Detection of weak directional coupling: phase-dynamics approach versus state-space approach

Phase-based causality analysis with partial mutual information from mixed embedding

Instantaneous phases extracted from multivariate time series can retain information about the relationships between the underlying mechanisms that generate the series. Although phases have been widely used in the study of nondirectional coupling and connectivity, they have not found similar appeal in the study of causality. Herein, we present a new method for phase-based causality analysis, which combines ideas from the mixed embedding technique and the information-theoretic approach to causality in coupled oscillatory systems. We then use the introduced method to investigate causality in simulated datasets of bivariate, unidirectionally paired systems from combinations of Rössler, Lorenz, van der Pol, and Mackey-Glass equations. We observe that causality analysis using the phases can capture the true causal relation for coupling strength smaller than the analysis based on the amplitudes can capture. On the other hand, the causality estimation based on the phases tends to have larger variability, which is attributed more to the phase extraction process than the actual phase-based causality method. In addition, an application on real electroencephalographic data from an experiment on elicited human emotional states reinforces the usefulness of phases in causality identification.

Phase-dynamic causalities within dynamical effects framework

This work investigates numerics of several widely known phase-dynamic quantifiers of directional (causal) couplings between oscillatory systems: transfer entropy (TE), differential quantifier, and squared-coefficients quantifier based on an evolution map. The study is performed on the system of two stochastic Kuramoto oscillators within the framework of dynamical causal effects. The quantifiers are related to each other and to an asymptotic effect of the coupling on phase diffusion. Several novel findings are listed as follows: (i) for a non-synchronous regime and high enough noise levels, the TE rate multiplied by a certain characteristic time (called here reduced TE) equals twice an asymptotic effect of a directional coupling on phase diffusion; (ii) "information flow" expressed by the TE rate unboundedly rises with the coupling coefficient even in the domain of effective synchronization; (iii) in any effective synchronization regime, the reduced TE is equal to 1/8 n.u. in each direction for equal coupling coefficients and equal noise intensities, and it is in general a simple function of the ratio of noise intensities and the ratio of coupling coefficients.

Transfer entropies within dynamical effects framework

Transfer entropy (TE) is widely used in time-series analysis to detect causal couplings between temporally evolving objects. As a coupling strength quantifier, the TE alone often seems insufficient, raising the question of its further interpretations. Here the TE is related to dynamical causal effects (DCEs) which quantify long-term responses of a coupling recipient to variations in a coupling source or in a coupling itself: Detailed relationships are established for a paradigmatic stochastic dynamical system of bidirectionally coupled linear overdamped oscillators, their practical applications and possible extensions are discussed. It is shown that two widely used versions of the TE (original and infinite-history) can become qualitatively distinct, diverging to different long-term DCEs.

The Human Organism as an Integrated Interaction Network: Recent Conceptual and Methodological Challenges

The field of Network Physiology aims to advance our understanding of how physiological systems and sub-systems interact to generate a variety of behaviors and distinct physiological states, to optimize the organism's functioning, and to maintain health. Within this framework, which considers the human organism as an integrated network, vertices are associated with organs while edges represent time-varying interactions between vertices. Likewise, vertices may represent networks on smaller spatial scales leading to a complex mixture of interacting homogeneous and inhomogeneous networks of networks. Lacking adequate analytic tools and a theoretical framework to probe interactions within and among diverse physiological systems, current approaches focus on inferring properties of time-varying interactions-namely strength, direction, and functional form-from time-locked recordings of physiological observables. To this end, a variety of bivariate or, in general, multivariate time-series-analysis techniques, which are derived from diverse mathematical and physical concepts, are employed and the resulting time-dependent networks can then be further characterized with methods from network theory. Despite the many promising new developments, there are still problems that evade from a satisfactory solution. Here we address several important challenges that could aid in finding new perspectives and inspire the development of theoretic and analytical concepts to deal with these challenges and in studying the complex interactions between physiological systems.

