  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.
