Stochastic FitzHugh-Nagumo equations on networks with impulsive noise

Stefano Bonaccorsi,Carlo Marinelli,Giacomo Ziglio

doi:10.1214/ejp.v13-532

Abstract

We consider a system of nonlinear partial differential equations with stochastic dynamical boundary conditions that arises in models of neurophysiology for the diffusion of electrical potentials through a finite network of neurons. Motivated by the discussion in the biological literature, we impose a general diffusion equation on each edge through a generalized version of the FitzHugh-Nagumo model, while the noise acting on the boundary is described by a generalized stochastic Kirchhoff law on the nodes. In the abstract framework of matrix operators theory, we rewrite this stochastic boundary value problem as a stochastic evolution equation in infinite dimensions with a power-type nonlinearity, driven by an additive Lévy noise. We prove global well-posedness in the mild sense for such stochastic partial differential equation by monotonicity methods.

Highlights

In this paper we study a system of nonlinear diffusion equations on a finite network in the presence of an impulsive noise acting on the nodes of the system
Electric signaling by neurons has been studied since the 50s, starting with the classical Hodgkin-Huxley model [16] for the diffusion of the transmembrane electrical potential in a neuronal cell
Successive simplifications of the model, trying to capture the key phenomena of the Hodgkin-Huxley model, lead to the reduced FitzHugh-Nagumo equation, which is a scalar equation with two stable states

Summary

Introduction

In this paper we study a system of nonlinear diffusion equations on a finite network in the presence of an impulsive noise acting on the nodes of the system. Electric signaling by neurons has been studied since the 50s, starting with the classical Hodgkin-Huxley model [16] for the diffusion of the transmembrane electrical potential in a neuronal cell. First we prove existence and uniqueness of mild solution for the problem under Lipschitz conditions on the nonlinear term (theorem 3.6) This result (essentially already known) is used to obtain existence and uniqueness in the mild sense for the SPDE with a locally Lipschitz continuous dissipative drift of FitzHugh-Nagumo type by techniques of monotone operators

Setting of the problem

Well-posedness of the linear deterministic problem

The stochastic Cauchy problem

The stochastic convolution process

Existence and uniqueness in the Lipschitz case

FitzHugh-Nagumo type nonlinearity