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Recurrent networks of artificial spiking neurons trained to perform target functions are a perspective tool for understanding dynamic principles of information processing in computational neuroscience. Here, we develop a system of this type based on a map-based model of neural activity allowing for producing various biologically relevant regimes. Target signals used to supervisely train the network are sinusoid functions of different frequencies. Impacts of individual neuron dynamics, coupling strength, network size and other key parameters on the learning error are studied. Our findings suggest, among others, that firing rate heterogeneity as well as mixing of spiking and nonspiking regimes of neurons comprising the network can improve its performance for a wider range of target frequencies. At a single neuron activity level, successful training gives rise to well separated domains with qualitatively different dynamics.

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

Research on the synchronization of delay-coupled oscillator networks is reviewed. A number of key research approaches using different models and methods are described, and major results obtained through their use are presented and generalized. The most characteristic properties of time-delay coupled systems are discussed.

We study the relaxation dynamics of the model of a neuron with complex threshold excitation, which is subjected to the action of a slowly increasing external current. The characteristics of the neuron response to an arbitrarily slow evolution of the current in time are found to be changed compared with those in the quasistatic case. It is shown that the spike-sequence appearance time is, first, uniquely determined by the injection-current intensity at the initial time (memory effect) and, second, occurs with a time delay compared with the quasistatic case of the current increase (delay effect). The dynamic mechanism of the spike-oscillation appearance is studied. It is established that it is based on the dynamic subcritical Andronov–Hopf bifurcation and nonlocal oscillation properties of the model, in the formation of which an important role is played by the behavior of the stable separatrix surfaces of the saddle manifold in phase space.

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.

We propose a model of an adaptive network of spiking neurons that gives rise to a hypernetwork of its dynamic states at the upper level of description. Left to itself, the network exhibits a sequence of transient clustering which relates to a traffic in the hypernetwork in the form of a random walk. Receiving inputs the system is able to generate reproducible sequences corresponding to stimulus-specific paths in the hypernetwork. We illustrate these basic notions by a simple network of discrete-time spiking neurons together with its FPGA realization and analyze their properties., Comment: 10 pages, 4 figures

In this work, a new type of relaxation systems is considered. Their prominent feature is that they comprise two distinct epochs, one is slow regular motion and another is fast chaotic motion. Unlike traditionally studied slow-fast systems that have smooth manifolds of slow motions in the phase space and fast trajectories between them, in this new type one observes, apart the same geometric objects, areas of transient chaos. Alternating periods of slow regular motions and fast chaotic ones as well as transitions between them result in a specific chaotic attractor with chaos on a fast time scale. We formulate basic properties of such attractors in the framework of discrete-time systems and consider several examples. Finally, we provide an important application of such systems, the neuronal electrical activity in the form of chaotic spike-burst oscillations.

Rings of oscillators with delayed pulse coupling are studied analytically, numerically, and experimentally. The basic regimes observed in such rings are rotating waves with constant interspike intervals and phase lags between the neighbors. We show that these rotating waves may destabilize leading to the so-called jittering waves. For these regimes, the interspike intervals are no more equal but form a periodic sequence in time. Analytic criterion for the emergence of jittering waves is derived and confirmed by the numerical and experimental data. The obtained results contribute to the hypothesis that the multijitter instability is universal in systems with pulse coupling.

We propose a discrete model of the olivo-cerebellar system. The model consists of three layers of interacting elements, namely, inferior olive neurons, Purkinje cells, and deep cerebellar nuclear neurons combined into a structure by axonal connections. Each element of the structure is described by a two-dimensional map with an individual set of parameters for each type of neurons. Dynamic properties of different types of neurons are described and spontaneous and stimulusinduced dynamics of the system is explored. Unlike the previously proposed models, this study takes into account the axonal interaction of neurons of different layers, as well as the interaction of the inferior olive neurons through electrical synapses with the property of plasticity. It is shown that the inclusion of these factors plays a significant role in the formation of spatio-temporal activity of the inferior olive neurons.

We present a new approach for the description and analysis of sequential dynamics of excitatory neural networks. It is based on the reduction of dynamics of networks to the dynamics of cellular automata (CA) on the graph of connections. The attractors of the CA determine the different regimes of sequential dynamics of the original neural network. We illustrate our approach through Morris–Lecar and Hodgkin–Huxley neural networks.

Dynamics of the ensemble of two model neurons interacting through electrical synapse is investigated. Both neurons are described by two-dimensional discontinuous map. It is shown that in four-dimensional phase space a chaotic attractor of relaxation type exists corresponding to spike-bursting chaotic oscillations. A new effect of recurrent transitory chaotic oscillations underlies a dynamical mechanism of chaotic bursts formation. It is shown that, under coupling, the transient from chaotic bursts generation into rest state occurs with a time delay. A new characteristic estimating the degree of spike-bursting synchronization is introduced. Dependence of the synchronism degree on the coupling strength is shown for some coupling interval where only activity synchronization occurs. A probabilistic study provides a dynamical explanation of these phenomena.

