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The purpose of this study is to acquaint the reader with one of the effective approaches to describing processes in adaptive networks, built in the framework of the well-known Kuramoto model. Methods. The solution to this problem is based on the analysis of the results of works devoted to the study of the dynamics of oscillatory networks with adaptive couplings. Main classes of models of dynamical couplings used in the description of adaptive networks are considered, and the dynamical and structural effects caused by the presence of the corresponding law of coupling adaptation are analyzed. Results. Principles of constructing models of adaptive networks based on the phase description developed by Kuramoto are presented. Materials presented in the review show that the Kuramoto system with dynamic couplings demonstrates a wide range of fundamentally new phenomena and modes. Considered networks include well-known models of dynamical couplings that implement various laws of adaptation of inter-element interactions depending on the states of the elements, in particular, on their relative phase difference. For each network model, a class of possible solutions is established, and general properties of collective dynamics are identified, due to the presence of adaptability of couplings. One of the features of such networks is the multistability of behavior, determined by the possibility of the formation in the network of many different cluster states, including chimera ones. It was found that the implemented coupling adaptation mechanism affects not only the configuration of clusters formed in the network, but also the nature of phase distributions within them. The processes of cluster formation are accompanied by a restructuring of the interaction topology, leading to the formation of hierarchical and modular structures. Conclusion. In conclusion, we briefly summarize the results presented in the review.

The effect of a parametric change in the excitation threshold on processes of generation of the spike activity in an ensemble of FitzHugh–Nagumo neurons has been analyzed. It has been shown that the periodic change in the excitation threshold results in the existence of chaotic regimes of generation of the spike activity with various statistical properties. It has been found that chaos in the model of an individual neuron appears through Pomeau–Manneville intermittence. It has been shown that the information characteristics of the neuron can be controlled by varying the modulation depth and the frequency of parametric action. It has been found that chaotic spike sequences forming fractal-like spatiotemporal patterns appear when neurons are joined in an ensemble.

In this work, we consider recurrent neural networks of firing rate neurons supervisely trained to generate multidimensional sequences of given configurations. We study dynamical objects in the network multidimensional phase space underlying successfully trained outputs and analyze spatiotemporal neural activity and its features in three cases. First, we consider autonomous generation of complex sequences by output units driven by a recurrent network. Second, we study how input pulses can trigger different output units. Third, we explore the case where input pulses allow us to switch between different sequential activities of output units.

We considered a network consisting of two populations of phase oscillators, the interaction of which is determined by different rules for the coupling adaptation. The introduction of various adaptation rules leads to the suppression of splay states and the emergence of each population complex non-stationary behavior called transient circulant clusters. In such states, each population contains a pair of anti-phase clusters whose size and composition slowly change over time as a result of successive transitions of oscillators between clusters. We show that an increase in the mismatch of the adaptation rules makes it possible to stop the process of rearrangement of clusters in one or both populations of the network. Transitions to such modes are always preceded by the appearance of solitary states in one of the populations.

We report the results of study of two models of power grids with hub cluster topology based on the second-order Kuramoto system. The first model considered is the small grid consisting of a consumer and two generators. The second model is the Nizhny Novgorod power grid. The areas in the parameter spaces of the grids that corresponds to different modes, including working synchronous one, of their operation are obtained. The dynamic stability of synchronous mode in the Nizhny Novgorod power grid model to transient disturbances of the power at its elements is tested. We show that the stability of peripheral elements of the grid to disturbances depends significantly on the lengths of their connections to the rest of the grid

Reduction of collective dynamics of large heterogeneous populations to low-dimensional mean-field models is an important task of modern theoretical neuroscience. Such models can be derived from microscopic equations, for example with the help of Ott-Antonsen theory. An often used assumption of the Lorentzian distribution of the unit parameters makes the reduction especially efficient. However, the Lorentzian distribution is often implausible as having undefined moments, and the collective behavior of populations with other distributions needs to be studied. In the present Letter we propose a method which allows efficient reduction for an arbitrary distribution and show how it performs for the Gaussian distribution. We show that a reduced system for several macroscopic complex variables provides an accurate description of a population of thousands of neurons. Using this reduction technique we demonstrate that the population dynamics depends significantly on the form of its parameter distribution. In particular, the dynamics of populations with Lorentzian and Gaussian distributions with the same center and width differ drastically.

