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Division of Neuroscience, University of Alberta, Edmonton, Alberta T6G 2S2, Canada
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ABSTRACT |
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 Abstract
 Introduction
Methods
Results
Discussion
References
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Prochazka, Arthur, Deborah Gillard, and David J. Bennett. Implications of positive feedback in the control of movement. J. Neurophysiol. 77: 3237-3251, 1997. In this paper we review some theoretical aspects of positive feedback in the control of movement. The focus is mainly on new theories regarding the reflexive role of sensory signals from mammalian tendon organ afferents. In static postures these afferents generally mediate negative force feedback. But in locomotion there is evidence of a switch to positive force feedback action. Positive feedback is often associated with instability and oscillation, neither of which occur in normal locomotion. We address this paradox with the use of analytic models of the neuromuscular control system. It is shown that positive force feedback contributes to load compensation and is surprisingly stable because the length-tension properties of mammalian muscle provide automatic gain control. This mechanism can stabilize control even when positive feedback is very strong. The models also show how positive force feedback is stabilized by concomitant negative displacement feedback and, unexpectedly, by delays in the positive feedback pathway. Other examples of positive feedback in animal motor control systems are discussed, including the 
-fusimotor system, which mediates positive feedback of displacement. In general it is seen that positive feedback reduces the sensitivity of the controlled extremities to perturbations of posture and load. We conclude that positive force feedback can provide stable and effective load compensation that complements the action of negative displacement and velocity feedback.
In the last few years there has been an accumulation of evidence for positive feedback in the control of motor tasks in animals. Positive feedback control of limb position or muscle length in different species has been discussed for many years (invertebrates: Bässler 1993 Mathematical modeling
We recognized from the outset that to be general, conclusions regarding closed-loop behavior should not depend too heavily on the particular structure of a model or on the choice of parameters within a given model. We therefore studied numerous models of mammalian neuromuscular systems, ranging from simple linear control loops to models that incorporated nonlinear elements such as conduction delays and nonlinear force-velocity and length-tension relationships. The modeling described in this paper was performed with the use of Matlab 4.2c and its associated graphics-based Simulink 1.3c control systems software. The "raw" Simulink versions of the models are detailed in the APPENDIX and may be replicated within minutes, allowing interested readers to verify our simulations and to explore and test the behavior of our reflex model. Stability was separately analyzed with numerous combinations of parameters with the use of Bode, Nyquist, and root locus plots, some of which are presented in the APPENDIX.
Model 1: simple reflex model
Figure 1 shows a highly simplified model of a reflex system. In this model, key elements of real neuromuscular systems such as tendon compliance, dynamic transfer functions of sensor, and the length and velocity dependence of muscle force production are omitted or simplified. These elements are represented more rigorously in the nonlinear model below. Yet, despite the basic structural differences between the two models, they exhibited essentially the same closed-loop behaviors. Other variants of these models not detailed in this paper also showed the same basic behaviors. The generality of the main features of closed-loop behavior in the face of significant parametric and structural variation is our strongest argument that our main conclusions are not model specific.
Model 2: nonlinear reflex model
Figure 3 shows a more representative nonlinear model of the mammalian neuromuscular system. As in the simple model, the major functional elements are represented independently in each block (e.g., active and passive components of muscle force generation, tendon compliance, load, etc.). In natural movements of the limbs, agonist and antagonist muscles are often coactive, which tends to cancel out the rectifying nature of nonlinear stiffness and damping in individual muscles. In our first set of experiments, described in the previous paper, the muscles were coactivated to mimic this situation. However, the recent evidence for positive force feedback comes from cat locomotion, where extensor muscles are not coactive with flexor muscles during weight bearing. We therefore chose to model this situation with a single muscle supporting an inertial load.
