Phase synchronization of chaotic oscillators

Synchronization of vectorial noise-sustained structures

The synchronization of vectorial, noise-sustained structures in nonlinear optical systems is discussed. In particular, the analysis is made for nondegenerate optical parametric oscillators with walk off. The interplay between walk off and noise fluctuations leads to the formation of noise-sustained transverse patterns in both the signal and idler fields. Despite the fact that both patterns are stochastic macroscopic structures driven by independent sources of noise, their correlation grows with time, finally leading to a spatially distributed time synchronization of noise-sustained structures. A physical explanation of this phenomenon is found by analyzing the linear instability process and the existence of exact nonlinear solutions that show the same correlation.

Phase synchronization of chaotic attractors with prescribed periodic signals

Given a chaotic attractor in a dynamical system with dense periodic windows (i.e., structurally unstable), is it possible to find a periodic driver that will phase synchronize the chaotic attractor? We conjecture that the answer is typically yes, and we give an example for a funneling chaotic attractor in the Roessler system.

Dynamic synchronization analysis of venous pressure-driven cardiac output in rainbow trout

Measurement of venous function in vivo is inherently difficult. In this study, we used the Hilbert transform to examine the dynamic relationships between venous pressure and cardiac output (CO) in rainbow trout whose blood volume was continuously increased and decreased by ramp infusion and withdrawal (I/W). The dorsal aorta and ductus Cuvier were cannulated percutaneously and connected to pressure transducers; a flow probe was placed around the ventral aorta. Whole blood from a donor was then I/W via the dorsal aortic cannula at a rate of 10% of the estimated blood volume per minute, and the duration of I/W was varied from 40, 60, 80, 90, 120, 230, 240, 260, 300, and 340 s. Compliance [change in (delta) blood vol/deltavenous pressure] was 2.8 +/- 0.2 ml x mmHg-1x g-1 (N = 25 measurements; 6 fish with closed pericardium) and 2.8 +/- 0.3 ml. mmHg-1x kg-1 (N = 19 measurements, 4 fish with open pericardium). Compliance was positively correlated with the duration of I/W, indicative of cardiovascular reflex responses at longer I/W durations. In trout with closed pericardium, CO followed venous pressure oscillations with an average time lag of 4.2 +/- 1.0 s (N = 9); heart rate (HR) was inversely correlated with CO. These studies show that CO is entrained by modulation of venous pressure, not by HR. Thus, although trout have a rigid pericardium, venous pressure (vis-a-tergo), not cardiac suction (vis-a-fronte), appears to be the primary determinant of CO. Estimation of venous compliance by ramp-modulation of venous pressure is faster and less traumatic than classical capacitance measurements and appears applicable to a variety of vertebrate species, as does the Hilbert transform, which permits analysis of signals with disparate frequencies.

Estimation of interaction strength and direction from short and noisy time series

A technique for determination of character and intensity of interaction between the elements of complex systems based on reconstruction of model equations for phase dynamics is extended to the case of short and noisy time series. Corrections, which eliminate systematic errors of the estimates, and expressions for confidence intervals are derived. Analytic results are presented for a particular case of linear uncoupled systems, and their validity for a much wider range of situations is demonstrated with numerical examples. The technique should be useful for the analysis of nonstationary processes in real time, including the situations of significant noise and restrictions on the observation time.

Detection of mutual phase synchronization in multivariate signals and application to phase ensembles and chaotic data

This work presents a method for the detection of mutual phase synchronization in nonstationary time series. We show how the application of a cluster algorithm that considers spatiotemporal structures of data follows from the general condition of phase-synchronized data. In view of the topology of phasic data, we reformulate the K-means cluster algorithm on a flat torus and apply a segmentation index derived in an earlier work [A. Hutt and H. Riedel, Physica D 177, 203 (2003)]. This index is extended by means of averaging in order to reflect phase synchronization in ensembles of multivariate time series. The method is illustrated using simulated multivariate phase dynamics and arrays of chaotic systems, in which temporal segments of phase-synchronized states are registered. A comparison with results from an existing bivariate synchronization index reveals major advantages of our method.

