In this talk, I will present two or three questions arising in the characterization of the dynamics of randomly connected neural networks. I will show that the level of disorder in the connectivity coefficients tightly controls the dynamics and governs transitions towards collectively synchronized or chaotic dynamics. First, I will show that synchronization in networks with excitation and inhibition arises from mean-field properties of the system, except when excitation and inhibition are balanced, in which case synchronization is associated to the eigenvalue with largest real part of the connectivity matrix. I will then present microscopic mechanisms arising at the classical phase transition from trivial to chaotic behaviors in purely randomly connected networks (with no separation between excitation and inhibition), and show that this is associated to an exponential divergence of the number of equilibria of the dynamics. I will compute this explosion and show that the rate of divergence happens to be equal to the Lyapunov exponent of the chaotic dynamics, leading to a possible microscopic interpretation of the emergence of chaos in these systems. These are joint works with G. Hermann, G. Wainrib, K. Pakdaman and Luis Garcia del Molino.

INRIA et College de France