Phase synchronization of chaotic oscillators

Synchronization in coupled chaotic oscillators with a no-flux boundary condition

We investigate the synchronization of coupled chaotic oscillators with a no-flux boundary condition. We find that the spectrum of the coupling matrix is divided into two parts, the isolated part with a zero eigenvalue and the continuous one with the other N-1 eigenvalues falling onto a line. Based on the eigenvalue analysis, the stability of the synchronization in a coupled Lorenz system is explored thoroughly in the parameter space of the size of the system, the diffusion, and gradient coupling constants.

Phase synchronization in unidirectionally coupled chaotic ratchets

We study chaotic phase synchronization of unidirectionally coupled deterministic chaotic ratchets. The coupled ratchets were simulated in their chaotic states and perfect phase locking was observed as the coupling was gradually increased. We identified the region of phase synchronization for the ratchets and show that the transition to chaotic phase synchronization is via an interior crisis transition to strange attractor in the phase space.

A quantitative comparison of different methods to detect cardiorespiratory coordination during night-time sleep

Background

The univariate approaches used to analyze heart rate variability have recently been extended by several bivariate approaches with respect to cardiorespiratory coordination. Some approaches are explicitly based on mathematical models which investigate the synchronization between weakly coupled complex systems. Others use an heuristic approach, i.e. characteristic features of both time series, to develop appropriate bivariate methods.

Objective

In this study six different methods used to analyze cardiorespiratory coordination have been quantitatively compared with respect to their performance (no. of sequences with cardiorespiratory coordination, no. of heart beats coordinated with respiration). Five of these approaches have been suggested in the recent literature whereas one method originates from older studies.

Results

The methods were applied to the simultaneous recordings of an electrocardiogram and a respiratory trace of 20 healthy subjects during night-time sleep from 0:00 to 6:00. The best temporal resolution and the highest number of coordinated heart beats were obtained with the analysis of 'Phase Recurrences'. Apart from the oldest method, all methods showed similar qualitative results although the quantities varied between the different approaches. In contrast, the oldest method detected considerably fewer coordinated heart beats since it only used part of the maximum amount of information available in each recording.

Conclusions

The method of 'Phase Recurrences' should be the method of choice for the detection of cardiorespiratory coordination since it offers the best temporal resolution and the highest number of coordinated sequences and heart beats. Excluding the oldest method, the results of the heuristic approaches may also be interpreted in terms of the mathematical models.

"Dynamic" connectivity in neural systems: theoretical and empirical considerations

The study of functional interdependences between brain regions is a rapidly growing focus of neuroscience research. This endeavor has been greatly facilitated by the appearance of a number of innovative methodologies for the examination of neurophysiological and neuroimaging data. The aim of this article is to present an overview of dynamical measures of interdependence and contrast these with statistical measures that have been more widely employed. We first review the motivation, conceptual basis, and experimental approach of dynamical measures of interdependence and their application to the study of neural systems. A consideration of boot-strap "surrogate data" techniques, which facilitate hypothesis testing of dynamical measures, is then used to clarify the difference between dynamical and statistical measures of interdependence. An overview of some of the most active research areas such as the study of the "synchronization manifold," dynamical interdependence in neurophysiology data and the putative role of nonlinear desynchronization is then given. We conclude by suggesting that techniques based on dynamical interdependence--or "dynamical connectivity"--show significant potential for extracting meaningful information from functional neuroimaging data.

Phase synchronization from noisy univariate signals

We present methods for detecting phase synchronization of two unidirectionally coupled, self-sustained noisy oscillators from a signal of the driven oscillator alone. One method detects soft phase locking; another hard phase locking. Both are applied to the problem of detecting phase synchronization in von Kármán vortex flow meters.

Dynamical characterization of chaotic itinerancy in a three-mode laser subjected to frequency-shifted optical feedback

We investigated chaotic dynamics in a microchip three-mode solid-state laser subjected to frequency-shifted optical feedback. When the frequency shift was tuned to harmonic frequencies of the relaxation oscillation, a bifurcation from a periodic sustained relaxation oscillation ("soft-mode") state to a chaotic spiking ("hard-mode") state via a chaotic itinerancy was observed as the feedback intensity was increased. Dynamic characterizations of modal interplay and self-induced switching between the soft- and hard-mode chaotic states over times (i.e., chaotic itinerancy) were carried out by the information circulation analysis and joint time-frequency analysis of long-term experimental time series. Drastic changes in information transfer rates among oscillating modes and occasional frequency locking among periodicities of two chaotic states associated with switchings were identified in chaotic itinerancy. Essential dynamical behaviors were reproduced by numerical simulation.

