Higher-order interactions might play a significant role in the collective dynamics of the brain. With this motivation, we here consider a simplicial complex of neurons, in particular, studying the effects of pairwise and three-body interactions on the emergence of synchronization. We assume pairwise interactions to be mediated through electrical synapses, while for second-order interactions, we separately study diffusive coupling and nonlinear chemical coupling. For all the considered cases, we derive the necessary conditions for synchronization by means of linear stability analysis, and we compute the synchronization errors numerically. Our research shows that the second-order interactions, even if of weak strength, can lead to synchronization under significantly lower first-order coupling strengths. Moreover, the overall synchronization cost is reduced due to the introduction of three-body interactions if compared to pairwise interactions.

A memristor is a vital circuit element that can mimic biological synapses. This paper proposes the memristive version of a recently proposed map neuron model based on the phase space. The dynamic of the memristive map model is investigated by using bifurcation and Lyapunov exponents’ diagrams. The results prove that the memristive map can present different behaviors such as spiking, periodic bursting, and chaotic bursting. Then, a ring network is constructed by hybrid electrical and chemical synapses, and the memristive neuron models are used to describe the nodes. The collective behavior of the network is studied. It is observed that chemical coupling plays a crucial role in synchronization. Different kinds of synchronization, such as imperfect synchronization, complete synchronization, solitary state, two-cluster synchronization, chimera, and nonstationary chimera, are identified by varying the coupling strengths.