C. (1976) The improvement in linearity and the regulation of stiffness that results from the actions of the stretch reflex. J. Neurophysiol. 39, 119-142. 28. Oguztoreli M. N. and Stein R. B. (1983) A model for the spinal control of antagonistic muscles. J. theor. Neurohiol. In press. M. N. and Stein R. B. (1976) The effects of multiple reflex pathways on the oscillations in neuro-muscular 28a O&ztoreli systems. J. math. Biol. 3. 87-101. F. and Stein R. B. (1981) Nonlinear summation of contractions in cat muscles--II. The later facilitation 29. Parmiggiani and stiffness changes. J. gen. Physiol. 78, 295-31 I. of the Physiological 30. Phillips G. G. and Porter R. (1977) Corticospinal Neurones: Their Role in Mooement. Monograph Sot. No. 34. Academic Press, London. description of linear behavior of mammalian muscle spindles. 31. Poppele R. E. and Bowman R. J. (1980) Quantitative J. Neurophysiol. 33, 59-72. 32. Renshaw B. (1941) Influence of discharge of motoneurons upon excitation of neighboring motoneurons. J. Neurophvsiol. 4, 1677 183. mechanism as a basis for normal postural tremor. J. uppl. 33. Rietz R. R. and Stiles R. N. (1974) A viscoelastic-mass Phjxiol. 37, 852-860. 34. Rymer W. Z. and Hasan Z. (1980) Absence of force-feedback regulation in soleus muscle of the decerebrate cat. Brain Res. 184, 203-209. 35. Schafer S. S. (1967) The acceleration response of a primary muscle spindle ending to ramp stretch of the extrafusal muscle. Experientiu 23, 1026-1027. 36. Stein R. B. (1974) The peripheral control of movement. Physiol. Reo. 54, 215-243. 37. Stein R. B. (1982) What muscle variable(s) does the nervous system control in limb movements? Behuc. Bruin Sri. 5, 535-577. 38. Stein R. B. and Lee R. G. (1981) Tremor and clonus. In Motor Control (ed. Brooks V. B.) Am. Physiol. Sot. Handbook of Ph,vsio/ogy Sect. I, Vol. 2, pp. 325-343. Williams and Wilkins, Baltimore. 39. Stein R. B. and Oguztoreli M. N. (1976) Does the velocity sensitivity of muscle spindles stabilize the stretch reflex? Biol. Cyhern.. 23, 219-228. 40. Stein R. B. and Oguztoreli M. N. (1978) Reflex involvement in the generation and control of tremor and clonus. In Ph.vsiologicul Tremor. Pathological Tremors and Clonus (ed. Desmedt J. E.) Prog. clin. Neurophysiol. 5, 28-50. Karger, Basel. 41. Stein R. B. and Oguztoreli M. N. (1981) The role of y-motoneurons in mammalian reflex systems. Biol. C’yhern. 39, 171-179. 42. Stein R. B. and Oguztoreli M. N. (1983) A model of whole muscles incorporating functionally important nonlinearities. In Non-lineur Phenomena in Murhemarical Sciences (ed. Lakshimikantham V.) pp. 749-766. Academic Press, New York. ^ 43. Stein R. B. and Parnnggtam F. (11)X1) Nonhnear summation of contractions in cat muscles--I. The early depression. J. gen. Physiol. 78, 217-293.
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44. Vallbo A. B. (1971) Muscle spindle response at the onset of isometric voluntary contractions between fusimotor and skeletomotor effects. J. Physiol.. Land. 218, 405631. 45. Vallbo A. B. and Hulliger M. (1981) Independence of skeletomotor and fusimotor activiry 17G- 180. 46. Willis W. D. (1971) The case for the Renshaw cell. Bruin Behor. Ed. 4, 5--57.