Automatic classification of interference patterns in driven event series: application to single sympathetic neuron discharge forced by mechanical ventilation

This study proposes a method for the automatic classification of nonlinear interactions between a strictly periodical event series modelling the activity of an exogenous oscillator working at a fixed and well-known rate and an event series modelling the activity of a self-sustained oscillator forced by the exogenous one. The method is based on a combination of several well-known tools (probability density function of the cyclic relative phase, probability density function of the count of forced events per forcing cycle, conditional entropy of the cyclic relative phase sequence and a surrogate data approach). Classification is reached via a sequence of easily applicable decision rules, thus rendering classification virtually user-independent and fully reproducible. The method classifies four types of dynamics: full uncoupling, quasiperiodicity, phase locking and aperiodicity. In the case of phase locking, the coupling ratio (i.e. n: m) and the strength of the coupling are calculated. The method, validated on simulations of simple and complex phase-locking dynamics corrupted by different levels of noise, is applied to data derived from one anesthetized and artificially ventilated rat to classify the nonlinear interactions between mechanical ventilation and: (1) the discharges of two (contemporaneously recorded) single postganglionic sympathetic neurons innervating the caudal ventral artery in the tail and (2) arterial blood pressure. Under central apnea, the activity of the underlying sympathetic oscillators is perturbed by means of five different lung inflation rates (0.58, 0.64, 0.76, 0.95, 1.99 Hz). While ventilation and arterial pressure are fully uncoupled, ventilation is capable of phase locking sympathetic discharges, thus producing 40% of phase-locked patterns (one case of 2:5, 1:1, 3:2 and 2:2) and 40% of aperiodic dynamics. In the case of phase-locked patterns, the coupling strength is low, thus demonstrating that this pattern is sliding. Non-stationary interactions are observed in 20% of cases. The two discharges behave differently, suggesting the presence of a population of sympathetic oscillators working at different frequencies.

Synchronization of chaotic oscillators due to common delay time modulation

We have found a synchronization behavior between two identical chaotic systems when their delay times are modulated by a common irregular signal. This phenomenon is demonstrated both in two identical chaotic maps whose delay times are driven by a common chaotic or random signal and in two identical chaotic oscillators whose delay times are driven by a signal of another chaotic oscillator. We analyze the phenomenon by using the Lyapunov exponents and discuss it in relation to generalized synchronization.

Phase synchronization in ensembles of bursting oscillators

We study the effects of mutual and external chaotic phase synchronization in ensembles of bursting oscillators. These oscillators (used for modeling neuronal dynamics) are essentially multiple time scale systems. We show that a transition to mutual phase synchronization takes place on the bursting time scale of globally coupled oscillators, while on the spiking time scale, they behave asynchronously. We also demonstrate the effect of the onset of external chaotic phase synchronization of the bursting behavior in the studied ensemble by a periodic driving applied to one arbitrarily taken neuron. We also propose an explanation of the mechanism behind this effect. We infer that the demonstrated phenomenon can be used efficiently for controlling bursting activity in neural ensembles.

Adaptation through minimization of the phase lag in coupled nonidentical systems

We show that the internal control of adaptation can be obtained from the properties of the phase lag that results from phase synchronization of two nonidentical chaotic oscillators. The direction and magnitude of the phase lag depend upon the relative internal properties of the coupled units, and they can be used as indicators during the adjustment of dynamics, i.e., adaptation of the target unit to match that of the control. The properties of the phase lag are obtained using a method based on the estimation of properties of the distributions of relative event times of both (target and control) units. The phase lag dependent mechanism to control the adaptation process was applied to a system of nonidentical Rössler oscillators and a system of nonidentical Lorenz oscillators. We also elucidate its importance as a control mechanism of the changes of neuronal activity showing its application to neural adaptation.

