Abstract

The emergence of patterned activity in a neural network are key processes in human and animal brains. However, since they often arise from the interplay between a large number of cells, these mechanisms are very difficult to encompass whithout the use of simple, consistent and self-contained mathematical models. In this talk, I shall present some nonlinear partial differential models obtained from mean-field techniques applied to particle systems. I will be focussing in particular on the so-called Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model (a non-linear non-local Fokker-Planck-type equation) for which we have, my collaborators and myself, obtained some new results and insights about the convergence to stationary states (desynchronisation), the emergence and the shape of periodic self-sustained oscillations and the finite-time blow-up (synchronisation).