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