We report on the phenomenon of the emergence of mixed dynamics in a system of two adaptively coupled phase oscillators under the action of a harmonic external force. We show that in the case of mixed dynamics, oscillations in forward and reverse time become similar, especially at some specific frequencies of the external force. We demonstrate that the mixed dynamics prevents forced synchronization of a chaotic attractor. We also show that if an external force is applied to a reversible core formed in an autonomous case, the fractal dimension of the reversible core decreases. In addition, with increasing amplitude of the external force, the average distance between the chaotic attractor and the chaotic repeller on the global Poincare secant decreases almost to zero. Therefore, at the maximum intersection, we see a trajectory belonging approximately to a reversible core in the numerical simulation.

In this work, we study the phenomenon of disordered quenching in arrays of coupled Bautin oscillators, which are the normal form for bifurcation in the vicinity of the equilibrium point when the first Lyapunov coefficient vanishes and the second one is nonzero. For particular parameter values, the Bautin oscillator is in a bistable regime with two attractors—the equilibrium and the limit cycle—whose basins are separated by the unstable limit cycle. We consider arrays of coupled Bautin oscillators and study how they become quenched with increasing coupling strength. We analytically show the existence and stability of the dynamical regimes with amplitude disorder in a ring of coupled Bautin oscillators with identical natural frequencies. Next, we numerically provide evidence that disordered oscillation quenching holds for rings as well as chains with nonidentical natural frequencies and study the characteristics of this effect.

We study the interplay of global attractive coupling and individual noise in a system of identical active rotators in the excitable regime. Performing a numerical bifurcation analysis of the nonlocal nonlinear Fokker–Planck equation for the thermodynamic limit, we identify a complex bifurcation scenario with regions of different dynamical regimes, including collective oscillations and coexistence of states with different levels of activity. In systems of finite size, this leads to additional dynamical features, such as collective excitability of different types and noise-induced switching and bursting. Moreover, we show how characteristic quantities such as macroscopic and microscopic variability of interspike intervals can depend in a non-monotonous way on the noise level.

The impact of frequency-degree and amplitude-degree correlation is studied for heterogeneous networks of coupled Stuart–Landau oscillators. It is shown that increasing coupling strength gives rise to hierarchical processes of oscillation quenching. In case of frequency-degree correlated networks, higher-frequency oscillators gradually become almost quenched while low-frequency ones still remain oscillating. In case of amplitude-degree correlated networks, there appear three distinct domains, two contain low-amplitude oscillations with positive and negative means, and the third includes high-amplitude oscillations around the origin.

We continue to apply the notion of mean complexities to study dynamical networks. We show that the mean complexities can help to single out the nodes with similar features (and dynamical behavior) and to reveal some properties of the topology of the networks. We found that the nodes with the same degree (number of connections) have equal values of the mean complexities in the regime of full binding. At the same time, the mean complexities of nodes with different degree follow a descending order with respect to the degree.

We study the influence of the initial topology of connections on the organization of synchronous behavior in networks of phase oscillators with adaptive couplings. We found that networks with a random sparse structure of connections predominantly demonstrate the scenario as a result of which chimera states are formed. The formation of chimera states retains the features of the hierarchical organization observed in networks with global connections [D.V. Kasatkin, S. Yanchuk, E. Scholl, V.I. Nekorkin, Phys. Rev. E 96, 062211 (2017)], and also demonstrates a number of new properties due to the presence of a random structure of network topology. In this case, the formation of coherent groups takes a much longer time interval, and the sets of elements that form these groups can be significantly rearranged during the evolution of the network. We also found chimera states, in which along with the coherent and incoherent groups, there are subsets, whose different elements can be synchronized with each other for sufficiently long periods of time.

