Effects of Lévy noise on the Fitzhugh-Nagumo model: A perspective on the maximal likely trajectories

This paper considers about the escape paths of the FitzHugh-Nagumo neural system driven by symmetric α-stable Lévy noise (non-Gaussian noise). The existing research has shown that noise can make this system produce a spike pulse, which corresponds to a state transition. To analyze the effects of Lévy noise on the state transition, a novel statistical quantity called maximal likely trajectory, which is obtained by recording the maximizer of the probability density function at every moment, is used to characterize the escape paths of the equilibrium and revel the relationship between state transition and noise intensity or Lévy motion index. The numerical experiments show that for fixed Lévy motion index, the larger noise intensity can promote this neural system to an excitatory state. In addition, the influence of Lévy motion index on the state transition depends on the selection of noise intensity in this neural system. Meanwhile, as a comparison, the case driven by Brownian motion (Gaussian noise) is also taken into account, which shows that in some situations Lévy noise makes the FitzHugh-Nagumo system excited in shorter time. In addition, the maximal likely trajectory provides us with a new perspective to show the existence of a separatrix in the stochastic setting of the FitzHugh-Nagumo model and also depict the rough shape of the middle part of this separatrix.