Inferring Functional Neural Connectivity with Phase Synchronization Analysis: A Review of Methodology
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paper_title: Phase Synchronization of Chaotic Oscillators

paper_content:

We present the new effect of phase synchronization of weakly coupled self-sustained chaotic oscillators. To characterize this phenomenon, we use the analytic signal approach based on the Hilbert transform and partial Poincar\'e maps. For coupled R\"ossler attractors, in the synchronous regime the phases are locked, while the amplitudes vary chaotically and are practically uncorrelated. Coupling a chaotic oscillator with a hyperchaotic one, we observe another new type of synchronization, where the frequencies are entrained, while the phase difference is unbounded. A relation between the phase synchronization and the properties of the Lyapunov spectrum is studied.
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paper_title: Measuring synchronization in coupled model systems: A comparison of different approaches

paper_content:

The investigation of synchronization phenomena on measured experimental data such as biological time series has recently become an increasing focus of interest. Different approaches for measuring synchronization have been proposed that rely on certain characteristic features of the dynamical system under investigation. For experimental data the underlying dynamics are usually not completely known, therefore it is difficult to decide a priori which synchronization measure is most suitable for an analysis. In this study we use three different coupled model systems to create a ‘controlled’ setting for a comparison of six different measures of synchronization. All measures are compared to each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise. Results show that the measure to be applied to a certain task can not be chosen according to a fixed criterion but rather pragmatically as the measure which most reliably yields plausible information in test applications, although certain dynamical features of a system under investigation (e.g., power spectra, dimension) may render certain measures more suitable than others. c 2006 Elsevier B.V. All rights reserved.
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paper_title: The role of phase synchronization in memory processes

paper_content:

In this Review, Fell and Axmacher discuss how phase synchronization of neural oscillations facilitates neural communication and plasticity, and thereby promotes memory processes. They propose that working memory and long-term memory might interact through phase–phase and phase–amplitude synchronization in the hippocampus.
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paper_title: Analysis of auditory evoked potentials by magnitude-squared coherence

paper_content:

ABSTRACTEvoked potentials are usually analyzed in the time domain (voltage versus time). The most familiar frequency-domain measure, the power spectral density function, displays power as a function of frequency but doesn't distinguish signal power from noise power. The coherence function estimates,
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paper_title: Topics in Circular Statistics

paper_content:

Circular Probability Distributions Some Sampling Distributions Estimation of Parameters Tests for Mean Direction and Concentration Tests for Uniformity Nonparametric Testing Procedures Circular Correlation and Regression Predictive Inference for Directional Data Outliers and Related Problems Change-Point Problems Miscellaneous Topics Some Facts on Bessel Functions How to Use the CircStats Package.
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paper_title: Abnormal neural oscillations and synchrony in schizophrenia

paper_content:

Uhlhaas and Singer outline the evidence that schizophrenia is associated with deficits in neural oscillations. They discuss the possible underlying mechanisms and how these impairments might contribute to the pathophysiology of the disorder.
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paper_title: Detecting Phase Synchronization in a Chaotic Laser Array

paper_content:

Detection of phase synchronization of coupled chaotic oscillators is examined experimentally for the case of a linear laser array. Phase variables are computed by applying a Gaussian filter, peaked at a positive frequency, to the signal obtained from the intensity time series of the individual lasers. Relationships between different frequency components of the oscillator dynamics that are not otherwise apparent are unambiguously detected.
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paper_title: Phase Synchronization Analysis of EEG Signals: An Evaluation Based on Surrogate Tests

paper_content:

Phase synchronization (PS) analysis has been demonstrated to be a useful method to infer functional connectivity with multichannel neural signals, e.g., electroencephalography (EEG). Methodological problems on quantifying functional connectivity with PS analysis have been investigated extensively, but some of them have not been fully solved yet. For example, how long a segment of EEG signal should be used in estimating PS index? Which methods are more suitable to infer the significant level of estimated PS index? To address these questions, this paper performs an intensive computation study on PS analysis based on surrogate tests with 1) artificial surrogate data generated by shuffling the rank order, the phase spectra, or the instantaneous frequency of original EEG signals, and 2) intersubject EEG pairs under the assumption that the EEG signals of different subjects are independent. Results show that 1) the phase-shuffled surrogate method is workable for significance test of estimated PS index and yields results similar to those by intersubject EEG surrogate test; 2) generally, a duration of EEG waves covering about 3 16 cycles is suitable for PS analysis; and 3) the PS index based on mean phase coherence is more suitable for PS analysis of EEG signals recorded at relatively low sampling rate.
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paper_title: Spatio-temporal dynamics prior to neocortical seizures: amplitude versus phase couplings

paper_content:

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 (/spl Gt/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.
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paper_title: Pathological synchronization in Parkinson's disease: networks, models and treatments

paper_content:

Parkinson's disease is a common and disabling disorder of movement owing to dopaminergic denervation of the striatum. However, it is still unclear how this denervation perverts normal functioning to cause slowing of voluntary movements. Recent work using tissue slice preparations, animal models and in humans with Parkinson's disease has demonstrated abnormally synchronized oscillatory activity at multiple levels of the basal ganglia–cortical loop. This excessive synchronization correlates with motor deficit, and its suppression by dopaminergic therapies, ablative surgery or deep-brain stimulation might provide the basic mechanism whereby diverse therapeutic strategies ameliorate motor impairment in patients with Parkinson's disease. This review is part of the INMED/TINS special issue, Physiogenic and pathogenic oscillations: the beauty and the beast, based on presentations at the annual INMED/TINS symposium (http://inmednet.com/).
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paper_title: Detection of n:m Phase Locking from Noisy Data: Application to Magnetoencephalography

paper_content:

We use the concept of phase synchronization for the analysis of noisy nonstationary bivariate data. Phase synchronization is understood in a statistical sense as an existence of preferred values of the phase difference, and two techniques are proposed for a reliable detection of synchronous epochs. These methods are applied to magnetoencephalograms and records of muscle activity of a Parkinsonian patient. We reveal that the temporal evolution of the peripheral tremor rhythms directly reflects the time course of the synchronization of abnormal activity between cortical motor areas.
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paper_title: A Comparative Study of Synchrony Measures for the Early Diagnosis of Alzheimer’s Disease Based on EEG

paper_content:

Abstract It is well known that EEG signals of Alzheimer's disease (AD) patients are generally less synchronous than in age-matched control subjects. However, this effect is not always easily detectable. This is especially the case for patients in the pre-symptomatic phase, commonly referred to as mild cognitive impairment (MCI), during which neuronal degeneration is occurring prior to the clinical symptoms appearance. In this paper, various synchrony measures are studied in the context of AD diagnosis, including the correlation coefficient, mean-square and phase coherence, Granger causality, phase synchrony indices, information-theoretic divergence measures, state space based measures, and the recently proposed stochastic event synchrony measures. Experiments with EEG data show that many of those measures are strongly correlated (or anti-correlated) with the correlation coefficient, and hence, provide little complementary information about EEG synchrony. Measures that are only weakly correlated with the correlation coefficient include the phase synchrony indices, Granger causality measures, and stochastic event synchrony measures. In addition, those three families of synchrony measures are mutually uncorrelated, and therefore, they each seem to capture a specific kind of interdependence. For the data set at hand, only two synchrony measures are able to convincingly distinguish MCI patients from age-matched control patients, i.e., Granger causality (in particular, full-frequency directed transfer function) and stochastic event synchrony. Those two measures are used as features to distinguish MCI patients from age-matched control subjects, yielding a leave-one-out classification rate of 83%. The classification performance may be further improved by adding complementary features from EEG; this approach may eventually lead to a reliable EEG-based diagnostic tool for MCI and AD.
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paper_title: Spurious detection of phase synchronization in coupled nonlinear oscillators

paper_content:

Coupled nonlinear systems under certain conditions exhibit phase synchronization, which may change for different frequency bands or with the presence of additive system noise. In both cases, Fourier filtering is traditionally used to preprocess data. We investigate to what extent the phase synchronization of two coupled Rössler oscillators depends on (1) the broadness of their power spectrum, (2) the width of the bandpass filter, and (3) the level of added noise. We find that for identical coupling strengths, oscillators with broader power spectra exhibit weaker synchronization. Further, we find that within a broad bandwidth range, bandpass filtering reduces the effect of noise but can lead to a spurious increase in the degree of phase synchronization with narrowing bandwidth, even when the coupling between the two oscillators remains the same.
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paper_title: Electrophysiological Imaging of Brain Activity and Connectivity—Challenges and Opportunities

paper_content:

Unlocking the dynamic inner workings of the brain continues to remain a grand challenge of the 21st century. To this end, functional neuroimaging modalities represent an outstanding approach to better understand the mechanisms of both normal and abnormal brain functions. The ability to image brain function with ever increasing spatial and temporal resolution has made a significant leap over the past several decades. Further delineation of functional networks could lead to improved understanding of brain function in both normal and diseased states. This paper reviews recent advancements and current challenges in dynamic functional neuroimaging techniques, including electrophysiological source imaging, multimodal neuroimaging integrating fMRI with EEG/MEG, and functional connectivity imaging.
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paper_title: Online Control of a Brain-Computer Interface Using Phase Synchronization

paper_content:

Currently, almost all brain-computer interfaces (BCIs) ignore the relationship between phases of electroencephalographic signals detected from different recording sites (i.e., electrodes). The vast majority of BCI systems rely on feature vectors derived from e.g., bandpower or univariate adaptive autoregressive (AAR) parameters. However, ample evidence suggests that additional information is obtained by quantifying the relationship between signals of single electrodes, which might provide innovative features for future BCI systems. This paper investigates one method to extract the degree of phase synchronization between two electroencephalogram (EEG) signals by calculating the so-called phase locking value (PLV). In our offline study, several PLV-based features were acquired and the optimal feature set was selected for each subject individually by a feature selection algorithm. The online sessions with three trained subjects revealed that all subjects were able to control three mental states (motor imagery of left hand, right hand, and foot, respectively) with single-trial accuracies between 60% and 66.7% (33% would be expected by chance) throughout the whole session
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paper_title: Performance of different synchronization measures in real data: A case study on electroencephalographic signals

paper_content:

We agree with the Comment by Duckrow and Albano [Phys. Rev. E 67, 063901 (2003)] that mutual information, estimated with an optimized algorithm, can be a useful tool for studying synchronization in real data. However, we point out that the improvement they found is mainly due to an interesting but nonstandard embedding technique used, and not so much due to the algorithm used for the estimation of mutual information itself. We also address the issue of stationarity of electroencephalographic (EEG) data.
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paper_title: Broadband Criticality of Human Brain Network Synchronization

paper_content:

Self-organized criticality is an attractive model for human brain dynamics, but there has been little direct evidence for its existence in large-scale systems measured by neuroimaging. In general, critical systems are associated with fractal or power law scaling, long-range correlations in space and time, and rapid reconfiguration in response to external inputs. Here, we consider two measures of phase synchronization: the phase-lock interval, or duration of coupling between a pair of (neurophysiological) processes, and the lability of global synchronization of a (brain functional) network. Using computational simulations of two mechanistically distinct systems displaying complex dynamics, the Ising model and the Kuramoto model, we show that both synchronization metrics have power law probability distributions specifically when these systems are in a critical state. We then demonstrate power law scaling of both pairwise and global synchronization metrics in functional MRI and magnetoencephalographic data recorded from normal volunteers under resting conditions. These results strongly suggest that human brain functional systems exist in an endogenous state of dynamical criticality, characterized by a greater than random probability of both prolonged periods of phase-locking and occurrence of large rapid changes in the state of global synchronization, analogous to the neuronal “avalanches” previously described in cellular systems. Moreover, evidence for critical dynamics was identified consistently in neurophysiological systems operating at frequency intervals ranging from 0.05–0.11 to 62.5–125 Hz, confirming that criticality is a property of human brain functional network organization at all frequency intervals in the brain's physiological bandwidth.
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paper_title: Phase synchronization for the recognition of mental tasks in a brain-computer interface

paper_content:

Brain-computer interfaces (BCIs) may be a future communication channel for motor-disabled people. In surface electroencephalogram (EEG)-based BCIs, the extracted features are often derived from spectral estimates and autoregressive models. We examined the usefulness of synchronization between EEG signals for classifying mental tasks. To this end, we investigated the performance of features derived from the phase locking value (PLV) and from the spectral coherence and compared them to the classification rates resulting from the power densities in /spl alpha/, /spl beta//sub 1/, /spl beta//sub 2/, and 8-30-Hz frequency bands. Five recordings of 60 min, acquired from three subjects while performing three different mental tasks, were analyzed offline. No artifacts were removed or rejected. We noticed significant differences between PLV and mean spectral coherence. For sole use of synchronization measures, classification accuracies up to 62% were achieved. In general, the best result was obtained combining phase synchronization measures with /spl alpha/ power spectral density estimates. The results demonstrate that phase synchronization provides relevant information for the classification of spontaneous EEG during mental tasks.
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paper_title: A measure for brain complexity: relating functional segregation and integration in the nervous system

paper_content:

In brains of higher vertebrates, the functional segregation of local areas that differ in their anatomy and physiology contrasts sharply with their global integration during perception and behavior. In this paper, we introduce a measure, called neural complexity (CN), that captures the interplay between these two fundamental aspects of brain organization. We express functional segregation within a neural system in terms of the relative statistical independence of small subsets of the system and functional integration in terms of significant deviations from independence of large subsets. CN is then obtained from estimates of the average deviation from statistical independence for subsets of increasing size. CN is shown to be high when functional segregation coexists with integration and to be low when the components of a system are either completely independent (segregated) or completely dependent (integrated). We apply this complexity measure in computer simulations of cortical areas to examine how some basic principles of neuroanatomical organization constrain brain dynamics. We show that the connectivity patterns of the cerebral cortex, such as a high density of connections, strong local connectivity organizing cells into neuronal groups, patchiness in the connectivity among neuronal groups, and prevalent reciprocal connections, are associated with high values of CN. The approach outlined here may prove useful in analyzing complexity in other biological domains such as gene regulation and embryogenesis.
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paper_title: The brainweb: phase synchronization and large-scale integration.

paper_content:

The emergence of a unified cognitive moment relies on the coordination of scattered mosaics of functionally specialized brain regions. Here we review the mechanisms of large-scale integration that counterbalance the distributed anatomical and functional organization of brain activity to enable the emergence of coherent behaviour and cognition. Although the mechanisms involved in large-scale integration are still largely unknown, we argue that the most plausible candidate is the formation of dynamic links mediated by synchrony over multiple frequency bands.
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paper_title: Testing for phase synchronization

paper_content:

Phase synchronization analysis is frequently applied to data originating for example from Physics and Life Sciences. Statistical properties for quantities measuring phase synchronization have not been revealed. We derive an analytic significance level for a frequently used phase synchronization measure. Its performance is demonstrated for a system of coupled oscillators.
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paper_title: A unifying definition of synchronization for dynamical systems

paper_content:

We propose a unifying definition for synchronization between stationary finite dimensional deterministic dynamical systems. By example, we show that the synchronization phenomena discussed in the dynamical systems literature fits within the framework of this definition, and discuss problems with previous definitions of synchronization. We conclude with a discussion of possible extensions of the definition to infinite dimensional systems described by partial differential equations and/or systems where noise is present. (c) 2000 American Institute of Physics.
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paper_title: An approach to chaotic synchronization

paper_content:

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 Rossler systems and two coupled Chua’s circuits.
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paper_title: Unified framework for detecting phase synchronization in coupled time series

paper_content:

Phase synchronization (PS) has drawn increasing attention in recent years for its extensive applications in analyzing time series observed from coupled systems. In this paper, we examine the detection of PS in bivariate time series from the viewpoints of signal processing and circular statistics. Several definitions of instantaneous phase (IP) are first revisited and further unified into a framework, which defines IP as the argument of the signal with a specific bandpass filter applied. With this framework, the constraints for IP definition are discussed and the effect of noise in IP estimation is studied. The estimate error of IP, which is due to noise, is shown to obey a scale mixture of normal (SMN) distributions. Further, under the assumption that the SMN of IP error can be approximated by a particular normal distribution, the estimate of mean phase coherence of bivariate time series is shown to be degraded by a factor, which is determined by only the level of in-band noise. Finally, simulations are provided to support the theoretical results.
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paper_title: Enhancement of phase clustering in the EEG/MEG gamma frequency band anticipates transitions to paroxysmal epileptiform activity in epileptic patients with known visual sensitivity

paper_content:

A new analytical method for quantifying brain activity from magnetoelectroencephalogram (MEG) and electroencephalogram (EEG) recordings during periodic light stimulation is proposed. It consists in estimating the phase clustering of harmonically related frequency components of a subject's MEG/EEG responses evoked by the light stimulation. The method was developed to test the hypothesis that changes in the dynamics of brain systems in the course of intermittent photic stimulation (IPS) may precede the transition to seizure activity in photosensitive patients. We assumed that such changes would be reflected in the phase of harmonic components of the evoked responses. Thus, we determined the phase clustering for different harmonic components of these MEG/EEG signals. We found that the patients who develop epileptiform discharges during IPS present an enhancement of the phase clustering index at the gamma frequency band, compared with that at the driving frequency. We introduce a quantity-relative phase clustering index (rPCI)-by means of which this enhancement can be quantified. We argue that this quantity reflects the degree of excitability of the underlying dynamical system and it can indicate presence of nonlinear dynamics. rPCI can be applied to detect transitions to epileptic seizure activity in patients with known sensitivity to IPS.
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paper_title: Estimation of Effective and Functional Cortical Connectivity From Neuroelectric and Hemodynamic Recordings

paper_content:

In this paper, different linear and nonlinear methodologies for the estimation of cortical connectivity from neuroelectric and hemodynamic measurements are reviewed and applied on common data set in order to highlight similarities and differences in the results. Different effective and functional connectivity methods were applied to motor and cognitive data sets, including structural equation modeling (SEM), directed transfer function (DTF), partial directed coherence (PDC), and direct directed transfer function (dDTF). Comparisons were made between the results in order to understand if, for a same dataset, effective and functional connectivity estimators can return the same cortical connectivity patterns. An application of a nonlinear method [phase synchronization index (PSI)] to similar executed and imagined movements was also reviewed. Connectivity patterns estimated with the use of the neuroelectric information and of the information from the multimodal integration of neuroelectric and hemodynamic data were also compared. Results suggests that the estimation of the cortical connectivity patterns performed with the linear methods (SEM, DTF, PDC, dDTF) or with the nonlinear method (PSI) on movement related potentials returned similar cortical networks. Differences in cortical connectivity were noted between the patterns estimated with the use of multimodal integration and those estimated by using only the neuroelectric data.
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paper_title: Synchronization: A Universal Concept in Nonlinear Sciences

paper_content:

Preface 1. Introduction Part I. Synchronization Without Formulae: 2. Basic notions: the self-sustained oscillator and its phase 3. Synchronization of a periodic oscillator by external force 4. Synchronization of two and many oscillators 5. Synchronization of chaotic systems 6. Detecting synchronization in experiments Part II. Phase Locking and Frequency Entrainment: 7. Synchronization of periodic oscillators by periodic external action 8. Mutual synchronization of two interacting periodic oscillators 9. Synchronization in the presence of noise 10. Phase synchronization of chaotic systems 11. Synchronization in oscillatory media 12. Populations of globally coupled oscillators Part III. Synchronization of Chaotic Systems: 13. Complete synchronization I: basic concepts 14. Complete synchronization II: generalizations and complex systems 15. Synchronization of complex dynamics by external forces Appendix 1. Discovery of synchronization by Christiaan Huygens Appendix 2. Instantaneous phase and frequency of a signal References Index.
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paper_title: Short-window spectral analysis of cortical event-related potentials by adaptive multivariate autoregressive modeling: data preprocessing, model validation, and variability assessment

paper_content:

In this article we consider the application of parametric spectral analysis to multichannel event-related potentials (ERPs) during cognitive experiments. We show that with proper data preprocessing, Adaptive MultiVariate AutoRegressive (AMVAR) modeling is an effective technique for dealing with nonstationary ERP time series. We propose a bootstrap procedure to assess the variability in the estimated spectral quantities. Finally, we apply AMVAR spectral analysis to a visuomotor integration task, revealing rapidly changing cortical dynamics during different stages of task processing.
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paper_title: Neural Synchrony in Brain Disorders: Relevance for Cognitive Dysfunctions and Pathophysiology

paper_content:

Following the discovery of context-dependent synchronization of oscillatory neuronal responses in the visual system, novel methods of time series analysis have been developed for the examination of task- and performance-related oscillatory activity and its synchronization. Studies employing these advanced techniques revealed that synchronization of oscillatory responses in the β- and γ-band is involved in a variety of cognitive functions, such as perceptual grouping, attention-dependent stimulus selection, routing of signals across distributed cortical networks, sensory-motor integration, working memory, and perceptual awareness. Here, we review evidence that certain brain disorders, such as schizophrenia, epilepsy, autism, Alzheimer's disease, and Parkinson's are associated with abnormal neural synchronization. The data suggest close correlations between abnormalities in neuronal synchronization and cognitive dysfunctions, emphasizing the importance of temporal coordination. Thus, focused search for abnormalities in temporal patterning may be of considerable clinical relevance.
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paper_title: High Gamma Power Is Phase-Locked to Theta Oscillations in Human Neocortex

paper_content:

We observed robust coupling between the high- and low-frequency bands of ongoing electrical activity in the human brain. In particular, the phase of the low-frequency theta (4 to 8 hertz) rhythm modulates power in the high gamma (80 to 150 hertz) band of the electrocorticogram, with stronger modulation occurring at higher theta amplitudes. Furthermore, different behavioral tasks evoke distinct patterns of theta/high gamma coupling across the cortex. The results indicate that transient coupling between low- and high-frequency brain rhythms coordinates activity in distributed cortical areas, providing a mechanism for effective communication during cognitive processing in humans.
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paper_title: Synchronization of non-phase-coherent chaotic electrochemical oscillations

paper_content:

Experiments on phase and generalized synchronization of two coupled, nonidentical chaotic electrochemical oscillations are presented. We adapt measures of characterizing synchronization of a non-phase-coherent chaotic behavior and compare its properties and physicochemical mechanism to those of a phase-coherent behavior. Phase synchronization sets in along with the onset of generalized synchronization for the non-phase-coherent oscillations in contrast to phase-coherent oscillations in which the phase synchronization usually occurs at a weaker coupling strength.
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paper_title: Impaired neuronal synchrony after focal ischemic stroke in elderly patients

paper_content:

OBJECTIVE ::: This study investigated whether cortical synchrony derived from electroencephalography (EEG) in elderly patients is impaired and if the impairment might reflect long-term functional recovery after stroke. ::: ::: ::: METHODS ::: The scalp EEG signals of stroke patients (N=42) were collected within seven days after the onset of stroke and analyzed with phase synchronization (PS). Neurodeficit outcome was scored twice according to the National Institute of Health Stroke Scale (NIHSS): (1) at the same day of EEG recording and (2) two months after stroke. The correlation between cortical synchrony and NIHSS was analyzed. ::: ::: ::: RESULTS ::: The level of synchronization between lesion and intact areas in the ipsilateral hemisphere was reduced significantly after stroke, while the synchronization among intact areas increased to 114% among the control subjects. Furthermore, the patients with lower inter-hemispheric synchrony after stroke were observed to have a higher NIHSS two months after stroke. ::: ::: ::: CONCLUSIONS ::: Results indicated that the infarct broke down the cortical synchrony networks and affected large-scale neural communication. Inter-hemispheric synchrony was relevant to long-term functional recovery after stroke. ::: ::: ::: SIGNIFICANCE ::: The prognostic value of PS for functional recovery after stroke might be helpful in understanding the alteration of cortical networks after ischemic injury.
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paper_title: Phase Synchronization of Chaotic Oscillators

paper_content:

We present the new effect of phase synchronization of weakly coupled self-sustained chaotic oscillators. To characterize this phenomenon, we use the analytic signal approach based on the Hilbert transform and partial Poincar\'e maps. For coupled R\"ossler attractors, in the synchronous regime the phases are locked, while the amplitudes vary chaotically and are practically uncorrelated. Coupling a chaotic oscillator with a hyperchaotic one, we observe another new type of synchronization, where the frequencies are entrained, while the phase difference is unbounded. A relation between the phase synchronization and the properties of the Lyapunov spectrum is studied.
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paper_title: A product theorem for Hilbert transforms

paper_content:

Abstract : A mathematical theorem is presented for finding the Hilbert transform of a product of functions in a simplified fashion, under certain conditions. Engineering applications are discussed. (Author)
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paper_title: Measuring synchronization in coupled model systems: A comparison of different approaches

paper_content:

The investigation of synchronization phenomena on measured experimental data such as biological time series has recently become an increasing focus of interest. Different approaches for measuring synchronization have been proposed that rely on certain characteristic features of the dynamical system under investigation. For experimental data the underlying dynamics are usually not completely known, therefore it is difficult to decide a priori which synchronization measure is most suitable for an analysis. In this study we use three different coupled model systems to create a ‘controlled’ setting for a comparison of six different measures of synchronization. All measures are compared to each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise. Results show that the measure to be applied to a certain task can not be chosen according to a fixed criterion but rather pragmatically as the measure which most reliably yields plausible information in test applications, although certain dynamical features of a system under investigation (e.g., power spectra, dimension) may render certain measures more suitable than others. c 2006 Elsevier B.V. All rights reserved.
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paper_title: Detecting Phase Synchronization in a Chaotic Laser Array

paper_content:

Detection of phase synchronization of coupled chaotic oscillators is examined experimentally for the case of a linear laser array. Phase variables are computed by applying a Gaussian filter, peaked at a positive frequency, to the signal obtained from the intensity time series of the individual lasers. Relationships between different frequency components of the oscillator dynamics that are not otherwise apparent are unambiguously detected.
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paper_title: Definitions of Instantaneous Frequency under physical constraints

paper_content:

Abstract Ville's definition of Instantaneous Frequency (IF), even though widely used and accepted, fails in most cases of practical interest. This failure has been often reported, namely in the multicomponent case. In this paper, we will analyze the advantages and shortcomings of that definition, and determine when and why it fails, integrating all previously reported cases of failure in a simple unified theory. We will also be able to extrapolate and predict the behavior of the traditional definition for any type of multicomponent signal. Alternative definitions of IF are discussed.
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paper_title: Three Types of Transitions to Phase Synchronization in Coupled Chaotic Oscillators

paper_content:

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.
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paper_title: Performance of different synchronization measures in real data: A case study on electroencephalographic signals

paper_content:

We agree with the Comment by Duckrow and Albano [Phys. Rev. E 67, 063901 (2003)] that mutual information, estimated with an optimized algorithm, can be a useful tool for studying synchronization in real data. However, we point out that the improvement they found is mainly due to an interesting but nonstandard embedding technique used, and not so much due to the algorithm used for the estimation of mutual information itself. We also address the issue of stationarity of electroencephalographic (EEG) data.
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paper_title: Spectrographic measurement of instantaneous frequency and the time-dependent weighted average instantaneous frequency

paper_content:

Instantaneous frequency (IF) is an important signal characteristic arising in many fields. It is a concept intimately linked to time–frequency analysis, where it can be obtained from a time–frequency distribution (TFD) as the first conditional moment in frequency, suggesting that the IF is the average frequency at each time. However, this interpretation is questionable, since it is well known that the IF often ranges beyond the spectral support of the signal. In addition, to obtain the IF from a spectrogram (which is one possible TFD), a very wideband—and thus severely spectrally distorted—spectrogram must be used. More reasonable bandwidths are investigated, and, in particular, give the conditions by which the first conditional moment of a spectrogram can indeed be interpreted as the average frequency at each time. Under these conditions, namely when the spectrogram bandwidth is less than the frequency separation of the signal components in the time–frequency plane, the spectrogram yields not the usual I...
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paper_title: An approach to chaotic synchronization

paper_content:

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 Rossler systems and two coupled Chua’s circuits.
---
paper_title: Unified framework for detecting phase synchronization in coupled time series

paper_content:

Phase synchronization (PS) has drawn increasing attention in recent years for its extensive applications in analyzing time series observed from coupled systems. In this paper, we examine the detection of PS in bivariate time series from the viewpoints of signal processing and circular statistics. Several definitions of instantaneous phase (IP) are first revisited and further unified into a framework, which defines IP as the argument of the signal with a specific bandpass filter applied. With this framework, the constraints for IP definition are discussed and the effect of noise in IP estimation is studied. The estimate error of IP, which is due to noise, is shown to obey a scale mixture of normal (SMN) distributions. Further, under the assumption that the SMN of IP error can be approximated by a particular normal distribution, the estimate of mean phase coherence of bivariate time series is shown to be degraded by a factor, which is determined by only the level of in-band noise. Finally, simulations are provided to support the theoretical results.
---
paper_title: Synchronization: A Universal Concept in Nonlinear Sciences

paper_content:

