Phase synchronization of chaotic oscillators

Synchronous gamma activity: a review and contribution to an integrative neuroscience model of schizophrenia

Synchronous high frequency (Gamma band) activity has been proposed as a candidate mechanism for the integration or 'binding' of distributed brain activities. Since the first descriptions of schizophrenia, attempts to characterize this disorder have focused on disturbances in such integrative processing. Here, we review both micro- and macroscopic neuroscience research into Gamma synchrony, and its application to understanding schizophrenia. The review encompasses evidence from both animal and human studies for the functional significance of Gamma activity, the association between Gamma dysfunction and information processing disturbances, and the relevance of specific Gamma dysfunctions to the integration and extension of previous disconnection models of schizophrenia. Attention is given to the relationship between Gamma activity and the heterogeneous symptoms of schizophrenia. Existing studies show that measures of Gamma activity have the potential to explain far more of the variance in schizophrenia performance than previous neurophysiological measures. It is concluded that measures of Gamma synchrony offer a valuable window into the core integrative disturbance in schizophrenia cognition.

Noise-enhanced synchronization of homoclinic chaos in a CO2 laser

Many chaotic oscillators have rather coherent phase dynamics but strong fluctuation in the amplitudes. Conversely, homoclinic chaos is characterized by quite regular spikes but strong fluctuation in their time intervals. We study the effects of noise on the synchronization of homoclinic chaos to a weak periodic signal and demonstrate numerically and experimentally in a CO2 laser system that noise enhances synchronization of homoclinic chaos. The system exhibits both conventional resonance versus driving frequency and stochastic resonance with respect to noise intensity.

Competition between two frequencies for phase synchronization of a chaotic laser

Competition between two distinct driving frequencies to phase synchronize the intensity dynamics of a chaotic laser has been observed. The phase of the chaotic intensity signal is constructed using the complex analytic signal. Competing frequencies alternately show phase locking and phase slipping. Competition has been quantified by calculating the portion of time the laser phase locks to each of the driving frequencies and their average.

Locking-based frequency measurement and synchronization of chaotic oscillators with complex dynamics

We propose a method for the determination of a characteristic oscillation frequency for a broad class of chaotic oscillators generating complex signals. It is based on the locking of standard periodic self-sustained oscillators by an irregular signal. The method is applied to experimental data from chaotic electrochemical oscillators, where other approaches of frequency determination (e.g., based on Hilbert transform) fail. Using the method we characterize the effects of phase synchronization for systems with ill-defined phase by external forcing and due to mutual coupling.

Experimental characterization of the transition to phase synchronization of chaotic CO2 laser systems

We investigate the transition route to phase synchronization in a chaotic laser with external modulation. Such a transition is characterized by the presence of a regime of periodic phase synchronization, in which phase slips occur with maximal coherence in the phase difference between output signal and external modulation. We provide the first experimental evidence of such a regime and demonstrate that it occurs at the crossover point between two different scaling laws of the intermittent-type behavior of phase slips.

Phase synchronization and topological defects in inhomogeneous media

The influence of topological defects on phase synchronization and phase coherence in two-dimensional arrays of locally coupled, nonidentical, chaotic oscillators is investigated. The motion of topological defects leads to a breakdown of phase synchronization in the vicinities of the defects; however, the system is much more phase coherent as long as the coupling between the oscillators is strong enough to prohibit the continuous dynamical creation and annihilation of defects. The generic occurrence of topological defects in two and higher dimensions implies that the concept of phase synchronization has to be modified for these systems.

Phase synchronization in coupled chaotic oscillators with time delay

The phase synchronization (PS) of two Rössler oscillators with time-delayed signal coupling is studied. We find that time delay can always lead to PS even when the delay is very long. Moreover, with the increase of time delay, the coupling strength at the transition to PS undergoes a nearly periodic wave distribution. At some fixed time-delayed signal coupling, a PS region is followed by a non-PS region when the coupling strength increases. However, an increase of the coupling leads to the PS state again. This phenomenon occurs in systems with a relatively large PS transition point.

