Global solvability of a networked integrate-and-fire model of McKean-Vlasov type

TL;DR

This paper proves the existence and uniqueness of solutions over all time for a networked integrate-and-fire neuron model of McKean-Vlasov type when the interaction strength parameter is sufficiently small.

Contribution

It establishes global well-posedness results for a class of neuron interaction models in the regime of weak interactions, extending understanding of their mathematical behavior.

Findings

Existence and uniqueness of solutions for small interaction strength

Global well-posedness established for all time

Mathematical analysis of neuron interaction dynamics

Abstract

We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by $\alpha$, is of great importance as the resulting system is known to blow-up for large values of $\alpha$. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when $\alpha$ is small enough.