  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.
