We investigate a spike activity of a network of excitable FitzHugh-Nagumo neurons, which is under constant two-frequency auditory signals. The neurons are supplemented with linear frequency filters and nonlinear input signal converters. We show that it is possible to configure the network to recognize a specific frequency ratio (musical interval) by selecting the parameters of the neurons, input filters, and coupling between neurons. A set of appropriately configured subnetworks with different topologies and coupling strengths can serve as a classifier for musical intervals. We have found that the selective properties of the classifier are due to the presence of a specific topology of coupling between the neurons of the network.

Using methods of numerical simulation, we demonstrate the constructive role of memristive coupling in the context of the traveling wave formation and robustness in an ensemble of excitable oscillators described by the FitzHugh-Nagumo neuron model. First, the revealed aspects of the memristive coupling action are shown in an example of the deterministic model where the memristive properties of the coupling elements provide for achieving traveling waves at lower coupling strength as compared to non-adaptive diffusive coupling. In the presence of noise, the positive role of memristive coupling is manifested as significant, increasing a noise intensity critical value corresponding to the noise-induced destruction of traveling waves as compared to classical diffusive interaction. In addition, we point out the second constructive factor, the Lévy noise, whose properties provide for inducing traveling waves.

The spatio-temporal dynamics of networks of chaotic discrete-time systems is studied numerically in the presence of colored Gaussian noise-modulated nonlocal coupling between elements. As individual nodes of the networks, a modified Ricker map and a cubic map are used. Without noise, the networks demonstrate various types of chimera states, such as phase or double-well chimeras. The nonlocal coupling is modulated by either a common noise source or independent noise sources. The impact of noise-modulated coupling (coupling noise) on the dynamics of the networks is analyzed for a number of different random initial conditions and noise realizations. It is shown that the network dynamics and settled spatio-temporal structures can be controlled by varying the intensity and spectral characteristics of the coupling noise.

Models of two-layer multiplex networks of one-dimensional maps with memristive interlayer coupling are studied. In the absence of coupling, the layers exhibit different types of chimera states. Two types of maps are considered as the elements of networks: modified Ricker maps and cubic maps. The effects of complete and partial synchronization of the spatiotemporal dynamics of layers are observed in the case of a homogeneous and inhomogeneous network, respectively. The dependence of the synchronization threshold for the coupling parameter on the initial states of the memristors has been established. This dependence is partially preserved in the case of non-ideal memristors. Diagrams of the synchronization error are constructed on the plane of the initial state variable of memristors and the interlayer coupling coefficient. The effect of noise in memristive coupling elements on synchronization is considered.

We explore numerically the complex dynamics of multilayer networks (consisting of three and one hundred layers) of cubic maps in the presence of noise-modulated interlayer coupling (multiplexing noise). The coupling strength is defined by independent discrete-time sources of color Gaussian noise. Uncoupled layers can demonstrate different complex structures, such as double-well chimeras, coherent and spatially incoherent regimes. Regions of partial synchronization of these structures are identified in the presence of multiplexing noise. We elucidate how synchronization of a three-layer network depends on the initially observed structures in the layers and construct synchronization regions in the plane of multiplexing noise parameters "noise spectrum width - noise intensity". It is shown that in a hundred-layer network, clusters of synchronized layers can be formed at certain optimal values of multiplexing noise parameters. The performed numerical studies confidently indicate that the spatial dynamics of multilayer networks can be controlled by varying multiplexing noise parameters.

The interaction of two van der Pol–Mathieu oscillators, the simplest models of self-oscillatory systems with parametric excitation, is studied. The dynamics of an individual oscillator is determined either by its self-oscillatory or parametric component, therefore, all possible combinations are considered: both oscillators in a self-oscillatory regime; both oscillators in parametric regime; one oscillator in self-oscillatory regime, another one in parametric regime. To determine the nature of the system dynamics, both complete equations and averaged equations for amplitudes and phases are used, Lyapunov exponents, sections of phase trajectories, and oscillation power spectra are calculated. The behavior of the system under study is compared with the classical case of the interaction of two dissipatively coupled van der Pol oscillators.