Evaluation of Granger Causality Measures for Constructing Networks from Multivariate Time Series

Granger causality and variants of this concept allow the study of complex dynamical systems as networks constructed from multivariate time series. In this work, a large number of Granger causality measures used to form causality networks from multivariate time series are assessed. These measures are in the time domain, such as model-based and information measures, the frequency domain, and the phase domain. The study aims also to compare bivariate and multivariate measures, linear and nonlinear measures, as well as the use of dimension reduction in linear model-based measures and information measures. The latter is particular relevant in the study of high-dimensional time series. For the performance of the multivariate causality measures, low and high dimensional coupled dynamical systems are considered in discrete and continuous time, as well as deterministic and stochastic. The measures are evaluated and ranked according to their ability to provide causality networks that match the original coupling structure. The simulation study concludes that the Granger causality measures using dimension reduction are superior and should be preferred particularly in studies involving many observed variables, such as multi-channel electroencephalograms and financial markets.

Inferring directed networks using a rank-based connectivity measure

Inferring the topology of a network using the knowledge of the signals of each of the interacting units is key to understanding real-world systems. One way to address this problem is using data-driven methods like cross-correlation or mutual information. However, these measures lack the ability to distinguish the direction of coupling. Here, we use a rank-based nonlinear interdependence measure originally developed for pairs of signals. This measure not only allows one to measure the strength but also the direction of the coupling. Our results for a system of coupled Lorenz dynamics show that we are able to consistently infer the underlying network for a subrange of the coupling strength and link density. Furthermore, we report that the addition of dynamical noise can benefit the reconstruction. Finally, we show an application to multichannel electroencephalographic recordings from an epilepsy patient.

Transient and equilibrium causal effects in coupled oscillators

Two quite different types of causal effects are given by (i) changes in near future states of a driven system under changes in a current state of a driving system and (ii) changes in statistical characteristics of a driven system dynamics under changes in coupling parameters, e.g., under switching the coupling off. The former can be called transient causal effects and can be estimated from a time series within the well established framework of the Wiener-Granger causality, while the latter represent equilibrium (or stationary) causal effects which are often most interesting but generally inaccessible to estimation from an observed time series recorded at fixed coupling parameters. In this work, relationships between the two kinds of causal effects are found for unidirectionally coupled stochastic linear oscillators depending on their frequencies and damping factors. Approximate closed-form expressions for these relationships are derived. Their limitations and possible extensions are discussed, and their practical applicability to extracting equilibrium causal effects from time series is argued.

Causality, dynamical systems and the arrow of time

Using several methods for detection of causality in time series, we show in a numerical study that coupled chaotic dynamical systems violate the first principle of Granger causality that the cause precedes the effect. While such a violation can be observed in formal applications of time series analysis methods, it cannot occur in nature, due to the relation between entropy production and temporal irreversibility. The obtained knowledge, however, can help to understand the type of causal relations observed in experimental data, namely, it can help to distinguish linear transfer of time-delayed signals from nonlinear interactions. We illustrate these findings in causality detected in experimental time series from the climate system and mammalian cardio-respiratory interactions.

Directionality indices: Testing information transfer with surrogate correction

Directionality indices can be used as an indicator of the asymmetry in coupling between systems and have found particular application in relation to neurological systems. The directionality index between two systems is a function of measures of information transfer in both directions. Here we illustrate that before inferring the directionality of coupling it is first necessary to consider the use of appropriate tests of significance. We propose a surrogate corrected directionality index which incorporates such testing. We also highlight the differences between testing the significance of the directionality index itself versus testing the individual measures of information transfer in each direction. To validate the approach we compared two different methods of estimating coupling, both of which have previously been used to estimate directionality indices. These were the modeling-based evolution map approach and a conditional mutual information (CMI) method for calculating dynamic information rates. For the CMI-based approach we also compared two different methods for estimating the CMI, an equiquantization-based estimator and a k-nearest neighbors estimator.

Detection of coupling delay: A problem not yet solved

Nonparametric detection of coupling delay in unidirectionally and bidirectionally coupled nonlinear dynamical systems is examined. Both continuous and discrete-time systems are considered. Two methods of detection are assessed-the method based on conditional mutual information-the CMI method (also known as the transfer entropy method) and the method of convergent cross mapping-the CCM method. Computer simulations show that neither method is generally reliable in the detection of coupling delays. For continuous-time chaotic systems, the CMI method appears to be more sensitive and applicable in a broader range of coupling parameters than the CCM method. In the case of tested discrete-time dynamical systems, the CCM method has been found to be more sensitive, while the CMI method required much stronger coupling strength in order to bring correct results. However, when studied systems contain a strong oscillatory component in their dynamics, results of both methods become ambiguous. The presented study suggests that results of the tested algorithms should be interpreted with utmost care and the nonparametric detection of coupling delay, in general, is a problem not yet solved.

