Detection and characterization of changes of the correlation structure in multivariate time series

Stationary correlation pattern in highly non-stationary MEG recordings of healthy subjects and its relation to former EEG studies

In this study, we analyze magnetoencephalographic (MEG) recordings from 48 clinically healthy subjects obtained from the Human Connectome Project (HCP) while they performed a working memory task and a motor task. Our results reveal a well-developed, stable interrelation pattern that spans the entire scalp and is nearly universal, being almost task- and subject-independent. Additionally, we demonstrate that this pattern closely resembles a stationary correlation pattern (SCP) observed in EEG signals under various physiological and pathological conditions (the distribution of Pearson correlations are centered at about 0.75). Furthermore, we identify the most effective EEG reference for studying the brain's functional network derived from lag-zero cross-correlations. We contextualize these findings within the theory of complex dynamical systems operating near a critical point of a phase transition.

Harmonic cross-correlation decomposition for multivariate time series

We introduce harmonic cross-correlation decomposition (HCD) as a tool to detect and visualize features in the frequency structure of multivariate time series. HCD decomposes multivariate time series into spatiotemporal harmonic modes with the leading modes representing dominant oscillatory patterns in the data. HCD is closely related to data-adaptive harmonic decomposition (DAHD) [Chekroun and Kondrashov, Chaos 27, 093110 (2017)10.1063/1.4989400] in that it performs an eigendecomposition of a grand matrix containing lagged cross-correlations. As for DAHD, each HCD mode is uniquely associated with a Fourier frequency, which allows for the definition of multidimensional power and phase spectra. Unlike in DAHD, however, HCD does not exhibit a systematic dependency on the ordering of the channels within the grand matrix. Further, HCD phase spectra can be related to the phase relations in the data in an intuitive way. We compare HCD with DAHD and multivariate singular spectrum analysis, a third related correlation-based decomposition, and we give illustrative applications to a simple traveling wave, as well as to simulations of three coupled Stuart-Landau oscillators and to human EEG recordings.

Epileptic seizures in a heterogeneous excitatory network with short-term plasticity

Epilepsy involves a diverse group of abnormalities, including molecular and cellular disorders. These abnormalities prove to be associated with the changes in local excitability and synaptic dynamics. Correspondingly, the epileptic processes including onset, propagation and generalized seizure may be related with the alterations of excitability and synapse. In this paper, three regions, epileptogenic zone (EZ), propagation area and normal region, were defined and represented by neuronal population model with heterogeneous excitability, respectively. In order to describe the synaptic behavior that the strength was enhanced and maintained at a high level for a short term under a high frequency spike train, a novel activity-dependent short-term plasticity model was proposed. Bifurcation analysis showed that the presence of hyperexcitability could increase the seizure susceptibility of local area, leading to epileptic discharges first seen in the EZ. Meanwhile, recurrent epileptic activities might result in the transition of synaptic strength from weak state to high level, augmenting synaptic depolarizations in non-epileptic neurons as the experimental findings. Numerical simulation based on a full-connected weighted network could qualitatively demonstrate the epileptic process that the propagation area and normal region were successively recruited by the EZ. Furthermore, cross recurrence plot was used to explore the synchronization between neuronal populations, and the global synchronization index was introduced to measure the global synchronization. Results suggested that the synchronization between the EZ and other region was significantly enhanced with the occurrence of seizure. Interestingly, the desynchronization phenomenon was also observed during seizure initiation and propagation as reported before. Therefore, heterogeneous excitability and short-term plasticity are believed to play an important role in the epileptic process. This study may provide novel insights into the mechanism of epileptogenesis.

EMD-Based, Mean-Phase Coherence Analysis to Assess Instantaneous Phase-Synchrony Dynamics in Epilepsy Patients

In this paper, an adaptive, non-linear, analytical methodology is proposed in order to quantitatively evaluate the instantaneous phase-synchrony dynamics in epilepsy patients. A group of finite neuronal oscillators is extracted from a multichannel electrocorticographic (ECoG) data, using the empirical mode decomposition (EMD). The instantaneous phases of the extracted oscillators are measured using the Hilbert transform in order to be utilized in the mean-phase coherence analysis. Finally, the dynamical evolution of phase-synchrony among the extracted neuronal oscillators within 1-600 Hz frequency range is assessed using eigenvalue decomposition. A different phasesynchrony dynamics was observed in two patients with frontal vs. temporal lobe epilepsy, as their seizures evolve. However, experimental results demonstrated a hypersynchrony level at seizure offset for both types of epilepsy during the ictal periods. This result suggests that hypersynchronization of the epileptic network may be a crucial, self-regulatory mechanism by which the brain terminate seizures.

