Hopf bifurcation in a Mean-Field model of spiking neurons

We study a family of non-linear McKean-Vlasov SDEs driven by Poisson measure, modelling the mean-field asymptotic network generalized Integrate-and-Fire neurons. give sufficient conditions to have periodic solutions through Hopf bifurcation. Our spectral involve location roots an explicit holomorphic function. The proof relies on two main ingredients. First, we introduce discrete time Markov Chain modeling phases successive spikes neuron. invariant measure this is related shape solutions. Secondly, use Lyapunov-Schmidt method obtain self-consistent oscillations. illustrate result with toy model for which all can be analytically checked.