ASYMPTOTIC PROFILES FOR A TRAVELING FRONT SOLUTION OF A BIOLOGICAL EQUATION

作者：Chapuisat Guillemette^*; Joly Romain

来源：Mathematical Models and Methods in Applied Sciences, 2011, 21(10): 2155-2177.

DOI：10.1142/S0218202511005696

• Spreading depression
• reaction-diffusion equation
• traveling fronts
• Sturm-Liouville theory

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 vertical bar(vertical bar y vertical bar <= R) - alpha u1 vertical bar(vertical bar y vertical bar > R), with f a bistable nonlinearity taking effect only on the domain R x [-R,R], which represents the grey matter layer. We study the existence, the stability and the energy of nontrivial asymptotic profiles of the possible traveling fronts. For this purpose, we present dynamical systems techniques and graphic criteria based on Sturm-Liouville theory and apply them to the above equation. This yields three different types of behavior for the solution u after stimulation, depending on the thickness R of the grey matter. This may partly explain the difficulties to observe depolarization waves in the human brain.

• 出版日期2011-10