Mean Field Theory of Dynamical Systems Driven by External Signals

Our understanding of non linear dynamical systems and networks has made tremendous progress during the past decades. In most cases the autonomous dynamics is studied. The situation where the network is strongly driven by an external signal has so far been less investigated even though it arises in many different contexts in the natural and artificial world. Examples include networks of interacting chemicals (proteins, RNA) in a cell driven by unpredictable external chemical signals; networks of neurons driven by an external sensory input; artificial neural networks and their applications in machine learning; the response of population dynamics and ecological networks to changes in external conditions such as the weather; the responses of stock prices to economically significant news such as a company earnings, or unemployment numbers. In all these cases taking into account the external input is essential if one wants to understand correctly the dynamics, both because the external input is often large (it cannot be treated as a small perturbation), and because in some cases the systems itself has been selected according to its response to the fluctuating and unpredictable external variables. The aim of the present work is to show, through the study of a specific but important example, how mean field techniques can provide a detailed understanding of dynamical networks strongly driven by an external signal. In the mean field approach the average feedback of the variables on themselves is taken into account through a self consistent equation, while the correlations between individual variables are neglected. The apparently extremely complicated dynamics of the network is thus reduced to much simpler evolution equations for a few collective variables. Previous applications of the mean field approach to dynamical systems (but without including an external input), and in particular neural networks, include e.g.