The cost of the increased realism of nonlinear models such as that in Fig. 3 is an increase in the number of parameters. We experimented with many combinations of parameters and indeed numerous versions of the model itself. Our main conclusions regarding closed-loop behavior were robust across a large range of parametric and structural variations. The stability of the system was insensitive to large (2 orders of magnitude) variations in most of the parameters, as discussed in the APPENDIX. Figure 7 in the APPENDIX shows the version of the model that generated the representative simulations illustrated in Fig. 4. As in the simple model, simulations were performed with the use of the Runge-Kutta-3 simulation algorithm, although Euler and Linsim algorithms were also used. In Fig. 4 the initial conditions at time t = 0 are such that muscle force and length are zero at the threshold of the length-tension curve (i.e., muscle just slack). The step in external force at t = 0 causes rapid muscle stretch and overshoot, because the stiffness and damping of the contractile element rise relatively slowly in the initial portion of the length-tension curve. The response to the second (test) load increment at t = 1 is more damped because the contractile element impedance is greater at the longer muscle length. Fig. 4, left and right, differ only in the time of onset of force feedback. In Fig. 4, left, force feedback is activated at the same time as the test loading. This mimics a linkage of positive force feedback with the onset of the stance phase of the cat step cycle, which is one of the possibilities to be inferred from the recent cat data. In Fig. 4, right, feedback is activated before loading (at t = 0.5).
Our experiments and analysis verify that positive force feedback in the neuromuscular system can provide stable and effective load compensation. The analysis also shows that the conclusions regarding the stabilizing influence of muscle intrinsic properties, length feedback, and delays in positive feedback pathways were robust in the face of large parametric and structural variations in the systems considered (see APPENDIX). Stable behavior for large values of positive feedback gains was unexpected and initially quite puzzling. However, it became apparent that loop gain did not remain high, but rather it was automatically attenuated when muscles shortened and thereby reduced their force-producing capability.
Load compensation
Since the publication of the landmark papers by Pearson and Duysens (1976) Stability
It is a routine and essential aspect of the design of any control system to analyze the stability criteria. The two most common manifestations of instability are oscillation and monotonic unrestrained increases in the output variable. In a simple feedback loop, if the gain of the open-loop transfer function G is unity at a given frequency, the transfer function of the same loop when closed in positive feedback mode is 1/(1
Biological implications
What are the wider biological implications of the results? First, we feel that the paradox of homonymous excitatory reflex action from force receptors, with its underlying connotation of instability, is now partly resolved. As we have seen, the dynamic properties and nonlinearities of the mammalian neuromuscular system are well suited to compensate for the destabilizing tendencies of positive force feedback. Concomitant negative feedback of displacement and positive feedback of force is therefore a viable and effective configuration. At this stage the neurophysiological evidence for positive force feedback in mammals is largely restricted to the cat locomotor system (Conway et al. 1987 Concluding remarks
In this paper we provide a theoretical analysis of the hypothesis that positive force feedback confers useful load compensation in the control of animal movement. Several nonlinear properties of the neuromuscular system appear to provide stabilizing influences. The neurophysiological evidence from different species indicates that the sign and gain of force feedback is highly task dependent. This suggests that positive force feedback may be appropriate in some muscles and motor tasks but not in others. Models such as those presented in this paper may help in understanding the reason for this task dependence in the future.
INTRODUCTION
Abstract
Introduction
Methods
Results
Discussion
References
; Burrows and Pflueger 1988; Cruse et al. 1995; mammals: Grill and Rymer 1985; Houk 1972, 1979; Kouchtir et al. 1995). More recently, interest has shifted to the possibility of positive feedback control of muscle or limb force (invertebrates: Bässler 1993; Bässler and Nothof 1994; Cruse 1985; Cruse et al. 1995; cats: Brownstone et al. 1994; Conway et al. 1987; Gossard et al. 1994; Guertin et al. 1994, 1995; Pratt 1995; humans: Dietz et al. 1992). In addition, positive feedback loops have been posited within sensorimotor control areas of the mammalian brain (Houk et al. 1993).
; Roberts et al. 1995). However, the evidence for positive feedback connections in reflex systems controlling posture and movement has been something of a puzzle, precisely because of the potential for instability. On the one hand, it has been suggested that certain oscillatory behaviors such as rocking in stick insects (Bässler 1983) or paw shaking in cats (Prochazka et al. 1989) might reflect a "deliberate" augmentation of loop gain and therefore destabilization of stretch reflexes. Similarly, certain pathological tremors have been attributed to instability in the stretch reflex arc (Dimitrijevic et al. 1978; Jacks et al. 1988; Rack and Ross 1986) or in olivocerebellar and thalamocortical pathways (Lamarre 1984). In all these cases the neural loops are assumed to function as negative feedback systems under normal conditions, but they develop unstable positive feedback behavior because of a 180° net phase lag combined with an open-loop gain exceeding unity at a particular frequency. On the other hand, it has been argued that some reflex connections are "designed" to operate as positive feedback systems to provide nonoscillatory graded control of posture and movement (Cruse et al. 1995). It is here that the issue becomes problematic, for linear control theory predicts that to be efficacious in load compensation, the open-loop gain of positive force feedback (Gf) should be close to unity; but the closer it is to unity, the greater the risk of instability (Phillips and Harbor 1991).
