Dynamical behavior of delayed Hopfield neural networks with discontinuous activations

In this paper, a general class of neural networks with arbitrary constant delays is studied, whose neuron activations are discontinuous and may be unbounded or nonmonotonic. Based on the Leray–Schauder alternative principle and generalized Lyapunov approach, conditions are given under which there is a unique equilibrium of the neural network, which is globally asymptotically stable. Moreover, the existence and global asymptotic stability of periodic solutions are derived, where the neuron inputs are periodic. The obtained results extend previous works not only on delayed neural networks with Lipschitz continuous neuron activations, but also on delayed neural networks with discontinuous neuron activations.