We present numerical results on the effects of two presynaptic FitzHugh-Nagumo neurons on a postsynaptic neuron under unidirectional electrical coupling. The presynaptic neurons affect the postsynaptic neuron not simultaneously but with a certain time shift. We consider cases where the amplitudes of the presynaptic spikes can be both higher and lower than the excitation threshold level. The latter case receives the main attention in our work. We carefully examine the conditions under which the postsynaptic neuron is excited by the two asynchronous external spikes. With arbitrarily chosen parameters, the FitzHugh-Nagumo neuron is almost incapable of accumulating the energy of external signals, unlike, for example, the leaky integrate-and-fire neuron. In this case, the postsynaptic neuron only excites with a very short time delay between external impulses. However, we have discovered, for the first time, a parameter region where neuron excitation is possible even with significant time delays between presynaptic impulses with subthreshold amplitudes. We explain this effect in detail and describe the mechanism behind its occurrence. We identify the boundaries of this region in the parameter plane of time delay and coupling coefficient by varying the control parameter values of the neurons. The FitzHugh-Nagumo neuron has not previously been used as a node in spiking neural networks for training via spike-timing-dependent plasticity due to the lack of an integrate-and-fire effect. However, the detection of a certain range of parameters makes the potential application of this neuron for STDP training possible.

Typically, the period-doubling bifurcations exhibited by nonlinear dissipative systems are observed when varying systems' parameters. In contrast, the period-doubling bifurcations considered in the current research are induced by changing the initial conditions, whereas parameter values are fixed. Thus, the studied bifurcations can be classified as the period-doubling bifurcations without parameters. Moreover, we show a cascade of the period-doubling bifurcations without parameters, resulting in a transition to deterministic chaos. The explored effects are demonstrated by means of numerical modeling on an example of a modified Anishchenko-Astakhov self-oscillator where the ability to exhibit bifurcations without parameters is associated with the properties of a memristor. Finally, we compare the dynamics of the ideal-memristor-based oscillator with the behavior of a model taking into account the memristor forgetting effect.

The work is devoted to the study of biology-relevant neural networks, which are called spiking neural networks (SNN), and is aimed at classifying two-frequency auditory signals. We consider an ensemble of FitzHugh-Nagumo neurons. Each neuron is connected to a radiophysical circuit tuned to a certain signal frequency. Different clusters of neurons respond to signals with different ratios of frequencies applied to all circuits simultaneously.

Synchronization of traveling waves in two rings of FitzHugh–Nagumo neurons is studied. Coupling between neurons within each ring is dissipative, while one between rings is memristive. Complete synchronization of waves in identical rings in the presence of an initial phase shift between wave processes and partial synchronization of waves in the case of different coupling strengths inside the rings are considered. The influence of the initial states of memristive coupling on the synchronization of wave processes in the case of memristors with an infinitely long memory and with the forgetting effect is investigated.

We study numerically the spatio-temporal behavior of a two-layer multiplex network of bistable cubic maps when the interlayer coupling is modulated by noise. We refer to this interlayer coupling noise as a multiplexing noise. Various spatio-temporal structures can be established in the uncoupled layers depending on initial conditions. They may represent chimeras of different forms, spatially irregular patterns or spatially homogeneous states. The interlayer coupling strength between the mirror nodes of the layers is given by independent sources of Gaussian colored noise with identical characteristics. We explore the evolution of the spatio-temporal dynamics of the interacting layers when the multiplexing noise parameters are varied. The network dynamics is considered in the case of identical and nonidentical layers which differ in the intralayer coupling strength. It is shown that the low-frequency multiplexing noise produces the solitary states in the networks. The nodes which are in these solitary states have a special temporal dynamics which is revealed for the first time. It is demonstrated that complete and close to complete synchronization of the layers can be achieved at certain characteristics of the noise sources. We analyze how synchronization depends on the initial regimes of two layers. Diagrams are constructed that reflect the degree of layer synchronization in the plane of parameters that control the spectrum and intensity of noise sources.

The purpose of this work is to study the possibility of synchronization of wave processes in distributed excitable systems by means of noise modulation of the coupling strength between them. Methods. A simple model of a neural network, which consists of two coupled layers of excitable FitzHugh–Nagumo oscillators with a ring topology, is studied by numerical simulation methods. The connection between the layers has a random component, which is set for each pair of coupled oscillators by independent sources of colored Gaussian noise. Results. The possibility to obtain a regime close to full (in-phase) synchronization of traveling waves in the case of identical interacting layers and a regime of synchronization of wave propagation velocities in the case of non-identical layers differing in the values of the coefficients of intra-layer coupling is shown for certain values of parameters of coupling noise (intensity and correlation time). Conclusion. It is shown that the effects of synchronization of phases and propagation velocities of excitation waves in ensembles of neurons can be controlled using random processes of interaction of excitable oscillators set by statistically independent noise sources. In this case, both the noise intensity and its correlation time can serve as control parameters. The results obtained on a simple model can be quite general.