Complete synchronization and generalized synchronization of one-way coupled time-delay systems

The complete synchronization and generalized synchronization (GS) of one-way coupled time-delay systems are studied. We find that GS can be achieved by a single scalar signal, and its synchronization threshold for different delay times shows the parameter resonance effect, i.e., we can obtain stable synchronization at a smaller coupling if the delay time of the driven system is chosen such that it is in resonance with the driving system. Near chaos synchronization, the desynchronization dynamics displays periodic bursts with the period equal to the delay time of the driven system. These features can be easily applied to the recovery of time-delay systems.

Chaotic itinerancy

Chaotic itinerancy is universal dynamics in high-dimensional dynamical systems, showing itinerant motion among varieties of low-dimensional ordered states through high-dimensional chaos. Discovery, basic features, characterization, examples, and significance of chaotic itinerancy are surveyed.

Rapid convergence of time-averaged frequency in phase synchronized systems

Numerical and experimental evidences are presented to show that many phase synchronized systems of nonidentical chaotic oscillators, where the chaotic state is reached through a period-doubling cascade, show rapid convergence of the time-averaged frequency. The speed of convergence toward the natural frequency scales as the inverse of the measurement period. The results also suggest an explanation for why such chaotic oscillators can be phase synchronized.

Periodic phase synchronization in coupled chaotic oscillators

We investigate the characteristics of temporal phase locking states observed in the route to phase synchronization. It is found that before phase synchronization there is a periodic phase synchronization state characterized by periodic appearance of temporal phase-locking state and that the state leads to local negativeness in one of the vanishing Lyapunov exponents. By taking a statistical measure, we present the evidences of the phenomenon in unidirectionally and mutually coupled chaotic oscillators, respectively. And it is qualitatively discussed that the phenomenon is described by a nonuniform oscillator model in the presence of noise.

Experimental Chua-plasma phase synchronization of chaos

Experimental phase synchronization of chaos is demonstrated for two different chaotic oscillators: a plasma discharge and the Chua circuit. Our technique includes real-time capability for observing synchronization-desynchronization transitions. This capability results from a strong combination of synchronization and control, and allows tuning adjustments for search and stabilization of synchronous states. A power law is observed for the mean time between 2pi phase slips for different coupling strenghts. The experimental results are consistent with the numerical simulations.

On-off collective imperfect phase synchronization and bursts in wave energy in a turbulent state

A new type of synchronization, on-off collective imperfect phase synchronization, is found in a turbulent state. In the driver frame the nonlinear wave system can be transformed to a set of coupled oscillators moving in a potential related to the unstable steady wave. In "on" stages the oscillators in different spatial scales adjust themselves to collective imperfect phase synchronization, inducing strong bursts in the wave energy. The interspike intervals display a power-law distribution. In addition to the embedded saddle point, it is emphasized that the delocalization of the master mode also plays an important role in developing the on-off synchronization.

Three types of transitions to phase synchronization in coupled chaotic oscillators

We study the effect of noncoherence on the onset of phase synchronization of two coupled chaotic oscillators. Depending on the coherence properties of oscillations characterized by the phase diffusion, three types of transitions to phase synchronization are found. For phase-coherent attractors this transition occurs shortly after one of the zero Lyapunov exponents becomes negative. At rather strong phase diffusion, phase locking manifests a strong degree of generalized synchronization, and occurs only after one positive Lyapunov exponent becomes negative. For intermediate phase diffusion, phase synchronization sets in via an interior crises of the hyperchaotic set.

Transition from phase synchronization to complete synchronization in mutually coupled nonidentical Nd:YAG lasers

Using mutually coupled nonidentical continuous-wave Nd:YAG lasers, we experimentally confirmed the recently proposed transition route from phase synchronization to complete synchronization. As evidence of this transition we obtained the probability distribution of the intermittent synchronization time near the threshold of the complete synchronization transition.

Oscillatory and rotatory synchronization of chaotic autonomous phase systems

The existence of rotatory, oscillatory, and oscillatory-rotatory synchronization of two coupled chaotic phase systems is demonstrated in the paper. We find four types of transition to phase synchronization depending on coherence properties of motions, characterized by phase variable diffusion. When diffusion is small the onset of phase synchronization is accompanied by a change in the Lyapunov spectrum; one of the zero Lyapunov exponents becomes negative shortly before this onset. If the diffusion of the phase variable is strong then phase synchronization and generalized synchronization, occur simultaneously, i.e., one of the positive Lyapunov exponents becomes negative, or generalized synchronization even sets in before phase synchronization. For intermediate diffusion the phase synchronization appears via interior crisis of the hyperchaotic set. Soft and hard transitions to phase synchronization are discussed.

