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: 1soroor_behbahani@yahoo.com; 2Amir_h_jafari@rcstim.ir
Received 1 June 2009; revised 13 July 2009; accepted 15 July 2009.
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
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].
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
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).
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-
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.
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 ).
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
G. Fig-
ures 3I and 3J show that increasing the displacement
feedback gain is effective in specific limitation. While
G in comparison of the system has
more oscillating responses and the settling time is
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
and the settling time is still shorter.
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
Copyright © 2009 JBiSE
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.
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S. Behbahani et al. / J. Biomedical Science and Engineering 2 (2009) 480-483
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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.
[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.
[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,
[4] Frigon, A. and Rossignol, S., (2006) Experiments and
models of sensorimotor interactions during locomotion,
Springer Berlin/Heidelberg, Saturday, DOI: 10.1007/s
[5] Prochazka, A. and Gillard, D., (1997) Sensory control of
locomotion, Proceeding of American Control Confer-
[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
[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-