Analysis of positive feedback in the control of movement

doi:10.4236/jbise.2009.27069

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J. Biomedical Science and Engineering, 2009, 2, 480-483

doi: 10.4236/jbise.2009.27069 Published Online November 2009 (http://www.SciRP.org/journal/jbise/ JBiSE

Published Online November 2009 in SciRes. http://www.scirp.org/journal/jbise

Analysis of positive feedback in the control of movement

Soroor Behbahani1, Amir Homayoun Jafari2

1,2Biomedical Engineering Department, Science and Research Branch, Islamic Azad University, Tehran, Iran.

Email: 1[email protected]; 2[email protected]

Received 1 June 2009; revised 13 July 2009; accepted 15 July 2009.

ABSTRACT

Over the past three decades, neurophysiologists

studying the neural circuitry responsible for control

of skeletal muscles have developed several different

general theories of sensorimotor control. These have

usually invoked one or more of the sources of pro-

prioceptive signals (e.g. muscle spindle and Golgi

tendon organ afferents) in positive or negative feed-

back loops to the homonymous alpha motoneurones.

In this paper we consider to analyze the role of posi-

tive feedback in combination of negative feedback

due to important role of them in stabilizing the neu-

romuscular system.

Keywords: Positive Feedback; Negative Feedback; Hill

Model; Reflex Model

1. INTRODUCTION

The stretch reflex differs in decerebrate and intact ani-

mals and since Sherrington’s time it has come to be re-

alized that several CNS mechanisms may contribute

components of different latency to the stretch reflex re-

sponse. At the segmental level, muscle spindle Ia affer-

ents activated by muscle lengthening monosynaptically

excite homonymous alpha motoneurons which in turn

cause the muscle to resist the stretch. In static postures

Ib input generally results in homonymous inhibition, but

it has been shown that this switches to longer-latency

homonymous excitation during locomotion [1], at least

in cat extensor muscles. Group II input from muscle

spindles has also been implicated in long latency com-

ponents of stretch reflexes [2,3,4]. Ia homonymous ex-

citation represents negative displacement feedback,

which augments the intrinsic stiffness of active muscles

in the face of length perturbations. Ib homonymous feed-

back on the other hand represents positive force feed-

back. Positive feedback is synonymous with instability

and oscillation in engineering systems, but when mus-

cles are the actuators; their nonlinear lengthtension prop-

erties turn out to stabilize the positive feedback loop [5].

2. INTERACTIONS BETWEEN POSITIVE

AND NEGATIVEFEEDBACK

According to linear systems theory, positive feedback

loops alone produce instability, while negative loops

alone may require very high gains to be useful. The gen-

eral advantage of combining positive and negative feed-

back loops is that it reduces the sensitivity of system

response to unpredictable variations in any of its com-

ponents, for example, fatiguing of a muscle or increase

of a mechanical load [5], without requiring excessive

negative feedback gains.

On the other hand, Prochazka [6] demonstrated that

positive force feedback could also perform this role as-

sisted by negative length feedback.

Positive force feedback through group Ib afferents

may work in parallel to the positive length feedback

through the loop and/or group II afferent modulation of

activity [7,8]. In the case of Ib force feedback, the loop

is polysynaptic so that transmission through it is state

dependent. One possible advantage of loops is that they

influence and motoneurons monosynapticall y [6].

In current research we analyzed the role of positive

feedback in combination of negative feedback due to

important role of them in stabilizing the neuromuscular

system. We used a simple reflex model to show the be-

haviors of neuromuscular system. Whole of these mod-

els were done using MATLAB software.

2.1. Reflex Model

Figure 1 shows a highly simplified model of a reflex

system. In this model, key elements of real neuromuscu-

lar systems such as tendon compliance, dynamic transfer

functions of sensor, and length and velocity dependence

of muscle force production are omitted or simplified.

The load in this model is a mass of 1 kg.

In this model, muscle force is modeled by a first-order

active component and a viscoelastic parallel stiffness.

The force velocity relationship [9,10], the muscle length

tension curve and tendon compliance are all neglected.

12 are used to represent the image obtained by limb

leads.

S. Behbahani et al. / J. Biomedical Science and Engineering 2 (2009) 480-483 481

Figure 1. Reflex model.

The external force is summed with tendon force, the

Resultant force acting on the inertial load. Force feed-

back 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.

Many combinations of parameters were tested in

simulations. Figure 2 shows a small set of simulations.

All simulations were performed with the use of the

Runge-Kutta-3 simulation algorithm in the Simulink

program. Two variables are plotted: external force (per-

turbing input) and displacement (output).

3. RESULTS

Without displacement feedback or force feedback, the

system, comprising the viscoelastic muscle parallel

stiffness and the inertial load, showed a damped dis-

placement response (Figure 2-1) to the external force.

Adding positive force feedback at loop gains pro-

duced stable load co mpensation (yield in Figure 2B with

is about half that in Figure 2-1 without feed-

back).

5.0

In this model, was determined by inspection as

the multiple of the static gains around the force loop.

Because displacement is held constant for this purpose,

the only elements involved are the muscle contractile

element and the force feedback element. When

in Figure 2B, apart from the small transients, the mass

did not move, i.e., the system had infinite stiffness.

