Stationary Distribution of Stochastic Hopfield Neural Network with Cross-Diffusion Under Time-Varying Modified Markov Switching

In this paper, we introduce a class of stochastic Hopfield neural network with cross-diffusion (SHNNCD), and for the first time, investigate the existence of the ergodic stationary distribution for SHNNCD under time-varying modified Markov switching. In terms of the internal structure, we incorporate cross-diffusion to fully account for the coupling effects between different neuronal states. In terms of the external environment, compared with traditional Markov switching, we consider a class of Markov switching modified by time-varying functions, including periodic, decaying, and random types. Based on these innovations in both internal structure and external environment, we construct a global Lyapunov function for SHNNCD using graph theory under the consideration of cross-diffusion, and derive the criterion for the existence of the ergodic stationary distribution of SHNNCD. Finally, we verify the validity of our theoretical results through numerical simulations.