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We discuss the construction and approximation of solutions to a nonlinearMcKean-Vlasov equation driven by a singular self-excitatory interaction of themean-field type. Such an equation is intended to describe an infinitepopulation of neurons which interact with one another. Each time a proportionof neurons 'spike', the whole network instantaneously receives an excitatorykick. The instantaneous nature of the excitation makes the system singular andprevents the application of standard results from the literature. Making use ofthe Skorohod M1 topology, we prove that, for the right notion of a 'physical'solution, the nonlinear equation can be approximated either by a finiteparticle system or by a delayed equation. As a by-product, we obtain theexistence of 'synchronized' solutions, for which a macroscopic proportion ofneurons may spike at the same time.

Particle systems with a singular mean-field self-excitation. Application to neuronal networks