Robustness and versatility of a nonlinear interdependence method for directional coupling detection from spike trains

The detection of directional couplings between dynamics based on measured spike trains is a crucial problem in the understanding of many different systems. In particular, in neuroscience it is important to assess the connectivity between neurons. One of the approaches that can estimate directional coupling from the analysis of point processes is the nonlinear interdependence measure L. Although its efficacy has already been demonstrated, it still needs to be tested under more challenging and realistic conditions prior to an application to real data. Thus, in this paper we use the Hindmarsh-Rose model system to test the method in the presence of noise and for different spiking regimes. We also examine the influence of different parameters and spike train distances. Our results show that the measure L is versatile and robust to various types of noise, and thus suitable for application to experimental data.

Detecting switching and intermittent causalities in time series

During the last decade, complex network representations have emerged as a powerful instrument for describing the cross-talk between different brain regions both at rest and as subjects are carrying out cognitive tasks, in healthy brains and neurological pathologies. The transient nature of such cross-talk has nevertheless by and large been neglected, mainly due to the inherent limitations of some metrics, e.g., causality ones, which require a long time series in order to yield statistically significant results. Here, we present a methodology to account for intermittent causal coupling in neural activity, based on the identification of non-overlapping windows within the original time series in which the causality is strongest. The result is a less coarse-grained assessment of the time-varying properties of brain interactions, which can be used to create a high temporal resolution time-varying network. We apply the proposed methodology to the analysis of the brain activity of control subjects and alcoholic patients performing an image recognition task. Our results show that short-lived, intermittent, local-scale causality is better at discriminating both groups than global network metrics. These results highlight the importance of the transient nature of brain activity, at least under some pathological conditions.

Coupling strength versus coupling impact in nonidentical bidirectionally coupled dynamics

The understanding of interacting dynamics is important for the characterization of real-world networks. In general, real-world networks are heterogeneous in the sense that each node of the network is a dynamics with different properties. For coupled nonidentical dynamics symmetric interactions are not straightforwardly defined from the coupling strength values. Thus, a challenging issue is whether we can define a symmetric interaction in this asymmetric setting. To address this problem we introduce the notion of the coupling impact. The coupling impact considers not only the coupling strength but also the energy of the individual dynamics, which is conveyed via the coupling. To illustrate this concept, we follow a data-driven approach by analyzing signals from pairs of coupled model dynamics using two different connectivity measures. We find that the coupling impact, but not the coupling strength, correctly detects a symmetric interaction between pairs of coupled dynamics regardless of their degree of asymmetry. Therefore, this approach allows us to reveal the real impact that one dynamics has on the other and hence to define symmetric interactions in pairs of nonidentical dynamics.

Reconstructing links in directed networks from noisy dynamics

We address the long-standing challenge of how to reconstruct links in directed networks from measurements, and present a general method that makes use of a noise-induced relation between network structure and both the time-lagged covariance of measurements taken at two different times and the covariance of measurements taken at the same time. When the coupling functions have certain additional properties, we can further reconstruct the weights of the links.

Distinguishing between direct and indirect directional couplings in large oscillator networks: Partial or non-partial phase analyses?

We investigate the relative merit of phase-based methods for inferring directional couplings in complex networks of weakly interacting dynamical systems from multivariate time-series data. We compare the evolution map approach and its partialized extension to each other with respect to their ability to correctly infer the network topology in the presence of indirect directional couplings for various simulated experimental situations using coupled model systems. In addition, we investigate whether the partialized approach allows for additional or complementary indications of directional interactions in evolving epileptic brain networks using intracranial electroencephalographic recordings from an epilepsy patient. For such networks, both direct and indirect directional couplings can be expected, given the brain's connection structure and effects that may arise from limitations inherent to the recording technique. Our findings indicate that particularly in larger networks (number of nodes ≫10), the partialized approach does not provide information about directional couplings extending the information gained with the evolution map approach.

