Time averages of a metastable system of spiking neurons

Appears in collection : 2022 - T1 - WS2 - Mathematical modeling and statistical analysis in neuroscience

We study a stochastic system of spiking neurons in which the spikes of the neurons are represented by a family of interacting point processes on the positive real line. The model depends on a parameter gamma, representing the intensity of the natural leakage of the neurons. This model has already been proven to exhibit several interesting behaviors. Firstly it undergoes phase transition with respect to the parameter gamma [1]. Moreover the time of extinction of finite versions of the system have been proven to be asymptotically memory-less for small gamma [2,3], a characteristic property of metastable systems. Here we show that this last result actually holds in the whole subcritical region and that previous to extinction the finite versions of the system are in a regime which in some sense resemble stationarity. This is the second characteristic property of metastable dynamics. The main idea is to use a bypass through the theory of "Interacting particles systems".