Neuronal dynamics on branching-generated graphs: macroscopic limits and Self-Organized Criticality

• Grant number: 26/12625-5
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: July 30, 2026
• End date: July 29, 2027
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Probability
• Principal Investigator: Nancy Lopes Garcia
• Grantee: Denis Araujo Luiz
• Supervisor: Eva Loecherbach
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: École Polytechnique, France
• Associated to the scholarship: 25/02013-0 - Some spatial processes with contact dynamics and generalizations, BP.PD

Abstract

The probabilistic modeling of neuronal dynamics and information diffusion has been significantly advanced by the rigorous study of interacting particle systems. Recent breakthroughs have established propagation of chaos for nonlinear jumping stochastic differential equations driven by Poisson measures. However, these limits typically rely on mean-field regimes, assuming complete or homogeneous random graphs. In biological systems, structural connectomes exhibit sparse, modular, and hierarchical topologies. This project proposes a novel mathematical framework by placing a generalized Galves-L\"ocherbach neuronal model - incorporating deterministic refractory periods - onto a hierarchical random graph $\mathcal{G}_n$ generated by the leaves of a Galton-Watson branching process. By deriving the macroscopic limits of this delay-driven system via propagation of chaos techniques, we aim to rigorously investigate the emergence of Self-Organized Criticality (SOC), demonstrating how the structural hierarchy and the deterministic delay act as a restorative feedback mechanism. Furthermore, we will investigate the system's metastability and functional fluctuations under nearly stable noise, deriving Functional Central Limit Theorems (FCLT) for this structured dynamics. (AU)