Controlling the ultimate state of projective synchronization in chaotic systems of arbitrary dimension

The ultimate state of projective synchronization is hardly predictable. A control algorithm is thus proposed to manipulate the synchronization in arbitrary dimension. The control law derived from the Lyapunov stability theory with the aid of slack variables is effective to any initial conditions. The method allows us to amplify and reduce the synchronized dynamics in any desired scale with tiny control inputs. Applications are illustrated for seven- and ten-dimensional chaotic systems.

Event synchronization: a simple and fast method to measure synchronicity and time delay patterns

We propose a simple method to measure synchronization and time-delay patterns between signals. It is based on the relative timings of events in the time series, defined, e.g., as local maxima. The degree of synchronization is obtained from the number of quasisimultaneous appearances of events, and the delay is calculated from the precedence of events in one signal with respect to the other. Moreover, we can easily visualize the time evolution of the delay and synchronization level with an excellent resolution. We apply the algorithm to short rat electroencephalogram (EEG) signals, some of them containing spikes. We also apply it to an intracranial human EEG recording containing an epileptic seizure, and we propose that the method might be useful for the detection of epileptic foci. It can be easily extended to other types of data and it is very simple and fast, thus being suitable for on-line implementations.

Phase synchronization and its cluster feature in two-dimensional coupled map lattices

Due to a diffusive nearest-neighbor coupling, phase-synchronized states can emerge in two-dimensional chaotic coupled map lattices. By defining a direction phase (like a spin with up or down direction) as the direction of two sequential iterations of the logistic map, we find several novel kinds of phase synchronization which correspond to four different regions in a phase diagram. For the phase with partial phase synchronization, as the coupling strength epsilon increases to a critical threshold epsilon(c), a percolationlike transition is found in the cluster feature of the direction phases relating to the pattern formation. In addition, a scaling of the percolation probability rho approximately (epsilon-epsilon(c))(beta) with beta=2.1 is obtained. The spatial and time correlation functions of the phase clusters are also discussed.

Decreased neuronal synchronization during experimental seizures

Synchronization between CA1 pyramidal neurons was studied using dual-cell patch-clamp techniques simultaneous with an extracellular measurement of network activity. We explored various linear and nonlinear methods to detect weak synchronization in this network, using cross-correlation, mutual information in one and two dimensions, and phase correlation in both broad and narrow band. The linear and nonlinear methods demonstrated different patterns of sensitivity to detect synchrony in this network, depending on the dynamical state of the network. Bursts in 4-amino-pyridine (4AP) were highly synchronous events. Unexpectedly, seizure-like events in 4AP were desynchronous events, both in comparison with interictal periods preceding the seizure without bursts (cut Schaffer collateral tract) and in comparison with bursts preceding the seizures (intact Schaffer collateral tract). The finding that seizure-like events are associated with desynchronization in such networks is consistent with recent theoretical work, suggesting that asynchrony is necessary to maintain a high level of activity in neuronal networks for sustained periods of time and that synchrony may disrupt such activity.

Collective chaos synchronization of pairs of modes in a chaotic three-mode laser

We study chaos synchronization experimentally in a modulated globally coupled three-mode laser with different modal gains subjected to self-mixing Doppler-shifted feedback, which can apply the loss modulation to individual modes at Doppler-shift frequencies. Depending on the pump power, different forms of collective chaos synchronizations were found to appear when the laser was modulated at the highest relaxation oscillation frequency, reflecting the change in cross-saturation coefficient among modes. In the present experiment, each pair of modes exhibited phase, lag, or generalized synchronization collectively according to the inherent antiphase dynamics, where these types of synchronization have already been demonstrated in two coupled chaotic oscillators in different physical systems. Information flows among oscillating modes which are established in different forms of collective chaos synchronizations were characterized by information-circulation analysis of the experimental time series. (c) 2002 American Institute of Physics.

