Dynamics of moments of FitzHugh-Nagumo neuronal models and stochastic bifurcations

Abstract

For the study of the behavior of noisy neuronal models, Rodriguez and Tuckwell have introduced an elegant and systematic method which consists of replacing the system of stochastic differential equations with a system of deterministic equations representing the dynamics of the means, variances, and covariance of the state variables [R. Rodriguez and H.C. Tuckwell, Phys. Rev. E 54, 5585 (1996)]. In this work, we first report a modification of their method in the case of the FitzHugh-Nagumo model which enhances the accuracy of the approximation without including higher order moments. This method is then combined with a self-consistency argument in order to better characterize the behavior of the underlying stochastic processes through the computation of approximate auto- and cross-correlation functions of the state variables. Finally, we argue that the moments' equations can also reveal the existence of stochastic bifurcations, i.e., qualitative changes in the dynamics of stochastic systems.