Cluster synchronization and chimera in discrete-time memristive neuronal networks: A master stability approach

Understanding synchronization in neuronal networks is crucial for uncovering mechanisms of coherent information processing and complex brain dynamics. In particular, discrete-time memristive neurons offer a promising yet largely unexplored framework for studying such behaviors. This paper investigates cluster synchronization and chimera states in discrete-time memristive Chialvo neuronal networks using the Master Stability Function (MSF) framework. To assess synchronizability, the network is divided into two clusters with nonlocal and global coupling configurations and analyzed at both small (N=15) and large (N=100) scales. The stability of the cluster-synchronization manifold is established through Lyapunov analysis of the transverse perturbation modes, while numerical simulations validate the theoretical predictions via synchronization error calculations. The results reveal three distinct dynamical regimes: (i) asynchronous dynamics in both clusters, (ii) full cluster synchronization, and (iii) chimera states, where only one cluster synchronizes while the other remains incoherent with sine-like activity patterns. Cross-scale comparisons show that although the qualitative MSF structure is preserved, the critical coupling threshold for cluster synchronization decreases in larger networks and further diminishes when diffusive electrical synapses are replaced with inner-linking functions. These findings emphasize the role of coupling structure, network size, and interaction type in governing synchronization transitions, offering new insights into coherent and partially coherent dynamics in neuronal networks.