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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.

In this paper, we propose a model describing the dynamics of a neuron capable of storing the so-called short-term memory. From the dynamical viewpoint, the effect of short-term memory means that the neuron “ remembers” the fact of its short-pulse excitation and the action potential is periodically generated for a long time after it. This mechanism of memory storage is realized due to the property of afterdepolarization included in the model. This property is well known in real (live)neurons of cortex and hippocampus.

A model is presented that simulates the process of neuronal synchronization, formation of coherent activity clusters and their dynamic reorganization in the olivo-cerebellar system. Three coupled 2D lattices dealing with the main cellular groups in this neuronal circuit are used to model the dynamics of the excitatory feedforward loop linking the inferior olive (IO) neurons to the cerebellar nuclei (CN) via collateral axons that also proceed to terminate as climbing fiber afferents to Purkinje cells (PC). Inhibitory feedback from the CN-lattice fosters decoupling of units in a vicinity of a given IO neuron. It is shown that noise-sustained oscillations in the IO-lattice are capable to synchronize and generate coherent firing clusters in the layer accounting for the excitable collateral axons. The model also provides phase resetting of the oscillations in the IO-lattices with transient silent behavior. It is also shown that the CN-IO feedback leads to transient patterns of couplings in the IO and to a dynamic control of the size of clusters.

The olivo-cerebellar network plays a key role in the organization of vertebrate motor control. The oscillatory properties of inferior olive (IO) neurons have been shown to provide timing signals for motor coordination in which spatio-temporal coherent oscillatory neuronal clusters control movement dynamics. Based on the neuronal connectivity and electrophysiology of the olivo-cerebellar network we have developed a general-purpose control approach, which we refer to as a universal control system (UCS), capable of dealing with a large number of actuator parameters in real time. In this UCS, the imposed goal and the resultant feedback from the actuators specify system properties. The goal is realized through implementing an architecture that can regulate a large number of parameters simultaneously by providing stimuli-modulated spatio-temporal cluster dynamics.

In this paper, we consider the dynamics of two types of modular neural networks. The first network consists of two modules of non-interacting neurons while each neuron inhibits all the neurons of an opposite module. We explain the mechanism for emergence of anti-phase group bursts in the network and showed that the collective behavior underlies a regular response of the system to external pulse stimulation. The networks of the second type contain modules with complex topology which are connected by relatively sparse excitatory delayed coupling. We found a dual role of the inter-module coupling delay in the collective network dynamics. First, with increasing time delay, in-phase and anti-phase regimes, where individual spikes form rhythmic modular burst-like oscillations, alternate with each other. Second, the average frequency of the collective oscillations in each of these regimes decreases with increasing inter-module coupling delay.

We propose a model of neural network demonstrating variety of sequential activity modes. Unlike the previously known models of transient dynamics, in the present model transient sequential modes are formed by means of dynamical bifurcations and not directly related to the existence of heteroclinic channels. It is shown that network being initially at rest generates a sequence of metastable oscillatory states of activity in response to an external stimulus. We study the influence of the parameters characterized the initial times of synaptic activation processes caused by input information signals on the network dynamics. Numerical simulation of the model has shown, that these parameters determine not only the structure of the set of oscillatory metastable states and the sequence of transitions between them, but also the temporal characteristics of the transition sequences such as the time duration of the oscillatory metastable states.

Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillator's phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous "jittering" regimes with non-equal interspike intervals (ISIs). Each of these regimes corresponds to a periodic solution of the system with a period roughly proportional to the delay. The number of different "jittering" solutions emerging at the bifurcation point increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how a periodic solution exhibiting several distinct ISIs can imply the existence of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback.

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.

Pulse-mediated interactions are common in networks of different nature. Here we develop a general framework for simulation of networks with pulse delayed coupling. We introduce the discrete map governing the dynamics of such networks and describe the computation algorithm for its numerical simulation.

Oscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in the last years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. In the bifurcation point numerous regimes with non-equal interspike intervals emerge simultaneously. We show that this bifurcation is triggered by the steepness of the oscillator's phase resetting curve and that the number of the emerging, so-called "jittering" regimes grows exponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the "multi-jitter" bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phase-reduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit.

We carry out theoretical and experimental studies of cross-frequency synchronization of two pulse oscillators with time-delayed coupling. In the theoretical part of the paper we utilize the concept of phase resetting curves and analyze the system dynamics in the case of weak coupling. We construct a Poincar\'e map and obtain the synchronization zones in the parameter space for $m:n$ synchronization. To challenge the theoretical results we designed an electronic circuit implementing the coupled oscillators and studied its dynamics experimentally. We show that the developed theory predicts dynamical properties of the realistic system, including location of the synchronization zones and bifurcations inside them.

