We consider a power grid consisting of three synchronous generators supplying a common static load, in which one of the generators is located electrically much closer to the load than the others, due to a shorter transmission line with longitudinal inductance compensation. A reduced model is derived in the form of an ensemble with a star (hub) topology without parameter interdependence. We show that stable symmetric and asymmetric synchronous modes can be realized in the grid, which differ, in particular, in the ratio of currents through the second and third power supply paths. The modes of different types are not observed simultaneously, but the asymmetric modes always exist in pairs. A partition of the parameter space into regions with different dynamical regimes of the grid are obtained. Regions are highlighted where only synchronous operating modes can be established. It is shown that the grid can be highly multistable and, along with synchronous operating modes, have simultaneously various types of non-synchronous modes. We study non-local stability of the asymmetric synchronous modes and switchings between them under the influence one-time disturbances and additive noise fluctuations in the mechanical powers of the generators' turbines. The characteristics of one-time disturbances are obtained leading to either return the grid back to the initial synchronous mode or switching the grid to another synchronous mode or some non-synchronous mode. The characteristics of noise fluctuations are obtained, which provide either a more probable finding of the grid in the desirable quasi-synchronous mode, or switching to an undesirable one.

We consider several topologies of power grids and analyze how the addition of transmission lines affects their dynamics. The main example we are dealing with is a power grid that has a tree-like three-element motif at the periphery. We establish conditions where the addition of a transmission line in the motif enhances its stability or induces Braess's paradox and reduces stability of the entire grid. By using bifurcation theory and nonlocal stability analysis, we show that two scenarios for Braess's paradox are realized in the grid. The first scenario is well described and is associated with the disappearance of the synchronous mode. The second scenario has not been previously described and is associated with the reduction of nonlocal stability of the synchronous mode due to the appearance of asynchronous modes. The necessary conditions for stable operation of the grid, under the addition of a line, are derived. It is proved that the new scenario for Braess's paradox is realized in the grids with more complex topologies even when several lines are added in their bulks.

The main purpose of this paper is to investigate the dynamics of the power grid model with hub cluster topology based on the Kuramoto equations with inertia. It is essential to study the stability of synchronous grid operation mode and to find conditions of its global stability. The conditions that ensure establishment of the synchronous mode instead of coexisting asynchronous ones are considered. Methods. In this paper we use numerical modelling of different grid operation modes. Also we use an approach based on the second Lyapunov method, which allows to give an estimate of the area of safe perturbations that do not violate the synchronous mode. Results. Various power grid operation modes and boundaries of their existence in the parameter space are considered. An approach allowing to estimate the magnitude of safe disturbances that do not violate the synchronous mode, is described. Conclusion. The paper considers a power grid model with hub cluster topology. For hub-clusters of three and four elements, their parameter spaces are partitioned into areas corresponding to different operation modes. In particular, parameter areas with global asymptotic stability of synchronous modes that is with trouble-free operations under any initial conditions has been identified. To characterize the modes of hub clusters outside the areas of global asymptotic stability, estimates of the areas of safe perturbations that do not violate the synchronous grid operation mode is given.

The purpose of this work is to study the dynamics of a weakly inhomogeneous ensemble of three FitzHugh-Nagumo neurons with excitatory synaptic couplings. To single out main types of canard solutions of the system and obtain the regions in parameter space the solutions exist in. Methods. In this paper the dynamics of autonomous systems are studied by using methods based on geometric singular perturbation theory. To study the dynamics of non-autonomous systems we develop an approximate approach and use numerical methods such as obtaining of Poincare maps. Results. The dynamics of the second neuron was studied analytically provided that the first neuron is at rest. The existence of solutions corresponding to different types of periodic oscillations, including canard ones, was shown. To study non-autonomous system, describing the dynamics of third neuron, the approach based on construction of auxiliary autonomous system was developed. It was shown that in the system there exist different types of invariant torus, including canard ones, corresponding to the regimes of various subthreshold and spiking quasi-periodic oscillations of the third neuron. Torus-canards were discovered corresponding to various types of subthreshold-spiking oscillations. Conclusion. The dynamics of a weakly inhomogeneous ensemble of three FitzHugh-Nagumo neurons, the depolarization levels of which linearly increase with their index, and connected in series through chemical synapses, is considered. It was shown that the existence of torus-canards with simple and complex shape is possible in the ensemble. The torus-canards with simple shape correspond either to subthreshold or to spiking oscillations of the ensemble neurons. On the other hand, the torus-canards with complex shape correspond to the regimes of complex subthreshold-spiking oscillations.

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.