discussed the least disputable example of positive feedback in mammalian reflexes, the -loop. This comprises -skeletofusimotor neurons that exert fusimotor action on muscle spindle afferents, which reflexly excite the same -skeletofusimotor neurons. Houk (1972) argued that even if the open-loop gain of the -loop (G
) exceeded 1, the system remained stable because the -loop was nested inside a negative feedback loop controlling muscle displacement. In this view, activation of -motoneurons would cause the spindle-bearing extrafusal muscle to shorten, counteracting -fusimotor excitation of spindle afferents and thereby holding in check the reflex action of these afferents on the -motoneurons. This argument was based on a simple static feedback model that ignored muscle and receptor dynamics and nonlinearities. Some years later, Grill and Rymer (1985) inferred G from differences in spindle afferent stretch sensitivity before and after dorsal rhizotomy in decerebrate cats. The estimates of G were in the range of 0.2-0.8.
showed that adding negative force feedback to negative displacement feedback produced springlike behavior of muscle in the face of externally applied loads. The greater the force feedback gain, the more compliant the muscle would be to loading. Whether force feedback gain was actually modulated in normal animal behavior remained an open question (Crago et al. 1976; Hoffer and Andreassen 1981; Rymer and Hasan 1980).
; Harrison et al. 1983) to excitatory during locomotor behavior (Conway et al. 1987; Gossard et al. 1994; Guertin et al. 1995; Pearson and Collins 1993). Because Ia and Ib reflexes have the same sign under these circumstances, there has been a shift away from describing reflex action in terms of sensory modalities such as force and displacement, toward lumped descriptions such as "group I excitatory action," whereby afferents are classified according to conduction velocity rather than modality (Angel et al. 1995; Jankowska and McCrea 1983). Yet the functional consequences of separate modalities cannot be avoided. If signals from force-sensing afferents reflexly increase the force the receptors sense, this constitutes positive force feedback, and, for the reasons discussed above, the issue of stability must then be dealt with.
) we described experiments in human subjects whose muscles were controlled with electrical stimulation under positive force feedback control. These experiments showed that positive force feedback resulted in stable load compensation under a surprisingly wide range of feedback gains. The closed-loop behavior was characterized by reductions or even reversals in the yield that inertial loading would otherwise have caused. The reversals occurred when Gf had been set greater than unity. The very fact that the loop remained stable for Gf > 1 was a surprise, and further investigation suggested that the muscle shortening associated with negative yield automatically reduced Gf to unity. Delays in the feedback pathway similar to those seen in cat locomotion stabilized positive force feedback. This was also unexpected.
METHODS
Abstract
Introduction
Methods
Results
Discussion
References
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FIG. 1.
Highly simplified model of segmental reflex system used in 1st set of simulations. Graphic version of this model, as it appeared in Simulink interface, is presented in APPENDIX (Fig. 6) with representative parameters. This model ignored dynamic components of sensory transduction and various nonlinear features of muscle. It is presented because of its simplicity and because in simulations it exhibited all salient functional properties of more complex nonlinearmodels.
), the muscle length-tension curve (Rack and Westbury 1969), and tendon compliance are all neglected. The external force is summed with tendon force, the resultant force acting on the inertial load. Force feedback is represented without dynamic components, but an adjustable reflex delay is included because this was shown in the empiric work to stabilize positive force feedback. Feedback from muscle spindles is likewise represented without dynamics and without delays.
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FIG. 6.
A: system diagram of simple reflex model of Fig. 2 as it appears in Simulink graphic interface. Note that Matlab program does not allow an excess of zeros in a transfer function. This required inclusion of (s + 100) term in shadowed box and corresponding gain adjustment. B: root locus plot of static reflex model, showing how 25-ms delay in positive force feedback pathway stabilizes system. Without delay( 
), closed-loop poles cross into positive half plane for Gfs = 1.4. Inclusion of delay (- - -) pulls closed-loop roots back into negative half plane, thus allowing a higher gain (Gfs = 2.38) before instability occurs. Asterisk: element is a "slider" gain (i.e., an adjustable component).
