Dynamical behaviors of recurrently connected neural networks and linearly coupled networks with discontinuous right-hand sides

Abstract

The aim of this paper is to provide a systematic review on the framework to analyze dynamics in recurrently connected neural networks with discontinuous right-hand sides with a focus on the authors’ works in the past three years. The concept of the Filippov solution is employed to define the solution of the neural network systems by transforming them to differential inclusions. The theory of viability provides a tool to study the existence and uniqueness of the solution and the Lyapunov function (functional) approach is used to investigate the global stability and synchronization. More precisely, we prove that the diagonal-dominant conditions guarantee the existence, uniqueness, and stability of a general class of integro-differential equations with (almost) periodic self-inhibitions, interconnection weights, inputs, and delays. This model is rather general and includes the well-known Hopfield neural networks, Cohen-Grossberg neural networks, and cellular neural networks as special cases. We extend the absolute stability analysis of gradient-like neural network model by relaxing the analytic constraints so that they can be employed to solve optimization problem with non-smooth cost functions. Furthermore, we study the global synchronization problem of a class of linearly coupled neural network with discontinuous right-hand sides.