Detection of Coupling in Short Physiological Series by a Joint Distribution Entropy Method

OBJECTIVE

In this study, we developed a joint distribution entropy (JDistEn) method to robustly estimate the coupling in short physiological series.

METHODS

The JDistEn method is derived from a joint distance matrix which is constructed from a combination of the distance matrix corresponding to each individual data channel using a geometric mean calculation. A coupled Rössler system and a coupled dual-kinetics neural mass model were used to examine how well JDistEn performed, specifically, its sensitivity for detecting weak coupling, its consistency in gauging coupling strength, and its reliability in processing input of decreased data length. Performance of JDistEn in estimating physiological coupling was further examined with bivariate electroencephalography data from rats and RR interval and diastolic time interval series from human beings. Cross-sample entropy (XSampEn), cross-conditional entropy (XCE), and Shannon entropy of diagonal lines in the joint recurrence plots (JENT) were applied for purposes of comparison.

RESULTS

Simulation results suggest that JDistEn showed markedly higher sensitivity than XSampEn, XCE, and JENT for dynamics in weak coupling, although as the simulation models were more intensively coupled, JDistEn performance was comparable to the three others. In addition, this improved sensitivity was much more pronounced for short datasets. Experimental results further confirmed that JDistEn outperformed XSampEn, XCE, and JENT for detecting weak coupling, especially for short physiological data.

CONCLUSION

This study suggested that our proposed JDistEn could be useful for continuous and even real-time coupling analysis for physiological signals in clinical practice.

Causality networks from multivariate time series and application to epilepsy

Granger causality and variants of this concept allow the study of complex dynamical systems as networks constructed from multivariate time series. In this work, a large number of Granger causality measures used to form causality networks from multivariate time series are assessed. For this, realizations on high dimensional coupled dynamical systems are considered and the performance of the Granger causality measures is evaluated, seeking for the measures that form networks closest to the true network of the dynamical system. In particular, the comparison focuses on Granger causality measures that reduce the state space dimension when many variables are observed. Further, the linear and nonlinear Granger causality measures of dimension reduction are compared to a standard Granger causality measure on electroencephalographic (EEG) recordings containing episodes of epileptiform discharges.

Resting-state EEG coupling analysis of amnestic mild cognitive impairment with type 2 diabetes mellitus by using permutation conditional mutual information

OBJECTIVE

This study was meant to explore whether the coupling strength and direction of resting-state electroencephalogram (rsEEG) could be used as an indicator to distinguish the patients of type 2 diabetes mellitus (T2DM) with or without amnestic mild cognitive impairment (aMCI).

METHODS

Permutation conditional mutual information (PCMI) was used to calculate the coupling strength and direction of rsEEG signals between different brain areas of 19 aMCI and 20 normal control (NC) with T2DM on 7 frequency bands: Delta, Theta, Alpha1, Alpha2, Beta1, Beta2 and Gamma. The difference in coupling strength or direction of rsEEG between two groups was calculated. The correlation between coupling strength or direction of rsEEG and score of different neuropsychology scales were also calculated.

RESULTS

We have demonstrated that PCMI can calculate effectively the coupling strength and directionality of EEG signals between different brain regions. The significant difference in coupling strength and directionality of EEG signals was found between the patients of aMCI and NC with T2DM on different brain regions. There also existed significant correlation between sex or age and coupling strength or coupling directionality of EEG signals between a few different brain regions from all subjects.

CONCLUSIONS

The coupling strength or directionality of EEG signals calculated by PCMI are significantly different between aMCI and NC with T2DM.

SIGNIFICANCE

These results showed that the coupling strength or directionality of EEG signals calculated by PCMI might be used as a biomarker in distinguishing the aMCI from NC with T2DM.

