Synchronization of Rulkov neuron networks coupled by excitatory and inhibitory chemical synapses

Complete synchronization of three-layer Rulkov neuron network coupled by electrical and chemical synapses

This paper analyzes the complete synchronization of a three-layer Rulkov neuron network model connected by electrical synapses in the same layers and chemical synapses between adjacent layers. The outer coupling matrix of the network is not Laplacian as in linear coupling networks. We develop the master stability function method, in which the invariant manifold of the master stability equations (MSEs) does not correspond to the zero eigenvalues of the connection matrix. After giving the existence conditions of the synchronization manifold about the nonlinear chemical coupling, we investigate the dynamics of the synchronization manifold, which will be identical to that of a synchronous network by fixing the same parameters and initial values. The waveforms show that the transient chaotic windows and the transient approximate periodic windows with increased or decreased periods occur alternatively before asymptotic behaviors. Furthermore, the Lyapunov exponents of the MSEs indicate that the network with a periodic synchronization manifold can achieve complete synchronization, while the network with a chaotic synchronization manifold can not. Finally, we simulate the effects of small perturbations on the asymptotic regimes and the evolution routes for the synchronous periodic and the non-synchronous chaotic network.

Rulkov neural network coupled with discrete memristors

The features of memristive-coupled neural networks have been studied extensively in the continuous field. However, the particularities of the discrete domain are rarely mentioned. This paper constructs a discrete memristor with sine-type conductance and applies the discrete memristor to coupling the Rulkov neuron maps for the first time. The properties of the proposed memristive-coupled bi-neuron Rulkov map and multi-neuron Rulkov neural network are probed. In order to better characterize the discrete system, many numerical simulation methods are used. Such as the normalized mean synchronization error, bifurcation diagrams, phase portraits, spatiotemporal patterns and so on. Numerical studies have shown that in discrete memristor-coupled neural networks, both parameters and coupling factors affect the dynamics of the system, resulting in complex and interesting behavioural changes.

Stochastic transformations of multi-rhythmic dynamics and order-chaos transitions in a discrete 2D model

A problem of the analysis of stochastic effects in multirhythmic nonlinear systems is investigated on the basis of the conceptual neuron map-based model proposed by Rulkov. A parameter zone with diverse scenarios of the coexistence of oscillatory regimes, both spiking and bursting, was revealed and studied. Noise-induced transitions between basins of periodic attractors are analyzed parametrically by statistics extracted from numerical simulations and by a theoretical approach using the stochastic sensitivity technique. Chaos-order transformations of dynamics caused by random forcing are discussed.

Influence of Autapses on Synchronization in Neural Networks With Chemical Synapses

A great deal of research has been devoted on the investigation of neural dynamics in various network topologies. However, only a few studies have focused on the influence of autapses, synapses from a neuron onto itself via closed loops, on neural synchronization. Here, we build a random network with adaptive exponential integrate-and-fire neurons coupled with chemical synapses, equipped with autapses, to study the effect of the latter on synchronous behavior. We consider time delay in the conductance of the pre-synaptic neuron for excitatory and inhibitory connections. Interestingly, in neural networks consisting of both excitatory and inhibitory neurons, we uncover that synchronous behavior depends on their synapse type. Our results provide evidence on the synchronous and desynchronous activities that emerge in random neural networks with chemical, inhibitory and excitatory synapses where neurons are equipped with autapses.