1

The loop remained stable up to (Figure 2C),

exhibiting an affirming reaction (i.e., instead of yielding,

the mass now moved in the opposite direction of the

imposed force). The reason that the system remained

stable for values of between 1 and 1.4 is that the

parallel stiffness element provided negative displace-

ment feedback and velocity feedback, which had a stabi-

lizing effect (if the gain of this element was set to 0, th e

system went unstable at ).

4.1

1

Adding a 25-ms delay to the force feedback pathway

had a stabilizing effect (Figure 2D), allowing a higher

loop gain to be attained with a larger affirming reaction

(Figure 2F).

Note that the magnitude and sign of feedback gain are

separately specified in this paper.

Negative force feedback in combination with negative

displacement feedback resulted in a spring like response

to external loading, the stiffness of which increased as

increased and decreased (not illustrated). Com-

parison of Figures 3E and 3H shows another results

when the displacement feedback gain is exponential in-

stead of constant value. The stab ility of syste m increased

and the settling time will become shorter so the system

will be more rapid.

Figures 3F and 3G show that if we increase the force

feedback to2



G, the amplitude of oscillation parts

will increase and transient response of the system will

become more oscillating and the settling time will also

increase so increasing the for more than 1.4 will

lead to oscillating th e system, and it emphasize that max

gain for obtaining optimum response is

4.1



G. Fig-

ures 3I and 3J show that increasing the displacement

feedback gain is effective in specific limitation. While

5.0



G in comparison of the system has

more oscillating responses and the settling time is

longer.

1

But in comparison of Figures 3I and 3J we could in-

fer the optimum is approximately, because

increasing to value of 2 lead to oscillating the re-

sponses of the system. Although these oscillations are

smaller than the oscillations of system when

G1

5.0



and the settling time is still shorter.

4. DISCUSSIONS AND CONCLUSIONS

In the simple reflex model the muscle contractile ele-

ment is modeled as a first-order low-pass filter with a

cutoff frequency of 8 Hz, somewhat higher than the

SciRes

482 S. Behbahani et al. / J. Biomedical Science and Engineering 2 (2009) 480-483

Figure 2. Simulations based on static reflex model

of Figure 1.

Figure 3. Exponential displacement feedback.

S. Behbahani et al. / J. Biomedical Science and Engineering 2 (2009) 480-483

483

isometric frequency response characteristic of cat triceps

surae muscles of Rosenthal that positive force feedback

may be appropriate in some et al. 1970. A mass of 1 kg

represents the inertial load borne by a cat hind limb. A

force increment of 10 N represents the mean force de-

veloped by triceps surae dur ing the stance phase of slow

gait [11]. 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. It was clear that the

reflex mechanism in question represented positive feed-

back 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 sug-

gest that a combination of intrinsic muscle properties,

concomitant negative displacement feedback, and reflex

delays found in neuromuscular may provide this auto-

matic gain control.

JBiSE

[3] Donelan, J. M. and Pearson, K. G., (2004) Contribution

of sensory feedback to ongoing ankle extensor activity

during the stance phase of walking, Can. J. Physiol.

Pharmacol, 82, 589–598.

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, leng th feedback,

and delays in positive feedb ack pathways were robust in

the face of large parametric and structural variations in

the systems considered. 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.

REFERENCES

[1] Brownstone, R. M., Gossard, J. P., and Hultborn, H.,

(1994) Voltage-dependent excitation of motoneurones

from spinal locomotor centres in the cat, Springer-Verlag,

Exp. Brain Res., 102, 3444.

[2] Grey, M. J., Mazzaro, N., Nielsen, J. B., and Sinkjær, T.,

(2004) Ankle extensor proprioceptors contribute to the

enhancement of the soleus EMG during the stance phase

of human walking1, Can. J. Physiol. Pharmacol., 82,

610–616.

[4] Frigon, A. and Rossignol, S., (2006) Experiments and

models of sensorimotor interactions during locomotion,

Springer Berlin/Heidelberg, Saturday, DOI: 10.1007/s

00422-006-0129-x.

[5] Prochazka, A. and Gillard, D., (1997) Sensory control of

locomotion, Proceeding of American Control Confer-

ence.

[6] Prochazka, A., Gillard, D., and Bennet, D. J., (1997)

Implications of positive feedback in the control of

movement, the American Physiological Society

0022-3077/97, 5.00 Copyright.

[7] Taylor, A., Ellaway, P. H., Durbaba, R., and Rawlinson,

S., (2000) Distinctive patterns of static and dynamic

gamma motor activity during locomotion in the decere-

brate cat, Journal of Physiology, 529(3), pp. 825–836.

[8] Raya, J. G., Ramı´rez, A., and Mun˜oz-Martı´nez, E. J.,

(2004) Gamma 3 alpha linkage and persistent firing of Ia

fibers by pudendal nerve stimulation in the decerebrate

cat, J. Neurophysiol, 92, 387–394.

[9] Krakauer, J. W., MD, (2006) Spinal cord physiology:

Receptors and reflexes.

[10] Csercsik, D., (2006) Construction of simple dynamic

models of the gamma-loop mechanism, Research Report

SCL-001.

[11] Knutson, G. A., Edward, D. C., and Owens, F., MS, DC,

(2003) Active and passive characteristics of muscle tone

and their relationship to models of subluxation, 0008-

3194/2003/269–283/$2.00/\JCCA.