Lag times and parameter mismatches in synchronization of unidirectionally coupled chaotic external cavity semiconductor lasers

We report an analysis of synchronization between two unidirectionally coupled chaotic external cavity master and slave semiconductor lasers with two characteristic delay times, where the delay time in the coupling is different from the delay time in the coupled systems themselves. We demonstrate that parameter mismatches in photon decay rates for the master and slave lasers can explain the experimental observation that the lag time is equal to the coupling delay time.

Phase synchronization in bidirectionally coupled optothermal devices

We present the experimental observation of phase synchronization transitions in the bidirectional coupling of chaotic and nonchaotic oscillators. A variety of transitions are characterized and compared to numerical simulations of a time delayed model. The characteristic 2pi phase jumps usually appear during the transitions, specially in those clearly associated with a saddle-node bifurcation. The study is done with pairs of optothermal oscillators linearly coupled by heat transfer.

Stable heteroclinic cycles for ensembles of chaotic oscillators

We study the formation of synchronous clusters in ensembles of globally coupled chaotic oscillators. We reveal that at least three clusters of identical synchronization are formed in such a system for large enough values of coupling strength. Our main result is an unexpected intermittent process of clusterization. This process gives strong indication to the existence of a stable heteroclinic cycle.

Noise-enhanced phase synchronization of chaotic oscillators

The effects of noise on phase synchronization (PS) of coupled chaotic oscillators are explored. In contrast to coupled periodic oscillators, noise is found to enhance phase synchronization significantly below the threshold of PS. This constructive role of noise has been verified experimentally with chaotic electrochemical oscillators of the electrodissolution of Ni in sulfuric acid solution.

Routes to complete synchronization via phase synchronization in coupled nonidentical chaotic oscillators

We study the transition route to complete synchronization through phase synchronization in generic coupled nonidentical chaotic oscillators. Through numerical studies, two routes are found, i.e., one, via lag synchronization, the other, via the intermittent chaotic burst state without lag synchronization. We claim that these two routes are universal. As evidence, we analyze several examples on the basis of the conventional theory of intermittency in the presence of noise.

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.

Noise-induced phase synchronization and synchronization transitions in chaotic oscillators

Whether common noise can induce complete synchronization in chaotic systems has been a topic of great relevance and long-standing controversy. We first clarify the mechanism of this phenomenon and show that the existence of a significant contraction region, where nearby trajectories converge, plays a decisive role. Second, we demonstrate that, more generally, common noise can induce phase synchronization in nonidentical chaotic systems. Such a noise-induced synchronization and synchronization transitions are of special significance for understanding neuron encoding in neurobiology.

Formation of a spiraling line defect and its meandering transition in a period-2 medium

The instability of a period-1 spiral wave resulting in a period-2 spiral wave with a line defect is investigated for the first time in a laboratory system. At the very onset the transition proceeds by an emergence of a spiraling line defect, "breathing" intermittently while retaining its symmetry of a period-1 spiral wave. With a further change in a control parameter, the line defect undergoes a meandering transition producing a compound tip trajectory, following a dynamic shape transition. The observed transitions have a strong analogy to the phase synchronization transition of two coupled nonlinear oscillators and the meandering transition of a period-1 spiral wave.

Spatial periodic synchronization of chaos in coupled ring and linear arrays of chaotic systems

Dynamic behaviors of coupled ring and linear arrays of unidirectionally coupled Lorenz oscillators are studied numerically. It is found that the chaotic rotating waves generated from the ring propagate with spatial periodic synchronization along the linear array, that is to say, two chaotic oscillators in the linear array are synchronized if the number of oscillators (spatial distance) between them is a multiple of oscillator number in the ring. Numerically it is shown that the stabilities of the synchronized states are enhanced by chaos, and degraded when the oscillators are far from the ring.

