Lévy noise-induced escape in an excitable system**This work was partly supported by the NSF grant 1620449, and NSFC grants 11531006, 11371367 and 11271290.

Abstract

This paper considers the dynamics of escape in the stochastic FitzHugh–Nagumo (FHN) neuronal model driven by symmetric α-stable Lévy noise. External or internal stimulation may make the excitable system produce a pulse or not, which can be interpreted as an escape problem. A new method to analyse the state transition from the rest state to the excitatory state is presented. This approach consists of two deterministic indices: the first escape probability (FEP) and the mean first exit time (MFET). We find that higher FEP in the rest state (equilibrium) promotes such a transition and MFET reflects the stability of the rest state directly with the selected escape region. The developed two dimensional numerical simulation method to calculate FEP and MFET can not only avoid a dimension reduction, but is also applicable for the cases with large noise. In addition, FEP provides us with a new perspective to understand the seperatrix of the stochastic FHN model. It can be seen that smaller jumps of the Lévy motion and relatively small noise intensity are conducive to the production of spikes. In order to characterize the effect of noise on the selected escape region in which the equilibrium lies, the area of higher FEP and MFET in the escape region are calculated. Meanwhile, Brownian motion as a special case is also taken into account for comparison.