In this paper we suggest a new dynamical model of neuron excitability. It is based on the classical FitzHugh–Nagumo model in which we introduce the third variable for additional ionic current. By using the method of fast and slow motions we study the afterdepolarization, spontaneous secondary spiking and tonic spiking effects. We build regions in the parameter space that correspond to different dynamical regimes. The obtained results may be important for different problems of neuroscience, e.g. for the problem of working memory.

In this paper, we study the dynamics of a system of two model neurons interacting via the electrical synapse. Each neuron is described by a two-dimensional discontinuous map. A chaotic relaxational-type attractor, which corresponds to the spiking-bursting chaotic oscillations of neurons is shown to exist in a four-dimensional phase space. It is found that the dynamical mechanism of formation of chaotic bursts is based on a new phenomenon of generation of transient chaotic oscillations. It is demonstrated that transition from the chaotic-burst generation to the state of relative rest occurs with a certain time delay. A new characteristic which estimates the degree of synchronization of the spiking-bursting oscillations is introduced. The dependence of the synchronization degree on the strength of coupling of the ensemble elements is studied.

We present a detailed study of the dynamics of pulse oscillators with time-delayed coupling. We get the return maps, obtain strict solutions and analyze their stability. For the case of two oscillators, a periodical structure of synchronization regions is found in parameter space, and the regions corresponding to in-phase and antiphase regimes alternate with growth of time delay. Two types of switching between in-phase and antiphase regimes are studied. We also show that for different parameters coupling delay may have synchronizing or desynchronizing effect. Another novel result is that phase locked regimes exist for arbitrary large values. The specificity of system dynamics with large delay is studied.

Inverse stochastic resonance is a phenomenon where an oscillating system influenced by noise exhibits a minimal oscillation frequency at an intermediate noise level. We demonstrate a novel generic scenario for such an effect in a multi-timescale system, considering an example of emergent oscillations in two adaptively coupled active rotators with excitable local dynamics. The impact of plasticity turns out to be twofold. First, at the level of multiscale dynamics, one finds a range of intermediate adaptivity rates that give rise to multistability between the limit cycle attractors and the stable equilibria, a condition necessary for the onset of the effect. Second, applying the fast-slow analysis, we show that the plasticity also plays a facilitatory role on a more subtle level, guiding the fast flow dynamics to parameter domains where the stable equilibria become focuses rather than nodes, which effectively enhances the influence of noise. The described scenario persists for different plasticity rules, underlying its robustness in the light of potential applications to neuroscience and other types of cell dynamics.

The electronic circuit of the basic element of a neurodynamic memory system has been developed and implemented. The dynamic modes of the circuit and its responses to different external signals have been investigated experimentally. The circuit under study is shown to have all dynamic properties required for the memory system. The experimental results agree with those obtained in the previous theoretical studies of the neurodynamic memory system.

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.

Space–time dynamics of the system modeling collective behaviour of electrically coupled nonlinear units is investigated. The dynamics of a local cell is described by the FitzHugh–Nagumo system with complex threshold excitation. It is shown that such a system supports formation of two distinct kinds of stable two-dimensional spatially localized moving structures without any external stabilizing actions. These are regular and polymorphic structures. The regular structures preserve their shape and velocity under propagation while the shape and velocity as well as other integral characteristics of polymorphic structures show rather complex temporal behaviour. Both kinds of structures represent novel sorts of spatially temporal patterns which have not been observed before in typical two-component reaction–diffusion type systems. It is demonstrated that there exist two types of regular structures: single and bound states and three types of polymorphic structures: periodic, quasiperiodic and even chaotic ones. The partition of the parameter plane into regions corresponding to the existence of these different types of structures is carried out. High multistability of regular structures is indicated. The interaction of regular structures is investigated. The correspondence between the structures and trajectories in multidimensional phase space associated with the system is given. Bifurcation mechanisms leading to the loss of stability of regular structures as well as to a transition from one type of polymorphic structure to another are indicated. The mechanisms of formation of regular and polymorphic structures are discussed.

We propose a new model of synaptic transmission on the basis of a FitzHugh–Nagumo system with nonlinear recovery. It is shown that the Morrice–Lecar neuron ensemble formed by such couplings shows various structurally stable regimes of transient dynamics in the form of sequential transitions between various metastable oscillatory states of the system.

We consider a case study of perturbing a system with a boundary crisis of a chaotic attractor by periodic forcing. In the static case, the system exhibits persistent chaos below the critical value of the control parameter but transient chaos above the critical value. We discuss what happens to the system and particularly to the transient chaotic dynamics if the control parameter periodically oscillates. We find a non-exponential decaying behavior of the survival probability function, study the impact of the forcing frequency and amplitude on the escape rate, analyze the phase-space image of the observed dynamics, and investigate the influence of initial conditions.

