A mean-field model with discontinuous coefficients for neurons with spatial interaction

TL;DR

This paper develops a mean-field model for neurons with spatial interactions, characterized by SDEs with discontinuous coefficients, and analyzes the resulting nonlinear PDE and its well-posedness.

Contribution

It introduces a novel mean-field framework incorporating spatially dependent discontinuous coefficients for neuron models, with rigorous analysis of the PDE and SDE system.

Findings

Existence and uniqueness of solutions to the PDE and SDE system.

Derivation of a nonlinear Fokker-Planck type PDE with discontinuous coefficients.

Validation of the mean-field limit as the number of neurons tends to infinity.

Abstract

Starting from a microscopic model for a system of neurons evolving in time which individually follow a stochastic integrate-and-fire type model, we study a mean-field limit of the system. Our model is described by a system of SDEs with discontinuous coefficients for the action potential of each neuron and takes into account the (random) spatial configuration of neurons allowing the interaction to depend on it. In the limit as the number of particles tends to infinity, we obtain a nonlinear Fokker-Planck type PDE in two variables, with derivatives only with respect to one variable and discontinuous coefficients. We also study strong well-posedness of the system of SDEs and prove the existence and uniqueness of a weak measure-valued solution to the PDE, obtained as the limit of the laws of the empirical measures for the system of particles.