Detection and description of non-linear interdependence in normal multichannel human EEG data

OBJECTIVES

This study examines human scalp electroencephalographic (EEG) data for evidence of non-linear interdependence between posterior channels. The spectral and phase properties of those epochs of EEG exhibiting non-linear interdependence are studied.

METHODS

Scalp EEG data was collected from 40 healthy subjects. A technique for the detection of non-linear interdependence was applied to 2.048 s segments of posterior bipolar electrode data. Amplitude-adjusted phase-randomized surrogate data was used to statistically determine which EEG epochs exhibited non-linear interdependence.

RESULTS

Statistically significant evidence of non-linear interactions were evident in 2.9% (eyes open) to 4.8% (eyes closed) of the epochs. In the eyes-open recordings, these epochs exhibited a peak in the spectral and cross-spectral density functions at about 10 Hz. Two types of EEG epochs are evident in the eyes-closed recordings; one type exhibits a peak in the spectral density and cross-spectrum at 8 Hz. The other type has increased spectral and cross-spectral power across faster frequencies. Epochs identified as exhibiting non-linear interdependence display a tendency towards phase interdependencies across and between a broad range of frequencies.

CONCLUSIONS

Non-linear interdependence is detectable in a small number of multichannel EEG epochs, and makes a contribution to the alpha rhythm. Non-linear interdependence produces spatially distributed activity that exhibits phase synchronization between oscillations present at different frequencies. The possible physiological significance of these findings are discussed with reference to the dynamical properties of neural systems and the role of synchronous activity in the neocortex.

Transitions from partial to complete generalized synchronizations in bidirectionally coupled chaotic oscillators

Generalized synchronization in an array of mutually (bidirectionally) coupled nonidentical chaotic oscillators is studied. Coupled Lorenz oscillators and coupled Lorenz-Rossler oscillators are adopted as our working models. With increasing the coupling strengths, the system experiences a cascade of transitions from the partial to the global generalized synchronizations, i.e., different oscillators are gradually entrained through a clustering process. This scenario of transitions reveals an intrinsic self-organized order in groups of interacting units, which generalizes the idea of generalized synchronizations in drive-response systems.

Head-free reading of horizontally and vertically arranged texts

Parameters of eye and head movements and their coordination in reading horizontally and vertically arranged texts were compared. Reading was faster for horizontally arranged than for vertically arranged texts by 24%, primarily due to larger gaze amplitude for horizontal reading, and thus smaller numbers of saccades and fixations. The higher velocity of gaze saccades for given amplitudes in horizontal than vertical reading also contributed to the difference in reading speed. The horizontal bias in reading is at least partly due to the oculomotor system, because the higher velocity for given amplitude of horizontal saccade was also observed in a control experiment devoid of lexical load, in which a sequentially stepping laser target was tracked. The analysis of instantaneous phase of eye and head movements with a new metric derived by the Hilbert transform suggests that eye and head coupling is stronger for vertical than for horizontal direction in both reading and laser-tracking tasks. These results, combined with previous evidence that text familiarity modulates the timing and strength of head movement commands with respect to eye movements (Vis. Res. 39 (1999) 3761), indicate that the coupling strength between eye and head movements is variable depending on the direction of gaze shift and cognitive context.

Phase synchronization of chaotic attractors in the presence of two competing periodic signals

We discuss the situation where two periodic signals compete to phase synchronize a chaotic attractor. Depending on the relative position of the periods with respect to the synchronization tongue for a single frequency signal, we distinguish several different cases. We find that, depending on parameters, it is possible that one or the other signal will entrain exclusively, or that they will entrain alternately, at their average frequency, or not at all.

Information theoretic approach to quantify complete and phase synchronization of chaos

Based on an information theory approach we suggest a quantitative characteristic for evaluating the degree of chaotic synchronization. The proposed characteristic is tested for the cases of complete and phase synchronization of chaos. It is shown that this characteristic is stable with respect to the influence of small noise and nonlinear signal distortion.