Cloning of chimera states, which is a new effect caused by the short-term interaction in a multiplex network, has been described. This effect is observed when two ring networks of linearly coupled two-frequency (bistable) oscillators are combined into the multiplex network. At certain values of the strength and duration of the inter-ring (multiplex) interaction, a copy of a chimera state with accuracy to phases in the incoherent part is formed in the ring with an initially random phase distribution. It has been shown that the effect is structurally stable and is due to the competition of self-sustained oscillations in individual rings.

We study stability of a synchronous regime in hub clusters of the power networks, which are simulated by ensembles of phase oscillators. An approach allowing one to estimate the regions in the parameter space, which correspond to the global asymptotic stability of this regime, is presented. The method is illustrated by an example of a hub cluster consisting of one generator and two consumers.

Chimera states in a system with linear local connections have been studied. The system is a ring ensemble of analog bistable self-excited oscillators with a resistive coupling. It has been shown that the existence of chimera states is not due to the nonidentity of oscillators and noise, which is always present in real experiments, but is due to the nonlinear dynamics of the system on invariant tori with various dimensions.

We report on the phenomenon of intersection of a chaotic attractor and a chaotic repeller in a system of two adaptively coupled phase oscillators. This is a feature of the presence of the so-called mixed dynamics, which is a new type of chaos characterized by the fundamental inseparability of conservative and dissipative behavior. The considered system is the first example of a time-irreversible system in which this type of dynamics is observed. We show that a crucial factor in this effect is the detuning of the natural frequencies of phase oscillators.

In this paper, we study collective dynamics of the network of rate neurons which constitute a central element of a reservoir computing system. The main objective of the paper is to identify the dynamic behaviors inside the reservoir underlying the performance of basic machine learning tasks, such as generating patterns with specified characteristics. We build a reservoir computing system which includes a reservoir—a network of interacting rate neurons—and an output element that generates a target signal. We study individual activities of interacting rate neurons, while implementing the task and analyze the impact of the dynamic parameter—a time constant—on the quality of implementation.

Adaptive dynamical networks appear in various real-word systems. One of the simplest phenomenological models for investigating basic properties of adaptive networks is the system of coupled phase oscillators with adaptive couplings. In this paper, we investigate the dynamics of this system. We extend recent results on the appearance of hierarchical frequency-multi-clusters by investigating the effect of the time-scale separation. We show that the slow adaptation in comparison with the fast phase dynamics is necessary for the emergence of the multi-clusters and their stability. Additionally, we study the role of double antipodal clusters, which appear to be unstable for all considered parameter values. We show that such states can be observed for a relatively long time, i.e., they are metastable. A geometrical explanation for such an effect is based on the emergence of a heteroclinic orbit., 16 pages, 8 figures

A new phenomenon of the chimera states cloning in a large two-layer multiplex network with short-term couplings has been discovered and studied. For certain values of strength and time of multiplex interaction, in the initially disordered layer, a state of chimera is formed with the same characteristics (the same average frequency and amplitude distributions in coherent and incoherent parts, as well as an identical phase distribution in coherent part), as in the chimera which was set in the other layer. The mechanism of the chimera states cloning is examined. It is shown that the cloning is not related with synchronization, but arises from the competition of oscillations in pairs of oscillators from different layers.

This work is devoted to studying the processes of activation of the neurons whose excitation thresholds are not constant and vary in time (the so-called dynamic thresholds). The neuron dynamics is described by the FitzHugh–Nagumo model with nonlinear behavior of the recovery variable. The neuron response to the external pulsed activating action in the presence of a slowly varying synaptic current is studied within the framework of this model. The structure of the dynamic threshold is studied and its properties depending on the external-action parameters are established. It is found that the formation of the “folds” in the separatrix threshold manifold in the model phase space is a typical feature of the complex dynamic threshold. High neuron sensitivity to the action of the comparatively weak slow control signals is established. This explains the capability of the neurons to perform flexible tuning of their selective properties for detecting various external signals in sufficiently short times (of the order of duration of several spikes).