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FIG. 2.
Simulations based on static reflex model of Figs. 1 and 6. A: input (external) force. B: displacement (yield) of load due to muscle stiffness; no feedback. C: reduced yield with addition of low-gain positive force feedback, open-loop gain of positive force feedback (Gf) = 0.5. D: zero net yield with unity-gain positive force feedback. E: underdamped overcompensation of load (negative yield or "affirming reaction") with Gf = 1.4. F: improved stability with 25-ms delay in positive force feedback pathway. G: delay allows higher force loop gain (Gf = 2), with larger affirming reactions. H: adding negative displacement feedback attenuates affirming reaction. Displacement calibrations in B also apply to C-H. Gd, displacement feedback gain.
), negative force feedback in combination with negative displacement feedback resulted in a springlike response to external loading, the stiffness of which increased as Gd increased and Gf decreased (not illustrated).
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FIG. 3.
Nonlinear model of segmental reflex system used in 2nd set of simulations. Graphic version of this model is presented in APPENDIX (Fig. 7). This model includes tendon compliance, nonlinear force-velocity, and length-tension properties of mammalian muscle, dynamic components of sensory transduction, and -fusimotor action. Despite increase in complexity over simple model and significant differences in some of common parameters, nonlinear model exhibited similar functional properties in simulations.
model of muscle. This includes a passive parallel viscoelastic stiffness and a parallel contractile stiffness that depends on activation level, muscle length, and velocity, all based on relationships in the literature (Bawa et al. 1976; Rack and Westbury 1969; Winters 1990). The series compliance is equated to tendon compliance with the use of the parameters of Bennett et al. (1996). The muscle portion of the series compliance is neglected (Zajac 1989).
; Matthews and Stein 1969) and tendon organ Ib afferents (Anderson 1974; Houk and Simon 1967) are used. These have been found to give good fits of ensemble afferent responses in freely moving cats (Appenteng and Prochazka 1984; Gorassini et al. 1993). As alternatives, two variants of the Houk et al. (1981) nonlinear spindle Ia model were also tested (see APPENDIX).
-fusimotor pathway. The activation of 
- and -motoneurons is assumed to be identical. The -linked component of 
-fusimotor activity is lumped with the -signal. Task-related fusimotor action (Prochazka et al. 1985) is represented as changes in Ia gain.
-fusimotor reflex paths. The Ia delay is assumed to be small (e.g., 10-20 ms). Current evidence from cat data indicates that positive Ib feedback has a delay or rise time of 30-40 ms (Gorassini et al. 1994; Gossard et al. 1994; Guertin et al. 1995). Consequently we tested Ib delays of up to 100 ms, or, alternatively, a single-pole, unity-gain, low-pass filter block with a turning point in the range of 10-100 rad/s.
RESULTS
Abstract
Introduction
Methods
Results
Discussion
References
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FIG. 7.
Diagram of nonlinear reflex model of Fig. 3 as it appears in Simulink interface. As in Fig. 6A, because Matlab program does not allow an excess of zeros in a transfer function, extra poles and corresponding gain adjustments appear in shadowed boxes. In force feedback pathway we used either a time delay or a 1st-order low-pass filter element (dashed boxes at top left). Muscle spindle Ia responses were modeled either with a linear transfer function or nonlinear model described in text (dashed boxes at bottom left).
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FIG. 4.
Simulations based on nonlinear reflex model of Figs. 3 and 7. Positive force feedback is activated either at same time as test increment in external force (at t = 1 s, left) or earlier (at t = 0.5 s, right). A: external force. Two 10-N force increments are applied, the 1st at t = 0 to stretch muscle from slack to its operating length, the 2nd comprising test load. B: no feedback. Muscle displacement in response to force increments depends on inertial mass and muscle viscoelastic properties only. C: low-gain positive force feedback (Gf = 0.5) stiffens muscle, reducing displacement by test load. D: unity-gain positive force feedback. Displacement due to test load is very small when force feedback is activated at same time as loading (left), but some yielding is evident if force feedback is activated earlier (right) E: overcompensation (affirming reaction) to loading for Gf = 2. Underdamping suggests that system is close to instability. F: instability when Gf is raised to 4. G: stabilizing action of adding 40-ms delay in force feedback pathway. Similar effect is seen if a low-pass filter is used instead. H: adding negative displacement feedback to positive force feedback attenuates all of displacement responses, including affirming reaction. I: adding -skeletofusimotor action stabilized unstable configuration of F. Displacement calibrations in B also apply to C-I. G, open-loop gain of -loop.
