Bistability of operating modes and their switching in a three-machine power grid.

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.

A new scenario for Braess's paradox in power grids.

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.

Noise-induced dynamical regimes in a system of globally coupled excitable units.

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.

Transient circulant clusters in two-population network of Kuramoto oscillators with different rules of coupling adaptation.

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.

The role of timescale separation in oscillatory ensembles with competitive coupling.

We study a heterogeneous population consisting of two groups of oscillatory elements, one with attractive and one with repulsive coupling. Moreover, we set different internal timescales for the oscillators of the two groups and concentrate on the role of this timescale separation in the collective behavior. Our results demonstrate that it may significantly modify synchronization properties of the system, and the implications are fundamentally different depending on the ratio between the group timescales. For the slower attractive group, synchronization properties are similar to the case of equal timescales. However, when the attractive group is faster, these properties significantly change and bistability appears. The other collective regimes such as frozen states and solitary states are also shown to be crucially influenced by timescale separation.

On the intersection of a chaotic attractor and a chaotic repeller in the system of two adaptively coupled phase oscillators.

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.

Synchronization of chimera states in a multiplex system of phase oscillators with adaptive couplings.

We study the interaction of chimera states in multiplex two-layer systems, where each layer represents a network of interacting phase oscillators with adaptive couplings. A feature of this study is the consideration of synchronization processes for a wide range of chimeras with essentially different properties, which are achieved due to the use of different types of coupling adaptation within isolated layers. We study the effect of forced synchronization of chimera states under unidirectional action between layers. This process is accompanied not only by changes in the frequency characteristics of the oscillators, but also by the transformation of the structure of interactions within the slave layer that become close to the properties of the master layer of the system. We show that synchronization close to identical is possible, even in the case of interaction of chimeras with essentially different structural properties (number and size of coherent clusters) formed by means of a relatively large parameter mismatch between the layers. In the case of mutual action of the layers in chimera states, we found a number of new scenarios of the multiplex system behavior along with those already known, when identical or different chimeras appear in both layers. In particular, we have shown that a fairly weak interlayer coupling can lead to suppression of the chimera state when one or both layers of the system demonstrate an incoherent state. On the contrary, a strong interlayer coupling provides a complete synchronization of the layer dynamics, accompanied by the appearance of multicluster states in the system's layers.

Self-organized emergence of multilayer structure and chimera states in dynamical networks with adaptive couplings.

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.

Relating the sequential dynamics of excitatory neural networks to synaptic cellular automata.

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.

[A two-compartment phenomenological model of a dopaminergic neuron].

A two-compartment model of the dopaminergic neuron based on modified FitzHue-Nagumo oscillators for each compartment has been built. The compartments corresponded to the soma and dendrites and differed by the values of small parameters. The influence of stimuli (applied current for the somatic compartment and synaptic activation for the dendritic compartment) on the model has been studied. It has been shown that the activation of AMPA and NMDA synaptic currents lead to the generation of high-frequency bursts by the neuron. The mechanisms underlying the generation of the bursts have been investigated.

[Synchronization in a model of interacting inferior olive cells with variable electrotonic coupling].

Synchronization processes have been studied within the framework of a model describing the dynamics of two inferior olive cells coupled electrotonically through gap junctions surrounded by synaptic inhibitory terminals that block these couplings. A simple model of a chemical synaptic terminal based on the first-order kinetics has been constructed to describe the coupling break. It was found that different types of synchronous behavior exist in the system, depending on the parameters. These are 1:1 and 1:2 synchronization regimes, spike time binding, and others. It was demonstrated that even small changes in the coupling parameters (coupling strength and coupling break delay) may significantly affect the synchronous regimes of interacting cells. In some cases, the activity of one of inferior olive cells is suppressed due to collective dynamics, whereas the activity of the other cell is conserved.

Activity clusters in dynamical model of the working memory system.

A mathematical model of working memory is proposed in the form of a network of neuron-like units interacting via global inhibitory feedback. This network is capable of storing information items in the form of clusters of periodical spiking activity. Several sequentially excited clusters can coexist simultaneously, corresponding to several items stored in the memory. The capacity of the memory is studied as the function of the system parameters.

Chaotic oscillations in a map-based model of neural activity.