An approach to chaotic synchronization

This paper deals with the chaotic oscillator synchronization. An approach to the synchronization of chaotic oscillators has been proposed. This approach is based on the analysis of different time scales in the time series generated by the coupled chaotic oscillators. It has been shown that complete synchronization, phase synchronization, lag synchronization, and generalized synchronization are the particular cases of the synchronized behavior called "time-scale synchronization." The quantitative measure of chaotic oscillator synchronous behavior has been proposed. This approach has been applied for the coupled Rössler systems and two coupled Chua's circuits.

Predicting phase synchronization in a spiking chaotic CO2 laser

An approach is presented for the reconstruction of phase synchronization phenomena in a chaotic CO2 laser from experimental data. We analyze this laser system in a regime able to phase synchronize with a weak sinusoidal forcing. Our technique recovers the synchronization diagram of the experimental system from only few measurement data sets, thus allowing the prediction of the regime of phase synchronization as well as nonsynchronization in a broad parameter space of forcing frequency and amplitude without further experiments.

Coupling scheme for complete synchronization of periodically forced chaotic CO2 lasers

We present a way of coupling two nonautonomous, periodically forced, chaotic C O2 lasers in a master-slave configuration in order to achieve complete synchronization. The method consists of modulating the forcing of the slave laser by means of the difference between the intensities of the two lasers, and lends itself to a simple physical implementation. Experimental evidence of complete synchronization induced by a suitable coupling strength is shown, and a numerical model is used to achieve further insight of the synchronization phenomena. Finally, we describe a possible application of the investigated technique to the design of a digital communication system.

Noise-aided synchronization of coupled chaotic electrochemical oscillators

We report experimental and numerical results on noise-enhanced synchronization of two coupled chaotic oscillators. Enhanced synchronization is achieved through superimposing small-amplitude Gaussian noise on a common system parameter of the two chaotic oscillators. A resonancelike behavior is found: at an optimum level of noise, maximum synchronization is attained. The simulations show that the resonance behavior occurs with both identical and nonidentical oscillators. Noncommon (asymmetric and independent) noise does not enhance synchronization; common noise seems to enhance synchronization.

Fourier-, Hilbert- and wavelet-based signal analysis: are they really different approaches?

Spectral signal analysis constitutes one of the most important and most commonly used analytical tools for the evaluation of neurophysiological signals. It is not only the spectral parameters per se (amplitude and phase) which are of interest, but there is also a variety of measures derived from them, including important coupling measures like coherence or phase synchrony. After reviewing some of these measures in order to underline the widespread relevance of spectral analysis, this report compares the three classical spectral analysis approaches: Fourier, Hilbert and wavelet transform. Recently, there seems to be increasing acceptance of the notion that Hilbert- or wavelet-based analyses be in some way superior to Fourier-based analyses. The present article counters such views by demonstrating that the three techniques are in fact formally (i.e. mathematically) equivalent when using the class of wavelets that is typically applied in spectral analyses. Moreover, spectral amplitude serves as an example to show that Fourier, Hilbert and wavelet analysis also yield equivalent results in practical applications to neuronal signals.

Dynamic control by sinusoidal perturbation and by Gaussian noise of a system of two nonlinear oscillators: computation and experimental results

In this paper we report numerical and experimental studies of the dynamic control of the inter-anode plasma of a double electrical discharge and of a system of two coupled nonlinear oscillators modeling this plasma. We compare the transition between chaotic dynamics and periodic dynamics induced by a sinusoidal perturbation and by small-dispersion Gaussian noise. Besides considerable differences between the effect of the two types of perturbation we also find important similarities. For small amplitude, both the sinusoidal and the white noise perturbations can induce the system to change from chaotic to regular dynamics. In the case of sinusoidal perturbation, the transition time from the chaotic to regular state has a definite duration that depends on the values of the perturbation parameters. The suppression of the perturbation has no influence on the state - the system remains in the same regular state. Subsequent reinstatement of the same type of perturbation with the same amplitude does not change the periodic state of the system but, for considerably higher amplitude, the system is switched back to its chaotic state. For moderate-amplitude sinusoidal perturbation, intermittent transitions between the chaotic and regular states is observed. Most of these predictions of the model have been observed experimentally in a system of two coupled electrical discharges. Our results suggest practical methods that can be used for controlling the discharge plasma dynamics.

