Phase transition in globally coupled Rössler oscillators

Measuring multivariate phase synchronization with symbolization and permutation

Phase synchronization is an important mechanism for the information processing of neurons in the brain. Most of the current phase synchronization measures are bivariate and focus on the synchronization between pairs of time series. However, these methods do not provide a full picture of global interactions in neural systems. Considering the prevalence and importance of multivariate neural signal analysis, there is an urgent need to quantify global phase synchronization (GPS) in neural networks. Therefore, we propose a new measure named symbolic phase difference and permutation entropy (SPDPE), which symbolizes the phase difference in multivariate neural signals and estimates GPS according to the permutation patterns of the symbolic sequences. The performance of SPDPE was evaluated using simulated data generated by Kuramoto and Rössler model. The results demonstrate that SPDPE exhibits low sensitivity to data length and outperforms existing methods in accurately characterizing GPS and effectively resisting noise. Moreover, to validate the method with real data, it was applied to classify seizures and non-seizures by calculating the GPS of stereoelectroencephalography (SEEG) data recorded from the onset zones of ten epilepsy patients. We believe that SPDPE will improve the estimation of GPS in many applications, such as EEG-based brain-computer interfaces, brain modeling, and simultaneous EEG-fMRI analysis.

Suppression and frequency control of repetitive spiking in the FitzHugh-Nagumo model

The FitzHugh-Nagumo equation is a simple model equation that exhibits spiking. The output signal of a neuron is represented in the spiking frequency or firing rate. We consider a few control methods for spiking from the viewpoint of nonlinear dynamics. The repetitive spiking can be suppressed in the FitzHugh-Nagumo model by the periodic sinusoidal force with high frequency. We study the transition to the suppressed state numerically and perform a linear stability analysis to understand the suppression of the spiking. Next, we study coupled FitzHugh-Nagumo equations. We find that the periodic forcing makes the system chaotic, and the desynchronization induced by chaos weakens the total output of spiking in a certain parameter range. Finally, we propose a method of feedback control for the spiking frequency. We can get the desired spiking and bursting frequency using this feedback control. The feedback control method is analyzed using a mapping for the input.

Coupling-induced population synchronization in an excitatory population of subthreshold Izhikevich neurons

We consider an excitatory population of subthreshold Izhikevich neurons which exhibit noise-induced firings. By varying the coupling strength J, we investigate population synchronization between the noise-induced firings which may be used for efficient cognitive processing such as sensory perception, multisensory binding, selective attention, and memory formation. As J is increased, rich types of population synchronization (e.g., spike, burst, and fast spike synchronization) are found to occur. Transitions between population synchronization and incoherence are well described in terms of an order parameter [Formula: see text]. As a final step, the coupling induces oscillator death (quenching of noise-induced spikings) because each neuron is attracted to a noisy equilibrium state. The oscillator death leads to a transition from firing to non-firing states at the population level, which may be well described in terms of the time-averaged population spike rate [Formula: see text]. In addition to the statistical-mechanical analysis using [Formula: see text] and [Formula: see text], each population and individual state are also characterized by using the techniques of nonlinear dynamics such as the raster plot of neural spikes, the time series of the membrane potential, and the phase portrait. We note that population synchronization of noise-induced firings may lead to emergence of synchronous brain rhythms in a noisy environment, associated with diverse cognitive functions.

Synchronization and quorum sensing in an ensemble of indirectly coupled chaotic oscillators

The fact that the elements in some realistic systems are influenced by each other indirectly through a common environment has stimulated a new surge of studies on the collective behavior of coupled oscillators. Most of the previous studies, however, consider only the case of coupled periodic oscillators, and it remains unknown whether and to what extent the findings can be applied to the case of coupled chaotic oscillators. Here, using the population density and coupling strength as the tuning parameters, we explore the synchronization and quorum sensing behaviors in an ensemble of chaotic oscillators coupled through a common medium, in which some interesting phenomena are observed, including the appearance of the phase synchronization in the process of progressive synchronization, the various periodic oscillations close to the quorum sensing transition, and the crossover of the critical population density at the transition. These phenomena, which have not been reported for indirectly coupled periodic oscillators, reveal a corner of the rich dynamics inherent in indirectly coupled chaotic oscillators, and are believed to have important implications to the performance and functionality of some realistic systems.

