Abstract:
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. |