Abstract:
In mathematical neuroscience, the noisy integrate-and-fire model is often formulated as a first hitting time problem for a stochastic differential equation (SDE): the membrane potential of a given neuron is modelled by an SDE which accumulates (or “integrates”) inputs, until it first reaches a certain threshold, at which point the neuron emits (or “fires”) an action potential. As opposed to taking the SDE for granted, and only modelling the appearance of an action potential, it is of course much more interesting to consider a large system of coupled neurons, where various interactions give rise to the particular inputs, and where the action potential is felt by the rest of the system. In this talk, we will consider such a stochastic particle system, and we will discus the convergence to a well-posed mean-field limit as the number of neurons becomes infinite. The limiting object is a nonlocal and nonlinear stochastic partial differential equation of McKean–Vlasov type.