Asymptotic profiles for a travelling front solution of a biological equation

TL;DR

This paper investigates the existence and stability of traveling wave solutions in a biological brain model, revealing how grey matter thickness influences wave behavior and potentially explains observational and therapeutic challenges.

Contribution

It introduces a dynamical systems approach with Sturm-Liouville theory to analyze traveling fronts in a bistable reaction-diffusion model of brain waves.

Findings

Three distinct behaviors depending on grey matter thickness R.

Conditions for existence and stability of traveling fronts.

Insights into difficulties observing depolarization waves and therapeutic failures.

Abstract

We are interested in the existence of depolarization waves in the human brain. These waves propagate in the grey matter and are absorbed in the white matter. We consider a two-dimensional model $u_t=\Delta u + f(u) \1_{|y|\leq R} - \alpha u \1_{|y|>R}$, with $f$ a bistable nonlinearity taking effect only on the domain $\Rm\times [-R,R]$, which represents the grey matter layer. We study the existence, the stability and the energy of non-trivial asymptotic profiles of possible travelling fronts. For this purpose, we present dynamical systems technics and graphic criteria based on Sturm-Liouville theory and apply them to the above equation. This yields three different behaviours of the solution $u$ after stimulation, depending of the thickness $R$ of the grey matter. This may partly explain the difficulties to observe depolarization waves in the human brain and the failure of several therapeutic trials.