Metastability for systems of interacting neurons

We study a stochastic system of interacting neurons and its metastable properties. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to 0 and all other neurons receive an additional amount h / N of potential. In between successive spike times, each neuron looses potential at exponential speed. We study this system in the supercritical regime, that is, for sufficiently high values of the synaptic weight h. Under very mild conditions on the behavior of the spiking rate function in the vicinity of 0, is has been shown in Duarte and Ost (Markov Process. Related Fields 22 (2016) 37-52) that the only invariant distribution of the finite system is the trivial measure delta(0) corresponding to extinction of the process. We strengthen these conditions to prove that for large synaptic weights h, the extinction time arrives at exponentially late times in N, and discuss the stability of the equilibrium delta(0) for the non-linear mean-field limit process depending on the parameters of the dynamics. We then specify our study to the case of saturating spiking rates and show that, under suitable conditions on the parameters of the model, (1) the non-linear mean-field limit admits a unique and globally attracting non trivial equilibrium and (2) the rescaled exit times for the mean spiking rate of a finite system from a neighbourhood of the non-linear equilibrium rate converge in law to an exponential distribution, as the system size diverges. In other words, the system exhibits a metastable behavior.