A secure communication scheme based on the phase synchronization of chaotic systems

Phase synchronization of chaotic systems with both weak and strong couplings has recently been investigated extensively. Similar to complete synchronization, this type of synchronization can also be applied in secure communications. We develop a digital secure communication scheme that utilizes the instantaneous phase as the signal transmitted from the drive to the response subsystems. Simulation results show that the scheme is difficult to be broken by some traditional attacks. Moreover, it operates with a weak positive conditional Lyapunov exponent in the response subsystem.

Constructive effects of noise in homoclinic chaotic systems

Many chaotic oscillators have coherent phase dynamics but strong fluctuations in the amplitudes. At variance with such a behavior, homoclinic chaos is characterized by quite regular spikes but strong fluctuation in their time intervals due to the chaotic recurrence to a saddle point. We study influences of noise on homoclinic chaos. We demonstrate both numerically and experimentally on a CO2 laser various constructive effects of noise, including coherence resonance, noise-induced synchronization in uncoupled systems and noise-enhanced phase synchronization, deterministic resonance with respect to signal frequency, and stochastic resonance versus noise intensity in response to weak signals. The peculiar sensitivity of the system along the weak unstable manifold of the saddle point underlines the unified mechanism of these nontrivial and constructive noise-induced phenomena.

Measure synchronization in coupled phi4 Hamiltonian systems

Transitions to measure synchronization both in the quasiperiodic and chaotic cases are investigated based on numerical computation of two coupled phi(4) equations. Some relevant quantities such as the bare energies, the interaction energy, and the phase difference of the two oscillators are computed to clarify the characteristics of the transitions and the measure-synchronous states. A bifurcation with discontinuous bare energy and continuous interaction energy, which takes the maximum value at the critical point, is found for the transition from the desynchronous quasiperiodic state to the measure-synchronous quasiperiodic state, and the related power law scalings are deduced. Stick-slip and random-walk-like behavior of the phase difference is found for the chaotic measure-synchronous state, and this explains the monotonous increase of the interaction energy with an increase of coupling.

Complete synchronizability of chaotic systems: a geometric approach

Synchronizability of chaotic systems is studied in this contribution. Geometrical tools are used to understand the properties of vector fields in affine systems. The discussion is focused on synchronizability of chaotic systems with equal order. The analysis is based on the synchronous behavior of all states of the master/slave system (complete synchronization). We state sufficient and necessary conditions for complete synchronizability which are based on controllability and observability of nonlinear affine systems. In this sense, the synchronizability is studied for complete synchronization via state feedback control.

Phase synchronization in the perturbed Chua circuit

We show experimental and numerical results of phase synchronization between the chaotic Chua circuit and a small sinusoidal perturbation. Experimental real-time phase synchronized states can be detected with oscilloscope visualization of the attractor, using specific sampling rates. Arnold tongues demonstrate robust phase synchronized states for perturbation frequencies close to the characteristic frequency of the unperturbed Chua.

Spatio-temporal dynamics prior to neocortical seizures: amplitude versus phase couplings

The mechanisms underlying the transition of brain activity toward epileptic seizures remain unclear. Based on nonlinear analysis of both intracranial and scalp electroencephalographic (EEG) recordings, different research groups have recently reported dynamical smooth changes in epileptic brain activity several minutes before seizure onset. Such preictal states have been detected in populations of patients with mesial temporal lobe epilepsy (MTLE) and, more recently, with different neocortical partial epilepsies (NPEs). In this paper, we are particularly interested in the spatio-temporal organization of epileptogenic networks prior to seizures in neocortical epilepsies. For this, we characterize the network of two patients with NPE by means of two nonlinear measures of interdependencies. Since the synchronization of neuronal activity is an essential feature of the generation and propagation of epileptic activity, we have analyzed changes in phase synchrony between EEG time series. In order to compare the phase and amplitude dynamics, we have also studied the degree of association between pairs of signals by means of a nonlinear correlation coefficient. Recent findings have suggested changes prior to seizures in a wideband frequency range. Instead, for the examples of this study, we report a significant decrease of synchrony in the focal area several minutes before seizures (>>30 min in both patients) in the frequency band of 10-25 Hz mainly. Furthermore, the spatio-temporal organization of this preictal activity seems to be specifically related to this frequency band. Measures of both amplitude and phase coupling yielded similar results in narrow-band analysis. These results may open new perspectives on the mechanisms of seizure emergence as well as the organization of neocortical epileptogenic networks. The possibility of forecasting the onset of seizures has important implications for a better understanding, diagnosis and a potential treatment of the epilepsy.