A critical review: coupling and synchronization analysis methods of EEG signal with mild cognitive impairment

At present, the clinical diagnosis of mild cognitive impairment (MCI) patients becomes the important approach of evaluating early Alzheimer's disease. The methods of EEG signal coupling and synchronization act as a key role in evaluating and diagnosing MCI patients. Recently, these coupling and synchronization methods were used to analyze the EEG signals of MCI patients according to different angles, and many important discoveries have been achieved. However, considering that every method is single-faceted in solving problems, these methods have various deficiencies when analyzing EEG signals of MCI patients. This paper reviewed in detail the coupling and synchronization analysis methods, analyzed their advantages and disadvantages, and proposed a few research questions needed to solve in the future. Also, the principles and best performances of these methods were described. It is expected that the performance analysis of these methods can provide the theoretical basis for the method selection of analyzing EEG signals of MCI patients and the future research directions.

Involvement of thalamus in initiation of epileptic seizures induced by pilocarpine in mice

Studies have suggested that thalamus is involved in temporal lobe epilepsy, but the role of thalamus is still unclear. We obtained local filed potentials (LFPs) and single-unit activities from CA1 of hippocampus and parafascicular nucleus of thalamus during the development of epileptic seizures induced by pilocarpine in mice. Two measures, redundancy and directionality index, were used to analyze the electrophysiological characters of neuronal activities and the information flow between thalamus and hippocampus. We found that LFPs became more regular during the seizure in both hippocampus and thalamus, and in some cases LFPs showed a transient disorder at seizure onset. The variation tendency of the peak values of cross-correlation function between neurons matched the variation tendency of the redundancy of LFPs. The information tended to flow from thalamus to hippocampus during seizure initiation period no matter what the information flow direction was before the seizure. In some cases the information flow was symmetrically bidirectional, but none was found in which the information flowed from hippocampus to thalamus during the seizure initiation period. In addition, inactivation of thalamus by tetrodotoxin (TTX) resulted in a suppression of seizures. These results suggest that thalamus may play an important role in the initiation of epileptic seizures.

Global Synchronization Measurement of Multivariate Neural Signals with Massively Parallel Nonlinear Interdependence Analysis

The estimation of synchronization amongst multiple brain regions is a critical issue in understanding brain functions. There is a lack of an appropriate approach which is capable of 1) measuring the direction and strength of synchronization of activities of multiple brain regions, and 2) adapting to the quickly increasing sizes and scales of neural signals. Nonlinear Interdependence (NLI) analysis is an effective method for measuring synchronization direction and strength of bivariate neural signal. However, the method currently does not directly apply in handling multivariate signal. Its application in practice has also long been largely hampered by the ultra-high complexity of NLI algorithms. Aiming at these problems, this study 1) extends the conventional NLI to quantify the global synchronization of multivariate neural signals, and 2) develops a parallelized NLI method with general-purpose computing on the graphics processing unit (GPGPU), namely, G-NLI. The approach performs synchronization measurement in a massively parallel manner. The G-NLI has improved the runtime performance by more than 1000 times comparing to the original sequential NLI. Meanwhile, the G-NLI was employed to analyze 10-channel local field potential (LFP) recordings from a patient suffering from temporal lobe epilepsy. The results demonstrate that the proposed G-NLI method can support real-time global synchronization measurement and it could be successful in localization of epileptic focus.

Alteration of phase-phase coupling between theta and gamma rhythms in a depression-model of rats

Alterations in oscillatory brain activity are strongly correlated with cognitive performance in various physiological rhythms, especially the theta and gamma rhythms. In this study, we investigated the coupling relationship of neural activities between thalamus and medial prefrontal cortex (mPFC) by measuring the phase interactions between theta and gamma oscillations in a depression model of rats. The phase synchronization analysis showed that the phase locking at theta rhythm was weakened in depression. Furthermore, theta-gamma phase locking at n:m (1:6) ratio was found between thalamus and mPFC, while it was diminished in depression state. In addition, the analysis of coupling direction based on phase dynamics showed that the unidirectional influence from thalamus to mPFC was diminished in depression state only in theta rhythm, while it was partly recovered after the memantine treatment in a depression model of rats. The results suggest that the effects of depression on cognitive deficits are modulated via profound alterations in phase information transformation of theta rhythm and theta-gamma phase coupling.