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FIG. 8.
System block diagram showing how a portion of nonlinear model of Fig. 6 is used to compute force loop gain. Length is treated as a contingent parameter (staircase generator at bottom right). A 1-Hz sinusoidal signal is supplied to input of force-sensing element. This transmits a signal via feedback pathway to contractile element, which, after combination with muscle intrinsic properties, results in a force output. Input and output force signals and displacement signal are gathered together and plotted against time. Force loop gain Gf is computed as ratio of output force signal to input force signal.
1 the system equilibrates to the length and force at which loop gain is unity, regardless of the load. As is seen in Fig. 5, this automatic gain control is also evident in the length-tension trajectories for given values of Gfs. The loop remains stable up to Gfs = 2 (Fig. 4E). In Fig. 4E, left, where feedback is turned on at the onset of loading, the muscle shortens on loading (affirming reaction). In Fig. 4E, right, after feedback is activated, the system equilibrates to a length at which loop gain is unity, so the test load now causes a small length increase. For Gfs = 4, the system is unstable (Fig. 4F). Adding a 40-ms delay to the force feedback pathway stabilizes it (Fig. 4G). This stabilizing action of a delay in the positive feedback pathway is confirmed with the use of root locus techniques in the APPENDIX. The addition of negative displacement feedback to the control loop attenuates and further stabilizes the load reactions (Fig. 4H, Gd = 1 starting at t = 0). Interestingly, delayed positive Gd via the -pathway can stabilize positive force feedback in the same way as a delay in the force feedback pathway (Fig. 4I: G = 0.5, Gfs = 4, no delay in force feedback pathway, 20-ms delay in -pathway).
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FIG. 5.
A: length-tension curves due to intrinsic muscle properties alone, compared with those with positive force feedback. Family of open-loop length-tension curves radiating from origin obtained from nonlinear model, as described in text, is overlaid with portions of closed-loop length-tension curves where setpoint loop gain (Gfs) ranges from 0 to 2 (Gfs is force loop gain evaluated at an operating point of 7.3 mm and 10 N). Stiffness increases with increasing Gfs. Equilibrium length for a given force in Fig. 4 may be obtained from corresponding closed-loop length-tension curve in Fig. 5 (see text). Hollow bars show that along a closed-loop curve, a given increment in contractile element activation always produces same force increment, indicating that loop gain is held constant along each closed-loop curve. B: diagram illustrating predicted effect of activation of positive force feedback in extensor muscles of cat hindlimb. Extensor muscles shorten to a new equilibrium length, resulting in a more upright posture.
1, all points in the closed-loop trajectories turned out to correspond to a loop gain of unity. This was shown in separate analyses with the open-loop gain system of Fig. 8 in the APPENDIX. The constancy of gain in these trajectories is illustrated by the approximate constancy of the force increments for a given increment in contractile element activation, shown by the superimposed hollow bars in Fig. 5B (the bars on the Gfs = 1 and Gfs = 2 curves are not precisely the same because loop gain depends on contractile element output and muscle fiber length, the latter also depending on force and tendon compliance).
-mediated positive displacement feedback can all contribute to the stability of closed-loop operation.