Epileptic seizure prediction and control

Epileptic seizures are manifestations of epilepsy, a serious brain dynamical disorder second only to strokes. Of the world's approximately 50 million people with epilepsy, fully 1/3 have seizures that are not controlled by anti-convulsant medication. The field of seizure prediction, in which engineering technologies are used to decode brain signals and search for precursors of impending epileptic seizures, holds great promise to elucidate the dynamical mechanisms underlying the disorder, as well as to enable implantable devices to intervene in time to treat epilepsy. There is currently an explosion of interest in this field in academic centers and medical industry with clinical trials underway to test potential prediction and intervention methodology and devices for Food and Drug Administration (FDA) approval. This invited paper presents an overview of the application of signal processing methodologies based upon the theory of nonlinear dynamics to the problem of seizure prediction. Broader application of these developments to a variety of systems requiring monitoring, forecasting and control is a natural outgrowth of this field.

Direction of coupling from phases of interacting oscillators: an information-theoretic approach

A directionality index based on conditional mutual information is proposed for application to the instantaneous phases of weakly coupled oscillators. Its abilities to distinguish unidirectional from bidirectional coupling, as well as to reveal and quantify asymmetry in bidirectional coupling, are demonstrated using numerical examples of quasiperiodic, chaotic, and noisy oscillators, as well as real human cardiorespiratory data.

Stochastic phase resetting of two coupled phase oscillators stimulated at different times

A model of two coupled phase oscillators is presented, where the oscillators are subject to random forces and are stimulated at different times. Transient phase dynamics, synchronization, and desynchronization, which are stimulus locked (i.e., tightly time locked to a repetitively administered stimulus), are investigated. Complex coordinated responses, in terms of a noise-induced switching across trials between qualitatively different responses, may occur when the two oscillators are reset close to an unstable fixed point of their relative phases. This can be achieved with an appropriately chosen delay between the two stimuli. The switching of the responses shows up as a coordinated cross-trial (CT) response clustering of the oscillators, where the two oscillators produce two different pairs of responses. By varying noise amplitude and coupling strength we observe a stochastic resonance and a coupling-mediated resonance of the CT response clustering, respectively. The presented data analysis method makes it possible to detect such processes in numerical and experimental signals. Its time resolution is enormous, since it is only restricted by the time resolution of the preprocessing necessary for extracting the phases from experimental data. In contrast, standard data analysis tools applied across trials relative to stimulus onset, such as CT averaging (where an ensemble of poststimulus responses is simply averaged), CT standard deviation, and CT cross correlation, fail in detecting complex coordinated responses and lead to severe misinterpretations and artifacts. The consequences for the analysis of evoked responses in medicine and neuroscience are significant and are discussed in detail.

On-off intermittency in chaotic rotation induced in liquid crystals by competition between spin and orbital angular momentum of light

We observed on-off intermittency in the chaotic rotation induced by a cw laser beam in a thin liquid crystal film where the spin and the orbital angular momentum of light compete in reorienting the sample. We found that the azimuthal angle phi(t) of the molecular director increased linearly in time on large time scales but, occasionally, it exhibited large fluctuations about its average value omega(0)t, so that its angular velocity phi;(t) undergoes an on-off intermittent motion. The intermittent signal omega(t)=phi;(t)-omega(0) obeyed the scaling laws of on-off intermittency, including the symmetry between laminar and burst phases. The chaotic rotations were observed only when the spin and the angular momentum of light were transferred simultaneously to the sample.