Steady-state visual evoked potentials and phase synchronization in migraine patients

We investigate phase synchronization in EEG recordings from migraine patients. We use the analytic signal technique, based on the Hilbert transform, and find that migraine brains are characterized by enhanced alpha band phase synchronization in the presence of visual stimuli. Our findings show that migraine patients have an overactive regulatory mechanism that renders them more sensitive to external stimuli.

Synchronization measures of bursting data: application to the electrocorticogram of an auditory event-related experiment

Synchronization measures have become an important tool for exploring the relationships between time series. We review three recently proposed nonlinear synchronization measures and expand their definitions in a straightforward way to apply to an ensemble of measurements. We also develop a synchronization measure in which nearest neighbors are determined across the ensemble. We compare these four nonlinear synchronization measures and show that our measure succeeds in physically motivated examples where the other methods fail. We apply the synchronization measure to human electrocorticogram data collected during an auditory event-related potential experiment. The results suggest a crude model of cortical connectivity.

Logarithmic periodicities in the bifurcations of type-I intermittent chaos

The critical relations for statistical properties on saddle-node bifurcations are shown to display undulating fine structure, in addition to their known smooth dependence on the control parameter. A piecewise linear map with the type-I intermittency is studied and a log-periodic dependence is numerically obtained for the average time between laminar events, the Lyapunov exponent, and attractor moments. The origin of the oscillations is built in the natural probabilistic measure of the map and can be traced back to the existence of logarithmically distributed discrete values of the control parameter giving Markov partition. Reinjection and noise effect dependences are discussed and indications are given on how the oscillations are potentially applicable to complement predictions made with the usual critical exponents, taken from data in critical phenomena.

Nearest neighbor phase synchronization as a measure to detect seizure activity from scalp EEG recordings

The author presents results from the application of a particular measure for synchronization between brain areas (i.e., phase synchronization) in its behavior to detect epileptic seizure activity from scalp EEG recordings. The primary motivation for the current study was to contribute to the development of physiologic measures that both transform the EEG to a visual domain that allows a more intuitive interpretation of the interictal and ictal EEG and allows automated analysis, both relevant for real-time monitoring. EEGs from 16 patients experiencing temporal lobe and generalized seizures were analyzed. Nearest neighbor phase synchronization (NNPS) values for several frequency bands were determined. Additional analysis of the NNPS in the delta band, using different thresholds, allows construction of receiver operating characteristics (ROC) curves for each EEG analyzed. The common value for the sensitivity and the specificity, Q*, was used as a measure for the test accuracy, with values of Q* near 1.0, indicating ROC curves with sensitivity and specificity both approaching 1.0. It was found that Q* = 0.48 to 0.87, depending on the EEG analyzed, indicating that the proposed method allows seizure detection in a significant portion of the EEGs studied. Nearest neighbor phase synchronization was typically increased during seizure activity and seems to be a promising method to detect seizure activity from scalp EEG recordings. The proposed visualization allows an intuitive interpretation of the EEG and may assist in real-time monitoring.

Breaking a secure communication scheme based on the phase synchronization of chaotic systems

A security analysis of a recently proposed secure communication scheme based on the phase synchronization of chaotic systems is presented. It is shown that the system parameters directly determine the cipher text waveform, hence it can be readily broken by system parameter estimation from the cipher text signal.

Spatial patterns of desynchronization bursts in networks

We adapt a previous model and analysis method (the master stability function), extensively used for studying the stability of the synchronous state of networks of identical chaotic oscillators, to the case of oscillators that are similar but not exactly identical. We find that bubbling induced desynchronization bursts occur for some parameter values. These bursts have spatial patterns, which can be predicted from the network connectivity matrix and the unstable periodic orbits embedded in the attractor. We test the analysis of bursts by comparison with numerical experiments. In the case that no bursting occurs, we discuss the deviations from the exactly synchronous state caused by the mismatch between oscillators.