We study numerically the spatio-temporal dynamics of a ring network of nonlocally coupled logistic maps when the coupling strength is modulated by colored Gaussian noise. Two cases of noise modulation are considered: 1) when the coupling coefficients characterizing the influence of neighbors on different elements are subjected to independent noise sources, and 2) when the coupling coefficients for all the network elements are modulated by the same stochastic signal. Without noise, the ring of chaotic maps exhibits a chimera state. The impact of noise-modulated coupling between the ring elements is explored when the parameter, which controls the correlation time and the spectral width of colored noise, and the noise intensity are varied. We investigate how the spatio-temporal structures observed in the ring evolve as the noise parameters change. The numerical results obtained are used to construct regime diagrams for the two cases of noise modulation. Our findings show the possibility of controlling the spatial structures in the ring in the presence of noise. Depending on the type of noise modulation, the spectral properties and intensity of colored noise, one can suppress the incoherent clusters of chimera states, and induce the regime of solitary states or synchronize chaotic oscillations of all the ring elements.

We study the simplest artificial model of a power-network operation. The original model has a ring topology consisting of locally coupled power generators alternating with power consumers. Each node of the network is represented as a phase oscillator such as the Kuramoto oscillator with inertia. The network dynamics equations are transformed in accordance with the method of effective network model proposed in [1] and are numerically studied. The purpose of the work is to analyze the possible regimes of the nonuniform-network behavior in the presence of a reactive power in the system. We also study the joint influence of the reactive power and nonlinear dissipation of oscillators. The coefficient of inertia, which is the same for all oscillators, and the reactive power of one of the oscillators are considered as the network control parameters. The regime maps on the plane of control parameters, which were obtained for constant and nonlinear dissipation of the oscillators, are compared. The results show that the reactive power complicates the network behavior and reduces the phase-locked region. On the contrary, the nonlinear dissipation has a positive effect, leading to synchronization of the network oscillators even in the presence of a reactive power.

We demonstrate how the pitchfork, transcritical and saddle-node bifurcations of steady states observed in dynamical systems with a finite number of isolated equilibrium points occur in systems with lines of equilibria. The exploration is carried out by using the numerical simulation and linear stability analysis applied to a model of a memristor-based circuit. All the discussed bifurcation scenarios are considered in the context of models with the piecewise-smooth memristor current-voltage characteristic (Chua’s memristor), as well as on examples of oscillators with the memristor nonlinearity that is smooth everywhere. Finally, we compare the dynamics of ideal-memristor-based oscillators with the behavior of models taking into account the memristor forgetting effect. The presented results are obtained for electronic circuit models, but the studied bifurcation phenomena can be exhibited by systems with lines of equilibria of any nature.

We explore numerically inter-layer synchronization of spatiotemporal patterns in a heterogeneous bilayer network of nonlocally coupled R & ouml;ssler oscillators. Both layers exhibit phase chimera regimes with similar cluster structures, but there is a mean frequency mismatch between the layers. We show that an entrainment of the mean frequency in the interacting layers is observed already for sufficiently weak inter-layer coupling even when the frequency mismatch is large. At the same time, frequency synchronization does not mean structure synchronization. Our numerical study indicates that inter-layer synchronization takes place within noticeably shorter ranges of the frequency mismatch and the inter-layer coupling strength than the frequency synchronization regime. Besides, we show that the weakest chaotic behavior is observed when both frequency and structure synchronizations occur, while the most developed chaotic oscillations are typical when the spatiotemporal structures in the layers are desynchronized. (c) 2021 Elsevier Ltd. All rights reserved.

We explore numerically the synchronization effects in a heterogeneous two-layer network of two-dimensional (2D) lattices of van der Pol oscillators. The inter-layer coupling of the multiplex network has an attractive character. One layer of 2D lattices is characterized by attractive coupling of oscillators and demonstrates a spiral wave regime for both local and nonlocal interactions. The oscillators in the second layer are coupled through active elements and the interaction between them has repulsive character. We show that the lattice with the repulsive type of coupling demonstrates complex spatiotemporal cluster structures, which can be called labyrinth-like structures. We show for the first time that this multiplex network with fundamentally various types of intra-layer coupling demonstrates mutual synchronization and a competition between two types of structures. Our numerical study indicates that the synchronization threshold and the type of spatiotemporal patterns in both layers strongly depend on the ratio of the intra-layer coupling strength of the two lattices. We also analyze the impact of intra-layer coupling ranges on the synchronization effects.

