On the Global Dynamics of an Electroencephalographic Mean Field Model of the Neocortex

TL;DR

This paper analyzes the mathematical properties and dynamics of a mean field model of the neocortex's EEG activity, establishing well-posedness, solution bounds, and discussing the complexity of the model's behavior.

Contribution

It provides a rigorous mathematical analysis of the model's solutions, including existence, uniqueness, regularity, and conditions for nonnegativity, advancing understanding of its global dynamics.

Findings

Solutions are well-posed in appropriate function spaces.

Bounded absorbing sets exist for all biophysical parameters.

The model may lack a global attractor due to non-compactness in certain parameter regimes.

Abstract

This paper investigates the global dynamics of a mean field model of the electroencephalogram developed by Liley et al., 2002. The model is presented as a system of coupled ordinary and partial differential equations with periodic boundary conditions. Existence, uniqueness, and regularity of weak and strong solutions of the model are established in appropriate function spaces, and the associated initial-boundary value problems are proved to be well-posed. Sufficient conditions are developed for the phase spaces of the model to ensure nonnegativity of certain quantities in the model, as required by their biophysical interpretation. It is shown that the semigroups of weak and strong solution operators possess bounded absorbing sets for the entire range of biophysical values of the parameters of the model. Challenges towards establishing a global attractor for the model are discussed and it is shown that there exist parameter values for which the constructed semidynamical systems do not possess a compact global attractor due to the lack of the asymptotic compactness property. Finally, using the theoretical results of the paper, instructive insights are provided into the complexity of the behavior of the model and computational analysis of the model.