Linear and nonlinear interrelations show fundamentally distinct network structure in preictal intracranial EEG of epilepsy patients

Resection of the seizure generating tissue can be highly beneficial in patients with drug-resistant epilepsy. However, only about half of all patients undergoing surgery get permanently and completely seizure free. Investigating the dependences between intracranial EEG signals adds a multivariate perspective largely unavailable to visual EEG analysis, which is the current clinical practice. We examined linear and nonlinear interrelations between intracranial EEG signals regarding their spatial distribution and network characteristics. The analyzed signals were recorded immediately before clinical seizure onset in epilepsy patients who received a standardized electrode implantation targeting the mesiotemporal structures. The linear interrelation networks were predominantly locally connected and highly reproducible between patients. In contrast, the nonlinear networks had a clearly centralized structure, which was specific for the individual pathology. The nonlinear interrelations were overrepresented in the focal hemisphere and in patients with no or only rare seizures after surgery specifically in the resected tissue. Connections to the outside were predominantly nonlinear. In all patients without worthwhile improvement after resective treatment, tissue producing strong nonlinear interrelations was left untouched by surgery. Our findings indicate that linear and nonlinear interrelations play fundamentally different roles in preictal intracranial EEG. Moreover, they suggest nonlinear signal interrelations to be a marker of epileptogenic tissue and not a characteristic of the mesiotemporal structures. Our results corroborate the network-based nature of epilepsy and suggest the application of network analysis to support the planning of resective epilepsy surgery.

The Influence of EEG References on the Analysis of Spatio-Temporal Interrelation Patterns

The characterization of the functional network of the brain dynamics has become a prominent tool to illuminate novel aspects of brain functioning. Due to its excellent time resolution, such research is oftentimes based on electroencephalographic recordings (EEG). However, a particular EEG-reference might cause crucial distortions of the spatiotemporal interrelation pattern and may induce spurious correlations as well as diminish genuine interrelations originally present in the dataset. Here we investigate in which manner correlation patterns are affected by a chosen EEG reference. To this end we evaluate the influence of 7 popular reference schemes on artificial recordings derived from well controlled numerical test frameworks. In this respect we are not only interested in the deformation of spatial interrelations, but we test additionally in which way the time evolution of the functional network, estimated via some bi-variate interrelation measures, gets distorted. It turns out that the median reference as well as the global average show the best performance in most situations considered in the present study. However, if a collective brain dynamics is present, where most of the signals get correlated, these schemes may also cause crucial deformations of the functional network, such that the parallel use of different reference schemes seems advisable.

Noise-Assisted Multivariate EMD-Based Mean-Phase Coherence Analysis to Evaluate Phase-Synchrony Dynamics in Epilepsy Patients

Spatiotemporal evolution of synchrony dynamics among neuronal populations plays an important role in decoding complicated brain function in normal cognitive processing as well as during pathological conditions such as epileptic seizures. In this paper, a non-linear analytical methodology is proposed to quantitatively evaluate the phase-synchrony dynamics in epilepsy patients. A set of finite neuronal oscillators was adaptively extracted from a multi-channel electrocorticographic (ECoG) dataset utilizing noise-assisted multivariate empirical mode de-composition (NA-MEMD). Next, the instantaneous phases of the oscillatory functions were extracted using the Hilbert transform in order to be utilized in the mean-phase coherence analysis. The phase-synchrony dynamics were then assessed using eigenvalue decomposition. The extracted neuronal oscillators were grouped with respect to their frequency range into wideband (1-600 Hz), ripple (80-250 Hz), and fast-ripple (250-600 Hz) bands in order to investigate the dynamics of ECoG activity in these frequency ranges as seizures evolve. Drug-refractory patients with frontal and temporal lobe epilepsy demonstrated a reduction in phase-synchrony around seizure onset. However, the network phase-synchrony started to increase toward seizure end and achieved its maximum level at seizure offset for both types of epilepsy. This result suggests that hyper-synchronization of the epileptic network may be an essential self-regulatory mechanism by which the brain terminates seizures.

