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We study a family of non-linear McKean-Vlasov SDEs driven by a Poissonmeasure, modelling the mean-field asymptotic of a network of generalizedIntegrate-and-Fire neurons. We give sufficient conditions to have periodicsolutions through a Hopf bifurcation. Our spectral conditions involve thelocation of the roots of an explicit holomorphic function. The proof relies ontwo main ingredients. First, we introduce a discrete time Markov Chain modelingthe phases of the successive spikes of a neuron. The invariant measure of thisMarkov Chain is related to the shape of the periodic solutions. Secondly, weuse the Lyapunov-Schmidt method to obtain self-consistent oscillations. Weillustrate the result with a toy model for which all the spectral conditionscan be analytically checked.

Hopf bifurcation in a Mean-Field model of spiking neurons