Synchronization and information processing by an on-off coupling

This paper proposes an on-off coupling process for chaos synchronization and information processing. An in depth analysis for the net effect of a conventional coupling is performed. The stability of the process is studied. We show that the proposed controlled coupling process can locally minimize the smoothness and the fidelity of dynamical data. A digital filter expression for the on-off coupling process is derived and a connection is made to the Hanning filter. The utility and robustness of the proposed approach is demonstrated by chaos synchronization in Duffing oscillators, the spatiotemporal synchronization of noisy nonlinear oscillators, the estimation of the trend of a time series, and restoration of the contaminated solution of the nonlinear Schrödinger equation.

Collective phase locked states in a chain of coupled chaotic oscillators

We discuss the emergence of a collective phase locked state in an open chain of N unidirectionally weakly coupled nonidentical chaotic oscillators. Such a regime is characterized by a Lyapunov spectrum where N-1 exponents that were zero in the uncoupled regime assume negative values as the coupling strength increases. The dynamics of such collective state is studied, and a comparison is drawn with the case of phase synchronization of a pair of coupled chaotic oscillators. In particular, it is shown that a full phase synchronized state cannot be constructed without at least partial correlation in the chaotic amplitudes.

Medium-range oscillatory network and the 20-Hz sensorimotor induced potential

Although synchronously oscillating neuronal assemblies have been the subject of many studies, a clear identification of the spatiotemporal characteristics of a medium-range oscillatory network is still lacking. Herein, we present a method for the extraction of a new waveform, namely the mean induced potential (IP), which allows the identification of the spatiotemporal characteristics of induced EEG responses. The IP calculation was applied to the 20-Hz component of the sensorimotor rhythm in order to obtain a 20-Hz sensorimotor induced potential (20-Hz SIP). The spatiotemporal characteristics of the 20-Hz bursts seen after median nerve stimulation and self-paced finger movements were extracted by means of current source density reconstruction and synchronization analysis. A cortical network including the controlateral primary motor cortex, the supplementary motor area, and the controlateral supramarginalis gyrus was found to generate the 20-Hz bursts, and the various activated areas were found to be highly synchronized. Our results demonstrate for the first time the existence of a medium-range cortical network in the human sensorimotor region whose constituents oscillate synchronously.

Synchronization of chaos due to linear response in optically driven semiconductor lasers

This paper presents physical aspects on chaos synchronization in semiconductor lasers (SLs) by studying synchronization from a fundamental standpoint of driven damped oscillators. We investigate the simple configuration, a chaotic master SL with optical feedback and a solitary slave SL. The point we emphasize is that the slave laser is regarded as a damped oscillator with relaxation oscillation. Linear stability analysis demonstrates that strong injection can enhance the damping of the slave SL. Consequently, the slave SL can have broad and nearly flat spectral characteristics in its driven response, which is sufficient for covering the broadband chaotic driving signal from the master SL. Numerical simulations verify that the slave SL subject to such strong injection synchronizes well with the chaotic driving signal. We consider that the synchronization phenomenon results from a quasilinear driven response of the slave SL with the remarkable spectral characteristics. Moreover, we discuss this type of chaos synchronization in comparison to anticipating-chaos synchronization occurring in our case from conventional complete synchronization theories, and clarify the different physical aspects of the chaos synchronization scheme. We also show that our analysis agrees well with the earlier experiments that could not have been explained by complete synchronization theory.

