On a finite-size neuronal population equation

TL;DR

This paper investigates a finite-size population equation for spiking neurons, proving its well-posedness and stability for certain models, and providing simulation algorithms for practical use.

Contribution

It extends population equations to finite networks, proving mathematical properties and offering simulation methods for neuroscience applications.

Findings

Proves well-posedness and stability of the finite-size population equation for LIF neurons with escape noise.

Develops efficient algorithms for simulating the finite-size population dynamics.

Bridges theoretical analysis with practical simulation tools for finite neural populations.

Abstract

Population equations for infinitely large networks of spiking neurons have a long tradition in theoretical neuroscience. In this work, we analyze a recent generalization of these equations to populations of finite size, which takes the form of a nonlinear stochastic integral equation. We prove that, in the case of leaky integrate-and-fire (LIF) neurons with escape noise and for a slightly simplified version of the model, the equation is well-posed and stable in the sense of Br\'emaud-Massouli\'e. The proof combines methods from Markov processes taking values in the space of positive measures and nonlinear Hawkes processes. For applications, we also provide efficient simulation algorithms.