Universal scaling of Lyapunov exponents in coupled chaotic oscillators

We have uncovered a phenomenon in coupled chaotic oscillators where a subset of Lyapunov exponents, which are originally zero in the absence of coupling, can become positive as the coupling is increased. This occurs for chaotic attractors having multiple scrolls, such as the Lorenz attractor. We argue that the phenomenon is due to the disturbance to the relative frequencies with which a trajectory visits different scrolls of the attractor. An algebraic scaling law is obtained which relates the Lyapunov exponents with the coupling strength. The scaling law appears to be universal.

Generalized phase synchronization in unidirectionally coupled chaotic oscillators

We investigate phase synchronization between two identical or detuned response oscillators coupled to a slightly different drive oscillator. Our result is that phase synchronization can occur between response oscillators when they are driven by correlated (but not identical) inputs from the drive oscillator. We call this phenomenon generalized phase synchronization and clarify its characteristics using Lyapunov exponents and phase difference plots.

Cerebral information transfer during word processing: where and when does it occur and how fast is it?

Different regions of the brain have to interact to perform language processing. Such neural integration processes can be studied by measuring synchronization of oscillations. Coherence is the best known algorithm to study synchronization between two sites of the cortex with regard to selected oscillation frequencies. Phase coherence quantifies phase synchronization, in particular. In addition, cross phase (or simply phase) gives information about the direction and speed of the spread of oscillations. We use a new method to study short-time phases between different sites of the cortex in order to explore transient neural networks during word processing. Particularly, processing of abstract and concrete nouns was investigated by dynamic phase analysis of the alpha1 frequency band (8-10 Hz). Abstract and concrete nouns turned out to induce different dynamic networks of information transfer. Whereas processing of concrete nouns excites a widespread network for about 800 msec, the interhemispheric interactions during abstract noun processing are restricted to 300-500 msec after word presentation. Further, the direction of cerebral information transfer differs for the two word categories. Additionally, it could be shown that the propagation speed of information is slower for concrete nouns than for abstract nouns.

Using nonisochronicity to control synchronization in ensembles of nonidentical oscillators

We investigate the transition to synchronization in ensembles of coupled oscillators with quenched disorder. We find that small coupling is able to increase the frequency disorder and to induce a spread of oscillator frequencies. This new effect of anomalous desynchronization is studied with numerical and analytical means in a large class of systems including Rössler, Lotka-Volterra, Landau-Stuart, and Van-der-Pol oscillators. We show that anomalous effects arise due to an interplay between nonisochronicity and natural frequency of each oscillator and can either increase or inhibit synchronization in the ensemble. This provides a novel possibility to control the synchronization transition in nonidentical systems by suitably distributing the disorder among system parameters. We conjecture that our results are of relevance for biological systems.

Noise-induced synchronization and coherence resonance of a Hodgkin-Huxley model of thermally sensitive neurons

We study nontrivial effects of noise on synchronization and coherence of a chaotic Hodgkin-Huxley model of thermally sensitive neurons. We demonstrate that identical neurons which are not coupled but subjected to a common fluctuating input (Gaussian noise) can achieve complete synchronization when the noise amplitude is larger than a threshold. For nonidentical neurons, noise can induce phase synchronization. Noise enhances synchronization of weakly coupled neurons. We also find that noise enhances the coherence of the spike trains. A saddle point embedded in the chaotic attractor is responsible for these nontrivial noise-induced effects. Relevance of our results to biological information processing is discussed.

Role of unstable periodic orbits in phase and lag synchronization between coupled chaotic oscillators

An increase of the coupling strength in the system of two coupled Rössler oscillators leads from a nonsynchronized state through phase synchronization to the regime of lag synchronization. The role of unstable periodic orbits in these transitions is investigated. Changes in the structure of attracting sets are discussed. We demonstrate that the onset of phase synchronization is related to phase-lockings on the surfaces of unstable tori, whereas transition from phase to lag synchronization is preceded by a decrease in the number of unstable periodic orbits.

Anomalous phase synchronization in populations of nonidentical oscillators

We report the phenomenon of anomalous phase synchronization in interacting oscillator systems with randomly distributed parameters. We show that coupling is first able to enlarge the frequency disorder leading to maximal decoherence for intermediate levels of coupling strength before reaching synchronization. Anomalous synchronization arises when the natural frequency covaries with nonisochronicity and allows for synchronization control by adjustment of system parameters.

