Abstract
This study focuses on the synchronization analysis of Hindmarsh–Rose neurons coupled through a common memristor (coupled mHRN). Initially, we thoroughly examine the synchronization of two mHRNs coupled via a common memristor before exploring synchronization in a network of mHRNs. The stability of the proposed model is analyzed in three cases, demonstrating the existence of a single equilibrium point whose stability is influenced by external stimuli. The stable and unstable regions are investigated using eigenvalues. Through bifurcation analysis and the determination of maximum Lyapunov exponents, we identify chaotic and hyperchaotic trajectories. Additionally, using the next-generation matrix method, we calculate the chaotic number C0, demonstrating the influence of coupling strength on the chaotic and hyperchaotic behavior of the system. The exponential stability of the synchronous mHRN is derived analytically using Lyapunov theory, and our results are verified through numerical simulations. Furthermore, we explore the impact of initial conditions and memristor synapses, as well as the coupling coefficient, on the synchronization of coupled mHRN. Finally, we investigate a network consisting of n number of mHRNs and observe various collective behaviors, including incoherent, coherent, traveling patterns, traveling wave chimeras, and imperfect chimeras, which are determined by the memristor coupling coefficient.
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