Generating macroscopic chaos in a network of globally coupled phase oscillators

We consider an infinite network of globally coupled phase oscillators in which the natural frequencies of the oscillators are drawn from a symmetric bimodal distribution. We demonstrate that macroscopic chaos can occur in this system when the coupling strength varies periodically in time. We identify period-doubling cascades to chaos, attractor crises, and horseshoe dynamics for the macroscopic mean field. Based on recent work that clarified the bifurcation structure of the static bimodal Kuramoto system, we qualitatively describe the mechanism for the generation of such complicated behavior in the time varying case.

Synchronization in interacting populations of heterogeneous oscillators with time-varying coupling

In many networks of interest (including technological, biological, and social networks), the connectivity between the interacting elements is not static, but changes in time. Furthermore, the elements themselves are often not identical, but rather display a variety of behaviors, and may come in different classes. Here, we investigate the dynamics of such systems. Specifically, we study a network of two large interacting heterogeneous populations of limit-cycle oscillators whose connectivity switches between two fixed arrangements at a particular frequency. We show that for sufficiently high switching frequency, this system behaves as if the connectivity were static and equal to the time average of the switching connectivity. We also examine the mechanisms by which this fast-switching limit is approached in several nonintuitive cases. The results illuminate novel mechanisms by which synchronization can arise or be thwarted in large populations of coupled oscillators with nonstatic coupling.

Emergence of coherence in complex networks of heterogeneous dynamical systems

We present a general theory for the onset of coherence in collections of heterogeneous maps interacting via a complex connection network. Our method allows the dynamics of the individual uncoupled systems to be either chaotic or periodic, and applies generally to networks for which the number of connections per node is large. We find that the critical coupling strength at which a transition to synchrony takes place depends separately on the dynamics of the individual uncoupled systems and on the largest eigenvalue of the adjacency matrix of the coupling network. Our theory directly generalizes the Kuramoto model of equal strength all-to-all coupled phase oscillators to the case of oscillators with more realistic dynamics coupled via a large heterogeneous network.

Onset of synchronization in systems of globally coupled chaotic maps

We study the transition to coherence of an ensemble of globally coupled chaotic maps allowing for ensembles of nonidentical maps and for noise. The transition coupling strength is determined from a kind of transfer function of the perturbation evolution. We present analytical results, and we test these results using numerical experiments for several large systems consisting of ensembles of many coupled maps. The later includes ensembles of identical noiseless maps, identical maps subject to noise, and ensembles of nonidentical maps. One of our examples suggests that the validity of the perturbation theory approach can be problematic for an ensemble of noiseless identical maps if the maps are nonhyperbolic. However, for such a case, noise and/or parameter spread seems to have a regularizing effect restoring the validity of perturbation theory.

Collective dynamics of chaotic chemical oscillators and the law of large numbers

Experiments on the nontrivial collective dynamics and phase synchronization of populations of nonidentical chaotic electrochemical oscillators are presented. Without added coupling no deviation from the law of large numbers is observed. Deviations do arise with weak global or short-range coupling; large, irregular, and periodic mean field oscillations occur along with (partial) phase synchronization.

Transition to coherence in populations of coupled chaotic oscillators: a linear response approach

We consider the collective dynamics in an ensemble of globally coupled chaotic maps. The transition to the coherent state with a macroscopic mean field is analyzed in the framework of the linear response theory. The linear response function for the chaotic system is obtained using the perturbation approach to the Frobenius-Perron operator. The transition point is defined from this function by virtue of the self-excitation condition for the feedback loop. Analytical results for the coupled Bernoulli maps are confirmed by the numerics.

Ion motion synchronization in an ion-trap resonator

Using a new type of ion trap, we demonstrate that the length of a packet of charged particles oscillating between two electrostatic mirrors will remain constant under special conditions. The effect can be understood in terms of phase synchronization, where, in a rather counterintuitive way, the repulsive Coulomb interaction between the ions actually holds the packet together. Application of this effect for mass spectrometry is discussed.