On a kinetic FitzHugh-Nagumo model of neuronal network

TL;DR

This paper studies a complex mathematical model of neuronal networks using a McKean-Vlasov PDE, establishing existence, uniqueness, and stability of solutions under certain conditions.

Contribution

It provides the first rigorous analysis of existence, uniqueness, and stability for a hypoelliptic, nonlocal FitzHugh-Nagumo neuronal network model.

Findings

Existence of solutions to the PDE was proven.

Non-trivial stationary solutions were identified.

Exponential stability was demonstrated in the small connectivity regime.

Abstract

We investigate existence and uniqueness of solutions of a McKean-Vlasov evolution PDE representing the macroscopic behaviour of interacting Fitzhugh-Nagumo neurons. This equation is hypoelliptic, nonlocal and has unbounded coefficients. We prove existence of a solution to the evolution equation and non trivial stationary solutions. Moreover, we demonstrate uniqueness of the stationary solution in the weakly nonlinear regime. Eventually, using a semigroup factorisation method, we show exponential nonlinear stability in the small connectivity regime.