Abstract: We consider a network of integrate-and-fire models for neurons subject to common noise and analyse its mean field limit. The neurons are described by their membrane potential, which diffuses until a threshold is reached, when the neuron fires, that is it resets to a resting potential and transmits voltage to other neurons. We show that in the mean field limit, the evolution of the limiting membrane potential density can be described by a stochastic non-local non-linear Fokker-Planck equation.  We investigate when there can be synchronization of the firing of non-trivial proportions of the network.

Bio-Sketch: Ben Hambly is Professor of Mathematics at University of Oxford. His research interests are in probability, stochastic processes, financial mathematics and fractals.