Detecting correlation changes in multivariate time series: A comparison of four non-parametric change point detection methods

Change point detection in multivariate time series is a complex task since next to the mean, the correlation structure of the monitored variables may also alter when change occurs. DeCon was recently developed to detect such changes in mean and\or correlation by combining a moving windows approach and robust PCA. However, in the literature, several other methods have been proposed that employ other non-parametric tools: E-divisive, Multirank, and KCP. Since these methods use different statistical approaches, two issues need to be tackled. First, applied researchers may find it hard to appraise the differences between the methods. Second, a direct comparison of the relative performance of all these methods for capturing change points signaling correlation changes is still lacking. Therefore, we present the basic principles behind DeCon, E-divisive, Multirank, and KCP and the corresponding algorithms, to make them more accessible to readers. We further compared their performance through extensive simulations using the settings of Bulteel et al. (Biological Psychology, 98 (1), 29-42, 2014) implying changes in mean and in correlation structure and those of Matteson and James (Journal of the American Statistical Association, 109 (505), 334-345, 2014) implying different numbers of (noise) variables. KCP emerged as the best method in almost all settings. However, in case of more than two noise variables, only DeCon performed adequately in detecting correlation changes.

Spatially extended versus frontal cerebral near-infrared spectroscopy during cardiac surgery: a case series identifying potential advantages

Stroke due to hypoperfusion or emboli is a devastating adverse event of cardiac surgery, but early detection and treatment could protect patients from an unfavorable postoperative course. Hypoperfusion and emboli can be detected with transcranial Doppler of the middle cerebral artery (MCA). The measured blood flow velocity correlates with cerebral oxygenation determined clinically by near-infrared spectroscopy (NIRS) of the frontal cortex. We tested the potential advantage of a spatially extended NIRS in detecting critical events in three cardiac surgery patients with a whole-head fiber holder of the FOIRE-3000 continuous-wave NIRS system. Principle components analysis was performed to differentiate between global and localized hypoperfusion or ischemic territories of the middle and anterior cerebral arteries. In one patient, we detected a critical hypoperfusion of the right MCA, which was not apparent in the frontal channels but was accompanied by intra- and postoperative neurological correlates of ischemia. We conclude that spatially extended NIRS of temporal and parietal vascular territories could improve the detection of critically low cerebral perfusion. Even in severe hemispheric stroke, NIRS of the frontal lobe may remain normal because the anterior cerebral artery can be supplied by the contralateral side directly or via the anterior communicating artery.

On the estimation of phase synchronization, spurious synchronization and filtering

Phase synchronization, viz., the adjustment of instantaneous frequencies of two interacting self-sustained nonlinear oscillators, is frequently used for the detection of a possible interrelationship between empirical data recordings. In this context, the proper estimation of the instantaneous phase from a time series is a crucial aspect. The probability that numerical estimates provide a physically relevant meaning depends sensitively on the shape of its power spectral density. For this purpose, the power spectrum should be narrow banded possessing only one prominent peak [M. Chavez et al., J. Neurosci. Methods 154, 149 (2006)]. If this condition is not fulfilled, band-pass filtering seems to be the adequate technique in order to pre-process data for a posterior synchronization analysis. However, it was reported that band-pass filtering might induce spurious synchronization [L. Xu et al., Phys. Rev. E 73, 065201(R), (2006); J. Sun et al., Phys. Rev. E 77, 046213 (2008); and J. Wang and Z. Liu, EPL 102, 10003 (2013)], a statement that without further specification causes uncertainty over all measures that aim to quantify phase synchronization of broadband field data. We show by using signals derived from different test frameworks that appropriate filtering does not induce spurious synchronization. Instead, filtering in the time domain tends to wash out existent phase interrelations between signals. Furthermore, we show that measures derived for the estimation of phase synchronization like the mean phase coherence are also useful for the detection of interrelations between time series, which are not necessarily derived from coupled self-sustained nonlinear oscillators.

Spurious cross-frequency amplitude-amplitude coupling in nonstationary, nonlinear signals

Recent studies of brain activities show that cross-frequency coupling (CFC) play an important role in memory and learning. Many measures have been proposed to investigate the CFC phenomenon, including the correlation between the amplitude envelopes of two brain waves at different frequencies - cross-frequency amplitude-amplitude coupling (AAC). In this short communication, we describe how nonstationary, nonlinear oscillatory signals may produce spurious cross-frequency AAC. Utilizing the empirical mode decomposition, we also propose a new method for assessment of AAC that can potentially reduce the effects of nonlinearity and nonstatonarity and, thus, help to avoid the detection of artificial AACs. We compare the performances of this new method and the traditional Fourier-based AAC method. We also discuss the strategies to identify potential spurious AACs.

