2010/05/11 by Guillemette Chapuisat, Chapuisat, Guillemette, Romain Joly +1
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.1005.1729
openalex publication_date 2010/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ut=Δu + f(u) \1|y|≤ R - αu \1|y|>R, with f a bistable nonlinearity taking effect only on the domain \Rm× [-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.