On November 30, 2020, at the age of 78, Vadim S. Anishchenko suddenly died. He was Head of the Department of Radiophysics and Nonlinear Dynamics, Saratov State University (SSU), Doctor of Physics and Mathematics, Professor, Honored Scientist of the Russian Federation, Laureate of the International Science Prize of the Alexander von Humboldt Foundation, Honorary Worker of Education of the Russian Federation, Honorary Professor of SSU, world-class specialist in the field of nonlinear dynamics, founder of one of the leading scientific schools in radiophysics and nonlinear dynamics. The memories of people who knew Vadim S. Anishchenko closely at different periods of his life are published, as a tribute to the memory of an outstanding personality.

The effect of complete synchronization of chaotic oscillations and complex spatial structures in systems with nonlinear inertial coupling of the memristive type is studied. Three models of interacting systems with complex dynamics are considered: a system of two radio engineering chaotic self-oscillators, a system of two logistic maps, and a two-layer multiplex network of logistic maps with a dissipative nonlocal intra-layer coupling and a nonlinear inertial local coupling between the layers. For all three models, the influence of memristive coupling is shown. It consists in the dependence of the synchronization boundary on the initial state of the coupling elements. It has been established that this property is retained with inertial nonlinear coupling close to memristive (with strong inertiality).

Background and Objectives: One of the actual problems in nonlinear dynamics is the formation and interaction of complex spatial structures such as chimeras and solitary states arising in multicomponent systems. Chimera states are typical for ensembles of identical oscillators with regular, chaotic, and even stochastic behavior in a case of nonlocal interaction of the elements. They represent cluster structures, including groups of elements with synchronous and non-synchronous oscillations. Chimeras were discovered and investigated in real experiments, that indicates the possibility of observing such regimes in complex systems in living nature and in technology. Solitary states are less studied today. The regime of solitary states is characterized by the synchronous behavior of most elements of the ensemble, while individual oscillators behave in a “special state”. In the present work, an ensemble of phase oscillators with inertia (rotators) is chosen as the basic model for investigation. Such ensembles with a specific coupling topology are widely used in modeling the operation of energy networks. Ensembles of rotators with nonlocal coupling are characterized by both chimera states and solitary state regimes. The problem of interaction of ensembles of rotators with nonlocal coupling and synchronization of complex spatial structures (chimeras and solitary states) formed in them has not been studied yet. Materials and Methods: A two-layer multiplex network of rotators with a nonlocal character of intralayer interactions is considered. Each layer consists of 100 elements with the same value of the coupling coefficient and coupling phase shift for each element within one layer. The interlayer coupling is symmetric. At the initial stage, with a random choice of initial conditions, steady regimes (chimeras or solitary states) in non-interacting layers were found. Next, the interlayer coupling was introduced and the evolution of the layer dynamics in the selected initial regimes was studied. Four cases of interaction with various initial states of the layers were considered. In the first case, the two layers are completely identical and demonstrate slightly different chimera structures without interlayer coupling. Their evolution with the introduction and growth of the interlayer coupling is considered for two values of the coupling phase shift. It is shown that, starting from a certain threshold value of the interlayer coupling coefficient, the complete synchronization regime is established in the layers, and the coupling phase shift significantly affects the value of the synchronization threshold. In the second case, the previous experiment is reproduced for the two layers with a frequency mismatch. Chimera states established without interlayer interaction are characterized by significantly different average frequencies of the elements in the two layers. In the presence of non-identity of the layers (in this case, frequency mismatch), the regime of complete synchronization of spatial structures is impossible. However, with an increase in the interlayer coupling coefficient, effective synchronization can be obtained which corresponds to a slight difference in the phases of rotators in the interacting layers with full frequency synchronization. In the third case, we consider the interaction between the layers in the solitary state regimes with different spatial structures. In this case, a frequency mismatch is also introduced for the elements of the two layers. For solitary states, the effective synchronization regime with an increase in the interlayer coupling is also established. In both layers the same configurations of solitary states are realized and frequency synchronization is observed. In the fourth case, a heterogeneous multiplex network is considered, in which one layer is in the chimera state, the second layer shows the solitary state mode. With a certain strength of the interlayer coupling the complex structures are destroyed in both layers of the network and a spatially uniform regimes are established. In this case, all the rotators of the two layers rotate at the same frequency, and the difference in the regimes in the layers reduces to a small phase shift, the same for all pairs of coupled rotators of the two layers. Conclusion: The effects of synchronization in the multiplex network were established for two layers in the regimes of complex spatio-temporal dynamics, such as chimera states and solitary states. The influence of the frequency mismatch of the network elements and the phase shift in the interlayer coupling on the synchronization phenomena was studied.