Frequency and phase locking of noise-sustained oscillations in coupled excitable systems: array-enhanced resonances

We study the interplay among noise, weak driving signal and coupling in excitable FitzHugh-Nagumo neurons. Due to coupling, noise-sustained oscillations become locked to the signal as functions of both signal frequency and noise intensity. Higher order m:n locking tongues and various array-enhanced resonance features are demonstrated. This resonance and locking behavior due to a time scale matching between noise-sustained oscillations and the signal is fundamentally different from stochastic resonance in usual noisy threshold elements.

Frequency and phase synchronization in stochastic systems

The phenomenon of frequency and phase synchronization in stochastic systems requires a revision of concepts originally phrased in the context of purely deterministic systems. Various definitions of an instantaneous phase are presented and compared with each other with special attention paid to their robustness with respect to noise. We review the results of an analytic approach describing noise-induced phase synchronization in a thermal two-state system. In this context exact expressions for the mean frequency and the phase diffusivity are obtained that together determine the average length of locking episodes. A recently proposed method to quantify frequency synchronization in noisy potential systems is presented and exemplified by applying it to the periodically driven noisy harmonic oscillator. Since this method is based on a threshold crossing rate pioneered by Rice the related phase velocity is termed the Rice frequency. Finally, we discuss the relation between the phenomenon of stochastic resonance and noise-enhanced phase coherence by applying the developed concepts to the periodically driven bistable Kramers oscillator.

Noise enhanced phase synchronization and coherence resonance in sets of chaotic oscillators with weak global coupling

The effect of noise on phase synchronization in small sets and larger populations of weakly coupled chaotic oscillators is explored. Both independent and correlated noise are found to enhance phase synchronization of two coupled chaotic oscillators below the synchronization threshold; this is in contrast to the behavior of two coupled periodic oscillators. This constructive effect of noise results from the interplay between noise and the locking features of unstable periodic orbits. We show that in a population of nonidentical chaotic oscillators, correlated noise enhances synchronization in the weak coupling region. The interplay between noise and weak coupling induces a collective motion in which the coherence is maximal at an optimal noise intensity. Both the noise-enhanced phase synchronization and the coherence resonance numerically observed in coupled chaotic Rössler oscillators are verified experimentally with an array of chaotic electrochemical oscillators.

Characterizing direction of coupling from experimental observations

We demonstrate that the direction of coupling of two interacting self-sustained electronic oscillators can be determined from the realizations of their signals. In our experiments, two electronic generators, operating in a periodic or a chaotic state, were subject to symmetrical or unidirectional coupling. In data processing, first the phases have been extracted from the observed signals and then the directionality of coupling was quantitatively estimated from the analysis of mutual dependence of the phase dynamics.

Epileptic seizures are preceded by a decrease in synchronization

The exact mechanisms leading to the occurrence of epileptic seizures in humans are still poorly understood. It is widely accepted, however, that the process of seizure generation is closely associated with an abnormal synchronization of neurons. In order to investigate this process, we here measure phase synchronization between different regions of the brain using intracranial EEG recordings. Based on our preliminary finding of a preictal drop in synchronization, we investigate whether this phenomenon can be used as a sensitive and specific criterion to characterize a preseizure state and to distinguish this state from the interictal interval. Applying an automated technique for detecting decreased synchronization to EEG recordings from a group of 18 patients with focal epilepsy comprising a total of 117 h, we observe a characteristic decrease in synchronization prior to 26 out of 32 analyzed seizures at a very high specificity as tested on interictal recordings. The duration of this preictal state is found to range from several minutes up to a few hours. Investigation of the spatial distribution of preictal desynchronization indicates that the process of seizure generation in focal epilepsy is not necessarily confined to the focus itself but may instead involve more distant, even contralateral areas of the brain. Finally, we demonstrate an intrahemispheric asymmetry in the spatial dynamics of preictal desynchronization that is found in the majority of seizures and appears to be an immanent part of the mechanisms underlying the initiation of seizures in humans.

Persistent clusters in lattices of coupled nonidentical chaotic systems

Two-dimensional (2D) lattices of diffusively coupled chaotic oscillators are studied. In previous work, it was shown that various cluster synchronization regimes exist when the oscillators are identical. Here, analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters. In the analytical approach, the stability of almost-perfect synchronization regimes is proved via the Lyapunov function method for a wide class of systems, and the synchronization error is estimated. Examples include a 2D lattice of nonidentical Lorenz systems with scalar diffusive coupling. In the numerical study, it is shown that in lattices of Lorenz and Rossler systems the cluster synchronization regimes are stable and robust against up to 10%-15% parameter mismatch and against small noise.

