A Markov chain approximation of switched Fokker-Planck equations for a model of on-off intermittency in the postural control during quiet standing

Abstract

Intermittent on-off switching of feedback control has been considered as a major mechanism of generating human postural sway during quiet stance. Such an intermittent control model is described by a switched stochastic delay system with unstable subsystems, driven by additive Gaussian white noise. Dynamics of the model can be analyzed by the corresponding switched-type hybrid Fokker-Planck (FP) equations, describing time evolutions of probability density function (PDF) for a state point to be located at each point of the state space. Here, in order to perform detailed numerical analysis of the intermittent control model, we develop a comprehensive numerical recipe to represent and simulate hybrid FP equations, and apply it to the intermittent control model. To this end, the hybrid FP equations are approximated by a finite state Markov chain model under certain assumptions and by using the finite element method. Then, stochastic on-off switching dynamics of the Markov chain model, including time evolutions of PDFs, stationary PDFs and power spectral density functions of model-simulated postural sway are analyzed. We also investigate how the stationary PDF alters as values of important parameters of the model change. Dynamics of the Markov chain model are compared with Monte Carlo-based dynamics of the original intermittent control model, by which the developed numerical recipe and the resultant Markov chain model are validated.