in man. Time difference in man’? Bruin
Rrs. 223,
NOTIFICATION
~306-4522/84
1984
$3.00 + 0.00
Pcrgamon Press Ltd C 1984IBRO
OF MUSCLE RESPONSES CIRCUITRY
BY SPINAL
R. B. STEIN and M. N. OC~UZTGRELI Departments of Physiology and Mathematics, University of Alberta, Edmonton, Canada T6G 2H7 Abstract-Recently, we developed a model based on ex~~rncntal data, which includes a pair of antagonistic muscles, a general load against which the muscles act, feedback pathways from muscle sense organs and spinal inhibitory circuits involving IA interneurons and Renshaw cells. Descending inputs can activate the model through combinations of inputs to a-motoneurons, y-motoneurons (via intrafusal muscles and their feedback pathways) or the IA interneurons. The role of each of these connections is analysed here in terms of its effect on the response of the muscles to impulse inputs, with particular interest in the effects on the overall stability of the systems. Increasing muscle stiffness or feedback from muscle receptors tends to produce high frequency oscillations. Coactivation of c1- and ~-motoneurons can lead to cancellation of oscillations, because of delays in the effects of y-motoneurons on contraction. Connections of IA inhibitory interneurons onto antagonist motoneurons accentuate the oscillations, Inhibitory connections from Renshaw cells onto a-motoneurons tend to prevent oscillations, whereas the concentrations onto y-motoneurons may produce them
The motor
system is arranged
in a hierarchical
fash-
ion, Spinal reflexes modify the most peripheral elements (i.e. the muscles). The spinal cord can also produce some of the basic patterns, such as are required for locomotiont~ and scratching,6 which are in turn controlled from brain stem structures. Finally, higher centres such as the motor cortex have executive control of the whole motor system.‘O Although such general ideas are widely accepted, the details and specific roles remain obscure. For example, the role of even the simplest spinal pathway, the monosynaptic reflex remains controversial.” It has been suggested that this reflex is used as part of a follow-up length servo,25 as part of a servo-assisted system for control of length2’,‘” or together with force feedback in the control of stiffness.” The role of force feedback from Golgi tendon organs is even more obscure. its gain is close to zero in decerebrate preparations where it has been measured accurateJy,34 although it might be greater in other preparations. Further experimental work is required to settle some of these controversies, but detailed analysis of simple mathematical models of muscles and reflexes would be useful to examine possible roles of reflexes in modifying the control of muscles. Such a model was proposed in a recent paperZX and some results were derived analytically. However, the number of analytic results which can be obtained are limited, and further results must be derived from computer simulations. Various simulations will be presented in this paper for the whole model and some simplified versions of it, so that the role of each pathway on the stability of the system can be examined. A tendency for instability would be evident behaviorally as a tremor of the limb.
In order to keep this paper relatively short, the full equations and definitions of terms will not be included. Details can be found in Oguztiireli and Stein2* However, the assumptions underlying the model and the terminology will be discussed briefly before proceeding with the results.
MODEL
The model we have used consists of two parts: a pair of antagonistic muscles and the spinal circuitry controlling these muscles. The muscles are shown in Fig. 1 within dotted lines. Each consists of an active force-generating element a(t), parallel and internal series elastic elements (with stiffness k, and k,, respectively) and a dashpot with viscosity B. The muscles work against a generalized load consisting of a mass M, an external spring of stiffness k, and a dashpot with viscosity D. The active force generating ability decays exponentially following a neural input, so the muscles behave as linear second order systems (with a viscoelastic time constant and a time constant for the decay of the active state). Despite its simplicity, such a model accounts well for a variety of muscle responses.5 It has been suggested that reflexes are adapted to compensate for some nonlinearities in muscle properties,27 so exclusion of nonlinearities from the present model may seem like a serious omission. However, there are several reasons for this omission. Firstly, a nonlinear system will often behave linearly over a small range, such as would occur during normal, physiological tremor.‘* Secondly, despite recent experimental’9,29,43 and theoretical’5,28 efforts to characterize the nonlinearities, a completely
231
232
R. B. Stein and M. N. Oguztiireli
_. . ..__________________________-_____.___~_~_~._ the connections between r-motoneurons and Ren-_-.-_-_-_-_-_---.-_-_-_-_-_-_-_---_-.-.-.-_-.-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-.-_-_-. ____. . _ shaw cells has been well described’ and is included in I
I
I
Fig. I. A viscoelastic model of antagonistic muscles and a generalized load consisting of a mass M, a dashpot with viscosity D and a spring of stiffness k,. Each muscle consists of an active force-generating element a(t), a dashpot B and parallel k, and internal series k, elastic elements. The y(t) measure positions with respect to a fixed point, whereas the U(I) are deviations from the steady-state position. External forces,f(t) may also be applied. Further details in the text and in Oguztiireli and Stein.‘R
satisfactory nonlinear model is not available. Finally, the methods available for linear systems are much more powerful than those for nonlinear systems, and a linear analysis can serve as the first stage for a later nonlinear analysis. For example, a steady force will change the position of the external load b(t) in Fig. I] and the elements within the muscles yj(t), j = 1. 2. However, the deviations u(r) and u,(t) about the new steady-state positions, in response to a perturbation, will not be affected. Spinal connections to and from the muscles are shown in Fig. 2. Neural inputs n come directly from r-motoneurons and produce muscle responses which interact with the load to produce movement. Length and velocity information g is fed back via muscle spindles to the a-motoneurons and to the 1A interneurons. These interneurons inhibit (-) agonist a-motoneurons and hence provide a natural link between the muscle in addition to the mechanical linkage through the muscles acting on the load. Force feedback from Golgi tendon organs is ignored, because of uncertainties about its efficacy (see Introduction). Renshaw cells R provide negative feedback with a characteristic lag to the large a-motoneurons as well as to the smaller y-motoneurons.‘* The dynamics of
the model.” The y-motoneurons produce negligibly small amounts of force, but do influence contractions by their actions on intrafusal muscles (with properties z), which affect the feedback from muscle receptors. Other connections between spinal neurons could obviously be added to the model. For example. Miller and Scott” considered inhibitory connectiona between IA inhibitory interneurons and from Renshaw cells onto IA inhibitory interneurons. They felt that these connections might contribute to large, low frequency oscillations such as walking (but see Menzies of ul.“). These spinal connections have been omitted in the present paper. since their dynamics have not been determined. and such large oscillations would exceed the linear range considered here. The contractions of the two muscles will also be afPected by descending inputs which may be focussed in different combinations on r-motoneurons. y-motoneurons and IA inhibitory interneurons to one or both muscles. Since the system has been assumed to be linear for present purposes. a brief input (Dirac (3-function or unit impulse) is sutticient to characterize the response to any input. One point of particular interest is the effect of the central connections on the stability of the entire system. The strength of these connections are denoted in Fig. 2 by the 0,. We will begin by setting the strength of these connections to zero to examine responses of the muscles and load in isolation. Then non-zero values of the 0, will be systematically introduced in selected pathways to examine the effect of these pathways on the overall responses. Analytical solutions for the entire system and
Fig. 2. Muscle receptors compare the actual movement within each muscle with the signals from ;‘-motoneurons. which are filtered by the intrafusal muscle properties Z. The feedback signals excite I-motoneurons and IA interneurons in the spinal cord, which inhibit antagonist motoneurons. a-motoneurons excite Renshaw cells which in turn inhibit both a- and y-motoneurons. The 0, represent the strength of the spinal connections. Inputs from other parls of the central nervous system may act on either type of motoneuron or the IA interneurons. Detailed equations and analysis are given in O&rtiireli and Stein.”