Preface 1. Introduction Part I. Synchronization Without Formulae: 2. Basic notions: the self-sustained oscillator and its phase 3. Synchronization of a periodic oscillator by external force 4. Synchronization of two and many oscillators 5. Synchronization of chaotic systems 6. Detecting synchronization in experiments Part II. Phase Locking and Frequency Entrainment: 7. Synchronization of periodic oscillators by periodic external action 8. Mutual synchronization of two interacting periodic oscillators 9. Synchronization in the presence of noise 10. Phase synchronization of chaotic systems 11. Synchronization in oscillatory media 12. Populations of globally coupled oscillators Part III. Synchronization of Chaotic Systems: 13. Complete synchronization I: basic concepts 14. Complete synchronization II: generalizations and complex systems 15. Synchronization of complex dynamics by external forces Appendix 1. Discovery of synchronization by Christiaan Huygens Appendix 2. Instantaneous phase and frequency of a signal References Index.
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paper_title: Synchronization of non-phase-coherent chaotic electrochemical oscillations

paper_content:

Experiments on phase and generalized synchronization of two coupled, nonidentical chaotic electrochemical oscillations are presented. We adapt measures of characterizing synchronization of a non-phase-coherent chaotic behavior and compare its properties and physicochemical mechanism to those of a phase-coherent behavior. Phase synchronization sets in along with the onset of generalized synchronization for the non-phase-coherent oscillations in contrast to phase-coherent oscillations in which the phase synchronization usually occurs at a weaker coupling strength.
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paper_title: Detecting Phase Synchronization in a Chaotic Laser Array

paper_content:

Detection of phase synchronization of coupled chaotic oscillators is examined experimentally for the case of a linear laser array. Phase variables are computed by applying a Gaussian filter, peaked at a positive frequency, to the signal obtained from the intensity time series of the individual lasers. Relationships between different frequency components of the oscillator dynamics that are not otherwise apparent are unambiguously detected.
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paper_title: An approach to chaotic synchronization

paper_content:

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 Rossler systems and two coupled Chua’s circuits.
---
paper_title: A product theorem for Hilbert transforms

paper_content:

Abstract : A mathematical theorem is presented for finding the Hilbert transform of a product of functions in a simplified fashion, under certain conditions. Engineering applications are discussed. (Author)
---
paper_title: Computer measuring of frequency stability and the analytic signal

paper_content:

A computer-oriented method of frequency measurement based on the analytic signal is presented here. Its advantage over the common method for stability measurements as well as the relationship between the two methods is discussed. >
---
paper_title: Unified framework for detecting phase synchronization in coupled time series

paper_content:

Phase synchronization (PS) has drawn increasing attention in recent years for its extensive applications in analyzing time series observed from coupled systems. In this paper, we examine the detection of PS in bivariate time series from the viewpoints of signal processing and circular statistics. Several definitions of instantaneous phase (IP) are first revisited and further unified into a framework, which defines IP as the argument of the signal with a specific bandpass filter applied. With this framework, the constraints for IP definition are discussed and the effect of noise in IP estimation is studied. The estimate error of IP, which is due to noise, is shown to obey a scale mixture of normal (SMN) distributions. Further, under the assumption that the SMN of IP error can be approximated by a particular normal distribution, the estimate of mean phase coherence of bivariate time series is shown to be degraded by a factor, which is determined by only the level of in-band noise. Finally, simulations are provided to support the theoretical results.
---
paper_title: Estimation of Effective and Functional Cortical Connectivity From Neuroelectric and Hemodynamic Recordings

paper_content:

In this paper, different linear and nonlinear methodologies for the estimation of cortical connectivity from neuroelectric and hemodynamic measurements are reviewed and applied on common data set in order to highlight similarities and differences in the results. Different effective and functional connectivity methods were applied to motor and cognitive data sets, including structural equation modeling (SEM), directed transfer function (DTF), partial directed coherence (PDC), and direct directed transfer function (dDTF). Comparisons were made between the results in order to understand if, for a same dataset, effective and functional connectivity estimators can return the same cortical connectivity patterns. An application of a nonlinear method [phase synchronization index (PSI)] to similar executed and imagined movements was also reviewed. Connectivity patterns estimated with the use of the neuroelectric information and of the information from the multimodal integration of neuroelectric and hemodynamic data were also compared. Results suggests that the estimation of the cortical connectivity patterns performed with the linear methods (SEM, DTF, PDC, dDTF) or with the nonlinear method (PSI) on movement related potentials returned similar cortical networks. Differences in cortical connectivity were noted between the patterns estimated with the use of multimodal integration and those estimated by using only the neuroelectric data.
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paper_title: A Comparative Study of Synchrony Measures for the Early Diagnosis of Alzheimer’s Disease Based on EEG

paper_content:

Abstract It is well known that EEG signals of Alzheimer's disease (AD) patients are generally less synchronous than in age-matched control subjects. However, this effect is not always easily detectable. This is especially the case for patients in the pre-symptomatic phase, commonly referred to as mild cognitive impairment (MCI), during which neuronal degeneration is occurring prior to the clinical symptoms appearance. In this paper, various synchrony measures are studied in the context of AD diagnosis, including the correlation coefficient, mean-square and phase coherence, Granger causality, phase synchrony indices, information-theoretic divergence measures, state space based measures, and the recently proposed stochastic event synchrony measures. Experiments with EEG data show that many of those measures are strongly correlated (or anti-correlated) with the correlation coefficient, and hence, provide little complementary information about EEG synchrony. Measures that are only weakly correlated with the correlation coefficient include the phase synchrony indices, Granger causality measures, and stochastic event synchrony measures. In addition, those three families of synchrony measures are mutually uncorrelated, and therefore, they each seem to capture a specific kind of interdependence. For the data set at hand, only two synchrony measures are able to convincingly distinguish MCI patients from age-matched control patients, i.e., Granger causality (in particular, full-frequency directed transfer function) and stochastic event synchrony. Those two measures are used as features to distinguish MCI patients from age-matched control subjects, yielding a leave-one-out classification rate of 83%. The classification performance may be further improved by adding complementary features from EEG; this approach may eventually lead to a reliable EEG-based diagnostic tool for MCI and AD.
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paper_title: A Time-Frequency-Based Approach to Phase and Phase Synchrony Estimation

paper_content:

Time-varying phase synchrony is an important bivariate measure that quantifies the dynamics between nonstationary signals and has been widely used in many applications including chaotic oscillators in physics and multichannel electroencephalography recordings in neuroscience. Current state-of-the-art in time-varying phase estimation uses either the Hilbert transform or the complex wavelet transform of the signals. Both of these methods have some major drawbacks such as the assumption that the signals are narrowband for the Hilbert transform and the nonuniform time-frequency resolution inherent to the wavelet analysis. In this paper, a new phase estimation method based on the Rihaczek distribution and Reduced Interference Rihaczek distribution belonging to Cohen's class is proposed. These distributions offer phase estimates with uniformly high time-frequency resolution which can be used for defining time and frequency dependent phase synchrony. Properties of the phase estimator and the corresponding phase synchrony measure are evaluated both analytically and through simulations showing the effectiveness of the new measures compared to existing methods.
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paper_title: Adaptive Phase Extraction: Incorporating the Gabor Transform in the Matching Pursuit Algorithm

paper_content:

Short-time Fourier transform (STFT), Gabor transform (GT), wavelet transform (WT), and the Wigner-Ville distribution (WVD) are just some examples of time-frequency analysis methods which are frequently applied in biomedical signal analysis. However, all of these methods have their individual drawbacks. The STFT, GT, and WT have a time-frequency resolution that is determined by algorithm parameters and the WVD is contaminated by cross terms. In 1993, Mallat and Zhang introduced the matching pursuit (MP) algorithm that decomposes a signal into a sum of atoms and uses a cross-term free pseudo-WVD to generate a data-adaptive power distribution in the time-frequency space. Thus, it solved some of the problems of the GT and WT but lacks phase information that is crucial e.g., for synchronization analysis. We introduce a new time-frequency analysis method that combines the MP with a pseudo-GT. Therefore, the signal is decomposed into a set of Gabor atoms. Afterward, each atom is analyzed with a Gabor analysis, where the time-domain Gaussian window of the analysis matches that of the specific atom envelope. A superposition of the single time-frequency planes gives the final result. This is the first time that a complete analysis of the complex time-frequency plane can be performed in a fully data-adaptive and frequency-selective manner. We demonstrate the capabilities of our approach on a simulation and on real-life magnetoencephalogram data.
---
paper_title: A product theorem for Hilbert transforms

paper_content:

Abstract : A mathematical theorem is presented for finding the Hilbert transform of a product of functions in a simplified fashion, under certain conditions. Engineering applications are discussed. (Author)
---
paper_title: On the analytic signal, the Teager-Kaiser energy algorithm, and other methods for defining amplitude and frequency

paper_content:

This paper compares the Teager-Kaiser algorithm (TKA) and other similar local methods to the analytic signal (AS) procedure. The general concepts of the instantaneous amplitude and frequency are discussed. It is shown that only AS meets certain physical conditions for the amplitude, phase, and frequency (APF). The advantage of accuracy and simplicity of the AS is also demonstrated.
---
paper_title: Unified framework for detecting phase synchronization in coupled time series

paper_content:

Phase synchronization (PS) has drawn increasing attention in recent years for its extensive applications in analyzing time series observed from coupled systems. In this paper, we examine the detection of PS in bivariate time series from the viewpoints of signal processing and circular statistics. Several definitions of instantaneous phase (IP) are first revisited and further unified into a framework, which defines IP as the argument of the signal with a specific bandpass filter applied. With this framework, the constraints for IP definition are discussed and the effect of noise in IP estimation is studied. The estimate error of IP, which is due to noise, is shown to obey a scale mixture of normal (SMN) distributions. Further, under the assumption that the SMN of IP error can be approximated by a particular normal distribution, the estimate of mean phase coherence of bivariate time series is shown to be degraded by a factor, which is determined by only the level of in-band noise. Finally, simulations are provided to support the theoretical results.
---
paper_title: Topics in Circular Statistics

paper_content:

Circular Probability Distributions Some Sampling Distributions Estimation of Parameters Tests for Mean Direction and Concentration Tests for Uniformity Nonparametric Testing Procedures Circular Correlation and Regression Predictive Inference for Directional Data Outliers and Related Problems Change-Point Problems Miscellaneous Topics Some Facts on Bessel Functions How to Use the CircStats Package.
---
paper_title: Detecting Phase Synchronization in Noisy Systems

paper_content:

Abstract A quantitative method for automatic detection of phase synchronization in noisy experimental bivariate time series is proposed, based on the fact that instantaneous phases of phase-synchronized (sub) systems are mutually dependent in a specific way irrespective of a relation between the original time series. The level of dependence between the instantaneous phases is quantified by a statistical dependence parameter, which also reflects the strength of the systems' phase synchronization. Ranges of the parameter values, for which the detection of the phase synchronization can be considered reliable, are estimated by using the technique of surrogate data. Possible applications of the proposed method are demonstrated by using both numerically generated and real experimental data, namely solutions of two coupled Rossler systems, mammalian cardio-respiratory data, and long-term recordings of surface atmospheric temperature and sunspot numbers.
---
paper_title: Detection of n:m Phase Locking from Noisy Data: Application to Magnetoencephalography

paper_content:

We use the concept of phase synchronization for the analysis of noisy nonstationary bivariate data. Phase synchronization is understood in a statistical sense as an existence of preferred values of the phase difference, and two techniques are proposed for a reliable detection of synchronous epochs. These methods are applied to magnetoencephalograms and records of muscle activity of a Parkinsonian patient. We reveal that the temporal evolution of the peripheral tremor rhythms directly reflects the time course of the synchronization of abnormal activity between cortical motor areas.
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paper_title: Spurious detection of phase synchronization in coupled nonlinear oscillators

paper_content:

Coupled nonlinear systems under certain conditions exhibit phase synchronization, which may change for different frequency bands or with the presence of additive system noise. In both cases, Fourier filtering is traditionally used to preprocess data. We investigate to what extent the phase synchronization of two coupled Rössler oscillators depends on (1) the broadness of their power spectrum, (2) the width of the bandpass filter, and (3) the level of added noise. We find that for identical coupling strengths, oscillators with broader power spectra exhibit weaker synchronization. Further, we find that within a broad bandwidth range, bandpass filtering reduces the effect of noise but can lead to a spurious increase in the degree of phase synchronization with narrowing bandwidth, even when the coupling between the two oscillators remains the same.
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paper_title: On the Stability of the n:m Phase Synchronization Index

paper_content:

Synchronization analysis of multitrial EEG or (magneto encephalogram) MEG signals is an excellent approach to detect functional connectivity between different neuronal oscillators. In our current research, the n:m phase synchronization index (n:m PSI ) is of special interest. We prove the existence of stable and unstable synchronies dependent upon the analysis frequencies and show that they lie closely together in the frequency domain. Thus, a plot of the time-frequency plane of the n:m PSI automatically violates the sampling theorem and accordingly, the method cannot be considered as a black box. A frequency-tiling approach is presented that can detect robust synchronies while ignoring the unstable ones. The improved synchrony detection is evaluated in numerical experiments on using both simulated and real-life data. It can be demonstrated that the transient synchronization events between MEG oscillations in distant frequency ranges can be detected and that compactly textured EEG synchronization patterns can be reliably characterized.
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paper_title: Testing for phase synchronization

paper_content:

Phase synchronization analysis is frequently applied to data originating for example from Physics and Life Sciences. Statistical properties for quantities measuring phase synchronization have not been revealed. We derive an analytic significance level for a frequently used phase synchronization measure. Its performance is demonstrated for a system of coupled oscillators.
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paper_title: A new approach to uncover dynamic phase coordination and synchronization

paper_content:

Phase attractive maps are an essential mechanism of multi-stable systems such as found in coupled neuronal oscillators, An essential feature of this type of dynamic nonlinear coordination is dynamic phase synchronization. The identification of dynamic phase synchronizations is complicated due to changing frequency ratios of synchronized intervals, other nonstationarities, and noise. In order to overcome these problems the momentary phase relations and their statistics were analyzed by several authors. In this way phase synchronizations also in chaotic and noisy oscillating systems could be uncovered. We propose a novel method which avoids one essential limitation of these approaches, namely the necessity of presetting particular frequency ratios of interest. The proposed novel method was validated by its application to a simulated driven neuronal generator during the transition period between different synchronization modes and to dynamically coupled components of sympathetic nerve discharges.
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paper_title: Reorganization of Brain Networks in Aging and Age-related Diseases.

paper_content:

Aging is associated with reorganization of brain in both structure and function. In recent years, graph theoretical analysis of brain organization has drawn increasing attention, and reorganization of brain in aging has been investigated in terms of connectivity and networks in topology such as modular organization, global and local efficiency, and small-worldness. Beyond studying on abnormity in local brain regions, connectivity quantifies alternations of correlation between two regions that may be spatially far separated, and graph theoretical analysis of brain network examines the complex interactions among multiple regions. This article reviewed complex brain networks of human in normal aging or with age-related diseases such as stroke and Alzheimer's disease after a technical introduction of brain networks and graph theoretical analysis. We further discussed the relationship between the functional and the structural brain networks of subjects in aging or with age-related diseases. Finally, we proposed several interesting topics for future research in this field.
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paper_title: Brain Graphs: Graphical Models of the Human Brain Connectome

paper_content:

Brain graphs provide a relatively simple and increasingly popular way of modeling the human brain connectome, using graph theory to abstractly define a nervous system as a set of nodes (denoting anatomical regions or recording electrodes) and interconnecting edges (denoting structural or functional connections). Topological and geometrical properties of these graphs can be measured and compared to random graphs and to graphs derived from other neuroscience data or other (nonneural) complex systems. Both structural and functional human brain graphs have consistently demonstrated key topological properties such as small-worldness, modularity, and heterogeneous degree distributions. Brain graphs are also physically embedded so as to nearly minimize wiring cost, a key geometric property. Here we offer a conceptual review and methodological guide to graphical analysis of human neuroimaging data, with an emphasis on some of the key assumptions, issues, and trade-offs facing the investigator.
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paper_title: Phase Synchronization Analysis of EEG Signals: An Evaluation Based on Surrogate Tests

paper_content:

Phase synchronization (PS) analysis has been demonstrated to be a useful method to infer functional connectivity with multichannel neural signals, e.g., electroencephalography (EEG). Methodological problems on quantifying functional connectivity with PS analysis have been investigated extensively, but some of them have not been fully solved yet. For example, how long a segment of EEG signal should be used in estimating PS index? Which methods are more suitable to infer the significant level of estimated PS index? To address these questions, this paper performs an intensive computation study on PS analysis based on surrogate tests with 1) artificial surrogate data generated by shuffling the rank order, the phase spectra, or the instantaneous frequency of original EEG signals, and 2) intersubject EEG pairs under the assumption that the EEG signals of different subjects are independent. Results show that 1) the phase-shuffled surrogate method is workable for significance test of estimated PS index and yields results similar to those by intersubject EEG surrogate test; 2) generally, a duration of EEG waves covering about 3 16 cycles is suitable for PS analysis; and 3) the PS index based on mean phase coherence is more suitable for PS analysis of EEG signals recorded at relatively low sampling rate.
---
paper_title: Detecting Phase Synchronization in Noisy Systems

paper_content:

Abstract A quantitative method for automatic detection of phase synchronization in noisy experimental bivariate time series is proposed, based on the fact that instantaneous phases of phase-synchronized (sub) systems are mutually dependent in a specific way irrespective of a relation between the original time series. The level of dependence between the instantaneous phases is quantified by a statistical dependence parameter, which also reflects the strength of the systems' phase synchronization. Ranges of the parameter values, for which the detection of the phase synchronization can be considered reliable, are estimated by using the technique of surrogate data. Possible applications of the proposed method are demonstrated by using both numerically generated and real experimental data, namely solutions of two coupled Rossler systems, mammalian cardio-respiratory data, and long-term recordings of surface atmospheric temperature and sunspot numbers.
---
paper_title: Testing for phase synchronization

paper_content:

We present different tests for phase synchronization which improve the procedures currently used in the literature. This is accomplished by using a two-sample test setup and by utilizing insights and methods from directional statistics and bootstrap theory. The tests differ in the generality of the situation in which they can be applied as well as in their complexity, including computational cost. A modification of the resampling technique of the bootstrap is introduced, making it possible to fully utilize data from time series.
---
paper_title: Hierarchical Organization of Human Cortical Networks in Health and Schizophrenia

paper_content:

The complex organization of connectivity in the human brain is incompletely understood. Recently, topological measures based on graph theory have provided a new approach to quantify large-scale cortical networks. These methods have been applied to anatomical connectivity data on nonhuman species, and cortical networks have been shown to have small-world topology, associated with high local and global efficiency of information transfer. Anatomical networks derived from cortical thickness measurements have shown the same organizational properties of the healthy human brain, consistent with similar results reported in functional networks derived from resting state functional magnetic resonance imaging (MRI) and magnetoencephalographic data. Here we show, using anatomical networks derived from analysis of inter-regional covariation of gray matter volume in MRI data on 259 healthy volunteers, that classical divisions of cortex (multimodal, unimodal, and transmodal) have some distinct topological attributes. Although all cortical divisions shared nonrandom properties of small-worldness and efficient wiring (short mean Euclidean distance between connected regions), the multimodal network had a hierarchical organization, dominated by frontal hubs with low clustering, whereas the transmodal network was assortative. Moreover, in a sample of 203 people with schizophrenia, multimodal network organization was abnormal, as indicated by reduced hierarchy, the loss of frontal and the emergence of nonfrontal hubs, and increased connection distance. We propose that the topological differences between divisions of normal cortex may represent the outcome of different growth processes for multimodal and transmodal networks and that neurodevelopmental abnormalities in schizophrenia specifically impact multimodal cortical organization.
---
paper_title: Testing for phase synchronization

paper_content:

Phase synchronization analysis is frequently applied to data originating for example from Physics and Life Sciences. Statistical properties for quantities measuring phase synchronization have not been revealed. We derive an analytic significance level for a frequently used phase synchronization measure. Its performance is demonstrated for a system of coupled oscillators.
---
paper_title: Disrupted small-world networks in schizophrenia

paper_content:

The human brain has been described as a large, sparse, complex network characterized by efficient small-world properties, which assure that the brain generates and integrates information with high efficiency. Many previous neuroimaging studies have provided consistent evidence of dysfunctional connectivity among the brain regions in schizophrenia; however, little is known about whether or not this dysfunctional connectivity causes disruption of the topological properties of brain functional networks. To this end, we investigated the topological properties of human brain functional networks derived from resting-state functional magnetic resonance imaging (fMRI). Data was obtained from 31 schizophrenia patients and 31 healthy subjects; then functional connectivity between 90 cortical and sub-cortical regions was estimated by partial correlation analysis and thresholded to construct a set of undirected graphs. Our findings demonstrated that the brain functional networks had efficient small-world properties in the healthy subjects; whereas these properties were disrupted in the patients with schizophrenia. Brain functional networks have efficient small-world properties which support efficient parallel information transfer at a relatively low cost. More importantly, in patients with schizophrenia the small-world topological properties are significantly altered in many brain regions in the prefrontal, parietal and temporal lobes. These findings are consistent with a hypothesis of dysfunctional integration of the brain in this illness. Specifically, we found that these altered topological measurements correlate with illness duration in schizophrenia. Detection and estimation of these alterations could prove helpful for understanding the pathophysiological mechanism as well as for evaluation of the severity of schizophrenia.
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paper_title: Age‐related changes in topological organization of structural brain networks in healthy individuals

paper_content:

The aim of this study was to examine structural brain networks using regional gray matter volume, as well as to investigate changes in small-world and modular organization with normal aging. We constructed structural brain networks composed of 90 regions in young, middle, and old age groups. We randomly selected 350 healthy subjects for each group from a Japanese magnetic resonance image database. Structural brain networks in three age groups showed economical small-world properties, providing high global and local efficiency for parallel information processing at low connection cost. The small-world efficiency and node betweenness varied significantly and revealed a U- or inverted U-curve model tendency among three age groups. Results also demonstrated that structural brain networks exhibited a modular organization in which the connections between regions are much denser within modules than between them. The modular organization of structural brain networks was similar between the young and middle age groups, but quite different from the old group. In particular, the old group showed a notable decrease in the connector ratio and the intermodule connections. Combining the results of small-world efficiency, node betweenness and modular organization, we concluded that the brain network changed slightly, developing into a more distributed organization from young to middle age. The organization eventually altered greatly, shifting to a more localized organization in old age. Our findings provided quantitative insights into topological principles of structural brain networks and changes related to normal aging. Hum Brain Mapp, 2012. © 2011 Wiley Periodicals, Inc.
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paper_title: Testing for nonlinearity in time series : the method of surrogate data

paper_content:

We describe a statistical approach for identifying nonlinearity in time series. The method first specifies some linear process as a null hypothesis, then generates surrogate data sets which are consistent with this null hypothesis, and finally computes a discriminating statistic for the original and for each of the surrogate data sets. If the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data, then the null hypothesis is rejected and nonlinearity is detected. We discuss various null hypotheses and discriminating statistics. The method is demonstrated for numerical data generated by known chaotic systems, and applied to a number of experimental time series which arise in the measurement of superfluids, brain waves, and sunspots; we evaluate the statistical significance of the evidence for nonlinear structure in each case, and illustrate aspects of the data which this approach identifies.
---
paper_title: Bivariate surrogate techniques: Necessity, strengths, and caveats

paper_content:

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.
---
paper_title: Phase Synchronization Analysis of EEG Signals: An Evaluation Based on Surrogate Tests

paper_content:

Phase synchronization (PS) analysis has been demonstrated to be a useful method to infer functional connectivity with multichannel neural signals, e.g., electroencephalography (EEG). Methodological problems on quantifying functional connectivity with PS analysis have been investigated extensively, but some of them have not been fully solved yet. For example, how long a segment of EEG signal should be used in estimating PS index? Which methods are more suitable to infer the significant level of estimated PS index? To address these questions, this paper performs an intensive computation study on PS analysis based on surrogate tests with 1) artificial surrogate data generated by shuffling the rank order, the phase spectra, or the instantaneous frequency of original EEG signals, and 2) intersubject EEG pairs under the assumption that the EEG signals of different subjects are independent. Results show that 1) the phase-shuffled surrogate method is workable for significance test of estimated PS index and yields results similar to those by intersubject EEG surrogate test; 2) generally, a duration of EEG waves covering about 3 16 cycles is suitable for PS analysis; and 3) the PS index based on mean phase coherence is more suitable for PS analysis of EEG signals recorded at relatively low sampling rate.
---
paper_title: Twin surrogates to test for complex synchronisation

paper_content:

We present an approach to generate (multivariate) twin surrogates (TS) based on recurrence properties. This technique generates surrogates which correspond to an independent copy of the underlying system, i.e. they induce a trajectory of the underlying system starting at different initial conditions. We show that these surrogates are well suited to test for complex synchronisation and exemplify this for the paradigmatic system of Rossler oscillators. The proposed test enables to assess the statistical relevance of a synchronisation analysis from passive experiments which are typical in natural systems.
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paper_title: Surrogate data analysis for assessing the significance of the coherence function

paper_content:

In cardiovascular variability analysis, the significance of the coupling between two time series is commonly assessed by setting a threshold level in the coherence function. While traditionally used statistical tests consider only the parameters of the adopted estimator, the required zero-coherence level may be affected by some features of the observed series. In this study, three procedures, based on the generation of surrogate series sharing given properties with the original but being structurally uncoupled, were considered: independent identically distributed (IID), Fourier transform (FT), and autoregressive (AR). IID surrogates maintained the distribution of the original series, while FT and AR surrogates preserved the power spectrum. The ability of the three methods to define the threshold for zero coherence was validated and compared by computer simulations reproducing typical cardiovascular interactions. While the IID threshold depended only on record length and design parameters of the coherence estimator, FT and AR thresholds were frequency-dependent with peaks corresponding to the local maxima of the estimated coherence. FT and AR surrogates were able to compensate spurious coherence peaks due to equal-frequency but independent oscillations in the two series. The benefit of frequency-dependent thresholds was evident for short series with narrow-band oscillations. Thus, surrogates preserving the power spectrum of the original series are recommended to avoid false coupling detections in the presence of oscillations occurring at nearby frequencies but produced by different mechanisms, as may frequently happen in cardiovascular and cardiorespiratory regulation.
---
paper_title: Detecting Phase Synchronization in Noisy Systems

paper_content:

Abstract A quantitative method for automatic detection of phase synchronization in noisy experimental bivariate time series is proposed, based on the fact that instantaneous phases of phase-synchronized (sub) systems are mutually dependent in a specific way irrespective of a relation between the original time series. The level of dependence between the instantaneous phases is quantified by a statistical dependence parameter, which also reflects the strength of the systems' phase synchronization. Ranges of the parameter values, for which the detection of the phase synchronization can be considered reliable, are estimated by using the technique of surrogate data. Possible applications of the proposed method are demonstrated by using both numerically generated and real experimental data, namely solutions of two coupled Rossler systems, mammalian cardio-respiratory data, and long-term recordings of surface atmospheric temperature and sunspot numbers.
---
paper_title: Testing for nonlinearity in time series : the method of surrogate data

paper_content:

We describe a statistical approach for identifying nonlinearity in time series. The method first specifies some linear process as a null hypothesis, then generates surrogate data sets which are consistent with this null hypothesis, and finally computes a discriminating statistic for the original and for each of the surrogate data sets. If the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data, then the null hypothesis is rejected and nonlinearity is detected. We discuss various null hypotheses and discriminating statistics. The method is demonstrated for numerical data generated by known chaotic systems, and applied to a number of experimental time series which arise in the measurement of superfluids, brain waves, and sunspots; we evaluate the statistical significance of the evidence for nonlinear structure in each case, and illustrate aspects of the data which this approach identifies.
---
paper_title: Phase Synchronization Analysis of EEG Signals: An Evaluation Based on Surrogate Tests

paper_content:

Phase synchronization (PS) analysis has been demonstrated to be a useful method to infer functional connectivity with multichannel neural signals, e.g., electroencephalography (EEG). Methodological problems on quantifying functional connectivity with PS analysis have been investigated extensively, but some of them have not been fully solved yet. For example, how long a segment of EEG signal should be used in estimating PS index? Which methods are more suitable to infer the significant level of estimated PS index? To address these questions, this paper performs an intensive computation study on PS analysis based on surrogate tests with 1) artificial surrogate data generated by shuffling the rank order, the phase spectra, or the instantaneous frequency of original EEG signals, and 2) intersubject EEG pairs under the assumption that the EEG signals of different subjects are independent. Results show that 1) the phase-shuffled surrogate method is workable for significance test of estimated PS index and yields results similar to those by intersubject EEG surrogate test; 2) generally, a duration of EEG waves covering about 3 16 cycles is suitable for PS analysis; and 3) the PS index based on mean phase coherence is more suitable for PS analysis of EEG signals recorded at relatively low sampling rate.
---
paper_title: Testing for phase synchronization

paper_content:

We present different tests for phase synchronization which improve the procedures currently used in the literature. This is accomplished by using a two-sample test setup and by utilizing insights and methods from directional statistics and bootstrap theory. The tests differ in the generality of the situation in which they can be applied as well as in their complexity, including computational cost. A modification of the resampling technique of the bootstrap is introduced, making it possible to fully utilize data from time series.
---
paper_title: Testing for phase synchronization

paper_content:

Phase synchronization analysis is frequently applied to data originating for example from Physics and Life Sciences. Statistical properties for quantities measuring phase synchronization have not been revealed. We derive an analytic significance level for a frequently used phase synchronization measure. Its performance is demonstrated for a system of coupled oscillators.
---
paper_title: Phase Synchronization Analysis of EEG Signals: An Evaluation Based on Surrogate Tests

paper_content:

Phase synchronization (PS) analysis has been demonstrated to be a useful method to infer functional connectivity with multichannel neural signals, e.g., electroencephalography (EEG). Methodological problems on quantifying functional connectivity with PS analysis have been investigated extensively, but some of them have not been fully solved yet. For example, how long a segment of EEG signal should be used in estimating PS index? Which methods are more suitable to infer the significant level of estimated PS index? To address these questions, this paper performs an intensive computation study on PS analysis based on surrogate tests with 1) artificial surrogate data generated by shuffling the rank order, the phase spectra, or the instantaneous frequency of original EEG signals, and 2) intersubject EEG pairs under the assumption that the EEG signals of different subjects are independent. Results show that 1) the phase-shuffled surrogate method is workable for significance test of estimated PS index and yields results similar to those by intersubject EEG surrogate test; 2) generally, a duration of EEG waves covering about 3 16 cycles is suitable for PS analysis; and 3) the PS index based on mean phase coherence is more suitable for PS analysis of EEG signals recorded at relatively low sampling rate.
---
paper_title: Measuring synchronization in coupled model systems: A comparison of different approaches

paper_content:

The investigation of synchronization phenomena on measured experimental data such as biological time series has recently become an increasing focus of interest. Different approaches for measuring synchronization have been proposed that rely on certain characteristic features of the dynamical system under investigation. For experimental data the underlying dynamics are usually not completely known, therefore it is difficult to decide a priori which synchronization measure is most suitable for an analysis. In this study we use three different coupled model systems to create a ‘controlled’ setting for a comparison of six different measures of synchronization. All measures are compared to each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise. Results show that the measure to be applied to a certain task can not be chosen according to a fixed criterion but rather pragmatically as the measure which most reliably yields plausible information in test applications, although certain dynamical features of a system under investigation (e.g., power spectra, dimension) may render certain measures more suitable than others. c 2006 Elsevier B.V. All rights reserved.
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paper_title: Detecting phase synchronization in noisy data from coupled chaotic oscillators

paper_content:

Received 6 November 2007; published 17 April 2008Two schemes are proposed to detect phase synchronization from chaotic data contaminated by noise. Theﬁrst is a neighborhood-based method which links time delay embedding with instantaneous phase estimation.The second adopts the local projection method as a preprocessing ﬁlter to noisy data. Both schemes utilize thestate recurrence, an important feature of chaotic data. The proposed schemes are applied to data measured fromtwo typical chaotic systems, i.e., the coupled Rossler systems and the coupled Lorenz systems, respectively.The results show that phase synchronization, which may be buried by noise, is detected even when the noiselevel is high. Moreover, the overestimation of the degree of phase synchronization, which may be introducedby the Hilbert transform combined with a traditional linear bandpass ﬁlter, can be avoided when the data arecontaminated by only measurement noise.DOI: 10.1103/PhysRevE.77.046213 PACS number s : 05.45.Xt, 05.45.Pq, 05.40.Ca
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paper_title: Spurious detection of phase synchronization in coupled nonlinear oscillators

paper_content:

Coupled nonlinear systems under certain conditions exhibit phase synchronization, which may change for different frequency bands or with the presence of additive system noise. In both cases, Fourier filtering is traditionally used to preprocess data. We investigate to what extent the phase synchronization of two coupled Rössler oscillators depends on (1) the broadness of their power spectrum, (2) the width of the bandpass filter, and (3) the level of added noise. We find that for identical coupling strengths, oscillators with broader power spectra exhibit weaker synchronization. Further, we find that within a broad bandwidth range, bandpass filtering reduces the effect of noise but can lead to a spurious increase in the degree of phase synchronization with narrowing bandwidth, even when the coupling between the two oscillators remains the same.
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paper_title: Performance of different synchronization measures in real data: A case study on electroencephalographic signals

paper_content:

We agree with the Comment by Duckrow and Albano [Phys. Rev. E 67, 063901 (2003)] that mutual information, estimated with an optimized algorithm, can be a useful tool for studying synchronization in real data. However, we point out that the improvement they found is mainly due to an interesting but nonstandard embedding technique used, and not so much due to the algorithm used for the estimation of mutual information itself. We also address the issue of stationarity of electroencephalographic (EEG) data.
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paper_title: Unified framework for detecting phase synchronization in coupled time series

paper_content:

Phase synchronization (PS) has drawn increasing attention in recent years for its extensive applications in analyzing time series observed from coupled systems. In this paper, we examine the detection of PS in bivariate time series from the viewpoints of signal processing and circular statistics. Several definitions of instantaneous phase (IP) are first revisited and further unified into a framework, which defines IP as the argument of the signal with a specific bandpass filter applied. With this framework, the constraints for IP definition are discussed and the effect of noise in IP estimation is studied. The estimate error of IP, which is due to noise, is shown to obey a scale mixture of normal (SMN) distributions. Further, under the assumption that the SMN of IP error can be approximated by a particular normal distribution, the estimate of mean phase coherence of bivariate time series is shown to be degraded by a factor, which is determined by only the level of in-band noise. Finally, simulations are provided to support the theoretical results.
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paper_title: Characterizing and Approximating Infinite Scale Mixtures of Normals

paper_content:

The purpose of this study is to approximate and identify infinite scale mixtures of normals, SMN. A new method for approximating any infinite SMN with a known mixing measure by a finite SMN is presented. In the new method, the modulus of continuity of the normal family as a function of the scale is used to discretize the mixing measure. This method will be used to approximate univariate and multivariate SMN with mean 0. In the multivariate case, two different methods are used to approximate the infinite SMN. Several results related to SMN are proved and other known ones are presented. For example, SMN are characterized by their corresponding Laplace transforms.
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paper_title: Topics in Circular Statistics

paper_content:

Circular Probability Distributions Some Sampling Distributions Estimation of Parameters Tests for Mean Direction and Concentration Tests for Uniformity Nonparametric Testing Procedures Circular Correlation and Regression Predictive Inference for Directional Data Outliers and Related Problems Change-Point Problems Miscellaneous Topics Some Facts on Bessel Functions How to Use the CircStats Package.
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paper_title: An improved index of phase-synchronization for electrophysiological data in the presence of volume-conduction, noise and sample-size bias

paper_content:

Phase-synchronization is a manifestation of interaction between neuronal groups measurable from LFP, EEG or MEG signals, however, volume conduction can cause the coherence and the phase locking value to spuriously increase. It has been shown that the imaginary component of the coherency (ImC) cannot be spuriously increased by volume-conduction of independent sources. Recently, it was proposed that the phase lag index (PLI), which estimates to what extent the phase leads and lags between signals from two sensors are nonequiprobable, improves on the ImC. Compared to ImC, PLI has the advantage of being less influenced by phase delays. However, sensitivity to volume-conduction and noise, and capacity to detect changes in phase-synchronization, is hindered by the discontinuity of the PLI, as small perturbations turn phase lags into leads and vice versa. To solve this problem, we introduce a related index, namely the weighted phase lag index (WPLI). Differently from PLI, in WPLI the contribution of the observed phase leads and lags is weighted by the magnitude of the imaginary component of the cross-spectrum. We demonstrate two advantages of the WPLI over the PLI, in terms of reduced sensitivity to additional, uncorrelated noise sources and increased statistical power to detect changes in phase-synchronization. Another factor that can affect phase-synchronization indices is sample-size bias. We show that, when directly estimated, both PLI and the magnitude of the ImC have typically positively biased estimators. To solve this problem, we develop an unbiased estimator of the squared PLI, and a debiased estimator of the squared WPLI.
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paper_title: Phase Synchronization Analysis of EEG Signals: An Evaluation Based on Surrogate Tests

paper_content:

Phase synchronization (PS) analysis has been demonstrated to be a useful method to infer functional connectivity with multichannel neural signals, e.g., electroencephalography (EEG). Methodological problems on quantifying functional connectivity with PS analysis have been investigated extensively, but some of them have not been fully solved yet. For example, how long a segment of EEG signal should be used in estimating PS index? Which methods are more suitable to infer the significant level of estimated PS index? To address these questions, this paper performs an intensive computation study on PS analysis based on surrogate tests with 1) artificial surrogate data generated by shuffling the rank order, the phase spectra, or the instantaneous frequency of original EEG signals, and 2) intersubject EEG pairs under the assumption that the EEG signals of different subjects are independent. Results show that 1) the phase-shuffled surrogate method is workable for significance test of estimated PS index and yields results similar to those by intersubject EEG surrogate test; 2) generally, a duration of EEG waves covering about 3 16 cycles is suitable for PS analysis; and 3) the PS index based on mean phase coherence is more suitable for PS analysis of EEG signals recorded at relatively low sampling rate.
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paper_title: On the Recording Reference Contribution to EEG Correlation, Phase Synchorony, and Coherence

paper_content:

The degree of synchronization in electroencephalography (EEG) signals is commonly characterized by the time-series measures, namely, correlation, phase synchrony, and magnitude squared coherence (MSC). However, it is now well established that the interpretation of the results from these measures are confounded by the recording reference signal and that this problem is not mitigated by the use of other EEG montages, such as bipolar and average reference. In this paper, we analyze the impact of reference signal amplitude and power on EEG signal correlation, phase synchrony, and MSC. We show that, first, when two nonreferential signals have negative correlation, the phase synchrony and the absolute value of the correlation of the two referential signals may have two regions of behavior characterized by a monotonic decrease to zero and then a monotonic increase to one as the amplitude of the reference signal varies in [0, +∞). It is notable that even a small change of the amplitude may lead to significant impact on these two measures. Second, when two nonreferential signals have positive correlation, the correlation and phase-synchrony values of the two referential signals can monotonically increase to one (or monotonically decrease to some positive value and then monotonically increase to one) as the amplitude of the reference signal varies in [0, +∞). Third, when two nonreferential signals have negative cross-power, the MSC of the two referential signals can monotonically decrease to zero and then monotonically increase to one as reference signal power varies in [0, +∞). Fourth, when two nonreferential signals have positive cross-power, the MSC of the two referential signals can monotonically increase to one as the reference signal power varies in [0, +∞). In general, the reference signal with small amplitude or power relative to the signals of interest may decrease or increase the values of correlation, phase synchrony, and MSC. However, the reference signal with high relative amplitude or power will always increase each of the three measures. In our previous paper, we developed a method to identify and extract the reference signal contribution to intracranial EEG (iEEG) recordings. In this paper, we apply this approach to referential iEEG recorded from human subjects and directly investigate the contribution of recording reference on correlation, phase synchrony, and MSC. The experimental results demonstrate the significant impact that the recording reference may have on these bivariate measures.
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paper_title: EEG microstate sequences in healthy humans at rest reveal scale-free dynamics

paper_content:

Recent findings identified electroencephalography (EEG) microstates as the electrophysiological correlates of fMRI resting-state networks. Microstates are defined as short periods (100 ms) during which the EEG scalp topography remains quasi-stable; that is, the global topography is fixed but strength might vary and polarity invert. Microstates represent the subsecond coherent activation within global functional brain networks. Surprisingly, these rapidly changing EEG microstates correlate significantly with activity in fMRI resting-state networks after convolution with the hemodynamic response function that constitutes a strong temporal smoothing filter. We postulate here that microstate sequences should reveal scale-free, self-similar dynamics to explain this remarkable effect and thus that microstate time series show dependencies over long time ranges. To that aim, we deploy wavelet-based fractal analysis that allows determining scale-free behavior. We find strong statistical evidence that microstate sequences are scale free over six dyadic scales covering the 256-ms to 16-s range. The degree of long-range dependency is maintained when shuffling the local microstate labels but becomes indistinguishable from white noise when equalizing microstate durations, which indicates that temporal dynamics are their key characteristic. These results advance the understanding of temporal dynamics of brain-scale neuronal network models such as the global workspace model. Whereas microstates can be considered the “atoms of thoughts,” the shortest constituting elements of cognition, they carry a dynamic signature that is reminiscent at characteristic timescales up to multiple seconds. The scale-free dynamics of the microstates might be the basis for the rapid reorganization and adaptation of the functional networks of the brain.
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paper_title: A Time-Frequency-Based Approach to Phase and Phase Synchrony Estimation

paper_content:

Time-varying phase synchrony is an important bivariate measure that quantifies the dynamics between nonstationary signals and has been widely used in many applications including chaotic oscillators in physics and multichannel electroencephalography recordings in neuroscience. Current state-of-the-art in time-varying phase estimation uses either the Hilbert transform or the complex wavelet transform of the signals. Both of these methods have some major drawbacks such as the assumption that the signals are narrowband for the Hilbert transform and the nonuniform time-frequency resolution inherent to the wavelet analysis. In this paper, a new phase estimation method based on the Rihaczek distribution and Reduced Interference Rihaczek distribution belonging to Cohen's class is proposed. These distributions offer phase estimates with uniformly high time-frequency resolution which can be used for defining time and frequency dependent phase synchrony. Properties of the phase estimator and the corresponding phase synchrony measure are evaluated both analytically and through simulations showing the effectiveness of the new measures compared to existing methods.
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paper_title: Online Control of a Brain-Computer Interface Using Phase Synchronization

paper_content:

Currently, almost all brain-computer interfaces (BCIs) ignore the relationship between phases of electroencephalographic signals detected from different recording sites (i.e., electrodes). The vast majority of BCI systems rely on feature vectors derived from e.g., bandpower or univariate adaptive autoregressive (AAR) parameters. However, ample evidence suggests that additional information is obtained by quantifying the relationship between signals of single electrodes, which might provide innovative features for future BCI systems. This paper investigates one method to extract the degree of phase synchronization between two electroencephalogram (EEG) signals by calculating the so-called phase locking value (PLV). In our offline study, several PLV-based features were acquired and the optimal feature set was selected for each subject individually by a feature selection algorithm. The online sessions with three trained subjects revealed that all subjects were able to control three mental states (motor imagery of left hand, right hand, and foot, respectively) with single-trial accuracies between 60% and 66.7% (33% would be expected by chance) throughout the whole session
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paper_title: The statistical performance of some instantaneous frequency estimators

paper_content:

The authors examine the class of smoothed central finite difference (SCFD) instantaneous frequency (IF) estimators which are based on finite differencing of the phase of the analytic signal. These estimators are closely related to IF estimation via the (periodic) first moment, with respect to frequency of discrete time-frequency representations (TFRs) in L. Cohen's (1966) class. The authors determine the distribution of this class of estimators and establish a framework which allows the comparison of several other estimators such as the zero-crossing estimator and one based on linear regression on the signal phase. It is found that the regression IF estimator is biased and exhibits a large threshold for much of the frequency range. By replacing the linear convolution operation in the regression estimator with the appropriate convolution operation for circular data the authors obtain the parabolic SCFD (PSCFD) estimator, which is unbiased and has a frequency-independent variance, yet retains the optimal performance and simplicity of the original estimator. >
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paper_title: The brainweb: phase synchronization and large-scale integration.

paper_content:

The emergence of a unified cognitive moment relies on the coordination of scattered mosaics of functionally specialized brain regions. Here we review the mechanisms of large-scale integration that counterbalance the distributed anatomical and functional organization of brain activity to enable the emergence of coherent behaviour and cognition. Although the mechanisms involved in large-scale integration are still largely unknown, we argue that the most plausible candidate is the formation of dynamic links mediated by synchrony over multiple frequency bands.
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paper_title: The pairwise phase consistency: A bias-free measure of rhythmic neuronal synchronization

paper_content:

Oscillatory activity is a widespread phenomenon in nervous systems and has been implicated in numerous functions. Signals that are generated by two separate neuronal sources often demonstrate a consistent phase-relationship in a particular frequency-band, i.e., they demonstrate rhythmic neuronal synchronization. This consistency is conventionally measured by the PLV (phase-locking value) or the spectral coherence measure. Both statistical measures suffer from significant bias, in that their sample estimates overestimate the population statistics for finite sample sizes. This is a significant problem in the neurosciences where statistical comparisons are often made between conditions with a different number of trials or between neurons with a different number of spikes. We introduce a new circular statistic, the PPC (pairwise phase consistency). We demonstrate that the sample estimate of the PPC is a bias-free and consistent estimator of its corresponding population parameter. We show, both analytically and by means of numerical simulations, that the population statistic of the PPC is equivalent to the population statistic of the squared PLV. The variance and mean squared error of the PPC and PLV are compared. Finally, we demonstrate the practical relevance of the method in actual neuronal data recorded from the orbitofrontal cortex of rats that engage in a two-odour discrimination task. We find a strong increase in rhythmic synchronization of spikes relative to the local field potential (as measured by the PPC) for a wide range of low frequencies (including the theta-band) during the anticipation of sucrose delivery in comparison to the anticipation of quinine delivery.
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paper_title: Dysfunctional Long-Range Coordination of Neural Activity during Gestalt Perception in Schizophrenia

paper_content:

Recent theoretical and empirical research on schizophrenia converges on the notion that core aspects of the pathophysiology of the disorder may arise from a dysfunction in the coordination of distributed neural activity. Synchronization of neural responses in the β-band (15–30 Hz) and γ-band range (30–80 Hz) has been implicated as a possible neural substrate for dysfunctional coordination in schizophrenia. To test this hypothesis, we examined the electroencephalography (EEG) activity in 19 patients with a Diagnostic and Statistical Manual of Mental Disorder , edition IV criteria, diagnosis of schizophrenia and 19 healthy control subjects during a Gestalt perception task. EEG data were analyzed for phase synchrony and induced spectral power as an index of neural synchronization. Schizophrenia patients were impaired significantly in the detection of images that required the grouping of stimulus elements into coherent object representations. This deficit was accompanied by longer reaction times in schizophrenia patients. Deficits in Gestalt perception in schizophrenia patients were associated with reduced phase synchrony in the β-band (20–30 Hz), whereas induced spectral power in the γ-band (40–70 Hz) was mainly intact. Our findings suggest that schizophrenia patients are impaired in the long-range synchronization of neural responses, which may reflect a core deficit in the coordination of neural activity and underlie the specific cognitive dysfunctions associated with the disorder.
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paper_title: Unified framework for detecting phase synchronization in coupled time series

paper_content:

Phase synchronization (PS) has drawn increasing attention in recent years for its extensive applications in analyzing time series observed from coupled systems. In this paper, we examine the detection of PS in bivariate time series from the viewpoints of signal processing and circular statistics. Several definitions of instantaneous phase (IP) are first revisited and further unified into a framework, which defines IP as the argument of the signal with a specific bandpass filter applied. With this framework, the constraints for IP definition are discussed and the effect of noise in IP estimation is studied. The estimate error of IP, which is due to noise, is shown to obey a scale mixture of normal (SMN) distributions. Further, under the assumption that the SMN of IP error can be approximated by a particular normal distribution, the estimate of mean phase coherence of bivariate time series is shown to be degraded by a factor, which is determined by only the level of in-band noise. Finally, simulations are provided to support the theoretical results.
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paper_title: Estimation of Effective and Functional Cortical Connectivity From Neuroelectric and Hemodynamic Recordings

paper_content:

In this paper, different linear and nonlinear methodologies for the estimation of cortical connectivity from neuroelectric and hemodynamic measurements are reviewed and applied on common data set in order to highlight similarities and differences in the results. Different effective and functional connectivity methods were applied to motor and cognitive data sets, including structural equation modeling (SEM), directed transfer function (DTF), partial directed coherence (PDC), and direct directed transfer function (dDTF). Comparisons were made between the results in order to understand if, for a same dataset, effective and functional connectivity estimators can return the same cortical connectivity patterns. An application of a nonlinear method [phase synchronization index (PSI)] to similar executed and imagined movements was also reviewed. Connectivity patterns estimated with the use of the neuroelectric information and of the information from the multimodal integration of neuroelectric and hemodynamic data were also compared. Results suggests that the estimation of the cortical connectivity patterns performed with the linear methods (SEM, DTF, PDC, dDTF) or with the nonlinear method (PSI) on movement related potentials returned similar cortical networks. Differences in cortical connectivity were noted between the patterns estimated with the use of multimodal integration and those estimated by using only the neuroelectric data.
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paper_title: Short-window spectral analysis of cortical event-related potentials by adaptive multivariate autoregressive modeling: data preprocessing, model validation, and variability assessment

paper_content:

In this article we consider the application of parametric spectral analysis to multichannel event-related potentials (ERPs) during cognitive experiments. We show that with proper data preprocessing, Adaptive MultiVariate AutoRegressive (AMVAR) modeling is an effective technique for dealing with nonstationary ERP time series. We propose a bootstrap procedure to assess the variability in the estimated spectral quantities. Finally, we apply AMVAR spectral analysis to a visuomotor integration task, revealing rapidly changing cortical dynamics during different stages of task processing.
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paper_title: EEG coherency: I: statistics, reference electrode, volume conduction, Laplacians, cortical imaging, and interpretation at multiple scales

paper_content:

Abstract Several methodological issues which impact experimental design and physiological interpretations in EEG coherence studies are considered, including reference electrode and volume conduction contributions to erroneous coherence estimates. A new measure, `reduced coherency', is introduced as the difference between measured coherency and the coherency expected from uncorrelated neocortical sources, based on simulations and analytic-statistical studies with a volume conductor model. The concept of reduced coherency is shown to be in semi-quantitative agreement with experimental EEG data. The impact of volume conduction on statistical confidence intervals for coherence estimates is discussed. Conventional reference, average reference, bipolar, Laplacian, and cortical image coherencies are shown to be partly independent measures of neocortical dynamic function at different spatial scales, due to each method's unique spatial filtering of intracranial source activity.
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paper_title: An improved index of phase-synchronization for electrophysiological data in the presence of volume-conduction, noise and sample-size bias

paper_content:

Phase-synchronization is a manifestation of interaction between neuronal groups measurable from LFP, EEG or MEG signals, however, volume conduction can cause the coherence and the phase locking value to spuriously increase. It has been shown that the imaginary component of the coherency (ImC) cannot be spuriously increased by volume-conduction of independent sources. Recently, it was proposed that the phase lag index (PLI), which estimates to what extent the phase leads and lags between signals from two sensors are nonequiprobable, improves on the ImC. Compared to ImC, PLI has the advantage of being less influenced by phase delays. However, sensitivity to volume-conduction and noise, and capacity to detect changes in phase-synchronization, is hindered by the discontinuity of the PLI, as small perturbations turn phase lags into leads and vice versa. To solve this problem, we introduce a related index, namely the weighted phase lag index (WPLI). Differently from PLI, in WPLI the contribution of the observed phase leads and lags is weighted by the magnitude of the imaginary component of the cross-spectrum. We demonstrate two advantages of the WPLI over the PLI, in terms of reduced sensitivity to additional, uncorrelated noise sources and increased statistical power to detect changes in phase-synchronization. Another factor that can affect phase-synchronization indices is sample-size bias. We show that, when directly estimated, both PLI and the magnitude of the ImC have typically positively biased estimators. To solve this problem, we develop an unbiased estimator of the squared PLI, and a debiased estimator of the squared WPLI.
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paper_title: The volume conductor may act as a temporal filter on the ECG and EEG

paper_content:

The influence of the volume conductor on the EEG, MEG, fetal ECG and fetal MCG is studied by means of simulations. The assumption that the Maxwell equations can be used in a quasi-static approximation is reconsidered and the fact that the conductivity of human tissue is frequency dependent is taken into account. It is found that displacement currents have a substantial effect on the fetal ECG and to a lesser degree on the fetal MCG. Moreover, the frequency dependence of the conductivity of the tissues within the head may have a considerable effect on the EEG.
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paper_title: Synchronization of parietal and premotor areas during preparation and execution of praxis hand movements

paper_content:

Abstract Objective We sought to determine temporal patterns of functional connectivity between the parietal, premotor, and motor cortices during preparation and execution of praxis hand movements. Methods Normal subjects were instructed to perform six transitive (tool use) and intransitive (communicative gesture) self-paced pantomimes with the right hand while recording 64-channel electroencephalography (EEG) and electromyography (EMG) from right thumb and forearm flexors. Focusing on corticocortical coherence, we explored the time-course of synchronously active parietal and premotor circuits involved in these motor tasks. Trials were marked for EMG onset and averaged across subjects to determine changes in coherence relative to baseline between parietal, premotor, and motor areas. Results Coherence of homologous electrode pairs was similar when comparing transitive and intransitive movements. During preparation, beta band (18–22 Hz) coherence was maximal between electrodes over the left parietal lobe and left premotor electrodes. Additionally during preparation, the premotor area showed high coherence to the motor hand area and the parietal cortex. Electrodes over the supplementary motor area also showed coherence to the motor and parietal, but not the premotor area. Before and during execution, a second peak of high coherence increase was present in each area that demonstrated coherence increases during preparation. There was no coherence increase between parietal and motor areas. Coherence rapidly diminished 1.5–2.0 s after movement onset. Conclusions Patterns of increased corticocortical coupling within a parietal, premotor, and motor network are present during preparation and execution of praxis movements. Significance This study adds to evidence that parietofrontal networks may be critical for integrating preparatory and motor-related activity for praxis movements.
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paper_title: Phase lag index: Assessment of functional connectivity from multi channel EEG and MEG with diminished bias from common sources

paper_content:

Objective: ::: To address the problem of volume conduction and active reference electrodes in the assessment of functional connectivity, we propose a novel measure to quantify phase synchronization, the phase lag index (PLI), and compare its performance to the well-known phase coherence (PC), and to the imaginary component of coherency (IC). ::: ::: Methods: ::: The PLI is a measure of the asymmetry of the distribution of phase differences between two signals. The performance of PLI, PC, and IC was examined in (i) a model of 64 globally coupled oscillators, (ii) an EEG with an absence seizure, (iii) an EEG data set of 15 Alzheimer patients and 13 control subjects, and (iv) two MEG data sets. ::: ::: Results: ::: PLI and PC were more sensitive than IC to increasing levels of true synchronization in the model. PC and IC were influenced stronger than PLI by spurious correlations because of common sources. All measures detected changes in synchronization during the absence seizure. In contrast to PC, PLI and IC were barely changed by the choice of different montages. PLI and IC were superior to PC in detecting changes in beta band connectivity in AD patients. Finally, PLI and IC revealed a different spatial pattern of functional connectivity in MEG data than PC. ::: ::: Conclusion: ::: The PLI performed at least as well as the PC in detecting true changes in synchronization in model and real data but, at the same token and like-wise the IC, it was much less affected by the influence of common sources and active reference electrodes. Hum Brain Mapp 2007. © 2007 Wiley-Liss, Inc.
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paper_title: Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem.

paper_content:

In this paper basic mathematical and physical concepts of the biomagnetic inverse problem are reviewed with some new approaches. The forward problem is discussed for both homogeneous and inhomogeneous media. Geselowitz' formulae and a surface integral equation are presented to handle a piecewise homogeneous conductor. The special cases of a spherically symmetric conductor and a horizontally layered medium are discussed in detail. The non-uniqueness of the solution of the magnetic inverse problem is discussed and the difficulty caused by the contribution of the electric potential to the magnetic field outside the conductor is studied. As practical methods of solving the inverse problem, a weighted least-squares search with confidence limits and the method of minimum norm estimate are discussed.
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paper_title: Identifying true brain interaction from EEG data using the imaginary part of coherency

paper_content:

Abstract Objective : The main obstacle in interpreting EEG/MEG data in terms of brain connectivity is the fact that because of volume conduction, the activity of a single brain source can be observed in many channels. Here, we present an approach which is insensitive to false connectivity arising from volume conduction. Methods : We show that the (complex) coherency of non-interacting sources is necessarily real and, hence, the imaginary part of coherency provides an excellent candidate to study brain interactions. Although the usual magnitude and phase of coherency contain the same information as the real and imaginary parts, we argue that the Cartesian representation is far superior for studying brain interactions. The method is demonstrated for EEG measurements of voluntary finger movement. Results : We found: (a) from 5 s before to movement onset a relatively weak interaction around 20 Hz between left and right motor areas where the contralateral side leads the ipsilateral side; and (b) approximately 2–4 s after movement, a stronger interaction also at 20 Hz in the opposite direction. Conclusions : It is possible to reliably detect brain interaction during movement from EEG data. Significance : The method allows unambiguous detection of brain interaction from rhythmic EEG/MEG data.
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paper_title: EEG phase synchrony differences across visual perception conditions may depend on recording and analysis methods

paper_content:

Objective: (1) To investigate the neural synchrony hypothesis by examining if there was more synchrony for upright than inverted Mooney faces, replicating a previous study; (2) to investigate whether inverted stimuli evoke neural synchrony by comparing them to a new scrambled control condition, less likely to produce face perception. Methods: Multichannel EEG was recorded via nose reference while participants viewed upright, inverted, and scrambled Mooney face stimuli. Gamma-range spectral power and inter-electrode phase synchrony were calculated via a wavelet-based method for upright stimuli perceived as faces and inverted/scrambled stimuli perceived as non-faces. Results: When the frequency of interest was selected from the upright condition exhibiting maximal spectral power responses (as in the previous study) greater phase synchrony was found in the upright than inverted/scrambled conditions. However, substantial synchrony was present in all conditions, suggesting that choosing the frequency of interest from the upright condition only may have been biased. In addition, artifacts related to nose reference contamination by micro-saccades were found to be differentially present across experimental conditions in the raw EEG. When frequency of interest was selected instead from each experimental condition and the data were transformed to a laplacian ‘reference free’ derivation, the between-condition phase synchrony differences disappeared. Spectral power differences were robust to the change in reference, but not the combined changes in reference and frequency selection criteria. Conclusions: Synchrony differences between face/non-face perceptions depend upon frequency selection and recording reference. Optimal selection of these parameters abolishes differential synchrony between conditions. Significance: Neural synchrony is present not just for face percepts for upright stimuli, but also for non-face percepts achieved for inverted/scrambled Mooney stimuli. q 2004 International Federation of Clinical Neurophysiology. Published by Elsevier Ireland Ltd. All rights reserved.
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paper_title: EEG coherency: I: statistics, reference electrode, volume conduction, Laplacians, cortical imaging, and interpretation at multiple scales

paper_content:

Abstract Several methodological issues which impact experimental design and physiological interpretations in EEG coherence studies are considered, including reference electrode and volume conduction contributions to erroneous coherence estimates. A new measure, `reduced coherency', is introduced as the difference between measured coherency and the coherency expected from uncorrelated neocortical sources, based on simulations and analytic-statistical studies with a volume conductor model. The concept of reduced coherency is shown to be in semi-quantitative agreement with experimental EEG data. The impact of volume conduction on statistical confidence intervals for coherence estimates is discussed. Conventional reference, average reference, bipolar, Laplacian, and cortical image coherencies are shown to be partly independent measures of neocortical dynamic function at different spatial scales, due to each method's unique spatial filtering of intracranial source activity.
---
paper_title: On the Recording Reference Contribution to EEG Correlation, Phase Synchorony, and Coherence

paper_content:

The degree of synchronization in electroencephalography (EEG) signals is commonly characterized by the time-series measures, namely, correlation, phase synchrony, and magnitude squared coherence (MSC). However, it is now well established that the interpretation of the results from these measures are confounded by the recording reference signal and that this problem is not mitigated by the use of other EEG montages, such as bipolar and average reference. In this paper, we analyze the impact of reference signal amplitude and power on EEG signal correlation, phase synchrony, and MSC. We show that, first, when two nonreferential signals have negative correlation, the phase synchrony and the absolute value of the correlation of the two referential signals may have two regions of behavior characterized by a monotonic decrease to zero and then a monotonic increase to one as the amplitude of the reference signal varies in [0, +∞). It is notable that even a small change of the amplitude may lead to significant impact on these two measures. Second, when two nonreferential signals have positive correlation, the correlation and phase-synchrony values of the two referential signals can monotonically increase to one (or monotonically decrease to some positive value and then monotonically increase to one) as the amplitude of the reference signal varies in [0, +∞). Third, when two nonreferential signals have negative cross-power, the MSC of the two referential signals can monotonically decrease to zero and then monotonically increase to one as reference signal power varies in [0, +∞). Fourth, when two nonreferential signals have positive cross-power, the MSC of the two referential signals can monotonically increase to one as the reference signal power varies in [0, +∞). In general, the reference signal with small amplitude or power relative to the signals of interest may decrease or increase the values of correlation, phase synchrony, and MSC. However, the reference signal with high relative amplitude or power will always increase each of the three measures. In our previous paper, we developed a method to identify and extract the reference signal contribution to intracranial EEG (iEEG) recordings. In this paper, we apply this approach to referential iEEG recorded from human subjects and directly investigate the contribution of recording reference on correlation, phase synchrony, and MSC. The experimental results demonstrate the significant impact that the recording reference may have on these bivariate measures.
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paper_title: Automatic Identification and Removal of Scalp Reference Signal for Intracranial EEGs Based on Independent Component Analysis

paper_content:

The pursuit of an inactive recording reference is one of the oldest technical problems in electroencephalography (EEG). Since commonly used cephalic references contaminate EEG and can lead to misinterpretation, extraction of the reference contribution is of fundamental interest. Here, we apply independent component analysis (ICA) to intracranial recordings and propose two methods to automatically identify and remove the reference based on the assumption that the scalp reference is independent from the local and distributed intracranial sources. This assumption, supported by our results, is generally valid because the reference scalp electrode is relatively electrically isolated from the intracranial electrodes by the skull's high resistivity. We point out that the linear model is underdetermined when the reference is considered as a source, and discuss one special underdetermined case for which a unique class of outputs can be separated. For this case most ICA algorithms can be applied, and we argue that intracranial or scalp EEGs follow this special case. We apply the two proposed methods to intracranial EEGs from three patients undergoing evaluation for epilepsy surgery, and compare the results to bipolar and average reference recordings. The proposed methods should have wide application in quantitative EEG studies.
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paper_title: The effect of a scalp reference signal on coherence measurements of intracranial electroencephalograms

paper_content:

Abstract Objective : To determine the effect of a scalp reference signal, such as that recorded from the mastoid or ear, on the coherence of referential intracranial electroencephalograms (EEGs). Methods : The relationship between reference signal power and magnitude squared coherence (MSC) was determined from the theoretical expression of the coherence of referential recordings, obtained under the assumption that the reference signal is not correlated with the signals being studied. The effect of a contaminated reference signal on the coherence of intracranial EEGs was determined by measuring the MSC of both a recording of background EEGs with a simulated contaminated reference signal and a contaminated recording of a seizure. Results : The MSC of referential intracranial EEGs is inflated due to the reference signal. This inflation is a function of the true MSC of the intracranial signals and the power of the reference and intracranial signals. The inflation is limited where reference signal power is smaller than the power of the intracranial signals; maximum inflation Conclusions : The findings presented here suggest a scalp reference signal is suitable, with careful monitoring of the reference signal, for coherence analysis of intracranial EEGs. A reference signal will have a limited effect on the coherence of intracranial EEGs except when it is contaminated. In the event the reference signal is contaminated it should be possible to detect this from the stereotyped features of the coherence spectrum.
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paper_title: A theoretical justification of the average reference in topographic evoked potential studies

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Abstract For multichannel evoked potential recording, several authors have proposed to use the ‘average reference.’ This paper tries to justify the choice of this reference. Indeed, it can be shown that the integral of the potential distribution over a sphere including current dipoles is null. The limitations of the use of such an average reference are discussed.
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paper_title: On the use of bipolar montages for time-series analysis of intracranial electroencephalograms

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Abstract Objective Bipolar montages are routinely employed for the interpretation of scalp and intracranial EEGs (icEEGs). In this manuscript we consider the assumptions that support the use of a bipolar montage and question the universal appropriateness of bipolar representation of icEEGs for the time-series analysis of these signals. Bipolar montages introduce an element of spatial processing into the observed time-series. In the case of icEEGs, we argue ambiguity may be introduced in some settings through this operation because of a lack of certifiability of local differentiability and continuity of the spatial structure of icEEGs, and their suboptimal spatial sampling. Methods Example icEEGs were collected from three patients being studied for possible resective epilepsy surgery. Referential and bipolar representations of these signals were subjected to different visual and time-series analysis. The time-series measures calculated were the power spectral density and magnitude squared coherence. Results Visual analysis and time-series measures revealed that the icEEG time-series was altered by the use of a bipolar montage. The changes resulted from either the introduction of unrelated information from the two referential time-series into the bipolar time-series, or from the removal or alteration of information common to the two referential time-series in the bipolar time-series. The changes could not be predicted without prior knowledge of the relationship between measurement sites that form the bipolar montage. Conclusions In certain settings, bipolar montages alter icEEGs and can confound the time-series analysis of these signals. In such settings, bipolar montages should be used with caution in the time-series analysis of icEEGs. Significance This manuscript addresses the representation of the intracranial EEG for time-series analysis. There may be contexts where the assumptions underpinning correct application of the bipolar montage to the intracranial EEG are not satisfied.
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paper_title: Graphical interaction models for multivariate time series1

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In this paper we extend the concept of graphical models for multivariate data to multivariate time series. We define a partial correlation graph for time series and use the partial spectral coherence between two components given the remaining components to identify the edges of the graph. As an example we consider multivariate autoregressive processes. The method is applied to air pollution data.
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paper_title: Detecting directional coupling in the human epileptic brain: Limitations and potential pitfalls

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We study directional relationships-in the driver-responder sense-in networks of coupled nonlinear oscillators using a phase modeling approach. Specifically, we focus on the identification of drivers in clusters with varying levels of synchrony, mimicking dynamical interactions between the seizure generating region (epileptic focus) and other brain structures. We demonstrate numerically that such an identification is not always possible in a reliable manner. Using the same analysis techniques as in model systems, we study multichannel electroencephalographic recordings from two patients suffering from focal epilepsy. Our findings demonstrate that--depending on the degree of intracluster synchrony--certain subsystems can spuriously appear to be driving others, which should be taken into account when analyzing field data with unknown underlying dynamics.
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paper_title: Frequency flows and the time-frequency dynamics of multivariate phase synchronization in brain signals

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Abstract The quantification of phase synchrony between brain signals is of crucial importance for the study of large-scale interactions in the brain. Current methods are based on the estimation of the stability of the phase difference between pairs of signals over a time window, within successive frequency bands. This paper introduces a new approach to study the dynamics of brain synchronies, Frequency Flows Analysis (FFA). It allows direct tracking and characterization of the nonstationary time-frequency dynamics of phase synchrony among groups of signals. It is based on the use of the one-to-one relationship between frequency locking and phase synchrony, which applies when the concept of phase synchrony is not taken in an extended ‘statistical’ sense of a bias in the distribution of phase differences, but in the sense of a continuous phase difference conservation during a short period of time. In such a case, phase synchrony implies identical instantaneous frequencies among synchronized signals, with possible time varying frequencies of synchronization. In this framework, synchronous groups of signals or neural assemblies can be identified as belonging to common frequency flows, and the problem of studying synchronization becomes the problem of tracking frequency flows. We use the ridges of the analytic wavelet transforms of the signals of interest in order to estimate maps of instantaneous frequencies and reveal sustained periods of common instantaneous frequency among groups of signal. FFA is shown to track complex dynamics of synchrony in coupled oscillator models, reveal the time-frequency and spatial dynamics of synchrony convergence and divergence in epileptic seizures, and in MEG data the large-scale ongoing dynamics of synchrony correlated with conscious perception during binocular rivalry.
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paper_title: Visual Working Memory Load-Related Changes in Neural Activity and Functional Connectivity

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BACKGROUND ::: Visual working memory (VWM) helps us store visual information to prepare for subsequent behavior. The neuronal mechanisms for sustaining coherent visual information and the mechanisms for limited VWM capacity have remained uncharacterized. Although numerous studies have utilized behavioral accuracy, neural activity, and connectivity to explore the mechanism of VWM retention, little is known about the load-related changes in functional connectivity for hemi-field VWM retention. ::: ::: ::: METHODOLOGY/PRINCIPAL FINDINGS ::: In this study, we recorded electroencephalography (EEG) from 14 normal young adults while they performed a bilateral visual field memory task. Subjects had more rapid and accurate responses to the left visual field (LVF) memory condition. The difference in mean amplitude between the ipsilateral and contralateral event-related potential (ERP) at parietal-occipital electrodes in retention interval period was obtained with six different memory loads. Functional connectivity between 128 scalp regions was measured by EEG phase synchronization in the theta- (4-8 Hz), alpha- (8-12 Hz), beta- (12-32 Hz), and gamma- (32-40 Hz) frequency bands. The resulting matrices were converted to graphs, and mean degree, clustering coefficient and shortest path length was computed as a function of memory load. The results showed that brain networks of theta-, alpha-, beta-, and gamma- frequency bands were load-dependent and visual-field dependent. The networks of theta- and alpha- bands phase synchrony were most predominant in retention period for right visual field (RVF) WM than for LVF WM. Furthermore, only for RVF memory condition, brain network density of theta-band during the retention interval were linked to the delay of behavior reaction time, and the topological property of alpha-band network was negative correlation with behavior accuracy. ::: ::: ::: CONCLUSIONS/SIGNIFICANCE ::: We suggest that the differences in theta- and alpha- bands between LVF and RVF conditions in functional connectivity and topological properties during retention period may result in the decline of behavioral performance in RVF task.
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paper_title: Detecting synchronization clusters in multivariate time series via coarse-graining of Markov chains

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Synchronization cluster analysis is an approach to the detection of underlying structures in data sets of multivariate time series, starting from a matrix R of bivariate synchronization indices. A previous method utilized the eigenvectors of R for cluster identification, analogous to several recent attempts at group identification using eigenvectors of the correlation matrix. All of these approaches assumed a one-to-one correspondence of dominant eigenvectors and clusters, which has however been shown to be wrong in important cases. We clarify the usefulness of eigenvalue decomposition for synchronization cluster analysis by translating the problem into the language of stochastic processes, and derive an enhanced clustering method harnessing recent insights from the coarse-graining of finite-state Markov processes. We illustrate the operation of our method using a simulated system of coupled Lorenz oscillators, and we demonstrate its superior performance over the previous approach. Finally we investigate the question of robustness of the algorithm against small sample size, which is important with regard to field applications.
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paper_title: Measuring Phase Synchrony in Brain Signals