Oscillatory systems driven by noise: frequency and phase synchronization

The phenomenon of effective phase synchronization in stochastic oscillatory systems can be quantified by an average frequency and a phase diffusion coefficient. A different approach to compute the noise-averaged frequency is put forward. The method is based on a threshold crossing rate pioneered by Rice. After the introduction of the Rice frequency for noisy systems we compare this quantifier with those obtained in the context of other phase concepts, such as the natural and the Hilbert phase, respectively. It is demonstrated that the average Rice frequency <omega>R typically supersedes the Hilbert frequency <omega>H, i.e. <omega>R > or = <omega>H. We investigate next the Rice frequency for the harmonic and the damped, bistable Kramers oscillator, both without and with external periodic driving. Exact and approximative analytic results are corroborated by numerical simulation results. Our results complement and extend previous findings for the case of noise-driven inertial systems.

Limits to the experimental detection of nonlinear synchrony

Chaos synchronization is often characterized by the existence of a continuous function between the states of the components. However, in coupled systems without inherent symmetries, the synchronization set can be extremely complicated. We describe and illustrate three typical complications that can arise, and we discuss how existing methods for detecting synchronization will be hampered by the presence of these features.

Identification of coupling direction: application to cardiorespiratory interaction

We consider the problem of experimental detection of directionality of weak coupling between two self-sustained oscillators from bivariate data. We further develop the method introduced by Rosenblum and Pikovsky [Phys. Rev. E 64, 045202 (2001)], suggesting an alternative approach. Next, we consider another framework for identification of directionality, based on the idea of mutual predictability. Our algorithms provide directionality index that shows whether the coupling between the oscillators is unidirectional or bidirectional, and quantifies the asymmetry of bidirectional coupling. We demonstrate the efficiency of three different algorithms in determination of directionality index from short and noisy data. These techniques are then applied to analysis of cardiorespiratory interaction in healthy infants. The results reveal that the direction of coupling between cardiovascular and respiratory systems varies with the age within the first 6 months of life. We find a tendency to change from nearly symmetric bidirectional interaction to nearly unidirectional one (from respiration to the cardiovascular system).

Shadows of artistry: cortical synchrony during perception and imagery of visual art

Functional and topographical differences between two groups, artists and non-artists, during the performances of visual perception and imagery of paintings were presented by means of EEG phase synchrony analysis. In artists as compared with non-artists, significantly higher phase synchrony was found in the high frequency beta and gamma bands during the perception of the paintings; in the low frequency bands (primarily delta), phase synchrony was mostly enhanced during imagery. Strong decreases in phase synchrony of alpha were found primarily in artists for both tasks. The right hemisphere was found to present higher synchrony than the left in artists, whereas hemispheric asymmetry was less significant in non-artists. In the artists, enhanced synchrony in the high frequency band is most likely due to their enhanced binding capabilities of numerous visual attributes, and enhanced synchrony in the low frequency band seems to be due to the higher involvement of long-term visual memory mostly in imagery. Thus, the analysis of phase synchrony from EEG signals yields new information about the dynamical co-operation between neuronal assemblies during the cognition of visual art.

Spatiotemporal coherence resonance of phase synchronization in weakly coupled chaotic oscillators

We study phase synchronization (PS) of coupled chaotic oscillators as a result of an interplay between local coupling and global noise. In the weak coupling region, noise first significantly enhances spatiotemporal PS, but it spoils the phase coherence of chaotic oscillation and reduces the degree of synchronization at large intensities. The spatiotemporal behavior exhibits coherence resonance. Noise enhanced phase synchronization is of great relevance to ecology.

Performance of different synchronization measures in real data: a case study on electroencephalographic signals

We study the synchronization between left and right hemisphere rat electroencephalographic (EEG) channels by using various synchronization measures, namely nonlinear interdependences, phase synchronizations, mutual information, cross correlation, and the coherence function. In passing we show a close relation between two recently proposed phase synchronization measures and we extend the definition of one of them. In three typical examples we observe that except mutual information, all these measures give a useful quantification that is hard to be guessed beforehand from the raw data. Despite their differences, results are qualitatively the same. Therefore, we claim that the applied measures are valuable for the study of synchronization in real data. Moreover, in the particular case of EEG signals their use as complementary variables could be of clinical relevance.