DISCUSSION
Abstract
Introduction
Methods
Results
Discussion
References
and Conway et al. (1987) demonstrating reflex excitation of load-bearing muscles by their own load sensors during cat locomotion, most of the functionally related discussion in the mammalian literature has centered on the potential of such reflexes to reinforce load compensation, a role previously relegated to intrinsic muscle properties and reflexes mediated by muscle spindles (Feldman 1966, 1986; Marsden et al. 1977). Yet it was clear that the reflex mechanism in question represented positive feedback and this was normally associated with instability. It was tacitly assumed that the nervous system would somehow always limit positive force feedback gain within a range consistent with stability. Our results suggest that a combination of intrinsic muscle properties, concomitant negative displacement feedback, and reflex delays found in neuromuscular systems may provide this automatic gain control. The characteristic affirming reaction that results for loop gains that are initially greater than unity can only occur in a permissive framework such as this, and as far as we know, it has not been described before. From a functional point of view, the affirmation of ground contact, described many years ago as Rademaker's "magnet reaction" (Roberts 1979), has an obvious weight-supporting role during the stance phase of gait. It may also contribute to load compensation in other motor tasks and systems. In a more general sense the affirmation of contact is a possible solution for the problem of "contact bounce" encountered with the use of negative force feedback in robotic actuators (An et al. 1988; Hogan 1985). In a recent robotics study, Tarn et al. (1996) showed that positive acceleration feedback was useful in eliminating contact bounce.
G), which has an infinite value at that frequency (Phillips and Harbor 1991). This in turn implies infinite amplitude oscillations in closed-loop operation. Monotonic instability occurs in cases where positive feedback G = 1 in the steady state (frequency = 0). In practice, unstable oscillations are constrained by nonlinearities to a finite range: "limit cycles," provided the actuator survives.
).
in relation to a robotic manipulator designed to avoid contact bounce. In a similar vein, Houk (1972) and Grill and Rymer (1985) argued that positive feedback mediated by the -fusimotor loop would automatically be held below a loop gain of 1 by virtue of being nested within the negative displacement feedback loop mediated by muscle spindles. Indeed it has been suggested that under suitable conditions the very presence of multiple reflex pathways may have a stabilizing action (Oguztöreli and Stein 1976). The repeated observation that spinal interneurons and motoneurons tested during fictive locomotion receive excitation from both group Ia spindle and Ib tendon organ afferents (Angel et al. 1995) is strong evidence for concomitant positive force and negative displacement feedback, at least in the cat.
; Rack et al. 1984). Stein and Kearney (1995) have pointed out another possible gain-attenuating nonlinearity in shortening muscles: the rectification of reflex responses. This property was not taken into account either in our experimental study or in our simulations. Whether it would stabilize or destabilize positive force feedback is not known. Furthermore, if a movement were being controlled by cocontracting antagonist muscles, the shortening of one muscle and the resulting reduction in its force loop gain would be counteracted by lengthening of the antagonist and a resulting increase in its force loop gain. In muscles that encounter immovable loads and contract isometrically, positive force feedback with gains exceeding unity could become unstable, even in the presence of the stabilizing influence of loop delays and the gain-attenuating effect of series tendon compliance. If muscles such as these had reflex connections that mediated positive force feedback, the strength of these connections would either have to be low or under the context-dependent control of the CNS. It may therefore be the case that positive force feedback will only be found in muscles, such as the leg extensors, that are inertially rather than isometrically loaded and are activated without their antagonists.
View this table:
TABLE 1.
Qualitative summary of the interaction between displacement and force feedback
; Guertin et al. 1995; Pearson and Collins 1993), although some evidence for a static postural role in cats and humans has been adduced (Dietz et al. 1992; Pratt 1995). There is also a body of evidence from invertebrates suggesting positive force feedback in certain behavioral states (Bässler 1993; Burrows and Pflueger 1988; Lindsey and Gerstein 1977; Pearson 1993).
; Crago et al. 1976; Gottlieb and Agarwal 1980), the evidence is either indirect (e.g., Matthews 1984) or obtained in reduced preparations in which the nervous system may not be responding normally (e.g., Hoffer and Andreassen 1981). However, in light of the recent evidence in the cat locomotor system, this issue now deserves to be reexamined. In particular, the contribution of tendon organ Ib input should be sought in the medium- and long-latency (40-100 ms) components of stretch reflexes, given the stabilizing effect of such latencies in positive feedback pathways. There are some indications that the pathways mediating negative force feedback in some muscles may be active at the same time that positive force feedback is active in other muscles (Pratt 1995). Furthermore, force feedback and displacement feedback are distributed across limb segments in ways that suggest global control strategies tailored to the motor task (Bonasera and Nichols 1994; Harrison et al. 1983; Nichols 1989). The effect on overall stability and load compensation of these distributed reflexes is beyond the scope of the present study, although clearly it is of basic importance in understanding the control of multisegmented limbs.
