A constructive mean field analysis of multi population neural networks with random synaptic weights and stochastic inputs

TL;DR

This paper introduces a constructive mean field framework for multi-population neural networks with random weights and stochastic inputs, enabling effective analysis and computation of neural dynamics across scales.

Contribution

It provides a new constructive method for solving mean field equations in multi-population neural networks, with proven convergence and complexity analysis.

Findings

The method converges to the unique solution of the mean field equations.

Numerical experiments demonstrate the framework's effectiveness in exploring neural behaviors.

The analysis offers insights into neural mass models like Jansen and Rit's, revealing richer dynamics.

Abstract

We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials. They are coupled by synaptic connections acting on their resulting activity, a nonlinear function of their membrane potential. At the second (mesoscopic) scale, interacting populations of neurons are described individually by similar equations. The equations describing the dynamical and the stationary mean field behaviors are considered as functional equations on a set of stochastic processes. Using this new point of view allows us to prove that these equations are well-posed on any finite time interval and to provide a constructive method for effectively computing their unique solution. This method is proved to converge to the unique solution and we characterize its complexity and convergence rate. We also provide partial results for the stationary problem on infinite time intervals. These results shed some new light on such neural mass models as the one of Jansen and Rit \cite{jansen-rit:95}: their dynamics appears as a coarse approximation of the much richer dynamics that emerges from our analysis. Our numerical experiments confirm that the framework we propose and the numerical methods we derive from it provide a new and powerful tool for the exploration of neural behaviors at different scales.