Classical Solutions for a nonlinear Fokker-Planck equation arising in Computational Neuroscience

TL;DR

This paper establishes the existence and properties of classical solutions to a nonlinear Fokker-Planck equation modeling neural network behavior, transforming it into a Stefan-like problem and analyzing stability and blow-up conditions.

Contribution

It introduces a novel transformation of a nonlinear Fokker-Planck equation into a Stefan-like free boundary problem and provides global existence results for inhibitory networks.

Findings

Global classical solutions for inhibitory neural networks.

Local well-posedness and blow-up criteria for excitatory networks.

Spectral analysis related to network stability and asymptotic behavior.

Abstract

In this paper we analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a source term given by a delta function. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Finally, we will also study the spectrum for the linear problem corresponding to uncoupled networks and its relation to Poincar\'e inequalities for studying their asymptotic behavior.