233
Spinal modification of muscle various simplifications of it have been published inclined previously,28 and the more mathematically reader should consult the previous paper for these solutions, as well as for mathematical details of the model. This paper will deal only with numerical computations of two types: firstly, from the transfer function for the system, the roots of the equation can first be determined numerically using methods which have been described.” Secondly, the inverse Laplace transforms can be computed to give the response to an impulse input. The roots of the transfer function can take several forms: a real number p, which corresponds to a solution of the form u = uOep’
(1)
where u0 is the initial value of u and the response grows or decays exponentially, depending on whether the rate constant p is positive or negative. Alternatively, the root may be a complex number, which corresponds to a solution u = uOeP’sin (2rrqt + I$)
(2)
where q is the frequency in Hz and C#Jis the phase of an oscillation, which will grow or decay exponentially depending on the sign of p, as above. More complex forms are also possible, if more than one root has the same value. For example, if there are two roots with a value p, the response will be of the form u = u,te”’ rather than the form given by (1) above. examples are given in standard texts.’
(3) Other
RESULTS
Inputs to a-motoneurons. We will begin with the simplest example and systematically add complexity to examine the effects of each element of the system on its overall performance. The simplest example consists of a single impulse (Dirac a-function) to one set of motoneurons (c(, in Fig. 2) in the absence of feedback. Feedback can be eliminated from the model by setting the gain of the pathway from muscle receptors (g) equal to zero, as well as the values of all the 0,. This abolishes central feedback from the Renshaw cells, for example, as well as peripheral feed back. If the load attached to the muscle contains a significant mass as well as a spring, there is some tendency for a damped oscillation to occur in response to a single impulse to the a-motoneurons (compare Fig. 3B, where M = 0.3 kg to Fig. 3A, where the mass is negligible). The parameters of muscle 1 have been set to standard values obtained from experimental data for cat plantaris muscles.28” In Fig. 3, the stiffness of the antagonist muscle 2 is varied from zero (Fig. 3B) to a value five times as large as that of muscle 1 (Fig. 3D). Note that if the
stiffness of muscle 2 is zero, it essentially decouples this muscle from the load (i.e. no force will be transmitted, however much the load moves). The responses of single muscles with and without feedback connections have been considered extensively in the past.40 Increasing the stiffness of the antagonist muscle in Fig. 3 increases the tendency for oscillations to occur at a high frequency (about 25 Hz). Such highfrequency mass-spring oscillations have been observed for example in hand tremor.33 The stiffness of isolated muscles does vary greatly with the level of activity,29 so the effects seen in Fig. 3 will be observed as the steady level of activity in muscle 2 varies. Muscle viscosity may also vary with the level of activity. However, if both the viscosity and stiffness of muscle are increased in parallel, the increased tendency for high frequency oscillations still occurs for a large range of parameters. Feedback ,fiom muscle receptors. In all subsequent calculations the stiffness of muscle 2 will be fixed at a value half that of muscle 1. Although the choice of this particular ratio is somewhat arbitrary, muscle 2 works with gravity while muscle 1 is an antigravity muscle (Fig. 1). In general, antigravity muscles are stronger simply because they must lift the mass of the body against the force of gravity. The form of the feedback used in this paper is
Gl + F,, t,[ - zi,(t- t,) + +I,(t - t,)]
g, 0) = FO,]-u,
(t - t,) + m,(t -
g2(t) = Fo2]uz(t - t2) + m2(r -
+
F0272[SZ(t
-
(4)
fdl f2)
+
h,(t
-
tJ1. (9
This form is somewhat simplified from that considered analytically2* in that we are only including (1) a single time delay r, for each pathway and (2) length and velocity (e.g. ti,) feedback. The delay used in all the simulations is t, = t, = 30 ms, corresponding to a spinal pathway. Longer latency pathway might also be considered.28” However, Eklund et al.” have shown experimentally that many phenomena, previously attributed to longer latency central pathways, can be accounted for by bursts of repetitive activity in muscle receptors acting through shorter latency, spinal pathways. In all the simulations, the velocity sensitivity of the feedback [the parameters T, and z2 in equations (4) and (5)] was set to 0.1 s. This corresponds to the value measured experimentally,23,3’ although greater stability could be achieved with smaller values of this parameter. 39 The effect of higher order derivatives, such as acceleration terms, have also been considered previously. 39 Muscle receptors do show an apparent acceleration sensitivity under some circumstances,35 but its nature is complex’ and its magnitude is small enough that it will not affect the general conclusions reached here. The feedback in equations (4) and (5) is based on
