title: Mean-field theory of echo state networks.
abstract: Dynamical systems driven by strong external signals are ubiquitous in nature and engineering. Here we study "echo state networks," networks of a large number of randomly connected nodes, which represent a simple model of a neural network, and have important applications in machine learning. We develop a mean-field theory of echo state networks. The dynamics of the network is captured by the evolution law, similar to a logistic map, for a single collective variable. When the network is driven by many independent external signals, this collective variable reaches a steady state. But when the network is driven by a single external signal, the collective variable is non stationary but can be characterized by its time averaged distribution. The predictions of the mean-field theory, including the value of the largest Lyapunov exponent, are compared with the numerical integration of the equations of motion.
authors.affiliation: Laboratoire d’Information Quantique, CP 225, Universit´e libre de Bruxelles (U.L.B.), Av. F. D. Roosevelt 50, B-1050 Bruxelles, Belgium.
authors.forename: Marc
authors.initials: M
authors.lastname: Massar
authors.forename: Serge
authors.initials: S
authors.lastname: Massar