Experimental control of coherence of a chaotic oscillator

We give experimental evidence that a delayed feedback control strategy is able to efficiently enhance the coherence of an experimental self-sustained chaotic oscillator obtained from a CO2 laser with electro-optical feedback. We demonstrate that coherence control is achieved for various choices of the delay time in the feedback control, including values that would lead to the stabilization of an unstable periodic orbit embedded within the chaotic attractor. The relationship between the two processes is discussed.

Statistical method for detection of phase-locking episodes in neural oscillations

In many networks of oscillatory neurons, synaptic interactions can promote the entrainment of units into phase-coupled groups. The detection of synchrony in experimental data, especially if the data consist of single-trial runs, can be problematic when, for example, phase entrainment is of short duration, buried in noise, or masked by amplitude fluctuations that are uncorrelated among the oscillating units. In the present study, we tackle the problem of detecting neural interactions from pairs of oscillatory signals in a narrow frequency band. To avoid the interference of amplitude fluctuations in the detection of synchrony, we extract a phase variable from the data and utilize statistical indices to measure phase locking. We use three different phase-locking indices based on coherence, entropy, and mutual information between the phase variables. Phase-locking indices are calculated over time using sliding analysis windows. By varying the duration of the analysis windows, we were able to inspect the data at different levels of temporal resolution and statistical reliability. The statistical significance of high index values was evaluated using four different surrogate data methods. We determined phase-locking indices using alternative methods for generating surrogate data and found that results are sensitive to the particular method selected. Surrogate methods that preserve the temporal structure of the individual phase time series decrease substantially the number of false positives when tested on a pair of independent signals.

Transmission of stimulus-locked responses in two coupled phase oscillators

A model of two n:m coupled phase oscillators is studied, where both oscillators are subject to random forces, but only one oscillator is repetitively stimulated with a pulsatile stimulus. The focus of the paper is on transmission of transient responses as well as transient synchronization and desynchronization, which are stimulus locked, i.e., tightly time locked to the stimulus. A bistability or multistability of stable synchronized states of the two-phase oscillators (modulo 2pi ) occurs due to the n:m coupling. Accordingly, after stimulation the two oscillators may tend to qualitatively different stable states, which leads to a cross-trial (CT) response clustering (i.e., a switching between qualitatively different poststimulus responses across trials) of either one of the oscillators or both. A stochastic CT phase resetting analysis allows one to detect such transient responses and provides a reliable estimation of the transmission time. In contrast, CT averaging (averaging over an ensemble of responses), CT standard deviation, and CT cross correlation fail in studying the transmission of such stimulus-locked responses, even in the simpler case of 1:1 coupling. In particular, even though being used as golden standard for the analysis of evoked responses in medicine and neuroscience, CT averaging typically causes severe artifacts and misinterpretations.

Irrational phase synchronization

We study the occurrence of physically observable phase locked states between chaotic oscillators and rotors in which the frequencies of the coupled systems are irrationally related. For two chaotic oscillators, the phenomenon occurs as a result of a coupling term which breaks the 2 pi invariance in the phase equations. In the case of rotors, a coupling term in the angular velocities results in very long times during which the coupled systems exhibit alternatively irrational phase synchronization and random phase diffusion. The range of parameters for which the phenomenon occurs contains an open set, and is thus physically observable.

Oscillations of heart rate and respiration synchronize during poetry recitation

The objective of this study was to investigate the synchronization between low-frequency breathing patterns and respiratory sinus arrhythmia (RSA) of heart rate during guided recitation of poetry, i.e., recitation of hexameter verse from ancient Greek literature performed in a therapeutic setting. Twenty healthy volunteers performed three different types of exercises with respect to a cross-sectional comparison: 1). recitation of hexameter verse, 2). controlled breathing, and 3). spontaneous breathing. Each exercise was divided into three successive measurements: a 15-min baseline measurement (S1), 20 min of exercise, and a 15-min effect measurement (S2). Breathing patterns and RSA were derived from respiratory traces and electrocardiograms, respectively, which were recorded simultaneously using an ambulatory device. The synchronization was then quantified by the index gamma, which has been adopted from the analysis of weakly coupled chaotic oscillators. During recitation of hexameter verse, gamma was high, indicating prominent cardiorespiratory synchronization. The controlled breathing exercise showed cardiorespiratory synchronization to a lesser extent and all resting periods (S1 and S2) had even fewer cardiorespiratory synchronization. During spontaneous breathing, cardiorespiratory synchronization was minimal and hardly observable. The results were largely determined by the extent of a low-frequency component in the breathing oscillations that emerged from the design of hexameter recitation. In conclusion, recitation of hexameter verse exerts a strong influence on RSA by a prominent low-frequency component in the breathing pattern, generating a strong cardiorespiratory synchronization.