R. B. Stein and M. N. Oguzttireli
234
2
derivatives are zero. This restriction will be removsed in the next section. Figure 4(A) shows the responses of the system to single impulses to the r,-motoneurons with and without feedback. With a feedback gain equal to zero, the response corresponds to the examples shown in Fig. 3. Increasing the feedback gains to both muscles clearly alfects the instability of the system (the tendency for it to oscillate). Initially. the feedback shortens the twitch response to a single impulse and decreases the tendency for mass-spring oscillations at high frequency (upper part of Pig. 4.4). With more feedback lower frequency reflex oscillations are observed, which may decay away or grow depending on the magnitude of the feedback gain (lower p;)rt 01 Fig. 4A). The stability of the system can be quantified from equation (2) using the parameter p. The more negative the value of p. the more rapidly will the oscillation decay. A positive value of p represents an exponentially growing oscillation (i.e. an unstable system). Figure 4(B) shows that the frequency of the reflex oscillation changes very little over a range of feedback gains, where the system goes from being very stable to very unstable. This agrees well with experimental observations on human elbow tremor,?“.?’
I
0 2
‘;‘ 0
’
ZO El Yz u L +D 5.8 A I
0 2
I
0
0
0.12 Time
0 24 ($1
Fig. 3. (A) A single muscle acting against a negiigibie mass produces a contraction in response to a signal impulse which rises and falls smoothly. (B) Increasing the mass to 0.3 kg produces a rapidly damped oscillation on the failing phase of the contraction. As the stiffness of the antagonist muscle is increased to a value (C) equal to or (D) 5 times as great as that of the agonist, these oscillations become increasingly prominent. All such mass-spring oscillations eventually die away in the absence of feedback. An impulse of unit area (Dirac ~-function) has been used as the input in this and subsequent Figures. Since the system is assumed to be linear larger or smaller responses could be obtained by simple scaling of the input.
the internal length changes of muscles U,(E), rather than the movement of the load u(t). This formulation is used because stimulation of muscles under isometric conditions [u(t) is constant] still produces large changes in feedback from muscle spindle receptors as a result of internal shortening and lengthening. An upward movement represents a shortening of muscle 1 and hence a negative sign is included in these equations and in Fig. 2. In contrast, an upward movement lengthens muscle 2, so the sign is positive. In the present calculations there are no inputs to y-motoneurons, so the values of m,. ml and their
Inputs IO ;‘-motorrrurons. Whether an input is given via the CL- or the y-pathway does not affect the stability of the system.” The stability is determined by the feedback pathway. rather than the inputs. However. the nature of the input will affect the speed and frequency components of the response. This is due to the delays in the feedback pathway, as well as the frequency response of the muscle receptors (equations 4 and 5) and the intrafusal muscle fibers Z. Figure 5 shows responses of the system to impulse inputs via the E- and y-pathways to the two muscles. Note that since the stiffness of the two muscles and hence the visco-elastic time constants are different. the responses to inputs in the x, and X: pathways have quite different time courses. However, in both cases the inputs to the y-pathways produce a slower. smoother response. The parameters of both intrafusal muscles (Z, and ZJ were equal and identical to those which were based on experiused previously,J’ mentally measured values.’ The gains of the y-pathways were adjusted to give a response of similar amplitude to that produced by the n-pathway. Also shovvn in Fig. 5 are the responses to coactivation which often occurs of both X- and y-pathways, naturally.“.‘” Because of the delays in the ;!-pathway. the oscillations it generates tend to be out of phase with those generated by the Y-pathway. Thus, coactivation of both pathways leads to a much less oscillatory response than activation of either alone. The implications of this result will be considered further in the Discussion. IA inhibitor! interncuror7.s. In recent years many of the same pathways which supply inputs to
Spinal
modification
-51
235
of muscle
-201 0
0 I6 Time
032
0
1.2 Feedback
(5)
2.4 goin
Fig, 4. (A) Feedback from muscle receptors can reduce the mass-spring osciliations and shorten a contraction (compare interrupted and solid lines in upper part of (A)). As the feedback gain is increased further, decaying or growing reflex oscillations can be produced (inter~pted and solid lines in lower part of (A)). All values are expressed as multiples of the gain that produces periodic oscillations (which neither decay or grow with time). The feedback gains to the two muscles were assumed to be equal. (B) The frequency of the reflex oscillations varies very little over a range of gains where the oscillation changes from having a rapidly decaying (emzO’) to a rapidly increasing (e2a1) rate constant.