Transition from intermittency to periodicity in lag synchronization in coupled Rössler oscillators

The dynamical and statistical behavior of lag synchronization in two coupled self-sustained chaotic Rössler oscillators is reexamined. The lack of uniqueness in the conventional characterization of lag synchronization based on the similarity function has caused much skepticism about the existence of lag synchronization. We provide an evidence that the emergence of lag synchronization is associated with the transition from on-off intermittency to a periodic structure in the laminar phase distribution.

n:m phase synchronization with mutual coupling phase signals

We generalize the n:m phase synchronization between two chaotic oscillators by mutual coupling phase signals. To characterize this phenomenon, we use two coupled oscillators to demonstrate their phase synchronization with amplitudes practically noncorrelated. We take the 1:1 phase synchronization as an example to show the properties of mean frequencies, mean phase difference, and Lyapunov exponents at various values of coupling strength. The phase difference increases with 2pi phase slips below the transition. The scaling rules of the slip near and away from the transition are studied. Furthermore, we demonstrate the transition to a variety of n:m phase synchronizations and analyze the corresponding coupling dynamics. (c) 2002 American Institute of Physics.

Reconstructing embedding spaces of coupled dynamical systems from multivariate data

A method for reconstructing dimensions of subspaces for weakly coupled dynamical systems is offered. The tool is able to extrapolate the subspace dimensions from the zero coupling limit, where the division of dimensions as per the algorithm is exact. Implementation of the proposed technique to multivariate data demonstrates its effectiveness in disentangling subspace dimensionalities also in the case of emergent synchronized motions, for both numerical and experimental systems.

Phase relationships between two or more interacting processes from one-dimensional time series. I. Basic theory

A general approach is developed for the detection of phase relationships between two or more different oscillatory processes interacting within a single system, using one-dimensional time series only. It is based on the introduction of angles and radii of return times maps, and on studying the dynamics of the angles. An explicit unique relationship is derived between angles and the conventional phase difference introduced earlier for bivariate data. It is valid under conditions of weak forcing. This correspondence is confirmed numerically for a nonstationary process in a forced Van der Pol system. A model describing the angles' behavior for a dynamical system under weak quasiperiodic forcing with an arbitrary number of independent frequencies is derived.

Nonlinear phase desynchronization in human electroencephalographic data

Ensembles of coupled nonlinear systems represent natural candidates for the modeling of brain dynamics. The objective of this study is to examine the complex signal produced by coupled chaotic attractors, to discuss their potential relevance to distributed processes in the brain, and to illustrate a method of detecting their contribution to human EEG morphology. Two measures of quantifying the behavior of coupled nonlinear systems are presented: a measure of phase synchrony and a novel measure of intermittent phase desynchronization. These are used to quantify the behavior of numerical examples of coupled chaotic attractors. Experimental evidence of their contribution to the morphology of the human alpha rhythm is then illustrated in a study of EEG recordings from 40 healthy human subjects. Amplitude-adjusted phase-randomized surrogate data is used to test the null hypothesis that the observed patterns of phase coherence can be described by purely linear methods. Statistical analysis reveals that this null hypothesis can be robustly rejected in a small number (approximately 4%) of EEG epochs. These findings are discussed with reference to the adaptive function and complex dynamics of the brain.

Amplitude envelope synchronization in coupled chaotic oscillators

A peculiar type of synchronization has been found when two Van der Pol-Duffing oscillators, evolving in different chaotic attractors, are coupled. As the coupling increases, the frequencies of the two oscillators remain different, while a synchronized modulation of the amplitudes of a signal of each system develops, and a null Lyapunov exponent of the uncoupled systems becomes negative and gradually larger in absolute value. This phenomenon is characterized by an appropriate correlation function between the returns of the signals, and interpreted in terms of the mutual excitation of new frequencies in the oscillators power spectra. This form of synchronization also occurs in other systems, but it shows up mixed with or screened by other forms of synchronization, as illustrated in this paper by means of the examples of the dynamic behavior observed for three other different models of chaotic oscillators.

