Around the complex interplay of noise, heterogeneity and excitability on collective dynamics of neuronal networks

Abstract: At functional scales, the brain processes information within very large networks of hundreds of thousands of cells, each cell receiving input from thousands of others. Neurons activity are characterized by high levels of noise. Moreover, neural networks are very heterogeneous (electrophysiological parameters of cells and connection amplitudes are highly variable), and some biological evidence tend to suggest that disorder levels may correlate with physiological or pathological behaviors. Understanding the dynamics of such large-scale networks in probabilistic settings is thus important to better understand the properties of brain networks. From the mathematical viewpoint, models of such networks raise a number of new deep questions in multiple domains. This talk will focus on a few open questions and partial solutions belonging to the domain of stochastic analysis. 

In particular, we will focus on three main aspects:

 (i) The role of noise in the spatio-temporal dynamics of large neural networks: using coupling techniques, I will show how to account for the specific multi-scale spatial distribution of cells and connectivity profiles. Studying the limit equations, I will exhibit the particularly surprising role of noise in these dynamics in synchronizing cells activity. 

 (ii) The role of disorder in the connections: Using large-deviations techniques, I will investigate the impact of random connections on the dynamics; here again, I will exhibit a mysterious and somewhat paradoxical role of disorder in synchronizing neuron’s activity (works in collaboration with T. Cabana)

 (iii) Excitable dynamics of cells: (i) and (ii) are done for simplified models of neurons. In reality, neurons show an excitable dynamics and communicate through sharp electrical impulses (spikes). Large-scale spiking neural networks constitute a challenging mathematical problem. If time allows, I will present recent developments on these questions (works in collaboration with Ph. Robert, S. Mischler and C. Quiñinao).