-
2
Q+Y 0
-2 ~
0
0.16
~ 0.32 Time
0
0.16
I .32
(s)
Fig. 5. Activating a-motoneurons (upper traces) to one muscle (A) or its antagonist (B) may produce a lesser or greater tendency for oscillation depending on muscle properties and feedback gain. Activation of y-motoneurons (central traces) to the same muscles produces delayed and somewhat smoother responses because of the properties of the intrafusal muscle fibers and feedback pathway. Note that the oscillations are mainly out of phase with those produced by the a-motoneurons so that coactivation of both c(- and y-motoneurons produces markedly less oscillatory responses (lower traces). NSC
I ,,‘I
M
236
R. B. Stein and M. N. O&tzt?ircii
motoneurons have been found to provide inputs to the IA interneurons.‘* Thus, these interneurons sum both central and peripheral influences. Inputs to the IA, interneurons in Fig. 2 will inhibit the x2-motoneurons to an extent which depends on the parameter 02, which has been zero in earlier sections. Similarly, the parameter 0, linking the IA, interneurons to the ~,-motoneurons has been zero. Figure 6 shows that increasing parameters 0: and 0, in parallel, while maintaining a constant impulse input to the x,-motoneurons, makes the system increasingly unstable. The frequency of the resultant oscillations is considerably less than for the direct pathway to r-motoneurons (compare Fig. 6B to Fig. 4R). Note however that the frequency has not been reduced to half as might occur because the pathway was lengthened by a factor of two to go to an antagonist motoneuron and back. The frequency changes relatively little over a range of values where the system goes from being quite stable to quite unstable (Fig. hC). The strength of the connections was assumed to be a simple constant in Fig. 6, because little is known of the dynamics of this pathway. Thus, descending inputs directly onto IA interneurons will produce responses in the model which are identical in form to those produced by inputs to the r-motoneurons they inhibit. There will merely be a change in polarity (because of the inhibition) and in amplitude (depending on the magnitude of the 0,). Coactivation of x-motoneurons together with their IA inhibitory interneurons will automatically produce reciprocal inhibition, which is a feature of many voluntary and rei3ex movements. On the other hand blocking the 1A inhibitory connections (setting the 0, = 0) would permit coactivation of antagonists, which is useful in dealing with unstable loads.‘,‘” Renshau~ inhibition. The functional role of Renshaw inhibition”-“’ IS less clear than that of the IA inhibition. The dynamic properties of the Renshaw pathway have been studied8 and have been included in the model.” To compare the etTects of the two types of inhibition. we have set the parameters (02 and 0,) of the IA inhibition to zero. and varied the parameters of the Renshaw inhibition (0, and 0,) to the ~-motoneurons in parallel. Figure 7 shows that increasing Renshaw inhibition stabilizes the system for all strengths of connections to x-motoneurons which were studied, cells have inhibitory etfects on Renshaw y-motoneurons as well as on r-motoneurons.” In Fig. 8 the effects of including connections to y-motoneurons (0, and 0,) in addition to those to u-motoneurons are shown. The strength of all connections from Renshaw cells were assumed to be equal. With strong connections a new low frequency oscillation is seen (Fig. 8B), whose rate constant increases monotonically. Thus, although the connections from Renshaw cells to ~-motoneurons improve the stability of the system, connections to
y-motoneurons may eventually compromise stability. However, the gain of the pathways from ;‘-motoneurons was somewhat arbitrarily set to a value that gave a similar amplitude of response to that of the pathway from g-motoneurons, as in Fig. 5. The strengths of the pathways from Renshaw cells to the two types of motoneurons were also assumed equal. Neither assumption is probably true in practice, as will be discussed. so the destabilizing effects of the pathway to ~-motoneurons may be small. DISCXJSSION
A number
of interesting
54
-5+
results have emerged
’
’
’
8
*
t
0
’
‘ 0.16
’
’
’
1
I
*
from
+ 0.32
Time(s) 20
j-----
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-t
0
20 Strength
of
connection
Fig. 6. (A) Increasing the sWength of connections (Qz = O,,) to IA inhibitory interneurons from 0 (dotted line) to 0.5 (interrupted line) to I.5 (solid line) increases the tendency for oscillation. (B) The frequency of oscillation is Iower than for the feedback directly into r-motoneurons (cf. Fig. 4) and does not vary greatly over a range where (C) the rate constant goes from quite negative (oscillation decays rapidly) to quite positive values (oscillation grows rapidly).
