Modeling neuronal networks as systems of interacting point processes with memory of variable length

Abstract

In this project I continue the research I developed in my PhD project and presented in my articles Laxa (2022) and Galves and Laxa (2022). Laxa (2022) studies a system of interacting point processes with memory of variable length modeling a finite but large network of spiking neurons with leakage. Associated to each neuron there are two point processes, describing its successive spiking and leakage times. We prove that the system has a metastable behavior as the population size diverges. This means that the time at which the system gets trapped by the list of null membrane potentials suitably re-scaled converges to a mean one exponential random time. Galves and Laxa (2022) studies a new model for a highly polarized social network. In this system, the point processes are marked and indicate the successive times in which a social actor express a "favorable" or "contrary" opinion on a certain subject. For this model, we prove that when the polarization coefficient diverges, the social network reaches instantaneous consensus and this consensus has a metastable behavior. The model studied in Galves and Laxa (2022) is an interesting mathematical extension of theclass of networks of spiking neurons introduced by Galves and Löcherbach (2013) and De Masi et al. (2014). One of the two main lines of research of RIDC Neuromat is the mathematical modeling and statistical analysis of neuronal networks. To contribute to the development of this direction of research on the class of models introduced by Galves and Löcherbach (2013) and De Masi et al. (2014), I intend to approach the following questions: metastability, phase transition, perfect simulation and estimation of the graph of interaction between neurons. (AU)