Detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble

Mutual nonlinear prediction as a tool to evaluate coupling strength and directionality in bivariate time series: comparison among different strategies based on k nearest neighbors

We compare the different existing strategies of mutual nonlinear prediction regarding their ability to assess the coupling strength and directionality of the interactions in bivariate time series. Under the common framework of k -nearest neighbor local linear prediction, we test three approaches based on cross prediction, mixed prediction, and predictability improvement. The measures of interdependence provided by these approaches are first evaluated on short realizations of bivariate time series generated by coupled Henon models, investigating also the effects of noise. The usefulness of the three mutual nonlinear prediction schemes is then assessed in a common physiological application during known conditions of interaction-i.e., the analysis of the interdependence between heart rate and arterial pressure variability in healthy humans during supine resting and passive head-up tilting. Based on both simulation results and physiological interpretability of cardiovascular results, we conclude that cross prediction is valuable to quantify the coupling strength and predictability improvement to elicit directionality of the interactions in short and noisy bivariate time series.

Symbolic transfer entropy

We propose to estimate transfer entropy using a technique of symbolization. We demonstrate numerically that symbolic transfer entropy is a robust and computationally fast method to quantify the dominating direction of information flow between time series from structurally identical and nonidentical coupled systems. Analyzing multiday, multichannel electroencephalographic recordings from 15 epilepsy patients our approach allowed us to reliably identify the hemisphere containing the epileptic focus without observing actual seizure activity.

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.

Dynamic causal modelling of induced responses

This paper describes a dynamic causal model (DCM) for induced or spectral responses as measured with the electroencephalogram (EEG) or the magnetoencephalogram (MEG). We model the time-varying power, over a range of frequencies, as the response of a distributed system of coupled electromagnetic sources to a spectral perturbation. The model parameters encode the frequency response to exogenous input and coupling among sources and different frequencies. The Bayesian inversion of this model, given data enables inferences about the parameters of a particular model and allows us to compare different models, or hypotheses. One key aspect of the model is that it differentiates between linear and non-linear coupling; which correspond to within and between-frequency coupling respectively. To establish the face validity of our approach, we generate synthetic data and test the identifiability of various parameters to ensure they can be estimated accurately, under different levels of noise. We then apply our model to EEG data from a face-perception experiment, to ask whether there is evidence for non-linear coupling between early visual cortex and fusiform areas.

Direction of coupling from phases of interacting oscillators: a permutation information approach

We introduce a directionality index for a time series based on a comparison of neighboring values. It can distinguish unidirectional from bidirectional coupling, as well as reveal and quantify asymmetry in bidirectional coupling. It is tested on a numerical model of coupled van der Pol oscillators, and applied to cardiorespiratory data from healthy subjects. There is no need for preprocessing and fine-tuning the parameters, which makes the method very simple, computationally fast and robust.

Inferring the directionality of coupling with conditional mutual information

Uncovering the directionality of coupling is a significant step in understanding drive-response relationships in complex systems. In this paper, we discuss a nonparametric method for detecting the directionality of coupling based on the estimation of information theoretic functionals. We consider several different methods for estimating conditional mutual information. The behavior of each estimator with respect to its free parameter is shown using a linear model where an analytical estimate of conditional mutual information is available. Numerical experiments in detecting coupling directionality are performed using chaotic oscillators, where the influence of the phase extraction method and relative frequency ratio is investigated.

Granger causality: cortico-thalamic interdependencies during absence seizures in WAG/Rij rats

Linear Granger causality was used to identify the coupling strength and directionality of information transport between frontal cortex and thalamus during spontaneous absence seizures in a genetic model, the WAG/Rij rats. Electroencephalograms were recorded at the cortical surface and from the specific thalamus. Granger coupling strength was measured before, during and after the occurrence of spike-wave discharges (SWD). Before the onset of SWD, coupling strength was low, but associations from thalamus-to-cortex were stronger than vice versa. The onset of SWD was associated with a rapid and significant increase of coupling strength in both directions. There were no changes in Granger causalities before the onset of SWD. The strength of thalamus-to-cortex coupling remained constantly high during the seizures. The strength of cortex-to-thalamus coupling gradually diminished shortly after the onset of SWD and returned to the pre-SWD level when SWD stopped. In contrast, the strength of thalamus-to-cortex coupling remained elevated even after cessation of SWD. The strong and sustained influence of thalamus-to-cortex may facilitate propagation and maintenance of seizure activity, while rapid reduction of cortex-to-thalamus coupling strength may prompt the cessation of SWD. However, the linear estimation of Granger coupling strength does not seem to be sufficient for predicting episodes with absence epilepsy.

