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