Phase synchronization of chaotic intermittent oscillations

We study phase synchronization effects of chaotic oscillators with a type-I intermittency behavior. The external and mutual locking of the average length of the laminar stage for coupled discrete and continuous in time systems is shown and the mechanism of this synchronization is explained. We demonstrate that this phenomenon can be described by using results of the parametric resonance theory and that this correspondence enables one to predict and derive all zones of synchronization.

A novel method for the topographic analysis of neural activity reveals formation and dissolution of 'Dynamic Cell Assemblies'

The study of synchronous oscillations in neural systems is a very active area of research. However, cognitive function may depend more crucially upon a dynamic alternation between synchronous and desynchronous activity rather than synchronous behaviour per se. The principle aim of this study is to develop and validate a novel method of quantifying this complex process. The method permits a direct mapping of phase synchronous dynamics and desynchronizing bursts in the spatial and temporal domains. Two data sets are analyzed: Numeric data from a model of a sparsely coupled neural cell assembly and experimental data consisting of scalp-recorded EEG from 40 human subjects. In the numeric data, the approach enables the demonstration of complex relationships between cluster size and temporal duration that cannot be detected with other methods. Dynamic patterns of phase-clustering and desynchronization are also demonstrated in the experimental data. It is further shown that in a significant proportion of the recordings, the pattern of dynamics exhibits nonlinear structure. We argue that this procedure provides a 'natural partitioning' of ongoing brain dynamics into topographically distinct synchronous epochs which may be integral to the brain's adaptive function. In particular, the character of transitions between consecutive synchronous epochs may reflect important aspects of information processing and cognitive flexibility.

Task-related coupling from high- to low-frequency signals among visual cortical areas in human subdural recordings

Cortical cooperativity during cognitive demands includes high- and low-frequency activities, which raises the question whether there are interdependencies between fast and slow processes and how they are reflected in electrical brain signals. We had the opportunity to record signals intracranially from occipital visual areas in an epileptic patient and quantified inter-areal signal coupling while the patient performed a visual delayed-match-to-sample task. We computed coherence, phase consistency and amplitude envelope correlation and we also determined inter-frequency coupling through correlation between low-frequency signal components and amplitude envelopes of high-frequency components. There was a pronounced task-related increase of correlation between gamma-band (28-70 Hz) signal envelopes from a superior (occipital) and low-frequency (0-3.5 Hz) signals from an inferior (occipital) visual area, lasting for approximately 1 s and possibly reflecting a short-term memory encoding process. The correlational delay between envelopes and low-frequency components was 40 ms. In contrast, coherence, phase consistency and envelope correlation showed event-, but no task-related changes of intra-areal and no changes of inter-areal coupling. Our data suggest a specific effect of gamma-activity in the superior onto low-frequency activity in the inferior area. We argue that temporal dispersion of conduction delays might prevent coherent transmission of high-frequency signals and thus account for the absence of gamma-coherence. As such dispersion is a general property of long-range projections, envelope-to-signal correlation possibly reflects a general neuronal mechanism. Hence, our method provides a powerful tool for detecting such inter-areal interactions not visible with conventional linear coupling measures.

Stochastic resonance and synchronization in the crayfish caudal photoreceptor

Stochastic resonance is the process by which noise added to a weak external stimulus can enhance encoding efficiency in the sensory periphery and thence in the central nervous system. Stochastic synchronization is the process by which noisy phase synchronization of two periodic (or aperiodic) signals can occur. Together with a brief review of both concepts, we illustrate their applications to the encoding of weak external hydrodynamic signals in the mechanosensory system of the crayfish.