) and stick insect (Bässler 1983) and also by the fact that many interneurons may be interposed in the long-latency and long-duration Ib pathways described by Gossard et al. (1994). No such task-dependent reversal has been found in short-latency Ia reflex action, although the magnitude of H reflexes is reduced when going from standing to walking and from walking to running (Capaday and Stein 1986, 1987).
), but it may reappear after upper motoneuron lesions (Eliasson et al. 1995). This suggests that the reflex connections are still present after infancy but that in adults they are under set-related control. The force of a grasp depends on prior evaluation of the load and also on sensory input related to the force of the load on the fingers and palm (Johansson and Westling 1988). In a series of experiments in which the load was suddenly altered during a pinch grip, Johansson et al. (1992a-c) showed that grip force was scaled to load force in two phases: a rapid "catchup" phase with electromyographic latencies in the range of 60-120 ms (Johansson and Westling 1988; Johansson et al. 1992c; Winstein et al. 1991) and a longer-latency and more gradual "tracking" phase (Johansson et al. 1992c). Local anesthesia of the thumb and finger tips disrupted but did not abolish these responses, suggesting cutaneous mediation with additional force-related sensory input from other receptors (Johansson et al. 1992c). Regardless of the sensory receptors involved, the scaling of grasp force to load is certainly consistent with segmental positive force feedback, both in latency and effect (D. Collins and A. Prochazka, unpublished data). One prediction would be that force-related medium-latency reflexes should be less subject to gating and may be exaggerated in amplitude in infantile and spastic grasp responses compared with those in normal adults.
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ACKNOWLEDGEMENTS |
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We are very grateful to Dr. Keir Pearson for valuable advice.
The study was supported by a grant from the Canadian Medical Research Council. The Alberta Heritage Foundation for Medical Research provided salary support.
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APPENDIX |
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Details of the models and root locus analysis
We analyzed several versions of the models presented in this paper with the use of Bode, root locus, and Nyquist plots. The idea was to see whether any generalizations could be made regarding the stability of the systems and the sensitivity of the main conclusions to the choice of parameters. Figures 6-9 illustrate the Simulink versions of the models and corresponding root locus plots. The models may be implemented and run without modification in the Simulink program. For convenience, they may be obtained as digital files from the authors or from the following web site: http://gpu.srv.ualberta.ca/~aprochaz/hpage.html with the use of the file names shown in the bottom right corner of each figure.
Simple model
Figure 6A shows the simple reflex model. The muscle contractile element is modeled as a first-order low-pass filter with a cutoff frequency of 8 Hz, somewhat higher than the isometric frequency response characteristic of cat triceps surae muscles of Rosenthal et al. 1970. A mass of 1 kg represents the inertial load borne by a cat hindlimb. A force increment of 10 N represents the mean force developed by triceps surae during the stance phase of slow gait (Walmsley et al. 1978). Inherent muscle properties are simplified to a linear viscoelastic element with a stiffness of 0.5 N/mm. The force feedback and displacement feedback signals are represented without dynamics. The graph element allows a real-time view of the simulation and the multiplexer element chooses which signals are displayed. The simulation parameters in this and the dynamic reflex model were set for a 5-s run time of the Runge-Kutta-3 algorithm, with minimum and maximum steps of 0.5 ms.
Nonlinear reflex model
The nonlinear reflex model is more complex because it includes the -skeletofusimotor loop, tendon compliance, and nonlinear force-velocity and length-tension relationships that are combined multiplicatively with the neural input to muscle. There is also an alternative nonlinear model of spindle Ia transduction. Even so, several known properties of the neuromuscular system are neglected, such as force-dependent tendon compliance, high-pass filtering in the central nervous portion of the Ia reflex arc, Renshaw cell inhibition, etc. The validity of any model must be balanced against complexity (Winters 1990). We feel that our nonlinear model captures enough of the properties of the system to provide insight into the mechanism of interaction of muscle properties and force feedback and displacement feedback without getting lost in superfluous detail. The basic validity of the conclusions drawn from the modeling is supported by the data from the previous paper, in which real muscles were part of the feedback system.
