# A Result of Metastability for an Infinite System of Spiking Neurons

## TL;DR

This paper demonstrates that a stochastic model of infinite interacting spiking neurons exhibits metastable behavior, with extinction times converging to an exponential distribution as the system size grows, especially for small leak rates.

## Contribution

It proves metastability in an infinite neuron system model, extending previous phase transition results to include metastable dynamics.

## Key findings

- Extinction times converge to exponential distribution for small leak rates.
- Metastable behavior observed in the neuron model.
- Extension of phase transition results to metastability context.

## Abstract

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 has a parameter $\gamma$, corresponding to the rate of the leaking times of the neurons and, as the title says, it was proven there to present a phase transition phenomenon with respect to this $\gamma$. Here we prove that this model also exhibit a metastable behavior. By this we mean that if $\gamma$ is small enough, then the re-normalized time of extinction converges toward an exponential random variable of mean 1 as the number of neurons goes to infinity.