Dichotomously switched phase flows

TL;DR: In this article, the effect of additive noise on the planar FitzHugh-Nagumo ordinary differential equations was examined and the statistical behavior of the oscillatory and direct transitions was examined.

Abstract: The general formalism for periodic dichotomous noise on nonpotential flows is considered. This uncorrelated noise process switches suddenly at integer values of period $\ensuremath{\tau}$. The effect of additive noise of this kind on the planar FitzHugh-Nagumo ordinary differential equations [R. FitzHugh, Biophys. J. 1, 445 (1961); J. Nagumo, S. Arimoto, and Y. Yoshikawa, Proc. IRE 50, 2061 (1962)] is examined. For large $\ensuremath{\tau}$, quasifractal attractors are observed, whereas for the white-noise limit, where $\ensuremath{\tau}$ is small, a Fokker-Planck equation describes the evolution. The magnitude of $\ensuremath{\tau}$ determines the smoothness of the transient evolution and equilibrium density of the system. Typically the stochastic equations give rise to two regions of high density near the stable fixed points of the underlying autonomous system. The stiffness parameter $\ensuremath{\varepsilon}$ in the differential equations determines the fast variable, its associated nullcline, and the resulting flow structure. For small $\ensuremath{\varepsilon}$ the cubic nullcline controls the motion and transitions between the high-density peaks occur along segments of a noisy limit cycle. For large $\ensuremath{\varepsilon}$ the linear nullcline governs the transitions and the peaks are joined by a single band. The statistical behavior of the oscillatory and direct transitions is examined.