Spectral properties of the temporal evolution of brain network structure

The temporal evolution properties of the brain network are crucial for complex brain processes. In this paper, we investigate the differences in the dynamic brain network during resting and visual stimulation states in a task-positive subnetwork, task-negative subnetwork, and whole-brain network. The dynamic brain network is first constructed from human functional magnetic resonance imaging data based on the sliding window method, and then the eigenvalues corresponding to the network are calculated. We use eigenvalue analysis to analyze the global properties of eigenvalues and the random matrix theory (RMT) method to measure the local properties. For global properties, the shifting of the eigenvalue distribution and the decrease in the largest eigenvalue are linked to visual stimulation in all networks. For local properties, the short-range correlation in eigenvalues as measured by the nearest neighbor spacing distribution is not always sensitive to visual stimulation. However, the long-range correlation in eigenvalues as evaluated by spectral rigidity and number variance not only predicts the universal behavior of the dynamic brain network but also suggests non-consistent changes in different networks. These results demonstrate that the dynamic brain network is more random for the task-positive subnetwork and whole-brain network under visual stimulation but is more regular for the task-negative subnetwork. Our findings provide deeper insight into the importance of spectral properties in the functional brain network, especially the incomparable role of RMT in revealing the intrinsic properties of complex systems.

Standing waves as an explanation for generic stationary correlation patterns in noninvasive EEG of focal onset seizures

Cerebral electrical activity is highly nonstationary because the brain reacts to ever changing external stimuli and continuously monitors internal control circuits. However, a large amount of energy is spent to maintain remarkably stationary activity patterns and functional inter-relations between different brain regions. Here we examine linear EEG correlations in the peri-ictal transition of focal onset seizures, which are typically understood to be manifestations of dramatically changing inter-relations. Contrary to expectations we find stable correlation patterns with a high similarity across different patients and different frequency bands. This skeleton of spatial correlations may be interpreted as a signature of standing waves of electrical brain activity constituting a dynamical ground state. Such a state could promote the formation of spatiotemporal neuronal assemblies and may be important for the integration of information stemming from different local circuits of the functional brain network.

Emerging spectra of singular correlation matrices under small power-map deformations

Correlation matrices are a standard tool in the analysis of the time evolution of complex systems in general and financial markets in particular. Yet most analysis assume stationarity of the underlying time series. This tends to be an assumption of varying and often dubious validity. The validity of the assumption improves as shorter time series are used. If many time series are used, this implies an analysis of highly singular correlation matrices. We attack this problem by using the so-called power map, which was introduced to reduce noise. Its nonlinearity breaks the degeneracy of the zero eigenvalues and we analyze the sensitivity of the so-emerging spectra to correlations. This sensitivity will be demonstrated for uncorrelated and correlated Wishart ensembles.

Distribution of the smallest eigenvalue in the correlated Wishart model

Wishart random matrix theory is of major importance for the analysis of correlated time series. The distribution of the smallest eigenvalue for Wishart correlation matrices is particularly interesting in many applications. In the complex and in the real case, we calculate it exactly for arbitrary empirical eigenvalues, i.e., for fully correlated Gaussian Wishart ensembles. To this end, we derive certain dualities of matrix models in ordinary space. We thereby completely avoid the otherwise unsurmountable problem of computing a highly nontrivial group integral. Our results are compact and much easier to handle than previous ones. Furthermore, we obtain a new universality for the distribution of the smallest eigenvalue on the proper local scale.

Genuine and spurious phase synchronization strengths during consciousness and general anesthesia

Spectral content in a physiological dataset of finite size has the potential to produce spurious measures of coherence. This is especially true for electroencephalography (EEG) during general anesthesia because of the significant alteration of the power spectrum. In this study we quantitatively evaluated the genuine and spurious phase synchronization strength (PSS) of EEG during consciousness, general anesthesia, and recovery. A computational approach based on the randomized data method was used for evaluating genuine and spurious PSS. The validity of the method was tested with a simulated dataset. We applied this method to the EEG of normal subjects undergoing general anesthesia and investigated the finite size effects of EEG references, data length and spectral content on phase synchronization. The most influential factor for genuine PSS was the type of EEG reference; the most influential factor for spurious PSS was the spectral content. Genuine and spurious PSS showed characteristic temporal patterns for each frequency band across consciousness and anesthesia. Simultaneous measurement of both genuine and spurious PSS during general anesthesia is necessary in order to avoid incorrect interpretations regarding states of consciousness.

