Pattern formation in reaction-diffusion neural networks with leakage delay

Due to the heterogeneity of the electromagnetic field in neural networks, the diffusion phenomenon of electrons exists inevitably. In this paper, we investigate pattern formation in a reaction-diffusion neural network with leakage delay. The existence of Hopf bifurcation, as well as the necessary and sufficient conditions for Turing instability, are studied by analyzing the corresponding characteristic equation. Based on the multiple-scale analysis, amplitude equations of the model are derived, which determine the selection and competition of Turing patterns. Numerical simulations are carried out to show the possible patterns and how these patterns evolve. In some cases, the stability performance of Turing patterns is weakened by leakage delay and synaptic transmission delay.