Topic. A review of the basic dynamical models of neural activity is presented and individual features of their behavior are discussed, which can be used as a basis for the subsequent development and construction of various configurations of neural networks. The work contains both new original results and generalization of already known ones published earlier in different journals. Aim is to familiarize the reader with the basic dynamical properties of neurons, such as the existence of a rest state and the generation of the action potential; to outline the dynamical mechanisms underlying these properties which are used in the development of neural models with various levels of detailing. Investigated models. From the mathematical point of view, neuron models are divided into two classes. The first class is represented by models with continuous time described by ordinary differential equations. The section devoted to continuous-time models starts from the most detailed Hodgkin-Huxley model, which is a canonical model for neural activity in nonlinear dynamics. Further we describe simplified models, such as a two-dimensional model of Morris-Lecar for spiking and a three-dimensional model of Hindmarsh-Rose for bursting. The FitzHugh-Nagumo model is described in detail, and detailed bifurcation analysis is presented. We also present models for neurons with specific properties, namely a neuron with afterdepolarization and an inferior olives neuron. The last and the simplest model is the "integrate-fire" model. The second class of neural models are systems with discrete time represented by discrete maps. Such models have recently gained increasing popularity due to the richness of the demonstrated dynamics and the ease of numerical simulations. We describe such models as the Chialvo model, the Izhikevich model, the Rulkov model, and the Courbage-Nekorkin model. Results. The basic physical principles underlying the construction of mathematical models of neural activity, based on ion transport, are outlined. Using the FitzHugh-Nagumo model as an example, the main properties and mechanisms of the emergence of multithreshold excitation regimes in neurons are described. The mechanism of formation of burst oscillations in the Hindmarsh-Rose model is outlined. A dynamic mechanism for temporal decline of the excitation threshold and the emergence of periodic oscillations in a neuron with afterdepolarization are described. The formation of (Ca2+)- and (Na2+)-dependent spikes in inferior olive neurons is described. Dynamic mechanisms of formation of the major regular and chaotic regimes of neural activity in discrete models of Chialvo, Izhikevich, Rulkov and Courbage-Nekorkin are described. Discussion. In the Conclusion we briefly summarize the content of the survey.

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.

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. A weak coupling of such neurons under Multisim Software can generate clusters based on the boundary conditions of the neurons and their initial conditions. For this study, we work in the region close to the fold bifurcation of limit cycles. In this region two limit cycles exist, one of the cycles is stable and another one is unstable.

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.

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.

An approach is presented for analyzing the 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 synaptic connections. We illustrate our approach on an example of network of Morris-Lecar neurons coupled by chemical synapses with short-term plasticity.

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.

Spatio-temporal dynamics of the chain system modeling collective behavior of electrically coupled nonlinear cells is investigated. The dynamics of a local cell is described by two component Morris Lecar model. It is shown that such a system supports formation of two stable types of anti-phase spiking patterns (ASP). These are travelling and oscillatory type. High multistability of the travelling ASP is indicated.

Dynamics of two types of one-dimensional networks of electrically coupled neuron-like oscillators implemented using analog electronic circuits are investigated. The networks mimic interacting excitable or bistable neurons which are coupled via gap junctions. The first network is composed of FitzHugh-Nagumo oscillators and the second one is composed of modified FitzHugh-Nagumo oscillators with additional conductance. It is forecasted theoretically and shown experimentally that in both types of networks there exist a variety of different propagating waves: fonts (kink and antikink), excitation pulses, periodic waves and solitary bound states. The fronts and pulses can annihilate or demonstrate particles-like behavior during the interaction with each other and borders of networks. It is shown that particle-like behavior can lead to formation of complex periodic spatiotemporal wave patterns. Besides the periodic patterns in modified FitzHugh-Nagumo network there exist chaotic fractal-like spatiotemporal patterns. It is demonstrated theoretically that emergence of complex patterns can be associated with the existence of a heteroclinic and/or gomoclinic contours in the phase space of corresponding traveling wave systems.

We have developed a new approach for the description of sequential dynamics of excitatory neural networks. Our approach is based on the dynamics of synapses possessing the short-term plasticity property. We suggest a model of such synapses in the form of a second-order system of nonlinear ODEs. In the framework of the model two types of responses are realized-the fast and the slow ones. Under some relations between their timescales a cellular automaton (CA) on the graph of connections is constructed. Such a CA has only a finite number of attractors and all of them are periodic orbits. The attractors of the CA determine the regimes of sequential dynamics of the original neural network, i.e., itineraries along the network and the times of successive firing of neurons in the form of bunches of spikes. We illustrate our approach on the example of a Morris-Lecar neural network.

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.