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This article presents, for the first time, a practical method for the direct quantification of frequency-specific synchronization (i.e., transient phase-locking) between two neuroelectric signals. The motivation for its development is to be able to examine the role of neural synchronies as a putative mechanism for long-range neural integration during cognitive tasks. The method, called phase-locking statistics (PLS), measures the significance of the phase covariance between two signals with a reasonable time-resolution (,100 ms). Unlike the more traditional method of spectral coherence, PLS separates the phase and amplitude components and can be directly interpreted in the framework of neural integration. To validate synchrony values against background fluctuations, PLS uses surrogate data and thus makes no a priori assumptions on the nature of the experimental data. We also apply PLS to investigate intracortical recordings from an epileptic patient performing a visual discrimination task. We find large-scale synchronies in the gamma band (45 Hz), e.g., between hippocampus and frontal gyrus, and local synchronies, within a limbic region, a few cm apart. We argue that whereas long-scale effects do reflect cognitive processing, short-scale synchronies are likely to be due to volume conduction. We discuss ways to separate such conduction effects from true signal synchrony. Hum Brain Mapping 8:194-208, 1999. r 1999 Wiley-Liss, Inc.
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paper_title: Partial Phase Synchronization for Multivariate Synchronizing Systems

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Graphical models applying partial coherence to multivariate time series are a powerful tool to distinguish direct and indirect interdependencies in multivariate linear systems. We carry over the concept of graphical models and partialization analysis to phase signals of nonlinear synchronizing systems. This procedure leads to the partial phase synchronization index which generalizes a bivariate phase synchronization index to the multivariate case and reveals the coupling structure in multivariate synchronizing systems by differentiating direct and indirect interactions. This ensures that no false positive conclusions are drawn concerning the interaction structure in multivariate synchronizing systems. By application to the paradigmatic model of a coupled chaotic Roessler system, the power of the partial phase synchronization index is demonstrated.
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paper_title: Measuring synchronization in coupled model systems: A comparison of different approaches

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The investigation of synchronization phenomena on measured experimental data such as biological time series has recently become an increasing focus of interest. Different approaches for measuring synchronization have been proposed that rely on certain characteristic features of the dynamical system under investigation. For experimental data the underlying dynamics are usually not completely known, therefore it is difficult to decide a priori which synchronization measure is most suitable for an analysis. In this study we use three different coupled model systems to create a ‘controlled’ setting for a comparison of six different measures of synchronization. All measures are compared to each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise. Results show that the measure to be applied to a certain task can not be chosen according to a fixed criterion but rather pragmatically as the measure which most reliably yields plausible information in test applications, although certain dynamical features of a system under investigation (e.g., power spectra, dimension) may render certain measures more suitable than others. c 2006 Elsevier B.V. All rights reserved.
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paper_title: An improved index of phase-synchronization for electrophysiological data in the presence of volume-conduction, noise and sample-size bias

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Phase-synchronization is a manifestation of interaction between neuronal groups measurable from LFP, EEG or MEG signals, however, volume conduction can cause the coherence and the phase locking value to spuriously increase. It has been shown that the imaginary component of the coherency (ImC) cannot be spuriously increased by volume-conduction of independent sources. Recently, it was proposed that the phase lag index (PLI), which estimates to what extent the phase leads and lags between signals from two sensors are nonequiprobable, improves on the ImC. Compared to ImC, PLI has the advantage of being less influenced by phase delays. However, sensitivity to volume-conduction and noise, and capacity to detect changes in phase-synchronization, is hindered by the discontinuity of the PLI, as small perturbations turn phase lags into leads and vice versa. To solve this problem, we introduce a related index, namely the weighted phase lag index (WPLI). Differently from PLI, in WPLI the contribution of the observed phase leads and lags is weighted by the magnitude of the imaginary component of the cross-spectrum. We demonstrate two advantages of the WPLI over the PLI, in terms of reduced sensitivity to additional, uncorrelated noise sources and increased statistical power to detect changes in phase-synchronization. Another factor that can affect phase-synchronization indices is sample-size bias. We show that, when directly estimated, both PLI and the magnitude of the ImC have typically positively biased estimators. To solve this problem, we develop an unbiased estimator of the squared PLI, and a debiased estimator of the squared WPLI.
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paper_title: Correntropy as a novel measure for nonlinearity tests

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Nonlinearity tests have become an essential step in system analysis and modeling due to the computational demands and complexity of analysis involved in nonlinear modeling. Standard nonlinear measures are either too complicated to estimate accurately (such as Lyapunov exponents and correlation dimension), or not able to capture sufficient but not necessary conditions of nonlinearity (such as time asymmetry). Correntropy is a kernel-based similarity measure which contains the information of both statistical and temporal structure of the underlying dataset. The capability of preserving nonlinear characteristics makes correntropy a suitable candidate as a measure for determining nonlinear dynamics. Moreover, since correntropy makes use of kernel methods, its estimation is computationally efficient. Using correntropy as the test statistic, nonlinearity tests based on the null hypothesis that signals of interest are realizations of linear Gaussian stochastic processes are carried out via surrogate data methods. Experiments performed on linear Gaussian, linear non-Gaussian, and nonlinear systems with varying in-band noise levels, data lengths, and kernel sizes confirm that correntropy can be employed as a discriminative measure for detecting nonlinear characteristics in time series. Results of tests performed on data collected from natural systems are in agreement with findings in time series analysis literature.
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paper_title: A Comparative Study of Synchrony Measures for the Early Diagnosis of Alzheimer’s Disease Based on EEG

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Abstract It is well known that EEG signals of Alzheimer's disease (AD) patients are generally less synchronous than in age-matched control subjects. However, this effect is not always easily detectable. This is especially the case for patients in the pre-symptomatic phase, commonly referred to as mild cognitive impairment (MCI), during which neuronal degeneration is occurring prior to the clinical symptoms appearance. In this paper, various synchrony measures are studied in the context of AD diagnosis, including the correlation coefficient, mean-square and phase coherence, Granger causality, phase synchrony indices, information-theoretic divergence measures, state space based measures, and the recently proposed stochastic event synchrony measures. Experiments with EEG data show that many of those measures are strongly correlated (or anti-correlated) with the correlation coefficient, and hence, provide little complementary information about EEG synchrony. Measures that are only weakly correlated with the correlation coefficient include the phase synchrony indices, Granger causality measures, and stochastic event synchrony measures. In addition, those three families of synchrony measures are mutually uncorrelated, and therefore, they each seem to capture a specific kind of interdependence. For the data set at hand, only two synchrony measures are able to convincingly distinguish MCI patients from age-matched control patients, i.e., Granger causality (in particular, full-frequency directed transfer function) and stochastic event synchrony. Those two measures are used as features to distinguish MCI patients from age-matched control subjects, yielding a leave-one-out classification rate of 83%. The classification performance may be further improved by adding complementary features from EEG; this approach may eventually lead to a reliable EEG-based diagnostic tool for MCI and AD.
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paper_title: Fourier-, Hilbert- and wavelet-based signal analysis: are they really different approaches?

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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.
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paper_title: Phase lag index: Assessment of functional connectivity from multi channel EEG and MEG with diminished bias from common sources

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Objective: ::: To address the problem of volume conduction and active reference electrodes in the assessment of functional connectivity, we propose a novel measure to quantify phase synchronization, the phase lag index (PLI), and compare its performance to the well-known phase coherence (PC), and to the imaginary component of coherency (IC). ::: ::: Methods: ::: The PLI is a measure of the asymmetry of the distribution of phase differences between two signals. The performance of PLI, PC, and IC was examined in (i) a model of 64 globally coupled oscillators, (ii) an EEG with an absence seizure, (iii) an EEG data set of 15 Alzheimer patients and 13 control subjects, and (iv) two MEG data sets. ::: ::: Results: ::: PLI and PC were more sensitive than IC to increasing levels of true synchronization in the model. PC and IC were influenced stronger than PLI by spurious correlations because of common sources. All measures detected changes in synchronization during the absence seizure. In contrast to PC, PLI and IC were barely changed by the choice of different montages. PLI and IC were superior to PC in detecting changes in beta band connectivity in AD patients. Finally, PLI and IC revealed a different spatial pattern of functional connectivity in MEG data than PC. ::: ::: Conclusion: ::: The PLI performed at least as well as the PC in detecting true changes in synchronization in model and real data but, at the same token and like-wise the IC, it was much less affected by the influence of common sources and active reference electrodes. Hum Brain Mapp 2007. © 2007 Wiley-Liss, Inc.
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paper_title: Performance of different synchronization measures in real data: A case study on electroencephalographic signals

paper_content:

We agree with the Comment by Duckrow and Albano [Phys. Rev. E 67, 063901 (2003)] that mutual information, estimated with an optimized algorithm, can be a useful tool for studying synchronization in real data. However, we point out that the improvement they found is mainly due to an interesting but nonstandard embedding technique used, and not so much due to the algorithm used for the estimation of mutual information itself. We also address the issue of stationarity of electroencephalographic (EEG) data.
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paper_title: Evaluation of different measures of functional connectivity using a neural mass model

paper_content:

Abstract 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.
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paper_title: An investigation of the phase locking index for measuring of interdependency of cortical source signals recorded in the EEG

paper_content:

The phase locking index (PLI) was introduced to quantify in a statistical sense the phase synchronization of two signals. It has been commonly used to process biosignals. In this article, we investigate the PLI for measuring the interdependency of cortical source signals (CSSs) recorded in the Electroencephalogram (EEG). To this end, we consider simple analytical models for the mapping of simulated CSSs into the EEG. For these models, the PLI is investigated analytically and through numerical simulations. An evaluation is made of the sensitivity of the PLI to the amount of crosstalk between the sources through biological tissues of the head. It is found that the PLI is a useful interdependency measure for CSSs, especially when the amount of crosstalk is small. Another common interdependency measure is the coherence. A direct comparison of both measures has not been made in the literature so far. We assess the performance of the PLI and coherence for estimation and detection purposes based on, respectively, a normalized variance and a novel statistical measure termed contrast. Based on these performance measures, it is found that the PLI is similar or better than the CM in most cases. This result is also confirmed through analysis of EEGs recorded from epileptic patients.
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paper_title: Enhancement of phase clustering in the EEG/MEG gamma frequency band anticipates transitions to paroxysmal epileptiform activity in epileptic patients with known visual sensitivity

paper_content:

A new analytical method for quantifying brain activity from magnetoelectroencephalogram (MEG) and electroencephalogram (EEG) recordings during periodic light stimulation is proposed. It consists in estimating the phase clustering of harmonically related frequency components of a subject's MEG/EEG responses evoked by the light stimulation. The method was developed to test the hypothesis that changes in the dynamics of brain systems in the course of intermittent photic stimulation (IPS) may precede the transition to seizure activity in photosensitive patients. We assumed that such changes would be reflected in the phase of harmonic components of the evoked responses. Thus, we determined the phase clustering for different harmonic components of these MEG/EEG signals. We found that the patients who develop epileptiform discharges during IPS present an enhancement of the phase clustering index at the gamma frequency band, compared with that at the driving frequency. We introduce a quantity-relative phase clustering index (rPCI)-by means of which this enhancement can be quantified. We argue that this quantity reflects the degree of excitability of the underlying dynamical system and it can indicate presence of nonlinear dynamics. rPCI can be applied to detect transitions to epileptic seizure activity in patients with known sensitivity to IPS.
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paper_title: Estimation of Effective and Functional Cortical Connectivity From Neuroelectric and Hemodynamic Recordings

paper_content:

In this paper, different linear and nonlinear methodologies for the estimation of cortical connectivity from neuroelectric and hemodynamic measurements are reviewed and applied on common data set in order to highlight similarities and differences in the results. Different effective and functional connectivity methods were applied to motor and cognitive data sets, including structural equation modeling (SEM), directed transfer function (DTF), partial directed coherence (PDC), and direct directed transfer function (dDTF). Comparisons were made between the results in order to understand if, for a same dataset, effective and functional connectivity estimators can return the same cortical connectivity patterns. An application of a nonlinear method [phase synchronization index (PSI)] to similar executed and imagined movements was also reviewed. Connectivity patterns estimated with the use of the neuroelectric information and of the information from the multimodal integration of neuroelectric and hemodynamic data were also compared. Results suggests that the estimation of the cortical connectivity patterns performed with the linear methods (SEM, DTF, PDC, dDTF) or with the nonlinear method (PSI) on movement related potentials returned similar cortical networks. Differences in cortical connectivity were noted between the patterns estimated with the use of multimodal integration and those estimated by using only the neuroelectric data.
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paper_title: Short-window spectral analysis of cortical event-related potentials by adaptive multivariate autoregressive modeling: data preprocessing, model validation, and variability assessment

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In this article we consider the application of parametric spectral analysis to multichannel event-related potentials (ERPs) during cognitive experiments. We show that with proper data preprocessing, Adaptive MultiVariate AutoRegressive (AMVAR) modeling is an effective technique for dealing with nonstationary ERP time series. We propose a bootstrap procedure to assess the variability in the estimated spectral quantities. Finally, we apply AMVAR spectral analysis to a visuomotor integration task, revealing rapidly changing cortical dynamics during different stages of task processing.
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Title: Inferring Functional Neural Connectivity with Phase Synchronization Analysis: A Review of Methodology

Section 1: Introduction
Description 1: Provide background information about neural connectivity, the importance of phase synchronization analysis, and the scope of the review.

Section 2: Definition of Instantaneous Phase
Description 2: Discuss the basic definition of instantaneous phase (IP) and various methods to define IP, such as using the Hilbert transform, Gaussian filter, and wavelet transform.

Section 3: Phase Synchronization Indexes
Description 3: Introduce different phase synchronization indexes (PSIs) including entropy-based PSI and mean phase coherence (MPC), and how these indexes are used to quantify functional connectivity.

Section 4: Significance Tests for PSI
Description 4: Describe methods for determining the significance of estimated PSI, including artificial surrogate tests, analytical significance tests, and surrogate tests based on intersubject signal pairs.

Section 5: Problems in Phase Synchronization Detection
Description 5: Discuss challenges in phase synchronization detection such as noise contamination, limited sample sizes, and the biased nature of PSI estimations.

Section 6: Effect of Noise in PS Detection
Description 6: Examine the influence of noise on the estimation of PSI and methods to mitigate its effects.

Section 7: Influence of Volume Conduction
Description 7: Discuss the problem of volume conduction in EEG/MEG signals and techniques to address it, such as using the imaginary part of coherency and phase lag index.

Section 8: Influence of Reference
Description 8: Address how different reference strategies for EEG signals can affect PSI estimation and suggest ways to mitigate these influences.

Section 9: Extensions of Phase Synchronization Analysis
Description 9: Introduce advanced methods for phase synchronization analysis, such as trial-based PSI and partial PSI, and discuss their applications.

Section 10: Discussions and Conclusions
Description 10: Summarize the main points discussed in the review, the advantages and limitations of PS analysis, and possible directions for future research.