Modeling inter-human movement coordination: synchronization governs joint task dynamics

Human interaction partners tend to synchronize their movements during repetitive actions such as walking. Research of inter-human coordination in purely rhythmic action tasks reveals that the observed patterns of interaction are dominated by synchronization effects. Initiated by our finding that human dyads synchronize their arm movements even in a goal-directed action task, we present a step-wise approach to a model of inter-human movement coordination. In an experiment, the hand trajectories of ten human dyads are recorded. Governed by a dynamical process of phase synchronization, the participants establish in-phase as well as anti-phase relations. The emerging relations are successfully reproduced by the attractor dynamics of coupled phase oscillators inspired by the Kuramoto model. Three different methods on transforming the motion trajectories into instantaneous phases are investigated and their influence on the model fit to the experimental data is evaluated. System identification technique allows us to estimate the model parameters, which are the coupling strength and the frequency detuning among the dyad. The stability properties of the identified model match the relations observed in the experimental data. In short, our model predicts the dynamics of inter-human movement coordination. It can directly be implemented to enrich human-robot interaction.

Finding weak directional coupling in multiscale time series

We propose a method to identify the direction of interactions between the components of a coupled system from observed data. This method tries to identify the direction of coupling for various severe conditions including for nonlinear dynamics, small sample size, weak coupling strength, noisy and multiscale data, and multistructure systems. Thus, it appears to be a promising method for real-world problems encountered in a variety of disciplines, such as physics, economics, and biology.

Decreased thalamo-cortical connectivity by alteration of neural information flow in theta oscillation in depression-model rats

Alterations in oscillatory brain activity are strongly correlated with cognitive performance in various physiological rhythms. The present study investigated whether the directionality of neural information flow (NIF) could be used to characterize the synaptic plasticity in thalamocortical (TC) pathway, and examined which frequency field oscillations were mostly related to the cognitive deficiency in depression. Two novel algorithms were employed to determine the coupling interaction between the LD thalamus and medial prefrontal cortex (mPFC) in five frequency bands, using the phase signals of local field potentials (LFP) in these two regions. The results showed that the power of neural activity in mPFC was increased in delta, theta and beta frequency bands in depression. However, the nonlinear characteristics of LFP activity were weakened in depression by means of sample entropy measurements. In the analysis of phase dynamics, the phase synchronization values were reduced in theta rhythm in stressed rats. Importantly, the coupling direction index d and the unidirectional influence from LD thalamus to mPFC were significantly reduced at the theta rhythm in rats in depression, and increased after memantine treatment, which were associated with the LTP alterations and cognitive impairment in our previous report. Moreover, the fact that the reduced entropy value was only found in mPFC might implicate postsynaptic effect involved in synaptic plasticity alteration in the depression model. The results suggest that the effects of depression on cognitive deficits are mediated via profound alterations in information flow in the TC pathway, and the directional index at theta rhythm could be used as a measurement of synaptic plasticity.

Assessing directed interactions from neurophysiological signals--an overview

The study of synchronization phenomena in coupled dynamical systems is an active field of research in many scientific disciplines including the neurosciences. Over the last decades, a number of time series analysis techniques have been proposed to capture both linear and nonlinear aspects of interactions. While most of these techniques allow one to quantify the strength of interactions, developments that resulted from advances in nonlinear dynamics and in information and synchronization theory aim at assessing directed interactions. Most of these techniques, however, assume the underlying systems to be at least approximately stationary and require a large number of data points to robustly assess directed interactions. Recent extensions allow assessing directed interactions from short and transient signals and are particularly suited for the analysis of evoked and event-related activity.

Detecting effective connectivity in networks of coupled neuronal oscillators

The application of data-driven time series analysis techniques such as Granger causality, partial directed coherence and phase dynamics modeling to estimate effective connectivity in brain networks has recently gained significant prominence in the neuroscience community. While these techniques have been useful in determining causal interactions among different regions of brain networks, a thorough analysis of the comparative accuracy and robustness of these methods in identifying patterns of effective connectivity among brain networks is still lacking. In this paper, we systematically address this issue within the context of simple networks of coupled spiking neurons. Specifically, we develop a method to assess the ability of various effective connectivity measures to accurately determine the true effective connectivity of a given neuronal network. Our method is based on decision tree classifiers which are trained using several time series features that can be observed solely from experimentally recorded data. We show that the classifiers constructed in this work provide a general framework for determining whether a particular effective connectivity measure is likely to produce incorrect results when applied to a dataset.

