How Can Anomalous-Diffusion Neural Networks Under Connectomics Generate Optimized Spatiotemporal Dynamics

Spatiotemporal dynamics in the brain have been recognized as strongly related to the formation of perceived and cognitive diseases, such as delusions and hallucinations in Alzheimer's disease. However, two practical considerations are rarely mentioned in related mechanism research: the connectomics networking and the anomalous diffusion generated by the complex medium between neurons and the complex topology of neural networks, respectively. Furthermore, how to optimize the corresponding dynamics behaviors has excellent implications for treating brain diseases. This article first realizes the networking under connectomics for an anomalous-diffusion single-neuron model and applies a nonlinear state feedback control to generate optimized dynamic behaviors, which provides a paradigm of nonequilibrium self-organization driven by anomalous diffusion. Then, by tracing the root distribution of the characteristic equation, some controlled conditions causing or inhibiting Turing instability and Hopf bifurcation are deduced, and the effects of self-diffusion and cross diffusion on Turing instability range are also revealed. At last, thorough numerical simulations are updated to illustrate the results. It is emphasized that delay, self-diffusion, cross diffusion, and fractional order occupy dominant positions in determining the network's spatiotemporal dynamics, and utilizing the control strategy can efficiently reduce Turing instability and delay Hopf bifurcation.