Chaos and Synchronisation in Large Systems of Interacting Particles with Random Connections

Abstract

I study systems of interacting particles indexed by a lattice. These have many applications, but in this talk I focus on applications in neuroscience. The particles are subject to white noise (Brownian motion), with random connections sampled from a probability distribution that is invariant under translations of the lattice. I study the limiting behaviour of system-wide averages (the “empirical measure”) as the size of the network asymptotes to infinity. I find a variety of different behaviours under different scalings of the connection strength. When the connection strength is scaled by $N^{-1∕2}$ ($N$ being the network size), the system becomes non-Markovian (meaning that the dynamics is influenced by the entire past). When the connection strength is unscaled but decays spatially, one obtains spatial correlations in the infinite size limit. I also find conditions under which particles with connections from an Erdos-Renyi random graph synchronize their oscillations in the large time limit.