A Result of Metastability for an Infinite System of Spiking Neurons

In 2018, Ferrari et al. wrote a paper called ``Phase Transition for Infinite Systems of Spiking Neurons{''} in which they introduced a continuous time stochastic model of interacting neurons. This model consists in a countable number of neurons, each of them having an integer-valued membrane potential, which value determine the rate at which the neuron spikes. This model has also a parameter gamma, corresponding to the rate of the leak times of the neurons, that is, the times at which the membrane potential of a given neuron is spontaneously reset to its resting value (which is 0 by convention). As its title says, it was proven in this previous article that this model presents a phase transition phenomenon with respect to gamma. Here we prove that this model also exhibits a metastable behavior. By this we mean that if gamma is small enough, then the re-normalized time of extinction of a finite version of this system converges toward an exponential random variable of mean 1 as the number of neurons goes to infinity. (AU)