Chaotic synchronization and evolution of optical phase in a bidirectional solid-state ring laser

We present results on experimental and theoretical studies of chaos in a solid-state ring laser with periodic pump modulation. We show that the synchronized chaos in the counter-propagating waves is observed for the values of pump modulation frequency fp satisfying the inequality f1 < fp < f2. The boundaries of this region, f1 and f2, depend on the pump-modulation depth. Inside the region of synchronized chaos we study not only dynamics of amplitudes of the counter-propagating waves but also the optical phases of them by mixing the fields of the counter-propagating waves and recording the intensity of the mixed signal. We demonstrate experimentally that in the regime of synchronized chaos the regular phase jumps appear during intervals between adjacent chaotic pulses. We improve the standard semi-classical model of a SSRL and consider an effect of spontaneous emission noise on the temporal evolution of intensities and phase dynamics in the regime of synchronized chaos. It is shown that at the parameters of the experimentally studied laser the noise strongly affects the temporal dependence of amplitudes of the counter-propagating waves.

Stochastic phase resetting of stimulus-locked responses of two coupled oscillators: transient response clustering, synchronization, and desynchronization

Transient phase dynamics, synchronization, and desynchronization which are stimulus-locked (i.e., tightly time-locked to a repetitively administered stimulus) are studied in two coupled phase oscillators in the presence of noise. The presented method makes it possible to detect such processes in numerical and experimental signals. The time resolution is enormous, since it is only restricted by the sampling rate. Stochastic stimulus locking of the phases or the n:m phase difference at a particular time t relative to stimulus onset is defined by the presence of one or more prominent peaks in the cross-trial distribution of the phases or the n:m phase difference at time t relative to stimulus onset in an ensemble of poststimulus responses. The oscillators' coupling may cause a transient cross-trial response clustering of the poststimulus responses. In particular, the mechanism by which intrinsic noise induces symmetric antiphase cross-trial response clustering in coupled detuned oscillators is a stochastic resonance. Unlike the presented approach, both cross-trial averaging (where an ensemble of poststimulus responses is simply averaged) and cross-trial cross correlation (CTCC) lead to severe misinterpretations: Triggered averaging cannot distinguish a cross-trial response clustering or decorrelation from a mean amplitude decrease of the single responses. CTCC not only depends on the oscillators' phase difference but also on their phases and, thus, inevitably displays "artificial" oscillations that are not related to synchronization or desynchronization.

Stochastic phase synchronization in the crayfish mechanoreceptor/photoreceptor system

The two light-sensitive neurons in the crayfish's abdominal sixth ganglion ("caudal photoreceptors," or CPRs), are both primary light sensors and secondary neurons in a mechanosensory pathway. Pei et al. (1996) demonstrated that light enhances the transduction of weak, periodic hydrodynamic stimuli (measured as an increase in the signal-to-noise ratio at the stimulus frequency in the power spectrum of the recorded neural spikes). This has been interpreted as a stochastic resonance effect, in which added light increases the noise intensity of the input to the photoreceptor (possibly through fluctuations in membrane potential), leading to an enhancement of the signal-to-noise ratio (SNR). Here, we discuss the recent demonstration (Bahar et al., 2002) of the correlation between a stochastic-resonance-like effect and an increase in stochastic phase synchronization between the neural response and a periodic mechanical stimulus. We also discuss a novel effect (Bahar et al., 2002) in which light increases the SNR of the second higher harmonic of a periodic input signal, effectively rectifying the input signal. This "second harmonic effect" can also be interpreted in terms of stochastic phase synchronization (Bahar et al., 2002). We review other recent results on the role of stochastic phase synchronization in mediating sensory responses in the crayfish nervous system.

Information encoding in homoclinic chaotic systems

We present a simple method for real-time encoding of information in the interspike intervals of a homoclinic chaotic system. The method has been experimentally tested on a CO2 laser with feedback displaying Sil'nikov chaos and synchronized with an external pulsed signal. Information is encoded by the length of the temporal intervals between consecutive pulses of the external signal. This length is varied each time a new pulse is generated.

