Existence and learning of oscillations in recurrent neural networks

In this paper we study a particular class of -node recurrent neural networks (RNN's). In the 3-node case we use monotone dynamical systems theory to show, for a well-defined set of parameters, that, generically, every orbit of the RNN is asymptotic to a periodic orbit. Then we investigate whether RNN's of this class can adapt their internal parameters so as to "learn" and then replicate autonomously (in feedback) certain external periodic signals. Our learning algorithm is similar to identification algorithms in adaptive control theory. The main feature of the algorithm is that global exponential convergence of parameters is guaranteed. We also obtain partial convergence results in the -node case.