Full-Dimensional Proportional-Derivative Control Technique for Turing Pattern and Bifurcation of Delayed Reaction-Diffusion Bidirectional Ring Neural Networks

In neural networks, the states of neural networks often exhibit significant spatio-temporal heterogeneity due to the diffusion effect of electrons and differences in the concentration of neurotransmitters. One of the macroscopic reflections of this time-spatial inhomogeneity is Turing pattern. However, most current research in reaction-diffusion neural networks has focused only on one-dimensional location information, and the remaining results considering two-dimensional location information are still limited to the case of two neurons. In this paper, we conduct the dynamic analysis and optimal control of a delayed reaction-diffusion neural network model with bidirectional loop structure. First, several mathematical descriptions are given for the proposed neural network model and the full-dimensional partial differential proportional-derivative (PD) controller is introduced. Second, by analyzing the characteristic equation, the conditions for Hopf bifurcation and Turing instability of the controlled network model are obtained. Furthermore, the amplitude equation of the controlled neural network is obtained based on the multiscale analysis method. Subsequently, we determine the key parameters affecting the formation of Turing pattern depending on the amplitude equation. Finally, multiple sets of computer simulations are carried out to support our theoretical results. It is found that the diffusion coefficients and time delays have significant effects on spatio-temporal dynamics of neural networks. Moreover, after reasonable parameter proportioning, the full-dimensional PD control method can alleviate the spatial heterogeneity caused by diffusion projects and time delays.