Reconstructing phase dynamics of oscillator networks

We generalize our recent approach to the reconstruction of phase dynamics of coupled oscillators from data [B. Kralemann et al., Phys. Rev. E 77, 066205 (2008)] to cover the case of small networks of coupled periodic units. Starting from a multivariate time series, we first reconstruct genuine phases and then obtain the coupling functions in terms of these phases. Partial norms of these coupling functions quantify directed coupling between oscillators. We illustrate the method by different network motifs for three coupled oscillators and for random networks of five and nine units. We also discuss nonlinear effects in coupling.

Using bivariate signal analysis to characterize the epileptic focus: the benefit of surrogates

The disease epilepsy is related to hypersynchronous activity of networks of neurons. While acute epileptic seizures are the most extreme manifestation of this hypersynchronous activity, an elevated level of interdependence of neuronal dynamics is thought to persist also during the seizure-free interval. In multichannel recordings from brain areas involved in the epileptic process, this interdependence can be reflected in an increased linear cross correlation but also in signal properties of higher order. Bivariate time series analysis comprises a variety of approaches, each with different degrees of sensitivity and specificity for interdependencies reflected in lower- or higher-order properties of pairs of simultaneously recorded signals. Here we investigate which approach is best suited to detect putatively elevated interdependence levels in signals recorded from brain areas involved in the epileptic process. For this purpose, we use the linear cross correlation that is sensitive to lower-order signatures of interdependence, a nonlinear interdependence measure that integrates both lower- and higher-order properties, and a surrogate-corrected nonlinear interdependence measure that aims to specifically characterize higher-order properties. We analyze intracranial electroencephalographic recordings of the seizure-free interval from 29 patients with an epileptic focus located in the medial temporal lobe. Our results show that all three approaches detect higher levels of interdependence for signals recorded from the brain hemisphere containing the epileptic focus as compared to signals recorded from the opposite hemisphere. For the linear cross correlation, however, these differences are not significant. For the nonlinear interdependence measure, results are significant but only of moderate accuracy with regard to the discriminative power for the focal and nonfocal hemispheres. The highest significance and accuracy is obtained for the surrogate-corrected nonlinear interdependence measure.

Reducing the bias of causality measures

Measures of the direction and strength of the interdependence between two time series are evaluated and modified to reduce the bias in the estimation of the measures, so that they give zero values when there is no causal effect. For this, point shuffling is employed as used in the frame of surrogate data. This correction is not specific to a particular measure and it is implemented here on measures based on state space reconstruction and information measures. The performance of the causality measures and their modifications is evaluated on simulated uncoupled and coupled dynamical systems and for different settings of embedding dimension, time series length, and noise level. The corrected measures, and particularly the suggested corrected transfer entropy, turn out to stabilize at the zero level in the absence of a causal effect and detect correctly the direction of information flow when it is present. The measures are also evaluated on electroencephalograms (EEG) for the detection of the information flow in the brain of an epileptic patient. The performance of the measures on EEG is interpreted in view of the results from the simulation study.

Inferring directional interactions from transient signals with symbolic transfer entropy

We extend the concept of symbolic transfer entropy to enable the time-resolved investigation of directional relationships between coupled dynamical systems from short and transient noisy time series. For our approach, we consider an observed ensemble of a sufficiently large number of time series as multiple realizations of a process. We derive an index that quantifies the preferred direction of transient interactions and assess its significance using a surrogate-based testing scheme. Analyzing time series from noisy chaotic systems, we demonstrate numerically the applicability and limitations of our approach. Our findings obtained from an analysis of event-related brain activities underline the importance of our method to improve understanding of gross neural interactions underlying cognitive processes.