237
Spinal modification of muscle
muscular contraction. With the mass of 0.3 kg used in Fig. 3 the oscillation was at a high frequency, as found for example in the mechanical oscillations of hand tremor.33 At more proximal joints with larger masses, mechanical oscillations may have a frequency at or even below the frequency of reflexly induced tremors.*’ Feedback from muscle receptors can reduce mechan ical oscillations, but tends to produce other oscillations. This result is in good agreement with experimental
0.16
0
0.32
Time(s)
c ,_
u+21
”
-2
”
”
1
1
1
1
I
1
40 Strength
of
,
t
1+
0.32
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’
! 80
connection
Fig. 7. Renshaw inhibition to a-motoneurons, in contrast to 1A inhibition, decreases the tendency for oscillation. Compare trace in (B), where 0, = 0, = 50, to that in (A) where the 0, = 0. The frequency of oscillations (C) is relatively unchanged while the rate constant (D) decreases monotonically (oscillations decay more rapidly).
the simulations presented here, which are listed below. Each will be discussed briefly in terms of available or obtainable experimental data. Increasing muscle stifSness increases the tendency for mechanical oscillations. This was illustrated clearly in Fig. 3, where the stiffness of the antagonist muscle was increased. The damping of a mass-spring system decreases with increasing stiffness, so this result represents an extension from an elementary system to the more complex system considered here. Any muscular contraction will increase muscle stiffness,29 so it will be important experimentally to distinguish mechanical oscillations from those produced by reflex or central mechanisms, which may also increase with
_ 7
”
Oi
’
’
’
’
’
-I I
’
’
D
Strength
of
_
connection
Fig. 8. If Renshsw inhibition to y-motoneurons is included, as well as that to a-motoneurons, slower oscillations are seen, when the connectivity is strong (compare the end of the trace in B to that in A). In fact, three frequencies of oscillation(C) contribute to the responses. The rate constant (D) for one decreases monotonically, while the others increase monotonically, as the strength of connections is increased. The higher frequency oscillation with monotonically decreasing rate constant arises from the a-pathway (cf. Fig. 7), while the lower frequency oscillations with monotonically increasing rate constant arise from the y-pathway.
23x
R. B. Stem and M. N. O@~~ttircli
observations. For example. Goodwin ef u/.” trained monkeys to generate constant forces with their jaw muscles. After cutting the sensory supply to these muscles. greater fluctuations in force were found at low frequencies. although oscillations at the presumed reflex frequency were eliminated. Reflex oscillations are quite stable in frequency (Fig. 4), which permits them to be distinguished experimentally from mechanical oscillations.“’ C’ouc~tkation o.wilirrtion.s
of’ r - und ;‘-nmtoneurons
which
would
he producwl
cm
cmc~el
h,~ uctiwting
ritlwr t.r’pe independently. This result may seem surprising initially, since we have shown analytically’“~” that activating one or other set of motoneurons does not affect the stability of the feedback pathways from muscle receptors (except insofar as there are strong connections from Renshaw cells back to ;,-motoneurons). However. coactivation reduces oscillations. because the signals are delayed in the :,-pathway so that they are out of phase with reflex oscillations resulting from activation of the r-pathway. Note that the dramatic cancellation seen in the bottom part of Fig. 5B results from choosing inputs to the two pathways which produced responses of similar amplitudes. Whether this degree of cancellation would occur in natural movements remains to be tested. Experimentally, both type of motoneurons do tend to be excited together.“” rather than having ;‘-motoneurons activated before r-motoneurons. as wax once thought. The role of coactivation in the control of movement has been questioned recently.“.“.” so it is worth considering a number 01 hypotheses. including the cancellation hypothesis discussed in this paragraph. IA
inhihitor~~
interneurons
1rrtion.r (It 11 lower ,freyuency rc+e.\-. This result
tend to produce than the normal
oscilstretch
can be seen clearly by comparing the traces in Figs 4 and 6. The frequency decreases. as the strength of the connections to IA inhibitory neurons is increased, although the change is rather modest (less than l5”;, over the range of values shown in Fig. 6B). One possible role for IA interneurons is that they are involved in regulating the degree of co-contraction. When pathways from IA interneurons are active, alternation of antagonist pairs of muscles will tend to occur. However. subjects cocontract voluntarily, when working against an un-
stable load.’ The co-contraction of antagonists will tend to Increase the stifrness about the joint for a given force level. Increasing stiffness by itself might tend to increase the instability about the joint (see Fig. I). However, if the effect of connections from I A interneurons is reduced by some means in producing co-contraction. stability will be enhanced. T/w
minectiom
fion1
Rrnshu,r~
WII.!
