This was part of Mean-Field Models for Interacting Agents

## Oscillations in a mean-field model of spiking neurons

Quentin Cormier, Princeton University
Wednesday, November 3, 2021

Abstract: We consider a model of interacting (biological) neurons. Each neuron is described by its membrane potential and is of the type « Integrate-and-Fire »: between two successive spikes the membrane potential (V_t) solves an ODE. The neuron « spikes » at the random rate f(V_t) (this rate only depends on the membrane potential of the neuron). At the spiking time, (V_t) is reset to a resting value. At the same time, the discharge is propagated to the other neurons with a jump in their membrane potential. We are interested in the limit where the number of neurons goes to infinity: a typical neuron in this mean-field asymptotic follows a McKean-Vlasov SDE. We study this equation: well-posedness an description of the invariant probability measures. We then study the (local) stability of these invariant probability measures as well as the existence of periodic solutions through a Hopf bifurcation.