Chimera state in neural network with the PID coupling

This study delves into the emergence of collective behaviors within a network comprising interacting cells. Each cell integrates a fixed number of neurons governed by an activation gradient based on Hopfield's model. The intra-cell interactions among neurons are local and directed, while inter-cell connections are facilitated through a PID (Proportional-Integral-Derivative) coupling mechanism. This coupling introduces an adaptable environmental variable, influencing the network dynamics significantly. Numerical simulations employing three neurons per cell across a network of fifty cells reveal diverse dynamics, including incoherence, coherence, synchronization, chimera states, and traveling wave. These phenomena are quantitatively assessed using statistical measures such as the order parameter, strength of incoherence, and discontinuity measure. Variations of the resistive, inductive, or capacitive couplings of the inter-cell environment are explored and their effects are analysed. Furthermore, the study identifies multistability in network dynamics, characterized by the coexistence of multiple stable states for the same set of parameters but with different initial conditions. A linear augmentation strategy is employed for its control.