to
As shown in Fig. 7, the action of the inhibitory pathway from Renshaw cells is opposite to that from IA interneurons. Renshaw cells have been suggested to play a role in the generation of physiological tremor.” although there is littlc or no direct experimental evidence for this. In fact, within the linear range investigated here, the effects of Renshaw cells will be to reduce the tendency for high frequency oscillations such as tremor. Adam et cd.’ showed experimentall) that Renshaw cells reduce the response or motoneurons to high frequency noise (50 Hz) and decouple the discharges of motoneurons on a millisecond time scale. Note that the pathway from Renshaw cells to ;*-motoneurons has an opposite effect to that from Renshaw cells to r-motoneurons. In Fig. 8 several different types of oscillations are seen when the connections to both J- and ;t-motoneurons are increased in parallel. The highest frequency oscillation is similar to that seen in Fig. 7 and decays increasingly rapidly as the strength of connections is increased. This oscillation is due to the connection to r-motoneurons. The other. lower frequency oscillations are due to the connection of Renshaw cells to y-motoneurons. This pathway has only recently been studied experimentally.” and its physiological role and strength, relative to the better known pathways to r-motoneurons, requires further investigation. It would be of interest to determine whether the conever become strong nections to y-motoneurons enough to produce oscillations under natural conditions. In conclusion, the model presented here gives a number of interesting results. which agree well with experimental data, to the extent that comparisons can be made. The model also suggests several ideas which should be experimentally testable, and it can be improved as more precise data on various spinal connections become available. x-mttoneuro~l.\
tend to prcr.cwt o.willutions.
REFERENCES
I. Adam D., Windhorst V. and Inbar G. F. (1978) The effects of recurrent inhibitlon on the cross-correlated tiring patterns of motoneurons (and their relation to signal transmission in the spinal cord&muscle channel). Viol C~~h~m. 29, 229-235. 2. Akazawa K.. Milner T. and Stein R. B. (1983) Modulation of reflex EMG and stiffness in response to stretch of human linger muscle. J. Neurophy.sio/. 49, 16-27. 3. Anderson B. F.. Lennerstrdnd G. and Thoden U. (1968) Response characteristics of muscle spindle endings at constant length to variations in fusimotor activation. Actor physiol. .srand. 74, 3Ol- 318. 4. Appenteng K., Prochazka A.. Proske V. and Wand P. (1982) Effect of fusimotor stimulation on la discharge during shortening of the cat soleus muscle at different speeds. J. Physiol.. Land. 329, 509.-526. 5. Bawa P., Mannard A. and Stein R. B. (1976) Predictions and experimental tests of a viaco-elastic muscle model using elastic and inertial loads. Biol C~&v-n. 22, 139.-145.
Spinal
modification
of muscle
239
during fictitious 6. Berkenbilt M. B., Deliagina T. G., Orlovsky G. N. and Feldman A. G. (1980) Activity of motoneurons scratch reflex in the cat. Brain Rex 193, 427-438. on the response 7. Brown M. C., Goodwin G. M. and Matthews P. B. C. (1969) After-effects of fusimotor stimulation of muscle spindle primary efferent endings. J. Physiol.. Lond. 205, 677-694. Viol. 8. Cleveland S. and Ross H.-G. (1977) Dynamic properties of Renshaw cells: frequency response characteristics. Cyhern. 27, 175-184. New York. 9. D’Azzo J. J. and Houpis C. H. (1966) Feedback Control System Analysis and Synthesis. McGraw-Hill, K.-E., Hagglund J. V. and Wallin E. V. (1982) The ‘late’ reflex responses to muscle stretch: the 10. Eklund G., Hagbarth ‘resonance hypothesis’ vs the ‘long-loop hypothesis’. J. Physiol., Lond. 326, 79-90. J. E. (1976) Motor-unit activity responsible for the 8 to 12Hz component of human II. Elble R. J. and Randall physiological finger tremor. J. Neurophysiol. 39, 370-383. of the recurrent inhibition of z- and y-motoneurones in the 12. Ellaway P. H. and Murphy P. R. (1981) A comparison cat. J. Physiol., Lond. 315, 43-58. 13. Goodwin G. M., Hoffman D. and Luschei E. S. (1978) The strength of the reflex response to sinusoidal stretch of monkey jaw closing muscles during voluntary contraction. J. Physiol., Lond. 279, 81-l I I. 14. Grillner S. (1981) Control of locomotion in bipeds, tetrapods and fish. In Motor Control (ed. Brooks V. B.) Am. Physiol. Sot. Handbook of Physiology Sect. I, Vol. 2, pp. 117991236. Williams and Wilkins, Baltimore. model of skeletal muscle. Biol. Cybern. 25, 103-I 19. 15. Hatze H. (1977) A myocybernetic of antagonist muscles. IEEE Trans. Auto. 16. Hogan N. (1983) Adaptive control of mechanical impedance by coactivation Conlrol. 