Seizure prediction using EEG spatiotemporal correlation structure

A seizure prediction algorithm is proposed that combines novel multivariate EEG features with patient-specific machine learning. The algorithm computes the eigenspectra of space-delay correlation and covariance matrices from 15-s blocks of EEG data at multiple delay scales. The principal components of these features are used to classify the patient's preictal or interictal state. This is done using a support vector machine (SVM), whose outputs are averaged using a running 15-minute window to obtain a final prediction score. The algorithm was tested on 19 of 21 patients in the Freiburg EEG data set who had three or more seizures, predicting 71 of 83 seizures, with 15 false predictions and 13.8 h in seizure warning during 448.3 h of interictal data. The proposed algorithm scales with the number of available EEG signals by discovering the variations in correlation structure among any given set of signals that correlate with seizure risk.

Evolution of genuine cross-correlation strength of focal onset seizures

To quantify the evolution of genuine zero-lag cross-correlations of focal onset seizures, we apply a recently introduced multivariate measure to broad band and to narrow-band EEG data. For frequency components below 12.5 Hz, the strength of genuine cross-correlations decreases significantly during the seizure and the immediate postseizure period, while higher frequency bands show a tendency of elevated cross-correlations during the same period. We conclude that in terms of genuine zero-lag cross-correlations, the electrical brain activity as assessed by scalp electrodes shows a significant spatial fragmentation, which might promote seizure offset.

Quickest detection of drug-resistant seizures: an optimal control approach

Epilepsy affects 50 million people worldwide, and seizures in 30% of the cases remain drug resistant. This has increased interest in responsive neurostimulation, which is most effective when administered during seizure onset. We propose a novel framework for seizure onset detection that involves (i) constructing statistics from multichannel intracranial EEG (iEEG) to distinguish nonictal versus ictal states; (ii) modeling the dynamics of these statistics in each state and the state transitions; you can remove this word if there is no room. (iii) developing an optimal control-based "quickest detection" (QD) strategy to estimate the transition times from nonictal to ictal states from sequential iEEG measurements. The QD strategy minimizes a cost function of detection delay and false positive probability. The solution is a threshold that non-monotonically decreases over time and avoids responding to rare events that normally trigger false positives. We applied QD to four drug resistant epileptic patients (168 hour continuous recordings, 26-44 electrodes, 33 seizures) and achieved 100% sensitivity with low false positive rates (0.16 false positive/hour). This article is part of a Supplemental Special Issue entitled The Future of Automated Seizure Detection and Prediction.

A phase-synchronization and random-matrix based approach to multichannel time-series analysis with application to epilepsy

We present a general method to analyze multichannel time series that are becoming increasingly common in many areas of science and engineering. Of particular interest is the degree of synchrony among various channels, motivated by the recognition that characterization of synchrony in a system consisting of many interacting components can provide insights into its fundamental dynamics. Often such a system is complex, high-dimensional, nonlinear, nonstationary, and noisy, rendering unlikely complete synchronization in which the dynamical variables from individual components approach each other asymptotically. Nonetheless, a weaker type of synchrony that lasts for a finite amount of time, namely, phase synchronization, can be expected. Our idea is to calculate the average phase-synchronization times from all available pairs of channels and then to construct a matrix. Due to nonlinearity and stochasticity, the matrix is effectively random. Moreover, since the diagonal elements of the matrix can be arbitrarily large, the matrix can be singular. To overcome this difficulty, we develop a random-matrix based criterion for proper choosing of the diagonal matrix elements. Monitoring of the eigenvalues and the determinant provides a powerful way to assess changes in synchrony. The method is tested using a prototype nonstationary noisy dynamical system, electroencephalogram (scalp) data from absence seizures for which enhanced cortico-thalamic synchrony is presumed, and electrocorticogram (intracranial) data from subjects having partial seizures with secondary generalization for which enhanced local synchrony is similarly presumed.

Multivariate singular spectrum analysis and the road to phase synchronization

We show that multivariate singular spectrum analysis (M-SSA) greatly helps study phase synchronization in a large system of coupled oscillators and in the presence of high observational noise levels. With no need for detailed knowledge of individual subsystems nor any a priori phase definition for each of them, we demonstrate that M-SSA can automatically identify multiple oscillatory modes and detect whether these modes are shared by clusters of phase- and frequency-locked oscillators. As an essential modification of M-SSA, here we introduce variance-maximization (varimax) rotation of the M-SSA eigenvectors to optimally identify synchronized-oscillator clustering.

