Global dynamics of winner-take-all networks

In this paper, we study the global dynamics of winner-take-all (WTA) networks. These networks generalize Hopfield's networks to the case where competitive behavior is enforced within clusters of neurons while the interaction between clusters is modeled by cluster-to-cluster connectivity matrices. Under the assumption of intracluster and intercluster symmetric connectivity, we show the existence of Lyapunov functions that allow us to draw rigorous results about the long-term behavior for both the iterated-map and continuous-time dynamics of the WTA network. Specifically, we show that the attractors of the synchronous, iterated-map dynamics are either fixed points or limit cycles of period 2. Moreover, if the network connectivity matrix satisfies a weakened form of positive definiteness, limit cycles can be ruled out. Furthermore, we show that the attractors of the continuous-time dynamics are only fixed points for any connectivity matrix. Finally, we generalize the WTA dynamics to distributed networks of clustered neurons where the only requirement is that the input-output mapping of each cluster be the gradient map of a convex potential.