Particle systems with a singular mean-field self-excitation. Application to neuronal networks

TL;DR

This paper models neuronal networks with a singular mean-field self-excitation using a McKean-Vlasov equation, proving solution existence and approximation methods, and revealing synchronized spiking behavior.

Contribution

It introduces a novel approach to handle singular self-excitation in neuronal models using Skorohod M1 topology, enabling solution construction and approximation.

Findings

Existence of solutions for the singular McKean-Vlasov equation.

Approximation of solutions via finite particle systems and delayed equations.

Identification of synchronized spiking solutions in the network.

Abstract

We discuss the construction and approximation of solutions to a nonlinear McKean-Vlasov equation driven by a singular self-excitatory interaction of the mean-field type. Such an equation is intended to describe an infinite population of neurons which interact with one another. Each time a proportion of neurons 'spike', the whole network instantaneously receives an excitatory kick. The instantaneous nature of the excitation makes the system singular and prevents the application of standard results from the literature. Making use of the Skorohod M1 topology, we prove that, for the right notion of a 'physical' solution, the nonlinear equation can be approximated either by a finite particle system or by a delayed equation. As a by-product, we obtain the existence of 'synchronized' solutions, for which a macroscopic proportion of neurons may spike at the same time.