Eigenvalue densities of real and complex Wishart correlation matrices

Wishart correlation matrices are the standard model for the statistical analysis of time series. The ensemble averaged eigenvalue density is of considerable practical and theoretical interest. For complex time series and correlation matrices, the eigenvalue density is known exactly. In the real case, a fundamental mathematical obstacle made it forbiddingly complicated to obtain exact results. We use the supersymmetry method to fully circumvent this problem. We present an exact formula for the eigenvalue density in the real case in terms of twofold integrals and finite sums.

Analyzing spatio-temporal patterns of genuine cross-correlations

In multivariate time series analysis, the equal-time cross-correlation is a classic and computationally efficient measure for quantifying linear interrelations between data channels. When the cross-correlation coefficient is estimated using a finite amount of data points, its non-random part may be strongly contaminated by a sizable random contribution, such that no reliable conclusion can be drawn about genuine mutual interdependencies. The random correlations are determined by the signals' frequency content and the amount of data points used. Here, we introduce adjusted correlation matrices that can be employed to disentangle random from non-random contributions to each matrix element independently of the signal frequencies. Extending our previous work these matrices allow analyzing spatial patterns of genuine cross-correlation in multivariate data regardless of confounding influences. The performance is illustrated by example of model systems with known interdependence patterns. Finally, we apply the methods to electroencephalographic (EEG) data with epileptic seizure activity.

Estimation of genuine and random synchronization in multivariate neural series

Synchronization is an important mechanism that helps in understanding information processing in a normal or abnormal brain. In this paper, we propose a new method to estimate the genuine and random synchronization indexes in multivariate neural series, denoted as GSI (genuine synchronization index) and RSI (random synchronization index), by means of a correlation matrix analysis and surrogate technique. The performance of the method is evaluated by using a multi-channel neural mass model (MNMM), including the effects of different coupling coefficients, signal to noise ratios (SNRs) and time-window widths on the estimation of the GSI and RSI. Results show that the GSI and the RSI are superior in description of the synchronization in multivariate neural series compared to the S-estimator. Furthermore, the proposed method is applied to analyze a 21-channel scalp electroencephalographic recording of a 35 year-old male who suffers from mesial temporal lobe epilepsy. The GSI and the RSI at different frequency bands during the epileptic seizure are estimated. The present results could be helpful for us to understand the synchronization mechanism of epileptic seizures.

Analysis of epileptogenic network properties during ictal activity

In the present study, we utilize methods from graph theory to analyze epileptogenic network properties during periods of ictal activity. Using these methods, we analyzed the DTF-based causal information flow in nine seizures recorded from two patients undergoing presurgical monitoring for the treatment of medically intractable epilepsy. From the results, we observed a high degree of correlation between the regions with a high amount of information outflow (termed the outdegree) and the cortical areas identified clinically as the generators of the ictal activity. We furthermore observe a frequency-dependent correlation between the co-localization of these "activated regions" and the clinical foci. These findings suggest that application of network analysis tools to ictal activity could provide clinically useful information concerning these epileptogenic networks.

Peri-ictal correlation dynamics of high-frequency (80-200 Hz) intracranial EEG

PURPOSE

To assess (1) how large-scale correlation of intracranial EEG signals in the high-frequency range (80-200Hz) evolves from the pre-ictal, through the ictal into the postictal state and (2) whether the contribution of local neuronal activity to large-scale EEG correlation differentiates epileptogenic from non-epileptogenic brain tissue.

METHODS

Large-scale correlation of intracranial EEG was assessed by the total correlation strength (TCS), a measure derived from the eigenvalue spectra of zero-lag correlation matrices computed in a time-resolved manner by using a moving window approach. The relative change of total correlation strength (Delta(j)) resulting from leaving out EEG channel j ("leave-one-out approach") was used to quantify the contribution of local neuronal activity to large-scale EEG correlation.

RESULTS

19 seizures of 3 patients were analyzed. On average, TCS increased significantly from the pre-ictal to the ictal, and from the ictal to the postictal state. In the pre-ictal state, Delta(j) was significantly more negative when EEG channels that recorded the electrical activity of brain tissue considered to be epileptogenic were left out; the identification of the epileptogenic area, that was subsequently surgically removed in two patients, was based on visual analysis. The spatio-temporal pattern of Delta(j) dramatically changed at seizure onsets and endings, revealing qualitative similarities between the seizures of different patients.

DISCUSSION

The evolution of large-scale EEG correlation in the high-frequency range is qualitatively similar to the one previously described for the low-frequency range. Because the two patients who underwent surgery became seizure free, our findings are consistent with the hypothesis that epileptogenic brain tissue may be characterized by its relatively increased contribution to pre-ictal large-scale correlation.