We study the collective dynamics of modular networks consisting of map-based neurons which generate irregular spike sequences. Three types of intramodule topology are considered: a random Erd\"os-R\'enyi network, a small-world Watts-Strogatz network, and a scale-free Barab\'asi-Albert network. The interaction between the neurons of different modules is organized by relatively sparse connections with time delay. For all the types of the network topology considered, we found that with increasing delay two regimes of module synchronization alternate with each other: inphase and antiphase. At the same time, the average rate of collective oscillations decreases within each of the time-delay intervals corresponding to a particular synchronization regime. A dual role of the time delay is thus established: controlling a synchronization mode and degree and controlling an average network frequency. Furthermore, we investigate the influence on the modular synchronization by other parameters: the strength of intermodule coupling and the individual firing rate.

The dynamics of a two-layer bistable coupled map lattice (CML) is investigated. A methodology for the construction of invariant domains in the phase space is developed and applied to the study of pattern formation and inter-layer synchronization. Sufficient conditions for global synchronization are obtained. The phenomena of pattern interaction and replication of form is also investigated. Finally, possible instabilities of the synchronization mode and on–off-diagonal attractors are analyzed.

A leaky-wave semiconductor laser diode has been developed based on the InGaAs/GaAs/InGaP heterostructure. This design made it possible to obtain a high radiation output in a narrow angular range (about 1°—2°) with an energy of 170 μJ in a laser with a cavity length of 0.8 mm and a stripe contact width of 360 μm, pumped by a single current pulse with an amplitude of 88 A and width of 5 μs.

The dynamics of a system composed of two nonlinearly coupled, drastically different nonlinear and eventually oscillatory elements is studied. The rich variety of attractors of the system is studied with the help of phase space analysis and Poincare maps.

A fiber-like lattice with resistively coupled electronic elements mimicking a 1-D discrete reaction-diffusion system is considered. The chosen unit or element in the fiber is the paradigmatic Chua’s circuit, capable of exhibiting bistable, excitable, oscillatory or chaotic behavior. Then the dynamics of a structure of two such interacting parallel active fibers is studied. Suitable conditions for the interaction to yield synchronization and other forms of collective behavior involving both fibers are obtained. They include wave front propagation, pulse reentry and pulse propagation failure, overcoming of propagation failure, and the appearance of a source of synchronized pulses. The possibility of designing controlled dynamic contacts by means of one or a few inter-fiber couplings is also discussed.

We study the dynamics of an assembly of globally coupled bistable elements. We show that bistability of elements results in some new features of clustering in the assembly when there is global coupling. We provide conditions for the existence of stable amplitude-phase clusters and splay-phase states.

In this paper, we study the phenomena of interaction of traveling pulses in a system of two coupled nerve fibers. Each fiber is modeled by a discrete chain of mutually coupled Fitz Hugh-Nagumo excitable elements. The interaction among the fibers can be distributed, homogeneous, and also localized at certain points in space. In the case of homogeneous interaction, we show the possibility of interfiber synchronization of pulses. For localized interactions, we study the dynamics of point and two-point contacts, allowing us to perform effective control over the propagation of excitation along the coupled fibers.

A system composed of two coupled lattices, hence a layered structure, is studied when the unit at each site is an active electronic circuit possessing two accessible stable steady states. In the absence of interlattice coupling, each lattice taken separately represents a discrete, reaction diffusion system. We show that, depending on the strength of the diffusion coefficient, each lattice may exhibit either a wide variety of stable steady patterns or a number of different wave patterns including rotating spirals. Moreover, for fixed reaction kinetics each lattice can exhibit spiral waves of both excitable and oscillatory type. For nonoscillating kinetics, the metastable periodiclike behavior of the unit is at the origin of the oscillatory spirals. From initially different global patterns or waves in each lattice, the interaction may lead to synchronization and hence a new (controlled) form and the replication of a given one. We also show how there is reentry of spiral waves between the two coupled layers associated with the ``competition'' of their oscillatory and excitable spiral wave properties.

The problem of reproduction of spatial images by lattices of oscillating neural units is discussed. We consider that each neuron can be at rest or can oscillate with fixed frequency and that the neurons are coupled electrically by a resistor. Then one layer of neurons, one lattice, is coupled to another similar layer. It is shown that for strong enough interlattice interaction relative to the intralattice diffusion, the shape of the pattern on one lattice is determined uniquely by the image of the other. The reproduction of a stimulus shape is possible even when the number of interlattice couplings is much smaller than the number of neurons in either lattice. Moreover, the spatial features of the images do not depend on the features of the eigenexcitations of the neural lattices, which are discrete, active nonequilibrium media.

The formation of spatio-temporal patterns is one of the most important forms of collective electrical activity of neural networks. Such forms of activity have been detected experimentally in different neural structures, including the structures in visual [4] and somatosensory [11] cortex, in the temporal lobe [7], in the inferior olives [5], etc. Modeling of the network structure and dynamics can be a possible way of identifying mechanisms of the pattern appearance and disappearance in such large-scale systems.

