Collective Behavior: Macroscopic versus Kinetic Descriptions

Macroscopic activity of spatially extended networks with singular multi-scale connectivity patterns: limits and noise-induced transitions

Jonathan Touboul

Collège de France, Paris

The cortex is a very large network characterized by a complex connectivity including at least two scales: a microscopic scale at which the interconnections are non-specific and very dense, while macroscopic connectivity patterns connecting different regions of the brain at larger scale are extremely sparse. This motivates to analyze the behavior of networks with multiscale coupling, in which a neuron is connected to its v(N) nearest-neighbors where v(N) = o(N), and in which the probability of macroscopic connection between two neurons vanishes. These are called singular multi- scale connectivity patterns. We introduce a class of such networks and derive their continuum limit. We show convergence in law and propagation of chaos in the thermodynamic limit. The limit equation obtained is an intricate non-local McKean-Vlasov equation with delays which is universal with respect to the type of micro-circuits and macro-circuits involved. In a specific case, the solutions to these equations are Gaussian, whose mean and standard deviation satisfy a closed-form integro-differential equation in which the noise is a parameter: this allows to uncover bifurcations arising as a function of noise levels.