The effect of graph connectivity on metastability in a stochastic system of spiking neurons

We consider a continuous-time stochastic model of spiking neurons originally introduced by Ferrari et al. in Ferrari et al. (2018). 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. In this model the spiking activity of any given neuron is represented by a point process, whose rate fluctuates between 1 and 0 over time depending on whether the membrane potential is positive or null. The membrane potential of each neuron evolves in time by integrating all the spikes of its adjacent neurons up to the last spike of the said neuron, so that when a neuron spikes, its membrane potential is reset to 0 while the membrane potential of each of its adjacent neurons is increased by one unit. Moreover, each neuron is exposed to a leakage effect, modeled as an abrupt loss of membrane potential which occurs at random times driven by a Poisson process of some fixed rate γ. It was previously proven that when the graph G is the infinite one-dimensional lattice, this model presents a phase transition with respect to the parameter γ. It was also proven that, when γ 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 does not hold anymore if γ is large enough, and in fact we prove that for γ>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 γ.