The effect of phase reset in a complex oscillatory system under the action of an external pulse is investigated. After the pulse action, the phase of oscillations in the system acquires a new value, which is virtually independent of the initial phase and is only determined by the pulse parameters (the amplitude and the duration). The phase reset is observed for pulses of different durations, including those considerably shorter than the characteristic scale of oscillations in the system. The effect occurs without any fundamental difference for the cases of regular and chaotic oscillations.

A scientific session of the Physical Sciences Division of the Russian Academy of Sciences (RAS) was held under the title "Methods of wave-based physics in neuroscience problems and applications" on October 31, 2007 in the conference hall of the P N Lebedev Physical Institute, RAS. The following reports were presented at the session: (1) Nekorkin V I (Institute of Applied Physics, RAS, Nizhny Novgorod) "Nonlinear oscillations and waves in neurodynamics"; (2) Bezruchko B P (Department of Nano- and Biomedical Technologies, Saratov State University, Saratov), Ponomarenko V I, Prokhorov M D, Smirnov D A (Saratov Branch of the Institute of Radioengineering and Electronics, RAS, Saratov), Tass P A (Juelich Research Center, Institute of Neuroscience and Biophysics 3-Medicine, Juelich, Germany) "Modeling nonlinear oscillatory systems and diagnostics of coupling between them using chaotic time series analysis: applications in neurophysiology". An abridged version of these reports is given below. • Nonlinear oscillations and waves in neurodynamics, V I Nekorkin Physics-Uspekhi, 2008, Volume 51, Number 3, Pages 295–304 • Modeling nonlinear oscillatory systems and diagnostics of coupling between them using chaotic time series analysis: applications in neurophysiology, B P Bezruchko, V I Ponomarenko, M D Prokhorov, D A Smirnov and P A Tass Physics-Uspekhi, 2008, Volume 51, Number 3, Pages 304–310

A control mechanism of the information processing in neural networks is investigated, based on the complex dynamic threshold of the neural excitation. The threshold properties are controlled by the slowly varying synaptic current. The dynamic threshold shows high sensitivity to the rate of the synaptic current variation. It allows both to realize flexible selective tuning of the network elements and to provide nontrivial regimes of neural coding.

The impact of modularity and time delay for spiking neural networks is considered in the first part of this report. We show for different complex topologies that time delay controls regimes of inter-module synchronization as well as the oscillatory rate of modules. In the second part of the work we study a paradigmatic model of the evolving neural network whose topology is influenced by nodal dynamics which results in generating different cluster sequences. We show the conditions under which the sequences are robust to small perturbations of initial conditions, parameter detuning, and noise, while at the same are selective to information stimuli.

We systematically study effects of external perturbations on models describing earthquake fault dynamics. The latter are based on the framework of the Burridge-Knopoff spring-block system, including the cases of a simple mono-block fault, as well as the paradigmatic complex faults made up of two identical or distinct blocks. The blocks exhibit relaxation oscillations, which are representative for the stick-slip behavior typical for earthquake dynamics. Our analysis is carried out by determining the phase response curves of first and second order. For a mono-block fault, we consider the impact of a single and two successive pulse perturbations, further demonstrating how the profile of phase response curves depends on the fault parameters. For a homogeneous two-block fault, our focus is on the scenario where each of the blocks is influenced by a single pulse, whereas for heterogeneous faults, we analyze how the response of the system depends on whether the stimulus is applied to the block having a shorter or a longer oscillation period., Comment: 13 two-column pages, 9 figures; accepted for publication in Chaos

The processes of synchronization and phase cluster formation are investigated in a complex network of dynamically coupled phase oscillators. Coupling weights evolve dynamically depending on the phase relations between the oscillators. It is shown that the network exhibits several types of behavior: the globally synchronized state, two-cluster and multi-cluster states, different synchronous states with a fixed phase relationship between the oscillators and chaotic desynchronized state.

A new type of chimera-like regime is reported that we call “jittering chimera”. The regime is observed in a ring of excitable units in which the excitation is invoked by an oscillator included into the ring. The jittering chimera is characterized by the presence of two domains, one with regular spiking and the other with irregular. A method to set and control desired chimera states in a physically implemented electronic circuit is developed.

The impact of connectivity and individual dynamics on the basin stability of the burst synchronization regime in small-world networks consisting of chaotic slow-fast oscillators is studied. It is shown that there are rewiring probabilities corresponding to the largest basin stabilities, which uncovers a reason for finding small-world topologies in real neuronal networks. The impact of coupling density and strength as well as the nodal parameters of relaxation or excitability are studied. Dynamic mechanisms are uncovered that most strongly influence basin stability of the burst synchronization regime.