Limit cycle behavior in spasticity: analysis and evaluation

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Abstract
We examined ankle clonus in four spastic subjects to determine whether this oscillatory behavior has the properties of a limit cycle, and whether it is driven by peripheral sensory input or by a spinal generator. Using Floquet Theory and Poincare sections to assess reflex stability, we found that cycle-to-cycle variability was small, such that the Floquet multipliers were always less than unity. Furthermore, the steady-state periodic orbit was not dependent on the initial position of the ankle. Both of these findings, coupled with strong correlations between the size of the applied load and the frequency of ankle movements and electromyogram burst frequency suggests that clonus behaves as a locally stable limit cycle driven from peripheral receptors. To better understand how nonlinear elements might produce stable oscillatory motion, we simulated the ankle stretch reflex response. We found that delays in the pathway caused the reflex to come on during the shortening phase of movement, so the additional reflex torque required to sustain oscillatory ankle movements was quite small. Furthermore, because the resistance to stretch is largely due to passive mechanics whose properties are quite stationary, the system is robust to small perturbations within the reflex pathway.

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