Stochastic FitzHugh-Nagumo equations on networks with impulsive noise

TL;DR

This paper models electrical potential diffusion in neural networks using stochastic FitzHugh-Nagumo equations with impulsive noise, proving global well-posedness of the resulting stochastic PDE system.

Contribution

It introduces a novel stochastic PDE framework for neural networks with impulsive noise, extending FitzHugh-Nagumo models to include stochastic boundary conditions and Levy noise.

Findings

Established global well-posedness of the stochastic PDE system

Extended FitzHugh-Nagumo models to stochastic boundary conditions

Developed a mathematical framework for neural diffusion with impulsive noise

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 L\'evy noise. We prove global well-posedness in the mild sense for such stochastic partial differential equation by monotonicity methods.