Global solvability of a networked integrate-and-fire model of McKean-Vlasov type

We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by $\alpha$, is of great importance as the resulting system is known to blow-up for large values of $\alpha$. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when $\alpha$ is small enough.