In phase and antiphase synchronization of coupled homoclinic chaotic oscillators

We numerically investigate the dynamics of a closed chain of unidirectionally coupled oscillators in a regime of homoclinic chaos. The emerging synchronization regimes show analogies with the experimental behavior of a single chaotic laser subjected to a delayed feedback.

Autocorrelation function and spectral linewidth of spiral chaos in a physical experiment

We present results of physical experiments where we measure the autocorrelation function (ACF) and the spectral linewidth of the basic frequency of a spiral chaotic attractor in a generator with inertial nonlinearity both without and in the presence of external noise. It is shown that the ACF of spiral attractors decays according to an exponential law with a decrement which is defined by the phase diffusion coefficient. It is also established that the evolution of the instantaneous phase can be approximated by a Wiener random process.

Detecting local synchronization in coupled chaotic systems

We introduce a technique to detect and quantify local functional dependencies between coupled chaotic systems. The method estimates the fraction of locally synchronized configurations, in a pair of signals with an arbitrary state of global synchronization. Application to a pair of interacting Rössler oscillators shows that our method is able to quantify the number of dynamical configurations where a local prediction task is possible, as well as in the absence of global synchronization features.

Effect of noise on the neutral direction of chaotic attractor

A chaotic attractor from a deterministic flow must necessarily possess a neutral direction, as characterized by a null Lyapunov exponent. We show that for a wide class of chaotic attractors, particularly those having multiple scrolls in the phase space, the existence of the neutral direction can be extremely fragile in the sense that it is typically destroyed by noise of arbitrarily small amplitude. A universal scaling law quantifying the increase of the Lyapunov exponent with noise is obtained. A way to observe the scaling law in experiments is suggested.

Toward a neurodynamical understanding of ictogenesis

Although considerable information on cellular and network mechanisms of epilepsy exists, it is still not understood why, how, and when the transition from interictal to ictal state takes place. The authors review their work on nonlinear EEG analysis and provide consistent evidences that dynamical changes in the neural activity allows the characterization of a preictal state several minutes before seizure onset. This new neurodynamical approach of ictogenesis opens new perspectives for studying the basic mechanisms in epilepsy as well as for possible therapeutic interventions.

Evaluation of different measures of functional connectivity using a neural mass model

We use a neural mass model to address some important issues in characterising functional integration among remote cortical areas using magnetoencephalography or electroencephalography (MEG or EEG). In a previous paper [Neuroimage (in press)], we showed how the coupling among cortical areas can modulate the MEG or EEG spectrum and synchronise oscillatory dynamics. In this work, we exploit the model further by evaluating different measures of statistical dependencies (i.e., functional connectivity) among MEG or EEG signals that are mediated by neuronal coupling. We have examined linear and nonlinear methods, including phase synchronisation. Our results show that each method can detect coupling but with different sensitivity profiles that depended on (i) the frequency specificity of the interaction (broad vs. narrow band) and (ii) the nature of the coupling (linear vs. nonlinear). Our analyses suggest that methods based on the concept of generalised synchronisation are the most sensitive when interactions encompass different frequencies (broadband analyses). In the context of narrow-band analyses, mutual information was found to be the most sensitive way to disclose frequency-specific couplings. Measures based on generalised synchronisation and phase synchronisation are the most sensitive to nonlinear coupling. These different sensitivity profiles mean that the choice of coupling measures can have dramatic effects on the cortical networks identified. We illustrate this using a single-subject MEG study of binocular rivalry and highlight the greater recovery of statistical dependencies among cortical areas in the beta band when mutual information is used.

Deterministic stochastic resonance in a Rössler oscillator

We discuss the characteristics of stochastic resonancelike behavior observed in a deterministic system. If a periodically forced Rössler oscillator strays from the phase locking state, it exhibits the intermittent behavior known as phase slips. When the periodic force is modulated by a weak signal, the phase slips synchronize with the weak signal statistically. We numerically demonstrate, in terms of interslip intervals and signal to noise ratio, that the maximum synchronization can be achieved with the optimum intensity of chaotic fluctuations. It is shown that the stochastic resonancelike behavior can be observed regardless of the choice of parameters. The frequency dependence of the signal indicates that there is an optimum frequency for the maximum resonance. The phase slip rate is derived based on the fact that the phase slips are caused by a boundary crisis caused by an unstable-unstable pair bifurcation. The interslip distributions obtained from the derived slip rate and the approximation theory of the time-dependent Poisson process agree with those obtained by numerical simulations. In addition, the maximum enhancement of a weak signal is shown to be achieved by adjusting the chaotic fluctuations even if a signal becomes mixed with noise.

