%0 Journal Article
%T Global solvability of a networked integrate-and-fire model of McKean-Vlasov type
%A Fran£¿ois Delarue
%A James Inglis
%A Sylvain Rubenthaler
%A Etienne Tanr¨¦
%J Mathematics
%D 2012
%I arXiv
%R 10.1214/14-AAP1044
%X 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.
%U http://arxiv.org/abs/1211.0299v5