Peculiarities of the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors in the presence of noise

We study the relaxation to an invariant probability measure on quasihyperbolic and nonhyperbolic chaotic attractors in the presence of noise. We also compare different characteristics of the rate of mixing and show numerically that the rate of mixing for nonhyperbolic chaotic attractors can significantly change under the influence of noise. A mechanism of the noise influence on mixing is presented, which is associated with the dynamics of the instantaneous phase of chaotic trajectories. We also analyze how the synchronization effect can influence the rate of mixing in a system of two coupled chaotic oscillators.

Characterizing instantaneous phase relationships in whole-brain fMRI activation data

Typically, fMRI data is processed in the time domain with linear methods such as regression and correlation analysis. We propose that the theory of phase synchronization may be used to more completely understand the dynamics of interacting systems, and can be applied to fMRI data as a novel method of detecting activation. Generalized synchronization is a phenomenon that occurs when there is a nonlinear functional relationship present between two or more coupled, oscillatory systems, whereas phase synchronization is defined as the locking of the phases while the amplitudes may vary. In this study, we developed an application of phase synchronization analysis that is appropriate for fMRI data, in which the phase locking condition is investigated between a voxel time series and the reference function of the task performed. A synchronization index is calculated to quantify the level of phase locking, and a nonparametric permutation test is used to determine the statistical significance of the results. We performed the phase synchronization analysis on the data from five volunteers for an event-related finger-tapping task. Functional maps were created that provide information on the interrelations between the instantaneous phases of the reference function and the voxel time series in a whole-brain fMRI activation data set. We conclude that this method of analysis is useful for revealing additional information on the complex nature of the fMRI time series.

Phase synchronization of chaotic rotators

We demonstrate the existence of phase synchronization of two chaotic rotators. Contrary to phase synchronization of chaotic oscillators, here the Lyapunov exponents corresponding to both phases remain positive even in the synchronous regime. Such frequency locked dynamics with different ratios of frequencies are studied for driven continuous-time rotators and for discrete circle maps. We show that this transition to phase synchronization occurs via a crisis transition to a band-structured attractor.

Controlling wake turbulence

This Letter introduces a control strategy for taming the wake turbulence behind a cylinder. An angular momentum injection scheme is proposed to synchronize the vertical velocity field. We show that the base suction, wake formation length, absolute instability, and the Kármán vortex street are effectively controlled by the angular momentum injection. A control equation is designed to implement the injection. The Navier-Stokes equations, along with the control equation, are solved. The occurrence of a new recirculation free zone is identified.

Self-induced slow-fast dynamics and swept bifurcation diagrams in weakly desynchronized systems

In systems close to the state of phase synchronization, the fast timescale of oscillations interacts with the slow timescale of the phase drift. As a result, "fast" dynamics is subjected to a slow modulation, due to which an autonomous system under fixed parameter values can imitate repeated bifurcational transitions. We demonstrate the action of this general mechanism for a set of two coupled autonomous chaotic oscillators and for a chaotic system perturbed by a periodic external force. In both cases, the Poincaré sections of phase portraits resemble bifurcation diagram of a logistic mapping with time-dependent parameter.

Surrogate analysis of coherent multichannel data

We present a method for generating surrogate data for multichannel time series. By preserving both the power spectra and the cross spectrum of the original data, we can determine if a given statistical test (such as the synchronization index) is being biased by the presence of coherence due to linear superposition of the separate measurements. Current methods based on phase randomization techniques are unsuitable for this particular task. This method is demonstrated on various canonical systems and numerical models. We will show that this algorithm is capable of properly preserving the power spectra and coherence function of the original data, and furthermore, that with the help of surrogate analysis the synchronization index measure is capable of distinguishing between coupled nonlinear oscillators and coherent superpositions of independent chaotic oscillators.