Local/global analysis of the stationary solutions of some neural field equations

TL;DR

This paper investigates how the stationary solutions of neural field equations depend on nonlinearity and input contrast, using advanced mathematical theories to predict their global and local behaviors, with applications to specific neural models.

Contribution

It introduces a combined use of degree and bifurcation theories in infinite-dimensional spaces to analyze neural field solutions and provides finite-dimensional approximations for detailed study.

Findings

Predictions of global and local behaviors of neural field solutions.

Analysis of neural mass and neural field models with heterogeneous connectivity.

New insights into the dependency of solutions on nonlinearity and input contrast.

Abstract

Neural or cortical fields are continuous assemblies of mesoscopic models, also called neural masses, of neural populations that are fundamental in the modeling of macroscopic parts of the brain. Neural fields are described by nonlinear integro-differential equations. The solutions of these equations represent the state of activity of these populations when submitted to inputs from neighbouring brain areas. Understanding the properties of these solutions is essential in advancing our understanding of the brain. In this paper we study the dependency of the stationary solutions of the neural fields equations with respect to the stiffness of the nonlinearity and the contrast of the external inputs. This is done by using degree theory and bifurcation theory in the context of functional, in particular infinite dimensional, spaces. The joint use of these two theories allows us to make new detailed predictions about the global and local behaviours of the solutions. We also provide a generic finite dimensional approximation of these equations which allows us to study in great details two models. The first model is a neural mass model of a cortical hypercolumn of orientation sensitive neurons, the ring model. The second model is a general neural field model where the spatial connectivity isdescribed by heterogeneous Gaussian-like functions.