Transitions to synchrony in coupled bursting neurons

Certain cells in the brain, for example, thalamic neurons during sleep, show spike-burst activity. We study such spike-burst neural activity and the transitions to a synchronized state using a model of coupled bursting neurons. In an electrically coupled network, we show that the increase of coupling strength increases incoherence first and then induces two different transitions to synchronized states, one associated with bursts and the other with spikes. These sequential transitions to synchronized states are determined by the zero crossings of the maximum transverse Lyapunov exponents. These results suggest that synchronization of spike-burst activity is a multi-time-scale phenomenon and burst synchrony is a precursor to spike synchrony.

Frequency entrainment of nonautonomous chaotic oscillators

We give evidence of frequency entrainment of dominant peaks in the chaotic spectra of two coupled chaotic nonautonomous oscillators. At variance with the autonomous case, the phenomenon is here characterized by the vanishing of a previously positive Lyapunov exponent in the spectrum, which takes place for a broad range of the coupling strength parameter. Such a state is studied also for the case of chaotic oscillators with ill-defined phases due to the absence of a unique center of rotation. Different phase synchronization indicators are used to circumvent this difficulty.

Data-driven optimal filtering for phase and frequency of noisy oscillations: Application to vortex flow metering

A method for measuring the phase of oscillations from noisy time series is proposed. To obtain the phase, the signal is filtered in such a way that the filter output has minimal relative variation in the amplitude over all filters with complex-valued impulse response. The argument of the filter output yields the phase. Implementation of the algorithm and interpretation of the result are discussed. We argue that the phase obtained by the proposed method has a low susceptibility to measurement noise and a low rate of artificial phase slips. The method is applied for the detection and classification of mode locking in vortex flow meters. A measure for the strength of mode locking is proposed.

Bivariate surrogate techniques: necessity, strengths, and caveats

The concept of surrogates allows testing results from time series analysis against specified null hypotheses. In application to bivariate model dynamics we here compare different types of surrogates, each designed to test against a different null hypothesis, e.g., an underlying bivariate linear stochastic process. Two measures that aim at a characterization of interdependence between nonlinear deterministic dynamics were used as discriminating statistics. We analyze eight different stochastic and deterministic models not only to demonstrate the power of the surrogates, but also to reveal some pitfalls and limitations.

Synchronization of fractional order chaotic systems

The chaotic dynamics of fractional order systems began to attract much attention in recent years. In this Brief Report, we study the master-slave synchronization of fractional order chaotic systems. It is shown that fractional order chaotic systems can also be synchronized.

A disturbance of nonlinear interdependence in scalp EEG of subjects with first episode schizophrenia

It has been proposed that schizophrenia arises through a disturbance of coupling between large-scale cortical systems. This "disconnection hypothesis" is tested by applying a measure of dynamical interdependence to scalp EEG data. EEG data were collected from 40 subjects with a first episode of schizophrenia and 40 matched healthy controls. An algorithm for the detection of dynamical interdependence was applied to six pairs of bipolar electrodes in each subject. The topographic organization of the interdependence was calculated and served as the principle measure of cortical integration. The rate of occurrence of dynamical interdependence did not statistically differ between subject groups at any of the sites. However, the topography across the scalp was significantly different between the two groups. Specifically, nonlinear interdependence tended to occur in larger concurrent "clusters" across the scalp in schizophrenia than in the healthy subjects. This disturbance was reflected most strongly in left intrahemispheric coupling and did not differ significantly according to symptomatology. Medication dose and subject arousal were not observed to be confounding factors. The study of dynamical interdependence in scalp EEG data does not support a straightforward interpretation of the disconnection hypothesis-that there is a decrease in the strength of functional coupling between adjacent cortical regions. Rather, it suggests a dysregulation in the organization of dynamical interactions across supraregional brain systems.