Modeling neuronal networks as systems of interacting point processes with memory of variable length: models comparison and mean-field limits

Abstract

My subject of research is the mathematical modeling and statistical analysis of biological neuronal networks. The framework for this study is the class of systems of spiking neurons introduced in discrete time by Galves and Löcherbach (2013) and in continuous time by De Masi et al. (2014). The models of this class are systems of interacting point processes with memory of variable length. The research that I am doing currently as a post-doctoral researcher intends to contribute to the development of this direction of research, studying questions such as metastability, phase transition, existence and perfect simulation and statistical inference of the graph of interaction between neurons. The aim of this internship abroad project is to work with Prof. Eva Löcherbach in the two following questions. First of all, we want to compare the linear versions of Galves and Löcherbach (GL) and Hawkes models for systems of spiking neurons. This is a very important issue, since the two models have been widely used to study systems of spiking neurons but no theoretical work has yet been proposed to compare them. Second, this project intends to study mean-field limits and phase transition for an extension of the GL model introduced in Galves and Laxa (2022). Prof. Eva Löcherbach is a distinguished foreigner researcher associated to the NeuroMat project. (AU)