The phenomenon of hidden heterogeneous extreme multistability is rarely reported in coupled neurons. This phenomenon is investigated in this contribution using a model of a 2D FitzHugh–Nagumo neuron coupled with a 3D Hindmarsh–Rose neuron through a multistable memristive synapse. The investigation of the equilibria revealed that the coupled neuron model is equilibrium free and, thus, displays a hidden dynamics. Some traditional nonlinear analysis tools are used to demonstrate that the heterogeneous neuron system is able to exhibit the coexistence of an infinite number of electrical activities involving both periodic and chaotic patterns. Of particular interest, a noninvasive control method is applied to suppress all the periodic coexisting activities, while preserving only the desired chaotic one. Finally, an electronic circuit of the coupled neurons is designed in the PSpice environment and used to further support some results of the theoretical investigations.

We perform a numerical study on the application of electromagnetic flux on a heterogeneous network of Chialvo neurons represented by a ring-star topology. Heterogeneities are realized by introducing additive noise modulations on both the central–peripheral and the peripheral–peripheral coupling links in the topology not only varying in space but also in time. The variation in time is understood by two coupling probabilities, one for the central–peripheral connections and the other for the peripheral–peripheral connections, respectively, that update the network topology with each iteration in time. We have further reported various rich spatiotemporal patterns like two-cluster states, chimera states, coherent, and asynchronized states that arise throughout the network dynamics. We have also investigated the appearance of a special kind of asynchronization behavior called “solitary nodes” that have a wide range of applications pertaining to real-world nervous systems. In order to characterize the behavior of the nodes under the influence of these heterogeneities, we have studied two different metrics called the “cross-correlation coefficient” and the “synchronization error.” Additionally, to capture the statistical property of the network, for example, how complex the system behaves, we have also studied a measure called “sample entropy.” Various two-dimensional color-coded plots are presented in the study to exhibit how these metrics/measures behave with the variation of parameters.

In this paper we have introduced and investigated the collective behavior of a network of memristive Hindmarsh-Rose (HR) neurons. The proposed model was built considering the memristive autapse of the traditional 2D HR neuron. Using the one-parameter bifurcation diagram and its corresponding maximal Lyapunov exponent graph, we showed that the proposed model was able to exhibit a reverse period doubling route to chaos, phenomenon of interior and exterior crises. Three different configurations of the ring-star network of the memristive HR neuron model, including ring-star, ring, and star, have been considered. The study of those network configurations revealed incoherent, coherent , chimera and cluster state behaviors. Coherent behavior is characterized by synchronization of the neurons of the network, while incoherent behaviors are characterized by the absence of synchronization. Chimera states refer to a differet state where there is a coexistence of synchroniaed and asynchronized nodes of the network. One of the interesting result of the paper is the prevalence of double-well chimera states in both ring and ring-star network and has been first mentioned in the case of memrisitve HR neuron model.