Nonlinear Fokker-Planck equations modelling large networks of neurons

PhD_course_Roux_0.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.