Strong error bounds for the convergence to its mean field limit for systems of interacting neurons in a diffusive scaling

TL;DR

This paper establishes explicit strong error bounds for the convergence of a stochastic system of interacting neurons to its mean field limit, improving understanding of the system's behavior as the number of neurons grows large.

Contribution

It provides the first strong convergence rate with explicit bounds for the neuron system's mean field limit in a diffusive scaling.

Findings

Proved strong convergence with explicit rate

Established coupling technique for jump process and Brownian motion

Enhanced understanding of neuron system dynamics in large populations

Abstract

We consider the stochastic system of interacting neurons introduced in De Masi et al. (2015) and in Fournier and L\"ocherbach (2016) and then further studied in Erny, L\"ocherbach and Loukianova (2021) in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order $ 1 / \sqrt{N}.$ In between successive spikes, each neuron's potential follows a deterministic flow. In a previous article we proved the convergence of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation. In the present article we complete this study by establishing a strong convergence result, stated with respect to an appropriate distance, with an explicit rate of convergence. The main technical ingredient of our proof is the coupling introduced in Koml\'os, Major and Tusn\'ady (1976) of the point process representing the small jumps of the particle system with the limit Brownian motion