Global asymptotic and exponential synchronization of ring neural network with reaction-diffusion term and unbounded delay

In this paper, we consider a ring neural network of coupled neurons with distributed and discrete time-varying delays along with the reaction-diffusion terms. We derive sufficient conditions that ensure the existence and uniqueness of the equilibrium point, synchronized asymptotic stability and exponential synchronization by using the theory of topological degree, properties of M-matrix, Lyapunov functional and analytic methods. The obtained results remove the assumption on the boundedness of activation functions. At the end, we give two examples to show the validity of our analysis.