On a Nonlinear Master Equation and the Haken-Kelso-Bunz Model

Abstract

Abstract.A nonlinear master equation (NLME) is proposed based on general information measures. Classical and cut-off solutions of the NLME are considered. In the former case, the NLME exhibits uniquely defined stationary distributions. In the latter case, there are multiple stationary distributions. In particular, for classical solutions, it is shown that transient solutions converge to stationary distributions that maximize information measures (H-theorem). Cut-off distributions are studied numerically for the Haken-Kelso-Bunz model. The Haken-Kelso-Bunz model is known to describe multistable human motor control systems. It is shown that a stochastic Haken-Kelso-Bunz model based on a NLME can exhibit multiple stationary cut-off distributions. In doing so, we illustrate that multistability in stochastic biological systems can be established by means of cut-off distributions.

Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

“…Frequently, nonlinear master equations have been studied as a tool to derive nonlinear Fokker-Planck equations (see, e.g., [14,[41][42][43][44]). However, nonlinear master equations have also been studied in their own merit (see, e.g., [45]).…”

“…The Kullback distance measure is defined on the basis of the Boltzmann-Gibbs-Shannon entropy, which is a fundamental entropy measure not only for equilibrium systems [50] but also for nonequilibrium systems [51]. It was suggested to exploit this link between the Boltzmann-Gibbs-Shannon entropy, the Kullback distance measure, and the ordinary master equation to define nonlinear master equations based on generalized entropy and generalized Kullback distance measures [9,45]. Accordingly, entropy measures of the form…”

“…This property is important for the master equation that will be defined in the following and guarantees that occupation probabilities ( ) solving the master equation do not become negative. The following master equation has been proposed [45]:…”

“…In Frank [45] the coordination of rhythmic movements of two limbs has been addressed exploiting the master equation model (67). It has been assumed that the physiological relevant variable, namely, the relative phase between the limbs, is appropriately described on a partitioned (i.e., discretevalue) state space.…”

Order Parameter Dynamics of Body-scaled Hysteresis and Mode Transitions in Grasping Behavior

“…The transitions between different behavioral modes of grasping are examples of the behavioral transitions that have been extensively studied in motor control systems [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. Behavioral transitions in motor control systems in turn are examples of transitions that naturally occur in complex non-equilibrium systems [30], as is shown in Fig.…”