Surrogate analysis of coherent multichannel data

A Model to Study Time Lagged Interactions, Source Connectivity and Source Activities Using Multi-channel EEG

A computational model to examine time lagged interactions; identify number of interacting pairs of neuronal sources; and determine source activities from multi-channel EEG measurements is described. It is based on the imaginary part of the cross spectrum between the EEG channels. The imaginary part of the cross spectrum between the EEG channels provides the most suitable property that reflects the presence of interacting sources. The model assumes that not all sources are activated simultaneously and that there is a time lag amongst some of them. A new analytical expression derived for the imaginary part of cross spectrum between channels shows that it is different from the zero lag case. A method is then proposed to identify time lag interactions, by studying its variation as a function of frequency. Assuming pair wise interaction between sources, the model shows that simultaneous diagonalization at different frequencies of symmetric matrices formed by multiplying the anti-symmetric matrix of the imaginary part of cross spectrum with its transpose will provide information on the number of interacting source pairs as a function of frequency. The matrix that simultaneously diagonalizes all the symmetric matrices is identified as the mixing matrix. This can be used to obtain the source activities.

Connectivity Analysis in EEG Data: A Tutorial Review of the State of the Art and Emerging Trends

Understanding how different areas of the human brain communicate with each other is a crucial issue in neuroscience. The concepts of structural, functional and effective connectivity have been widely exploited to describe the human connectome, consisting of brain networks, their structural connections and functional interactions. Despite high-spatial-resolution imaging techniques such as functional magnetic resonance imaging (fMRI) being widely used to map this complex network of multiple interactions, electroencephalographic (EEG) recordings claim high temporal resolution and are thus perfectly suitable to describe either spatially distributed and temporally dynamic patterns of neural activation and connectivity. In this work, we provide a technical account and a categorization of the most-used data-driven approaches to assess brain-functional connectivity, intended as the study of the statistical dependencies between the recorded EEG signals. Different pairwise and multivariate, as well as directed and non-directed connectivity metrics are discussed with a pros-cons approach, in the time, frequency, and information-theoretic domains. The establishment of conceptual and mathematical relationships between metrics from these three frameworks, and the discussion of novel methodological approaches, will allow the reader to go deep into the problem of inferring functional connectivity in complex networks. Furthermore, emerging trends for the description of extended forms of connectivity (e.g., high-order interactions) are also discussed, along with graph-theory tools exploring the topological properties of the network of connections provided by the proposed metrics. Applications to EEG data are reviewed. In addition, the importance of source localization, and the impacts of signal acquisition and pre-processing techniques (e.g., filtering, source localization, and artifact rejection) on the connectivity estimates are recognized and discussed. By going through this review, the reader could delve deeply into the entire process of EEG pre-processing and analysis for the study of brain functional connectivity and learning, thereby exploiting novel methodologies and approaches to the problem of inferring connectivity within complex networks.

Noise robust approach to reconstruction of van der Pol-like oscillators and its application to Granger causality

Van der Pol oscillators and their generalizations are known to be a fundamental model in the theory of oscillations and their applications. Many objects of a different nature can be described using van der Pol-like equations under some circumstances; therefore, methods of reconstruction of such equations from experimental data can be of significant importance for tasks of model verification, indirect parameter estimation, coupling analysis, system classification, etc. The previously reported techniques were not applicable to time series with large measurement noise, which is usual in biological, climatological, and many other experiments. Here, we present a new approach based on the use of numerical integration instead of the differentiation and implicit approximation of a nonlinear dissipation function. We show that this new technique can work for noise levels up to 30% by standard deviation from the signal for different types of autonomous van der Pol-like systems and for ensembles of such systems, providing a new approach to the realization of the Granger-causality idea.

A Simulation Framework for Benchmarking EEG-Based Brain Connectivity Estimation Methodologies

Due to its high temporal resolution, electroencephalography (EEG) is widely used to study functional and effective brain connectivity. Yet, there is currently a mismatch between the vastness of studies conducted and the degree to which the employed analyses are theoretically understood and empirically validated. We here provide a simulation framework that enables researchers to test their analysis pipelines on realistic pseudo-EEG data. We construct a minimal example of brain interaction, which we propose as a benchmark for assessing a methodology's general eligibility for EEG-based connectivity estimation. We envision that this benchmark be extended in a collaborative effort to validate methods in more complex scenarios. Quantitative metrics are defined to assess a method's performance in terms of source localization, connectivity detection and directionality estimation. All data and code needed for generating pseudo-EEG data, conducting source reconstruction and connectivity estimation using baseline methods from the literature, evaluating performance metrics, as well as plotting results, are made publicly available. While this article covers only EEG modeling, we will also provide a magnetoencephalography version of our framework online.

