  We consider a continuous-time stochastic model of spiking neurons. In this
model, we have a finite or countable number of neurons which are vertices in
some graph $G$ where the edges indicate the synaptic connection between them.
We focus on metastability, understood as the property for the time of
extinction of the network to be asymptotically memory-less, and we prove that
this model exhibits two different behaviors depending on the nature of the
specific underlying graph of interaction $G$ that is chosen. This model depends
on a leakage parameter $\gamma$, and it was previously proven that when the
graph $G$ is the infinite one-dimensional lattice, this model presents a phase
transition with respect to $\gamma$. It was also proven that, when $\gamma$ is
small enough, the renormalized time of extinction (the first time at which all
neurons have a null membrane potential) of a finite version of the system
converges in law toward an exponential random variable when the number of
neurons goes to infinity. The present article is divided into two parts. First
we prove that, in the finite one-dimensional lattice, this last result doesn't
hold if $\gamma$ is not small anymore, in fact we prove that for $\gamma > 1$
the renormalized time of extinction is asymptotically deterministic. Then we
prove that conversely, if $G$ is the complete graph, the result of
metastability holds for any positive $\gamma$.