Epilepsy and nonlinear dynamics

This overview summarizes findings obtained from analyzing electroencephalographic (EEG) recordings from epilepsy patients with methods from the theory of nonlinear dynamical systems. The last two decades have shown that nonlinear time series analysis techniques allow an improved characterization of epileptic brain states and help to gain deeper insights into the spatial and temporal dynamics of the epileptic process. Nonlinear EEG analyses can help to improve the evaluation of patients prior to neurosurgery, and with an unequivocal identification of precursors of seizures, they can be of great value in the development of seizure warning and prevention techniques.

Detecting directional coupling in the human epileptic brain: limitations and potential pitfalls

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.

Quantification of intra- and inter-cluster relations in nonstationary and noisy data

The interrelation between data channels of multivariate data sets may lead to cluster formation. Revealing the cluster structure can give important information about the underlying systems' properties. Here we investigate the features of a recent genuinely multivariate cluster detection algorithm that is suitable for time-resolved and unsupervised application to nonstationary and noisy time series. Using numerical test systems it is discussed under which conditions intra- and inter-cluster relations can be disentangled and quantified. In addition different types of errors occurring when channels are automatically attributed to clusters are investigated quantitatively. Finally, the algorithm is applied to nonstationary model time series and its time-dependent performance is compared to other cluster detection algorithms.

Synchronization Measurement of Multiple Neuronal Populations

The purpose of the present paper is to develop a method, based on equal-time correlation, correlation matrix analysis and surrogate resampling, that is able to quantify and describe properties of synchronization of population neuronal activity recorded simultaneously from multiple sites. Initially, Lorenz-type oscillators were used to model multiple time series with different patterns of synchronization. Eigenvalue and eigenvector decomposition was then applied to identify “clusters” of locally synchronized activity and to calculate a “global synchronization index.” This method was then applied to multichannel data recorded from an in vitro model of epileptic seizures. The results demonstrate that this novel method can be successfully used to analyze synchronization between multiple neuronal population series.

Detecting synchronization clusters in multivariate time series via coarse-graining of Markov chains

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.

Changes of EEG synchronization during low-frequency electric stimulation of the seizure onset zone

PURPOSE

To assess whether EEG synchronization changes during short-term low-frequency electrical stimulation of the seizure onset zone.

METHODS

In 10 patients (34+/-11 years) with pharmaco-resistant epilepsy the seizure onset zone (9 temporal lobe, 1 frontal lobe) was electrically stimulated at 1Hz for 5min via intracranial electrodes. Bipolar stimuli were applied and four pulse widths (0.05, 0.1, 0.5, and 1.0ms) were tested. Stimulation amplitudes were held fixed at 1mA for strip electrodes and at 2mA for depth electrodes. Changes of EEG synchronization were assessed by the eigenvalue dynamics of the cross-correlation matrix computed from a 2.5s sliding window.

RESULTS

37 stimulations were performed. We observed EEG desynchronization in 49% (18/37), an increase of EEG synchronization in 27% (10/37) and an EEG pattern with no significant change of synchronization in 24% (9/37). EEG synchronization most frequently occurred when stimulating with a pulse width of 0.5ms. In a patient with bilateral independent seizure onsets stimulation effects on EEG synchronization were different for each side. In the patient with the shortest duration of temporal lobe epilepsy, stimulation triggered periodic epileptic spikes phase-locked to stimulation. One patient experienced an aura during stimulation, which did not evolve into a seizure, and in one patient a sub-clincial seizure occurred.

DISCUSSION

Low-frequency stimulation of the seizure onset zone is associated with different changes of EEG synchronization and its effects depend on the widths of the stimulation pulses. It may be an appropriate stimulation technique for long-term studies assessing whether synchronized or desynchronized brain dynamics prevent seizure occurrence.

Increasing synchronization may promote seizure termination: Evidence from status epilepticus

OBJECTIVE

To test whether increasing synchronization of neuronal activity might be causally related to seizure termination.

METHODS

Neuronal synchronization was assessed by the relative changes of the eigenvalue spectrum of the equal-time correlation matrix computed from a short window sliding along multi-channel EEGs, recorded with either intracranial or surface electrodes.

RESULTS

Synchronization dynamics of six status epilepticus EEG recordings from six patients were assessed. In all six recordings EEG synchronization fluctuated around relatively low levels during ongoing epileptiform activity. Synchronization only persistently increased before, or in one case, at the end of status epilepticus. Ongoing seizure activity stopped without pharmacological intervention in one patient. In four of the five other cases, the persistent increase of synchronization followed administration of anticonvulsant drugs.

