Dynamics of large random recurrent neural networks: oscillations of 2-population model

We are interested in the asymptotic behavior of noisy discrete time neural networks with two populations of neurons. The couplings and thresholds are asymmetric and Gaussian. We use the large deviation techniques developed by Ben Arous and Guionnet (1995) to study the limit behavior of our networks when their size grows to infinity. We prove a propagation of chaos property, which is closely related to vanishing correlations of activation states. We are also able to compute the limit distribution of the activation potentials of the neurons in the thermodynamic limit. It is Gaussian and characterized by a set of dynamic mean field equations. The numerical study of these equations reveals a parametric domain where the mean of this limit law is subject to periodic oscillations. This property can be directly related to synchronization. Moreover, we prove a useful equation satisfied by the mean quadratic distance between two trajectories, which allows to predict the dynamics of the network