We study the dynamical regimes in the system of two identical interacting phase oscillators with plastic couplings. The joint evolution of the states of the elements and the interelement couplings is a feature of the system studied. It is shown that the introduction of plastic couplings leads to a multistable behavior of the system and emergence of the asynchronous regimes which are not observed for the considered parameter values in the case of static couplings. The parameter plane is divided into regions with different dynamic regimes of the system. In particular, the regions in which the system demonstrates bistable synchronous behavior and the region in which the coexistence of many various asynchronous regimes is observed are singled out.

We study a new type of attractor, the so-called reversible core, which is a mathematical image of mixed dynamics, in a strongly dissipative time-irreversible system of two adaptively coupled phase oscillators. The existence of mixed dynamics in this system was proved in our previous article [A. A. Emelianova and V. I. Nekorkin, Chaos 29, 111102 (2019)]. In this paper, we attempt to identify the dynamic mechanisms underlying the existence of mixed dynamics. We give the region of the existence of mixed dynamics on the parameter plane and demonstrate in what way, when a type of attractor changes, its main characteristics, such as its fractal dimension and the sum of Lyapunov exponents, transform. We demonstrate that when mixed dynamics appear in the system, the average frequencies of the oscillations in forward and reverse time begin to almost coincide, and its spectra gradually approach each other with an increase in the parameter responsible for the presence of mixed dynamics.

In this work, we consider two-layer multiplex networks of coupled Stuart-Landau oscillators. The first layer contains oscillators with amplitude heterogeneity and all-to-all adaptive links, while the second layer contains identical oscillators all-to-all coupled by links with constant weights. The links between different layers are adaptive and organized in a one-to-one manner. We study the evolution of one-layer and two-layer networks depending on intra- and interlayer coupling strengths and show hierarchical transitions between oscillatory and quenched regimes.

In a network of pulse-coupled oscillators with adaptive coupling, we discover a dynamical regime which we call an "itinerant chimera." Similarly as in classical chimera states, the network splits into two domains, the coherent and the incoherent. The drastic difference is that the composition of the domains is volatile, i.e., the oscillators demonstrate spontaneous switching between the domains. This process can be seen as traveling of the oscillators from one domain to another or as traveling of the chimera core across the network. We explore the basic features of the itinerant chimeras, such as the mean and the variance of the core size, and the oscillators lifetime within the core. We also study the scaling behavior of the system and show that the observed regime is not a finite-size effect but a key feature of the collective dynamics which persists even in large networks.

A system of two oscillators with delayed pulse coupling is studied analytically and numerically. The so-called jittering regimes with non-equal inter-spike intervals are observed. The analytical conditions for the emergence of in-phase and anti-phase jittering are derived. The obtained results suggest universality of the multi-jitter instability for systems with delayed pulse coupling.

In this paper, an experimental electronic neuron based on a complete Morris-Lecar model is presented, which is able to become an experimental unit tool to study collective association of coupled neurons. The circuit design is given according to the ionic currents of this model. The experimental results are compared with the theoretical prediction, leading to a good agreement between them, which therefore validate the circuit. The use of some parts of the circuit is also possible for other neurons models, namely for those based on ionic currents.

Using the neuronal-excitability model, we study an analog of the saddle-node bifurcation of the limit cycles in the case of slow variation in the control parameter, i.e., the so-called dynamic saddle-node bifurcation of cycles. It is shown that the stable oscillations in such a system occur on the two-dimensional invariant manifold which also exists after the passage through the “static” bifurcation value. It is found that oscillations disappear with a significant delay with respect to the oscillation-termination time which is predicted by the classical bifurcation theory. It is shown that the nonlocal oscillatory properties of the model and, in particular, the threshold properties of the two-dimensional invariant surface of the saddle trajectory play an important role in the delay formation.

In oscillatory systems with pulse coupling regular spiking regimes may destabilize via a peculiar scenario called “multi-jitter instability”. At the bifurcation point numerous so-called “jittering” regimes with distinct inter-spike intervals emerge simultaneously. Such regimes were first discovered in a single oscillator with delayed pulse feedback and later were found in networks of coupled oscillators. The present chapter reviews recent results on multi-jitter instability and discussed its features.

We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example we demonstrate how the complex bifurcation scenario of simultaneously emerging multi-jittering solutions can be transferred from a single oscillator with delayed pulse feedback to multi-jittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type.