Detecting triplet locking by triplet synchronization indices

We discuss the effect of triplet synchrony in oscillatory networks. In this state the phases and the frequencies of three coupled oscillators fulfill the conditions of a triplet locking, whereas every pair of systems remains asynchronous. We suggest an easy to compute measure, a triplet synchronization index, which can be used to detect such states from experimental data.

Nonrandomness, nonlinear dependence, and nonstationarity of electroencephalographic recordings from epilepsy patients

To derive tests for randomness, nonlinear-independence, and stationarity, we combine surrogates with a nonlinear prediction error, a nonlinear interdependence measure, and linear variability measures, respectively. We apply these tests to intracranial electroencephalographic recordings (EEG) from patients suffering from pharmacoresistant focal-onset epilepsy. These recordings had been performed prior to and independent from our study as part of the epilepsy diagnostics. The clinical purpose of these recordings was to delineate the brain areas to be surgically removed in each individual patient in order to achieve seizure control. This allowed us to define two distinct sets of signals: One set of signals recorded from brain areas where the first ictal EEG signal changes were detected as judged by expert visual inspection ("focal signals") and one set of signals recorded from brain areas that were not involved at seizure onset ("nonfocal signals"). We find more rejections for both the randomness and the nonlinear-independence test for focal versus nonfocal signals. In contrast more rejections of the stationarity test are found for nonfocal signals. Furthermore, while for nonfocal signals the rejection of the stationarity test increases the rejection probability of the randomness and nonlinear-independence test substantially, we find a much weaker influence for the focal signals. In consequence, the contrast between the focal and nonfocal signals obtained from the randomness and nonlinear-independence test is further enhanced when we exclude signals for which the stationarity test is rejected. To study the dependence between the randomness and nonlinear-independence test we include only focal signals for which the stationarity test is not rejected. We show that the rejection of these two tests correlates across signals. The rejection of either test is, however, neither necessary nor sufficient for the rejection of the other test. Thus, our results suggest that EEG signals from epileptogenic brain areas are less random, more nonlinear-dependent, and more stationary compared to signals recorded from nonepileptogenic brain areas. We provide the data, source code, and detailed results in the public domain.

Coupling of mesoscopic brain oscillations: recent advances in analytical and theoretical perspectives

Oscillatory brain activities have been traditionally studied in the context of how oscillations at a single frequency recorded from a single area could reveal functional insights. Recent advances in methodology used in signal analysis have revealed that cross-frequency coupling, within or between functional related areas, is more informative in determining the possible roles played by brain oscillations. In this review, we begin by describing the cellular basis of oscillatory field potentials and its theorized as well as demonstrated role in brain function. The recent development of mathematical tools that allow the investigation of cross-frequency and cross-area oscillation coupling will be presented and discussed in the context of recent advances in oscillation research based on animal data. Particularly, some pitfalls and caveats of methods currently available are discussed. Data generated from the application of examined techniques are integrated back into the theoretical framework regarding the functional role of brain oscillations. We suggest that the coupling of oscillatory activities at different frequencies between brain regions is crucial for understanding the brain from a functional ensemble perspective. Effort should be directed to elucidate how cross-frequency and area coupling are modulated and controlled. To achieve this, only the correct application of analytical tools may shed light on the intricacies of information representation, generation, binding, encoding, storage and retrieval in the brain.

Hypothesis test for synchronization: twin surrogates revisited

The method of twin surrogates has been introduced to test for phase synchronization of complex systems in the case of passive experiments. In this paper we derive new analytical expressions for the number of twins depending on the size of the neighborhood, as well as on the length of the trajectory. This allows us to determine the optimal parameters for the generation of twin surrogates. Furthermore, we determine the quality of the twin surrogates with respect to several linear and nonlinear statistics depending on the parameters of the method. In the second part of the paper we perform a hypothesis test for phase synchronization in the case of experimental data from fixational eye movements. These miniature eye movements have been shown to play a central role in neural information processing underlying the perception of static visual scenes. The high number of data sets (21 subjects and 30 trials per person) allows us to compare the generated twin surrogates with the "natural" surrogates that correspond to the different trials. We show that the generated twin surrogates reproduce very well all linear and nonlinear characteristics of the underlying experimental system. The synchronization analysis of fixational eye movements by means of twin surrogates reveals that the synchronization between the left and right eye is significant, indicating that either the centers in the brain stem generating fixational eye movements are closely linked, or, alternatively that there is only one center controlling both eyes.