Local cortical circuits appear highly non-random, but the underlying connectivity rule remains elusive. Here, we analyze experimental data observed in layer 5 of rat neocortex and suggest a model for connectivity from which emerge essential observed non-random features of both wiring and weighting. These features include lognormal distributions of synaptic connection strength, anatomical clustering, and strong correlations between clustering and connection strength. Our model predicts that cortical microcircuits contain large groups of densely connected neurons which we call clusters. We show that such a cluster contains about one fifth of all excitatory neurons of a circuit which are very densely connected with stronger than average synapses. We demonstrate that such clustering plays an important role in the network dynamics, namely, it creates bistable neural spiking in small cortical circuits. Furthermore, introducing local clustering in large-scale networks leads to the emergence of various patterns of persistent local activity in an ongoing network activity. Thus, our results may bridge a gap between anatomical structure and persistent activity observed during working memory and other cognitive processes.

The interaction of two coupled lattice dynamical systems of bistable elements is investigated. In particular, we give the critical value of the coupling strength and related variables for the mutual synchronization of regular and chaotic states.

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.

The spatio-temporal dynamics of discrete lattices of coupled bistable elements is considered. It is shown that both regular and chaotic spatial field distributions can be realized depending on parameter values and initial conditions. For illustration, we provide results for two lattice systems: the FitzHugh-Nagumo model and a network of coupled bistable oscillators. For the latter we also prove the existence of phase clusters, with phase locking of elements in each cluster.

Mathematical modeling of phenomena in living systems by using discrete-time systems has a long history. In particular, in the 1940s N. Wiener and A. Rosenblueth developed a cellular automaton system for modeling the propagation of excitation pulses in the cardiac tissue. Cellular automata are regular lattices of elements (cells), each having a finite number of specific states. These states are updated synchronously at discrete time moments, according to some fixed rule. Recently, a new class of discrete-time systems has aroused considerable interest for studying cooperative processes in large-scale neural networks: systems of the coupled nonlinear maps. The state of a map varies at discrete time moments as a cellular automaton, but unlike the latter, takes continuous values. Map-based models hold certain advantages over continuous-time models, i.e. differential equation systems. For example, for reproducing oscillatory properties in continuous-time systems one needs at least two dimensions, and at least three dimensions for chaotic behavior. In discrete time both types of dynamics can be described even in a one-dimensional map. Such a benefit is especially important when modeling complex activity regimes of individual neurons as well as large-scale neural circuits composed of various structural units interacting with each other. For example, to simulate in continuous time the regime of chaotic spike-bursting oscillations, one of the key neural behaviors, one needs to have at least a three-dimensional system of nonlinear differential equations. On the other hand, there are discrete-time two-dimensional systems [1, 2] adequately reproducing this oscillatory activity as well as many other dynamical regimes. For example, the model of Chialvo [3] allows to simulate, among other things, the so-called normal and supernormal excitability. The model of Rulkov [4, 5] has several modifications, one of which is adjusted to simulate different spiking and bursting oscillatory regimes, the other can generate the so-called sub-threshold oscillations, i.e., small-amplitude oscillations below a threshold of excitability. Here we propose the authors’ model [6–9] of neural activity in the form of a two-dimensional map and describe some activity modes which it can reproduce. We show that the model is fairly universal and generates many regimes of neuronal electrical activity. Next, we present the results of modeling the dynamics of a complex neural structure, the olivo-cerebellar system (OCS) of vertebrates, using our basic discrete-time model of neural activity.

International audience; Space-time dynamics of the network system modeling collective behavior of electrically coupled nonlinear cells is investigated. The dynamics of a local cell is described by the dimensionless Morris-Lecar system. It is shown that such a system yields a special class of traveling localized collective activity so called "anti-phase wave patterns". The mechanisms of formation of the patterns are discussed and the region of their existence is obtained by using the weakly coupled oscillators theory.

In this study, an experimental electronic neuron based on Morris-Lecar model is presented, able to become an experimental unit tool to study collective association of robust coupled neurons. The circuit design is given according to the ionic currents of this model. The experimental results are compared to the theoretical prediction, leading to validate this circuit.

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.

The spatiotemporal behavior of a chain of diffusively coupled bistable oscillators is investigated. It is stated that there is spatial disorder and its evolutionary character is demonstrated.

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.

The possible propagation of solitary waves in a one-dimensional array of inductively coupled Chua's circuits is considered. We show that in the long-wave limit, the problem can be reduced to the analysis of the homoclinic orbits of a dynamical system described by three coupled nonlinear ordinary differential equations modeling the individual dynamics of a single Chua's circuit. Analytical, numerical, and experimental results concerning the bifurcations associated with the appearance of homoclinic orbits and thus with the propagation of solitary waves are provided. >

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.

In recent years nonlinear wave processes are attracting growing interest in the studies of neuronal network dynamics and information processes in the brain. Waves of excitation, localized activity patterns, their propagation and interactions represent the key processes in the problem of inter-neuron communication, guiding the information flow and information processing in the neuronal networks of the brain.