The chapter is devoted to the dynamics of networks of oscillators with pulse time-delayed coupling. We develop a mathematical technique that allows to reduce the dynamics of such networks to multi-dimensional maps. With the help of these maps we consider networks of various configurations: a single oscillator with feedback, a feed-forward ring, a pair of oscillators with mutual coupling, a small network with heterogeneous delays, and a large network with all-to-all coupling. In all these examples we show that the role of the delay is significant and leads to the modification of the existing dynamical regimes and the emergence of new ones.

We report the phenomenon of self-organized emergence of hierarchical multilayered structures and chimera states in dynamical networks with adaptive couplings. This process is characterized by a sequential formation of subnetworks (layers) of densely coupled elements, the size of which is ordered in a hierarchical way, and which are weakly coupled between each other. We show that the hierarchical structure causes the decoupling of the subnetworks. Each layer can exhibit either a two-cluster state, a periodic traveling wave, or an incoherent state, and these states can coexist on different scales of subnetwork sizes.

Phase response curve (PRC) is an extremely useful tool for studying the response of oscillatory systems, e.g. neurons, to sparse or weak stimulation. Here we develop a framework for studying the response to a series of pulses which are frequent or/and strong so that the standard PRC fails. We show that in this case, the phase shift caused by each pulse depends on the history of several previous pulses. We call the corresponding function which measures this shift the phase response function (PRF). As a result of the introduction of the PRF, a variety of oscillatory systems with pulse interaction, such as neural systems, can be reduced to phase systems. The main assumption of the classical PRC model, i.e. that the effect of the stimulus vanishes before the next one arrives, is no longer a restriction in our approach. However, as a result of the phase reduction, the system acquires memory, which is not just a technical nuisance but an intrinsic property relevant to strong stimulation. We illustrate the PRF approach by its application to various systems, such as Morris-Lecar, Hodgkin-Huxley neuron models, and others. We show that the PRF allows predicting the dynamics of forced and coupled oscillators even when the PRC fails.

We analyze the emergent regimes and the stimulus-response relationship of a population of noisy map neurons by means of a mean-field model, derived within the framework of cumulant approach complemented by the Gaussian closure hypothesis. It is demonstrated that the mean-field model can qualitatively account for stability and bifurcations of the exact system, capturing all the generic forms of collective behavior, including macroscopic excitability, subthreshold oscillations, periodic or chaotic spiking and chaotic bursting dynamics. Apart from qualitative analogies, we find a substantial quantitative agreement between the exact and the approximate system, as reflected in matching of the parameter domains admitting the different dynamical regimes, as well as the characteristic properties of the associated time series. The effective model is further shown to reproduce with sufficient accuracy the phase response curves of the exact system and the assembly's response to external stimulation of finite amplitude and duration., Comment: 12 pages, 11 figures

We examine dynamics of a fault motion by analyzing behavior of a spring-slider model composed of 100 blocks where each block is coupled to a varying number of 2K neighboring units (1 $$\le $$ 2K $$\le $$ N, $$N=100$$ ). Dynamics of such model is studied under the effect of delayed interaction, variable coupling strength and random seismic noise. The qualitative analysis of stability and bifurcations is carried out by deriving an approximate deterministic mean-field model, which is demonstrated to accurately capture the dynamics of the original stochastic system. The primary effect concerns the direct supercritical Andronov–Hopf bifurcation, which underlies transition from equilibrium state to periodic oscillations under the variation of coupling delay. Nevertheless, the impact of delayed interactions is shown to depend on the coupling strength and the friction force. In particular, for loosely coupled blocks and low values of friction, observed system does not exhibit any bifurcation, regardless of the assumed noise amplitude in the expected range of values. It is also suggested that a group of blocks with the largest displacements, which exhibit nearly regular periodic oscillations analogous to coseismic motion for system parameters just above the bifurcation curve, can be treated as a representative of an earthquake hypocenter. In this case, the distribution of event magnitudes, defined as a natural logarithm of a sum of squared displacements, is found to correspond well to periodic (characteristic) earthquake model.