Pattern formation on the edge of chaos: experiments with CO oxidation on a Pt(110) surface under global delayed feedback

Experiments with catalytic oxidation of carbon monoxide on Pt(110) show that chemical turbulence in this system can be suppressed by application of appropriate global delayed feedback. Different spatiotemporal patterns, seen near the transition from turbulence to uniform oscillations, are investigated. Such patterns include intermittent turbulence, oscillatory standing waves, cellular structures, and phase clusters. Using a method based on the Hilbert transform, spatial distributions of local phase and amplitude in these patterns are reconstructed from the experimental data.

Synchronization tomography: a method for three-dimensional localization of phase synchronized neuronal populations in the human brain using magnetoencephalography

We present a noninvasive technique which allows the anatomical localization of phase synchronized neuronal populations in the human brain with magnetoencephalography. We study phase synchronization between the reconstructed current source density (CSD) of different brain areas as well as between the CSD and muscular activity. We asked four subjects to tap their fingers in synchrony with a rhythmic tone, and to continue tapping at the same rate after the tone was switched off. The phase synchronization behavior of brain areas relevant for movement coordination, inner voice, and time estimation changes drastically when the transition to internal pacing occurs, while their averaged amplitudes remain unchanged. Information of this kind cannot be derived with standard neuroimaging techniques like functional magnetic resonance imaging or positron emission tomography.

Automated detection of a preseizure state based on a decrease in synchronization in intracranial electroencephalogram recordings from epilepsy patients

The question whether information extracted from the electroencephalogram (EEG) of epilepsy patients can be used for the prediction of seizures has recently attracted much attention. Several studies have reported evidence for the existence of a preseizure state that can be detected using different measures derived from the theory of dynamical systems. Most of these studies, however, have neglected to sufficiently investigate the specificity of the observed effects or suffer from other methodological shortcomings. In this paper we present an automated technique for the detection of a preseizure state from EEG recordings using two different measures for synchronization between recording sites, namely, the mean phase coherence as a measure for phase synchronization and the maximum linear cross correlation as a measure for lag synchronization. Based on the observation of characteristic drops in synchronization prior to seizure onset, we used this phenomenon for the characterization of a preseizure state and its distinction from the remaining seizure-free interval. After optimizing our technique on a group of 10 patients with temporal lobe epilepsy we obtained a successful detection of a preseizure state prior to 12 out of 14 analyzed seizures for both measures at a very high specificity as tested on recordings from the seizure-free interval. After checking for in-sample overtraining via cross validation, we applied a surrogate test to validate the observed predictability. Based on our results, we discuss the differences of the two synchronization measures in terms of the dynamics underlying seizure generation in focal epilepsies.

Chaotic communications that are difficult to detect

Work on self-synchronizing systems for communications has had limited practicality because the chaotic signals were not as easy to detect in the presence of noise as conventional spread-spectrum signals. This difficulty may actually be an advantage in some cases, where one wants to conceal the existence of the communications signal. Conventional communications signals are cyclostationary; while they may look random, they have statistical properties that vary periodically. One may design chaotic communication signals that lack this cyclostationary property, and therefore are harder to detect.

Synchronization and desynchronization under the influence of quasiperiodic forcing

We study the influence of quasiperiodic forcing on synchronization and desynchronization using two coupled quasiperiodically forced logistic maps as a paradigm. We show that due to the forcing the synchronization region in parameter space shrinks. The loss of transverse stability of the synchronized attractors leads to desynchronization. Two types of such blowout bifurcations are described, namely, the blowout bifurcations of synchronized quasiperiodic motion on invariant curves and synchronized strange nonchaotic attractors, both yielding desynchronized chaotic attractors.

Self-organized and driven phase synchronization in coupled maps

We study the phase synchronization and cluster formation in coupled maps on different networks. We identify two different mechanisms of cluster formation: (a) self-organized phase synchronization which leads to clusters with dominant intracluster couplings and (b) driven phase synchronization which leads to clusters with dominant intercluster couplings. In the novel driven synchronization the nodes of one cluster are driven by those of the others. We also discuss the dynamical origin of these two mechanisms for small networks with two and three nodes.