Detecting couplings between interacting oscillators with time-varying basic frequencies: instantaneous wavelet bispectrum and information theoretic approach

In the natural world, the properties of interacting oscillatory systems are not constant, but evolve or fluctuating continuously in time. Thus, the basic frequencies of the interacting oscillators are time varying, which makes the system analysis complex. For studying their interactions we propose a complementary approach combining wavelet bispectral analysis and information theory. We show how these methods uncover the interacting properties and reveal the nature, strength, and direction of coupling. Wavelet bispectral analysis is generalized as a technique for detecting instantaneous phase-time dependence for the case of two or more coupled nonlinear oscillators whereas the information theory approach can uncover the directionality of coupling and extract driver-response relationships in complex systems. We generate bivariate time-series numerically to mimic typical situations that occur in real measured data, apply both methods to the same time-series and discuss the results. The approach is applicable quite generally to any system of coupled nonlinear oscillators.

A comparative study of pattern synchronization detection between neural signals using different cross-entropy measures

Cross-approximate entropy (X-ApEn) and cross-sample entropy (X-SampEn) have been employed as bivariate pattern synchronization measures for characterizing interdependencies between neural signals. In this study, we proposed a new measure, cross-fuzzy entropy (X-FuzzyEn), to describe the synchronicity of patterns. The performances of three statistics were first quantitatively tested using five different coupled systems including both deterministic and stochastic models, i.e., coupled broadband noises, Lorenz-Lorenz, Rossler-Rossler, Rossler-Lorenz, and neural mass model. All the measures were compared with each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise. The three measures were then applied to a real-life problem, pattern synchronization analysis of left and right hemisphere rat electroencephalographic (EEG) signals. Both simulated and real EEG data analysis results showed that the X-FuzzyEn provided an improved evaluation of bivariate series pattern synchronization and could be more conveniently and powerfully applied to different neural dynamical systems contaminated by noise.

Reliable detection of directional couplings using rank statistics

To detect directional couplings from time series various measures based on distances in reconstructed state spaces were introduced. These measures can, however, be biased by asymmetries in the dynamics' structure, noise color, or noise level, which are ubiquitous in experimental signals. Using theoretical reasoning and results from model systems we identify the various sources of bias and show that most of them can be eliminated by an appropriate normalization. We furthermore diminish the remaining biases by introducing a measure based on ranks of distances. This rank-based measure outperforms existing distance-based measures concerning both sensitivity and specificity for directional couplings. Therefore, our findings are relevant for a reliable detection of directional couplings from experimental signals.

Detection of couplings in ensembles of stochastic oscillators

The problem of detection and quantitative characterization of directional couplings in an ensemble of noisy oscillators from a time series is addressed. We suggest estimators for the strengths of couplings which are based on modeling the observed oscillations with a set of stochastic phase oscillators and easily interpreted from a physical viewpoint. Moreover, we present an analytic formula for a statistical significance level allowing to reveal an architecture of couplings reliably from a relatively short time series. The technique applies to weakly coupled nonsynchronized oscillators. It is introduced for oscillators with close basic frequencies but can be readily generalized to the case of arbitrary frequencies. Efficiency of the technique is demonstrated in numerical experiments.

Automated synchrogram analysis applied to heartbeat and reconstructed respiration

Phase synchronization between two weakly coupled oscillators has been studied in chaotic systems for a long time. However, it is difficult to unambiguously detect such synchronization in experimental data from complex physiological systems. In this paper we review our study of phase synchronization between heartbeat and respiration in 150 healthy subjects during sleep using an automated procedure for screening the synchrograms. We found that this synchronization is significantly enhanced during non-rapid-eye-movement (non-REM) sleep (deep sleep and light sleep) and is reduced during REM sleep. In addition, we show that the respiration signal can be reconstructed from the heartbeat recordings in many subjects. Our reconstruction procedure, which works particularly well during non-REM sleep, allows the detection of cardiorespiratory synchronization even if only heartbeat intervals were recorded.

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.

Measuring interdependences in dissipative dynamical systems with estimated Fokker-Planck coefficients

We propose a data-driven approach to measure interdependences between dissipative dynamical systems under the influence of noise. We estimate drift and diffusion coefficients of a Fokker-Planck equation and derive measures that allow one to quantify the asymmetry in coupling in a fully automated and computationally inexpensive and simple way. Our approach makes it possible to discriminate between interdependences in the deterministic and stochastic parts of the dynamics. We report results of numerical studies of exemplary time series from coupled stochastic and deterministic model systems and of an application to electroencephalographic recordings from epilepsy patients.