Choice of parameters
The mass and external force parameters are the same as those of the simple model. Muscle properties are modeled with five main elements: the contractile element, the force-velocity relationship, the length-tension relationship, passive parallel stiffness, and tendon compliance. The contractile element is modeled as a first-order low-pass filter with a cutoff frequency of 3.2 Hz. Feedback via the series (tendon) compliance adds a second pole, also at ~3 Hz, giving an overall second-order characteristic in line with the isometric data of Rosenthal et al. 1970. Tendon compliance is represented as a constant of 0.05 mm/N (Elek et al. 1990; Rack and Westbury 1984). Force dependence of tendon compliance is neglected. The length-tension curve is a sigmoidal relationship made up of two exponential functions. The baseline shape constants and offsets were chosen to mimic the ascending portion of the curves described by Rack and Westbury (1969). The mean stiffness of this element, assuming a unity value of output from the -motoneuron element, is set to a baseline value of 1 N/mm (Bawa et al. 1976). The force-velocity curve was based on the work of Winters (1990) and is made up of a Hill-type hyperbolic function for negative (shortening) velocities and an exponential function for positive (stretch) velocities. The peak force for rapid stretches was assumed to be 50% greater than the isometric force (Winters 1990). Numerous shape constants were explored to mimic the range of length-tension and force-velocity curves in the literature. Force feedback was represented by a linear transfer function. That of Houk and Simon (1967) is shown in Fig. 7, but that of Anderson (1974) was also tested. Spindle Ia feedback was modeled by a linear transfer function (Chen and Poppele 1978), or, alternatively by the Houk et al. (1981) nonlinear model modified in the following way
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was not specified for negative velocities).
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Parametric sensitivity
We studied the effect on closed-loop behavior of individually varying the main parameters of the nonlinear model from the baseline operating settings shown in the block elements of Fig. 7. Table 2 shows the results. The range shown in Table 2, third column, is that over which the parameters could be varied without instability occurring or without any fundamental change in the closed-loop behavior in relation to our main conclusions. For all simulations in this analysis a step size of 0.5 ms was used. Larger step sizes (e.g., 1 ms) sometimes produced instability for computational reasons. In two cases the settings of several parameters were changed, as indicated in Table 2, third column. This was to explore parametric sensitivity in situations of particular interest, such as force feedback alone. The shape parameters a-c of the force-velocity and length-tension curves are defined in the following equations used in the simulations
; Guertin et al. 1995) was modeled either as a simple time delay or as a single pole, p/(s + p), where the baseline setting of the turning point frequency p was 33 rad/s, giving a time constant of 33 ms.
View this table:
TABLE 2.
Parametric sensitivity analysis
(Length = 0 at the point of inflection of the sigmoidal length-tension curve. Note that in the model, the offset block sets this point of inflection at a muscle displacement of 8 mm.)
Determination of open-loop gain
Figure 8 shows the block diagram used to evaluate force loop gain in the nonlinear model. Spindle feedback and the -loop are removed. Because force loop gain is length dependent, two inputs are provided, force and length. The length input is provided by a "repeating sequence" element, (bottom right) set up to produce a staircase of unity increments at 2-s intervals. The input force signal (top right, including offset block) is a 1-Hz sinusoid that varies between 0 and 4. This frequency and amplitude were chosen arbitrarily. The force input signal is transmitted via the Ib transfer function, gain, and delay blocks to the contractile element block, resulting in an output that is multiplied by the outputs of the length-tension and force-velocity feedback blocks. In essence, all of the components within the dashed line comprise the muscle model. The two input signals and the force output signals are gathered together and displayed by the graph block. Force loop gain is computed as the ratio of the output force signal to the input force signal and varies with displacement.
Root locus analysis of linearized version of nonlinear reflex model Because there are no general methods for analyzing nonlinear systems, the feedback loop and the length-tension and force-velocity loops were removed from the nonlinear reflex model to facilitate a linear root locus analysis. The linearized version is shown in Fig. 9A. In addition, to represent the delay by a rational transfer function, a second-order Padé approximation was used.
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FOOTNOTES |
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Address for reprint requests: A. Prochazka, Division of Neuroscience, 507 HMRC, University of Alberta, Edmonton, Alberta T6G 2S2, Canada.
Received 22 February 1996; accepted in final form 3 March 1997.
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REFERENCES |
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Abstract
Introduction
Methods
Results
Discussion
References
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model) for motor control.
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