We propose a discrete time dynamical system (a map) as a phenomenological model of excitable and spiking-bursting neurons. The model is a discontinuous two-dimensional map. We find conditions under which this map has an invariant region on the phase plane, containing a chaotic attractor. This attractor creates chaotic spiking-bursting oscillations of the model. We also show various regimes of other neural activities (subthreshold oscillations, phasic spiking, etc.) derived from the proposed model.

Spiking dynamics of interacting oscillatory neurons.

Spiking sequences emerging from dynamical interaction in a pair of oscillatory neurons are investigated theoretically and experimentally. The model comprises two unidirectionally coupled FitzHugh-Nagumo units with modified excitability (MFHN). The first (master) unit exhibits a periodic spike sequence with a certain frequency. The second (slave) unit is in its excitable mode and responds on the input signal with a complex (chaotic) spike trains. We analyze the dynamic mechanisms underlying different response behavior depending on interaction strength. Spiking phase maps describing the response dynamics are obtained. Complex phase locking and chaotic sequences are investigated. We show how the response spike trains can be effectively controlled by the interaction parameter and discuss the problem of neuronal information encoding.

Self-referential phase reset based on inferior olive oscillator dynamics.

The olivo-cerebellar network is a key neuronal circuit that provides high-level motor control in the vertebrate CNS. Functionally, its network dynamics is organized around the oscillatory membrane potential properties of inferior olive (IO) neurons and their electrotonic connectivity. Because IO action potentials are generated at the peaks of the quasisinusoidal membrane potential oscillations, their temporal firing properties are defined by the IO rhythmicity. Excitatory inputs to these neurons can produce oscillatory phase shifts without modifying the amplitude or frequency of the oscillations, allowing well defined time-shift modulation of action potential generation. Moreover, the resulting phase is defined only by the amplitude and duration of the reset stimulus and is independent of the original oscillatory phase when the stimulus was delivered. This reset property, henceforth referred to as selfreferential phase reset, results in the generation of organized clusters of electrically coupled cells that oscillate in phase and are controlled by inhibitory feedback loops through the cerebellar nuclei and the cerebellar cortex. These clusters provide a dynamical representation of arbitrary motor intention patterns that are further mapped to the motor execution system. Being supplied with sensory inputs, the olivo-cerebellar network is capable of rearranging the clusters during the process of movement execution. Accordingly, the phase of the IO oscillators can be rapidly reset to a desired phase independently of the history of phase evolution. The goal of this article is to show how this selfreferential phase reset may be implemented into a motor control system by using a biologically based mathematical model.

Emergence of chaotic attractor and anti-synchronization for two coupled monostable neurons.

The dynamics of two coupled piece-wise linear one-dimensional monostable maps is investigated. The single map is associated with Poincare section of the FitzHugh-Nagumo neuron model. It is found that a diffusive coupling leads to the appearance of chaotic attractor. The attractor exists in an invariant region of phase space bounded by the manifolds of the saddle fixed point and the saddle periodic point. The oscillations from the chaotic attractor have a spike-burst shape with anti-phase synchronized spiking.

Olivo-cerebellar cluster-based universal control system.

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

Spiking patterns emerging from wave instabilities in a one-dimensional neural lattice.

The dynamics of a one-dimensional lattice (chain) of electrically coupled neurons modeled by the FitzHugh-Nagumo excitable system with modified nonlinearity is investigated. We have found that for certain conditions the lattice exhibits a countable set of pulselike wave solutions. The analysis of homoclinic and heteroclinic bifurcations is given. Corresponding bifurcation sets have the shapes of spirals twisting to the same center. The appearance of chaotic spiking patterns emerging from wave instabilities is discussed.

Wave front propagation failure in an inhomogeneous discrete Nagumo chain: theory and experiments.

The phenomenon of wave propagation failure in a discrete inhomogeneous Nagumo equation is investigated. It is found that the propagation failure occurs not only for small coupling coefficients but as well for an abrupt change of the interelement coupling. The investigation includes the study of the phase space of the system, numerical simulations, and real experiments with a nonlinear electrical lattice.

Theoretical and experimental study of two discrete coupled Nagumo chains.

We analyze front wave (kink and antikink) propagation and pattern formation in a system composed of two coupled discrete Nagumo chains using analytical and numerical methods. In the case of homogeneous interaction among the chains, we show the possibility of the effective control on wave propagation. In addition, physical experiments on electrical chains confirm all theoretical behaviors.