17. Houk J. C. and Rymer W. Z. (1981) Neural control of muscle length and tension. In Motor Control (ed. Brooks V. B.) Am. Plzysiol. So<. Handbook of Physiology Sect. I, Vol. 2, pp. 2577233. Williams and Wilkins, Baltimore. 18. Hultborn H. (1977) Transmission in the pathway of reciprocal Ia inhibition to motoneurones and its control during the tonic stretch reflex. Prog. Brain Res. 44, 235-254. 19. Huxley A. F. (1980) Reflections on Muscle: Sherrington Lectures XIV. Liverpool University Press. Liverpool. 20. Joyce G. C. and Rack P. M. H. (1974) The effects of load and force on tremor at the normal human elbow joint. J. Physiol., Lond. 240, 375-396. 21. Matthews P. B. C. (1972) Mammalian Muscle Receptors and Their Cen/ral Actions. Arnold, London. 22. Matthews P. B. C. and Muir R. B. (1980) Comparison of electromyogram spectra with force spectra during human elbow tremor. J. Physiol., Lond. 302, 427441. 23. Matthews P. B. C. and Stein R. B. (1969) The sensitivity of muscle spindle afferents to small sinusoidal changes in length. J. Physiol., Lond. 200, 723-743. 24. Menzies J. E., Albert C. P. and Jordan L. M. (1978) Testing a model for the spinal locomotor generator. Sot. Nrurosci. Abs. 4, 1219. 25. Merton P. A. (1953) Speculations on the servo-control of movement. In CIEA Found. Symp., The Spinul Cord (ed. Wolstenholme G. E. W.) pp. 2477255. Churchill, London. 26. Miller S. and Scott P. D. (1977) The spinal locomotor generator. Expl. Brain Res. 30, 387403. 27. Nichols T. R. and Houk J. C. (1976) The improvement in linearity and the regulation of stiffness that results from the actions of the stretch reflex. J. Neurophysiol. 39, 119-142. 28. Oguztoreli M. N. and Stein R. B. (1983) A model for the spinal control of antagonistic muscles. J. theor. Neurohiol. In press. M. N. and Stein R. B. (1976) The effects of multiple reflex pathways on the oscillations in neuro-muscular 28a O&ztoreli systems. J. math. Biol. 3. 87-101. F. and Stein R. B. (1981) Nonlinear summation of contractions in cat muscles--II. The later facilitation 29. Parmiggiani and stiffness changes. J. gen. Physiol. 78, 295-31 I. of the Physiological 30. Phillips G. G. and Porter R. (1977) Corticospinal Neurones: Their Role in Mooement. Monograph Sot. No. 34. Academic Press, London. description of linear behavior of mammalian muscle spindles. 31. Poppele R. E. and Bowman R. J. (1980) Quantitative J. Neurophysiol. 33, 59-72. 32. Renshaw B. (1941) Influence of discharge of motoneurons upon excitation of neighboring motoneurons. J. Neurophvsiol. 4, 1677 183. mechanism as a basis for normal postural tremor. J. uppl. 33. Rietz R. R. and Stiles R. N. (1974) A viscoelastic-mass Phjxiol. 37, 852-860. 34. Rymer W. Z. and Hasan Z. (1980) Absence of force-feedback regulation in soleus muscle of the decerebrate cat. Brain Res. 184, 203-209. 35. Schafer S. S. (1967) The acceleration response of a primary muscle spindle ending to ramp stretch of the extrafusal muscle. Experientiu 23, 1026-1027. 36. Stein R. B. (1974) The peripheral control of movement. Physiol. Reo. 54, 215-243. 37. Stein R. B. (1982) What muscle variable(s) does the nervous system control in limb movements? Behuc. Bruin Sri. 5, 535-577. 38. Stein R. B. and Lee R. G. (1981) Tremor and clonus. In Motor Control (ed. Brooks V. B.) Am. Physiol. Sot. Handbook of Ph,vsio/ogy Sect. I, Vol. 2, pp. 325-343. Williams and Wilkins, Baltimore. 39. Stein R. B. and Oguztoreli M. N. (1976) Does the velocity sensitivity of muscle spindles stabilize the stretch reflex? Biol. Cyhern.. 23, 219-228. 40. Stein R. B. and Oguztoreli M. N. (1978) Reflex involvement in the generation and control of tremor and clonus. In Ph.vsiologicul Tremor. Pathological Tremors and Clonus (ed. Desmedt J. E.) Prog. clin. Neurophysiol. 5, 28-50. Karger, Basel. 41. Stein R. B. and Oguztoreli M. N. (1981) The role of y-motoneurons in mammalian reflex systems. Biol. C’yhern. 39, 171-179. 42. Stein R. B. and Oguztoreli M. N. (1983) A model of whole muscles incorporating functionally important nonlinearities. In Non-lineur Phenomena in Murhemarical Sciences (ed. Lakshimikantham V.) pp. 749-766. Academic Press, New York. ^ 43. Stein R. B. and Parnnggtam F. (11)X1) Nonhnear summation of contractions in cat muscles--I. The early depression. J. gen. Physiol. 78, 217-293.
240
R. B. Stein and M. N. OguztCreli
44. Vallbo A. B. (1971) Muscle spindle response at the onset of isometric voluntary contractions between fusimotor and skeletomotor effects. J. Physiol.. Land. 218, 405631. 45. Vallbo A. B. and Hulliger M. (1981) Independence of skeletomotor and fusimotor activiry 17G- 180. 46. Willis W. D. (1971) The case for the Renshaw cell. Bruin Behor. Ed. 4, 5--57.
in man. Time difference in man’? Bruin
Rrs. 223,