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.

The investigation of new chaotic systems in spherical coordinates has been one of the present exciting research directions in exploring new chaotic systems. In this paper, a new system in spherical coordinates is presented. The appealing feature of the proposed system is that the dynamics of the system cannot pass through a sphere of a specific radius and stop as soon as the solution crosses the sphere in Cartesian coordinates. So, the system’s attractors are limited to be located on one side of the sphere and cannot touch it. Moreover, the reason for this phenomenon is that the velocity of a system’s variable becomes zero for a specific value of that variable. The proposed system has three unstable equilibrium points and four hidden attractors, including a limit cycle and a strange attractor inside and a limit cycle and a strange attractor outside the sphere. The system’s dynamical properties are investigated with the help of bifurcation diagrams and the calculation of Lyapunov exponents. The basin of attraction for the system’s attractors is also studied. Finally, the system is controlled or stabilized using the impulsive control theory.

<abstract> <p>Map-based neuronal models have received much attention due to their high speed, efficiency, flexibility, and simplicity. Therefore, they are suitable for investigating different dynamical behaviors in neuronal networks, which is one of the recent hottest topics. Recently, the memristive version of the Rulkov model, known as the m-Rulkov model, has been introduced. This paper investigates the network of the memristive version of the Rulkov neuron map to study the effect of the memristor on collective behaviors. Firstly, two m-Rulkov neuronal models are coupled in different cases, through electrical synapses, chemical synapses, and both electrical and chemical synapses. The results show that two electrically coupled memristive neurons can become synchronous, while the previous studies have shown that two non-memristive Rulkov neurons do not synchronize when they are coupled electrically. In contrast, chemical coupling does not lead to synchronization; instead, two neurons reach the same resting state. However, the presence of both types of couplings results in synchronization. The same investigations are carried out for a network of 100 m-Rulkov models locating in a ring topology. Different firing patterns, such as synchronization, lagged-phase synchronization, amplitude death, non-stationary chimera state, and traveling chimera state, are observed for various electrical and chemical coupling strengths. Furthermore, the synchronization of neurons in the electrical coupling relies on the network's size and disappears with increasing the nodes number.</p> </abstract>

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.

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