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The phenomenon of self-referential phase reset is investigated in ensembles of interacting FitzHugh–Nagumo neurons with different topologies of couplings. It is shown that the reset phase of neurons oscillation is independent of the initial phase and is determined by parameters of the stimulus. This process does not require direct influence on all elements of the ensemble and takes place when the stimulus is applied to one of the interacting neurons only. The influence of inter-neuron couplings and stimulus parameters on ensemble dynamics and phase reset phenomenon is studied.

The influence of coupling time delay on synchronization processes is studied within the framework of a model describing the dynamics of two inferior olive cells coupled electronically through gap-junctions surrounded by synaptic inhibitory terminals that block these couplings. It was found that different types of synchronous behavior exist in the system, depending on parameters, and that even small changes of coupling time delay may change a dynamic regime to another one. This may be the transition between the regimes of different synchronization types, spike time binding, nonsynchronous behavior or the regime of activity suppression of an inferior olive cell.

This tutorial reviews a new important class of mathematical phenomenological models of neural activity generated by iterative dynamical systems: the so-called map-based systems. We focus on 1-D and 2-D maps for the replication of many features of the neural activity of a single neuron. It was shown that such systems can reproduce the basic activity modes such as spiking, bursting, chaotic spiking-bursting, subthreshold oscillations, tonic and phasic spiking, normal excitability, etc. of the real biological neurons. We emphasize on the representation of chaotic spiking-bursting oscillations by chaotic attractors of 2-D models. We also explain the dynamical mechanism of formation of such attractors and transition from one mode to another. We briefly present some synchronization mehanisms of chaotic spiking-bursting activity for two coupled neurons described by 1-D maps.

A model of working memory system in the form of a network of neuron-like units is considered. The units are stimulated by a periodic subthreshold stimulus and interact via inhibitory feedback. The information storage in the system occurs in the form of periodic clusters of neural activity. Several sequentially firing clusters can coexist in the network simultaneously corresponding to several stored images. The influence of noise on cluster dynamics and noise resistance of the system are studied. Optimal parameters corresponding to maximal noise resistance of the system were found.

The problem of phase synchronization of Chua's chaotic oscillators is investigated. We consider Chua's circuit when it exhibits a chaotic attractor and apply a single pulse stimulus. It is shown that under certain conditions the system displays self-referential phase reset (SPR) phenomenon. This is a case when the reset phase of the chaotic oscillation is independent on the initial phase, hence on the time moment when the stimulus has been applied. In an ensemble of chaotic oscillators simultaneously stimulated, the SPR yields mutual phase coherence or synchronization between the units. We describe basic dynamical mechanisms of the effect and show how it can be used for controllable cluster formation and for the control of chaotic dynamics.

The dynamics of propagating interfaces in a bistable neural network is investigated. We consider the network composed of two coupled 1D lattices and assume that they interact in a local spatial point (pin contact). The network unit is modeled by the FitzHugh–Nagumo-like system in a bistable oscillator mode. The interfaces describe the transition of the network units from the rest (unexcited) state to the excited state where each unit exhibits periodic sequences of excitation pulses or action potentials. We show how the localized inter-layer interaction provides an "excitatory" or "inhibitory" action to the oscillatory activity. In particular, we describe the interface propagation failure and the initiation of spreading activity due to the pin contact. We provide analytical results, computer simulations and physical experiments with two-layer electronic arrays of bistable cells.

We study the propagation of nonlinear waves in a three-component reaction–diffusion system. The problem of the existence of the stationary pulse-like solutions is reduced to the analysis of homoclinic trajectories of a fourth-order system of nonlinear ODEs. We have obtained the parameter set corresponding to the homoclinic bifurcations that defines the velocity spectra of the traveling pulses. We have shown that the pulses behave like autowaves annihilating in head-on collision and like dissipative solitons crossing each other, reflecting at boundaries. We have provided a qualitative explanation for such a behavior.

A one-dimensional chain consisting of many locally coupled oscillatory units is considered when both diffusive (linear) and nonlinear interactions exist. The dynamics of units may also be altered by a weak white Gaussian noise. Two different collective evolutionary paths have been identified in the chain: (i) the onset of temporal phase clusters and their subsequent breakdown or reconstruction through phase resetting, and (ii) the formation of a phase wave during a process of selforganization. Such a wave may originate from an initially disordered pattern after appropriate reorganization of phases and amplitudes occurring on a time lapse that can be very long.

We show how wave motions propagate in a nonequilibrium discrete medium modeled by a one-dimensional array of diffusively coupled Chua's circuits. The problem of the existence of the stationary wave solutions is reduced to the analysis of bounded trajectories of a fourth-order system of nonlinear ODEs. Then, we study the homoclinic and heteroclinic bifurcations of the ODEs system. The lattice can sustain the propagation of solitary pulses, wave fronts and complex wave trains with periodic or chaotic profile.

Using analytical and numerical methods we analyze the phenomenon of spatial disorder and also travelling wave propagation in a ring of coupled bistable oscillators. We show the existence of space-homogeneous and space-inhomogeneous waves. We also provide the linear stability of these solutions. Moreover, we discuss the role of the bistability of the unit oscillator and point out a specific property, genuine of the discreteness of the chain.

In this paper, we investigate the possible propagation of travelling waves of a chaotic profile in an unbounded one-dimensional array of inductively-coupled Modified Chua’s Circuits. We show that the basic unit (cell) of our array is a relaxation-like chaotic oscillator, and its dynamics can be modeled by a two-dimensional system with hysteresis. This hysteresis system is studied via an associated 1D point map, and the existence of various distinct chaotic attractors is proved.

The possibility of travelling waves in a one-dimensional circular array of Chua's circuits is investigated. It is shown that the problem can be reduced to the analysis of the periodic orbits of a three-dimensional system of ordinary differential equations (ODEs) describing the individual dynamics of Chua's circuit. The results of analytical and numerical studies of the bifurcation associated with the appearance of the periodic orbits are presented. A criterion for stability of the travelling waves is also provided.

Numerical evidence is provided of (long lasting) solitary waves (traveling localized dissipative structures) and their bound states as well as periodic and “chaotic”, erratic wave trains of a dissipation-modified Korteweg-de Vries equation originally derived to account for long wavelength oscillatory Bénard-Marangoni convection.

Soliton dynamics in a perturbed sine-Gordon equation modeling a long Josephson junction is investigated. Solitons are found to exist in both simple and chaotic forms. Soliton synchronization by an alternating magnetic field is analysed. Current-voltage characteristics of Josephson junction are plotted.

Propagating dissipative (localized) structures like solitary waves, pulses or “solitons,” “bound solitons,” and “chaotic” wave trains are shown to be solutions of a dissipation-modified Korteweg-de Vries equation that in particular appears in Marangoni-Bénard convection when a liquid layer is heated from the air side and in the description of internal waves in sheared, stably stratified fluid layers.

An infinite set of stationary traveling waves are found in a discrete analog of the nonlinear diffusion Huxley equation. Their profiles are determined by the trajectories of Bernoulli scheme containing two symbols. It is proved that these waves are stable with respect to perturbations moving with the same speed. Thus it is shown that chaos of traveling waves is realized in this system.

Dynamics of a chain of coupled Chua's circuits is investigated. Existence of spatial disorder is revealed. Dynamics of wave fronts in the limiting case of a continuous medium is studied.