Periodic solutions to a mean-field model for electrocortical activity

TL;DR

This paper analyzes a continuum model of human cortical electrical activity, focusing on periodic solutions near a Hopf bifurcation, and computes standing wave solutions in a hyperbolic PDE framework.

Contribution

It introduces a detailed analysis of periodic solutions in a mean-field PDE model of electrocortical activity near a bifurcation point.

Findings

Identification of standing wave solutions near bifurcation

Analysis of stability of periodic solutions

Contribution to understanding cortical wave dynamics

Abstract

We consider a continuum model of electrical signals in the human cortex, which takes the form of a system of semilinear, hyperbolic partial differential equations for the inhibitory and excitatory membrane potentials and the synaptic inputs. The coupling of these components is represented by sigmoidal and quadratic nonlinearities. We consider these equations on a square domain with periodic boundary conditions, in the vicinity of the primary transition from a stable equilibrium to time-periodic motion through an equivariant Hopf bifurcation. We compute part of a family of standing wave solutions, emanating from this point.