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Hybrid dynamical systems combine evolution equations with state transitions. When the evolution equations are discrete-time (also called map-based), the result is a hybrid discrete-time system. A class of biological neural network models that has recently received some attention falls within this category: map-based neuron models connected by means of fast threshold modulation (FTM). FTM is a connection scheme that aims to mimic the switching dynamics of a neuron subject to synaptic inputs. The dynamic equations of the neuron adopt different forms according to the state (either firing or not firing) and type (excitatory or inhibitory) of their presynaptic neighbours. Therefore, the mathematical model of one such network is a combination of discrete-time evolution equations with transitions between states, constituting a hybrid discrete-time (map-based) neural network. In this paper, we review previous work within the context of these models, exemplifying useful techniques to analyse them. Typical map-based neuron models are low-dimensional and amenable to phase-plane analysis. In bursting models, fast-slow decomposition can be used to reduce dimensionality further, so that the dynamics of a pair of connected neurons can be easily understood. We also discuss a model that includes electrical synapses in addition to chemical synapses with FTM. Furthermore, we describe how master stability functions can predict the stability of synchronized states in these networks. The main results are extended to larger map-based neural networks.

A system consisting of two Rulkov map-based neurons coupled through reciprocal electrical synapses as a simple phenomenological example is discussed. When the electrical coupling is excitatory, the square-wave bursting can be well predicted by the bifurcation analysis of a comparatively simple low-dimensional subsystem embedded in the invariant manifold. While, when the synapses are inhibitory due to the artificial electrical coupling, a fast-slow analysis is carried out by treating the two slow variables as two different bifurcation parameters. The main result of this paper is to present a mechanism for the occurrence of a kind of special elliptic bursting. The mechanism for this kind of elliptic-like bursting is due to the interaction between two chaotic oscillations with different amplitudes. Moreover, the generation of antiphase synchronization of networks lies in the different switching orders between two pairs of different chaotic oscillations corresponding to the first neuron and the second neuron, respectively.

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. [ABSTRACT FROM AUTHOR]

In this paper, the synchronization problem of a chaotic Aihara neuron network is considered. Based on the master stability function (MSF) analysis, some necessary conditions are proposed to investigate the complete synchronization of the electrical coupled chaotic Aihara neuron networks. Especially, the two-dimensional parameter-space plots are obtained numerically to visualize the possible synchronized region. For the chaotic Rulkov neuron network, the numerical simulations of the MSF show that complete synchronization of the electrical coupled Aihara neuron network can be achieved when the coupling strength lies in a suitable interval, while it is highly improbable to attain complete synchronization when all neurons are in the chaotic bursting. Moreover, the complete synchronization cannot be reached with further increase or decrease of coupling strength. These results are probably common features as far as discrete-time neuron networks are concerned. Finally, based on the drive-response active control method, an LMI-based criterion is obtained to ensure complete synchronization of the available chaotic Aihara neuron network. The spatiotemporal patterns measured by the state distribution of neurons are employed to validate the effectiveness of our control scheme. [ABSTRACT FROM AUTHOR]