Kinetic Description of Social Dynamics: From Consensus to Flocking

Fokker Planck and neuroscience

Maria Schonbek

University of California Santa Cruz

Models for the collective behavior of a large body of interacting neurons are frequently represented by a stochastic differential equations. By Ito's rule, the corresponding equation, can be reduced to a deterministic partial differential equation which depict the evolution of a probability density. This equation has the structure of a nonlinear Fokker-Planck equation (FCE) with a source term. Surprisingly by appropriate changes the FCE can be transformed into a non standard Stefan-type free boundary value problem with a Dirac-delta source term. I will show that there are global classical solutions to this Stefan problem, in the case of inhibitory neural networks, while for excitatory networks there is local well-posedness of classical solutions together with a blow up criterium. The last part of the lecture will focus on an analysis of the spectrum for the linearized operator corresponding to uncoupled networks.

This work is joint with J. Carrillo, M. del M Gonzalez and M Gualdani.