CONCLUSIONS

Our findings support the hypothesis that increasing synchronization of neuronal activity may be considered as an emergent self-regulatory mechanism for seizure termination.

SIGNIFICANCE

The traditional concept is challenged that increasing neuronal synchronization during epileptic seizures is always pathological and should be suppressed. On the contrary, our findings imply that therapeutic interventions to increase synchronization during seizures might be beneficial.

The influence of static correlations on multivariate correlation analysis of the EEG

The choice of the EEG reference strongly influences the results derived from different correlation measures. Such a dependence may easily mislead the interpretation of the correlation structure of the brain activity. We provide a systematic study of the influence of the choice of reference on linear multivariate EEG correlation patterns as determined by sensitive correlation measures derived from the equal-time correlation matrix. In addition, an effective algorithm to extract the effect of static correlations is developed. The eigenvalues of the correlation matrix and their spacing statistics are studied for artificial time series with known correlation structure and for an epileptic EEG in various montages. The correction method proposed in this paper works with varying quality for different choices of the EEG reference. Furthermore, the optimal choice of the reference depends also on the correlation structure of the underlying system.

State-of-the-Art of Seizure Prediction

SUMMARY

Although there are numerous studies exploring basic neuronal mechanisms that are likely to be associated with seizures, to date no definite information is available as to how, when, or why a seizure occurs in humans. The fact that seizures occur without warning in the majority of cases is one of the most disabling aspects of epilepsy. If it were possible to identify preictal precursors from the EEG of epilepsy patients, therapeutic possibilities and quality of life could improve dramatically. The last three decades have witnessed a rapid increase in the development of new EEG analysis techniques that appear to be capable of defining seizure precursors. Since the 1970s, studies on seizure prediction have advanced from preliminary descriptions of preictal phenomena and proof of principle studies via controlled studies to studies on continuous multiday recordings. At present, it is unclear whether prospective algorithms can predict seizures. If prediction algorithms are to be used in invasive seizure intervention techniques in humans, they must be proven to perform considerably better than a random predictor. The authors present an overview of the field of seizure prediction, its history, accomplishments, recent controversies, and potential for future development.

Identifying phase synchronization clusters in spatially extended dynamical systems

We investigate two recently proposed multivariate time series analysis techniques that aim at detecting phase synchronization clusters in spatially extended, nonstationary systems with regard to field applications. The starting point of both techniques is a matrix whose entries are the mean phase coherence values measured between pairs of time series. The first method is a mean-field approach which allows one to define the strength of participation of a subsystem in a single synchronization cluster. The second method is based on an eigenvalue decomposition from which a participation index is derived that characterizes the degree of involvement of a subsystem within multiple synchronization clusters. Simulating multiple clusters within a lattice of coupled Lorenz oscillators we explore the limitations and pitfalls of both methods and demonstrate (a) that the mean-field approach is relatively robust even in configurations where the single-cluster assumption is not entirely fulfilled and (b) that the eigenvalue-decomposition approach correctly identifies the simulated clusters even for low coupling strengths. Using the eigenvalue-decomposition approach we studied spatiotemporal synchronization clusters in long-lasting multichannel EEG recordings from epilepsy patients and obtained results that fully confirm findings from well established neurophysiological examination techniques. Multivariate time series analysis methods such as synchronization cluster analysis, which account for nonlinearities in the data, are expected to provide complementary information which allows one to gain deeper insights into the collective dynamics of spatially extended complex systems.

Localized short-range correlations in the spectrum of the equal-time correlation matrix

We suggest a procedure to identify those parts of the spectrum of the equal-time correlation matrix C where relevant information about correlations of a multivariate time series is induced. Using an ensemble average over each of the distances between eigenvalues, all nearest-neighbor distributions can be calculated individually. We present numerical examples, where (a) information about cross correlations is found in the so-called "bulk" of eigenvalues (which generally is thought to contain only random correlations) and where (b) the information extracted from the lower edge of the spectrum of C is statistically more significant than that extracted from the upper edge. We apply the analysis to electroencephalographic recordings with epileptic events.

Quantification of cross correlations in complex spatiotemporal systems

We propose a design of the equal time correlation matrix suitable for the analysis of multivariate time series with ill-defined phases. We present the cross-correlation analysis of model data sets taken from coupled stochastic oscillators and compare the concept with the results obtained from a conventional correlation matrix analysis. We show that the concept provides a higher sensitivity combined with a better statistical significance when quantifying weak cross correlations.