The mean-field Limit of sparse networks of integrate and fire neurons

TL;DR

This paper develops a novel mean-field theory for large sparse networks of integrate-and-fire neurons, introducing new observables and hierarchies to handle non-exchangeable agents and singular interactions.

Contribution

It introduces a new framework for analyzing the mean-field limit of sparse neuron networks, including a novel notion of observables and a hierarchy extending BBGKY.

Findings

Derived a continuous macroscopic limit for neuron systems

Established stability estimates for the hierarchy

Addressed challenges of sparse, singular interactions

Abstract

We study the mean-field limit of a model of biological neuron networks based on the so-called stochastic integrate-and-fire (IF) dynamics. Our approach allows to derive a continuous limit for the macroscopic behavior of the system, the 1-particle distribution, for a large number of neurons with no structural assumptions on the connection map outside of a generalized mean-field scaling. We propose a novel notion of observables that naturally extends the notion of marginals to systems with non-identical or non-exchangeable agents. Our new observables satisfy a complex approximate hierarchy, essentially a tree-indexed extension of the classical BBGKY hierarchy. We are able to pass to the limit in this hierarchy as the number of neurons increases through novel quantitative stability estimates in some adapted weak norm. While we require non-vanishing diffusion, this approach notably addresses the challenges of sparse interacting graphs/matrices and singular interactions from Poisson jumps, and requires no additional regularity on the initial distribution.