Intermittent dynamics underlying the intrinsic fluctuations of the collective synchronization patterns in electrocortical activity

We investigate patterns of collective phase synchronization in brain activity in awake, resting humans with eyes closed. The alpha range of human electroencephalographic activity is characterized by ever-changing patterns, with strong fluctuations in both time and overall level of phase synchronization. The correlations of these patterns are reflected in power-law scaling of these properties. We present evidence that the dynamics underlying this fluctuation is type-I intermittency. We present a model study illustrating that the scaling property and the collective intermittent dynamics are emergent features of globally coupled phase oscillators near the critical point of entering global frequency locking.

swLORETA: a novel approach to robust source localization and synchronization tomography

Standardized low-resolution brain electromagnetic tomography (sLORETA) is a widely used technique for source localization. However, this technique still has some limitations, especially under realistic noisy conditions and in the case of deep sources. To overcome these problems, we present here swLORETA, an improved version of sLORETA, obtained by incorporating a singular value decomposition-based lead field weighting. We show that the precision of the source localization can further be improved by a tomographic phase synchronization analysis based on swLORETA. The phase synchronization analysis turns out to be superior to a standard linear coherence analysis, since the latter cannot distinguish between real phase locking and signal mixing.

Identifying interactions in mixed and noisy complex systems

We present a technique that identifies truly interacting subsystems of a complex system from multichannel data if the recordings are an unknown linear and instantaneous mixture of the true sources. The method is valid for arbitrary noise structure. For this, a blind source separation technique is proposed that diagonalizes antisymmetrized cross-correlation or cross-spectral matrices. The resulting decomposition finds truly interacting subsystems blindly and suppresses any spurious interaction stemming from the mixture. The usefulness of this interacting source analysis is demonstrated in simulations and for real electroencephalography data.

Nonlinear dynamical analysis of EEG and MEG: review of an emerging field

Many complex and interesting phenomena in nature are due to nonlinear phenomena. The theory of nonlinear dynamical systems, also called 'chaos theory', has now progressed to a stage, where it becomes possible to study self-organization and pattern formation in the complex neuronal networks of the brain. One approach to nonlinear time series analysis consists of reconstructing, from time series of EEG or MEG, an attractor of the underlying dynamical system, and characterizing it in terms of its dimension (an estimate of the degrees of freedom of the system), or its Lyapunov exponents and entropy (reflecting unpredictability of the dynamics due to the sensitive dependence on initial conditions). More recently developed nonlinear measures characterize other features of local brain dynamics (forecasting, time asymmetry, determinism) or the nonlinear synchronization between recordings from different brain regions. Nonlinear time series has been applied to EEG and MEG of healthy subjects during no-task resting states, perceptual processing, performance of cognitive tasks and different sleep stages. Many pathologic states have been examined as well, ranging from toxic states, seizures, and psychiatric disorders to Alzheimer's, Parkinson's and Cre1utzfeldt-Jakob's disease. Interpretation of these results in terms of 'functional sources' and 'functional networks' allows the identification of three basic patterns of brain dynamics: (i) normal, ongoing dynamics during a no-task, resting state in healthy subjects; this state is characterized by a high dimensional complexity and a relatively low and fluctuating level of synchronization of the neuronal networks; (ii) hypersynchronous, highly nonlinear dynamics of epileptic seizures; (iii) dynamics of degenerative encephalopathies with an abnormally low level of between area synchronization. Only intermediate levels of rapidly fluctuating synchronization, possibly due to critical dynamics near a phase transition, are associated with normal information processing, whereas both hyper-as well as hyposynchronous states result in impaired information processing and disturbed consciousness.

Nonlinear multivariate analysis of neurophysiological signals

Multivariate time series analysis is extensively used in neurophysiology with the aim of studying the relationship between simultaneously recorded signals. Recently, advances on information theory and nonlinear dynamical systems theory have allowed the study of various types of synchronization from time series. In this work, we first describe the multivariate linear methods most commonly used in neurophysiology and show that they can be extended to assess the existence of nonlinear interdependence between signals. We then review the concepts of entropy and mutual information followed by a detailed description of nonlinear methods based on the concepts of phase synchronization, generalized synchronization and event synchronization. In all cases, we show how to apply these methods to study different kinds of neurophysiological data. Finally, we illustrate the use of multivariate surrogate data test for the assessment of the strength (strong or weak) and the type (linear or nonlinear) of interdependence between neurophysiological signals.

Measuring phase synchronization of superimposed signals

Phase synchronization is an important phenomenon that occurs in a wide variety of complex oscillatory processes. Measuring phase synchronization can therefore help to gain fundamental insight into nature. In this Letter we point out that synchronization analysis techniques can detect spurious synchronization, if they are fed with a superposition of signals such as in electroencephalography or magnetoencephalography data. We show how techniques from blind source separation can help to nevertheless measure the true synchronization and avoid such pitfalls.

Bivariate surrogate techniques: necessity, strengths, and caveats

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

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

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

Scale-invariant fluctuations of the dynamical synchronization in human brain electrical activity

The dynamical properties of large-scale, long-term phase synchronization behavior in the alpha range of electroencephalographic signals were investigated. We observed dynamical phase synchronization and presented evidence of an underlying spatiotemporal ordering. Fluctuations in the duration of episodes of intermittent synchrony are scale-invariant. Moreover, the exponent used to describe this behavior is stable across different normal subjects. The results provide a new feature of self-organization in human brain activity and constitute a quantitative basis for modeling its dynamics.