Assessment of Granger causality by nonlinear model identification: application to short-term cardiovascular variability

A method for assessing Granger causal relationships in bivariate time series, based on nonlinear autoregressive (NAR) and nonlinear autoregressive exogenous (NARX) models is presented. The method evaluates bilateral interactions between two time series by quantifying the predictability improvement (PI) of the output time series when the dynamics associated with the input time series are included, i.e., moving from NAR to NARX prediction. The NARX model identification was performed by the optimal parameter search (OPS) algorithm, and its results were compared to the least-squares method to determine the most appropriate method to be used for experimental data. The statistical significance of the PI was assessed using a surrogate data technique. The proposed method was tested with simulation examples involving short realizations of linear stochastic processes and nonlinear deterministic signals in which either unidirectional or bidirectional coupling and varying strengths of interactions were imposed. It was found that the OPS-based NARX model was accurate and sensitive in detecting imposed Granger causality conditions. In addition, the OPS-based NARX model was more accurate than the least squares method. Application to the systolic blood pressure and heart rate variability signals demonstrated the feasibility of the method. In particular, we found a bilateral causal relationship between the two signals as evidenced by the significant reduction in the PI values with the NARX model prediction compared to the NAR model prediction, which was also confirmed by the surrogate data analysis. Furthermore, we found significant reduction in the complexity of the dynamics of the two causal pathways of the two signals as the body position was changed from the supine to upright. The proposed is a general method, thus, it can be applied to a wide variety of physiological signals to better understand causality and coupling that may be different between normal and diseased conditions.

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.

A new nonlinear similarity measure for multichannel signals

We propose a novel similarity measure, called the correntropy coefficient, sensitive to higher order moments of the signal statistics based on a similarity function called the cross-correntopy. Cross-correntropy nonlinearly maps the original time series into a high-dimensional reproducing kernel Hilbert space (RKHS). The correntropy coefficient computes the cosine of the angle between the transformed vectors. Preliminary experiments with simulated data and multichannel electroencephalogram (EEG) signals during behaviour studies elucidate the performance of the new measure versus the well-established correlation coefficient.

Detection of synchronization between chaotic signals: An adaptive similarity-based approach

We present an adaptive similarity-based approach to detect generalized synchronization (GS) with n:m phase synchronization (PS), where n and m are integers and one of them is 1. This approach is based on the similarity index (SI) and Gaussian mixture model with the minimum description length criterion. The clustering method, which is shown to be superior to the closeness and connectivity of a continuous function, is employed in this study to detect the existence of GS with n:m PS. We conducted a computer simulation and a finger-lifting experiment to illustrate the effectiveness of the proposed method. In the simulation of a Rössler-Lorenz system, our method outperformed the conventional SI, and GS with 2:1 PS within the coupled system was found. In the experiment of self-paced finger-lifting movement, cortico-muscular GS with 1:2 and 1:3 PS was found between the surface electromyogram signals on the first dorsal interossei muscle and the magnetoencephalographic data in the motor area. The GS with n:m PS ( n or m=1 ) has been simultaneously resolved from both simulation and experiment. The proposed approach thereby provides a promising means for advancing research into both nonlinear dynamics and brain science.

Evaluation of cortical connectivity during real and imagined rhythmic finger tapping

Accumulating evidence suggests the existence of a shared neural substrate between imagined and executed movements. However, a better understanding of the mechanisms involved in the motor execution and motor imagery requires knowledge of the way the co-activated brain regions interact to each other during the particular (real or imagined) motor task. Within this general framework, the aim of the present study is to investigate the cortical activation and connectivity sub-serving real and imaginary rhythmic finger tapping, from the analysis of multi-channel electroencephalogram (EEG) scalp recordings. A sequence of 250 auditory pacing stimuli has been used for both the real and imagined right finger tapping task, with a constant inter-stimulus interval of 1.5 s length. During the motor execution, healthy subjects were asked to tap in synchrony with the regular sequence of stimulus events, whereas in the imagery condition subjects imagined themselves tapping in time with the auditory cue. To improve the spatial resolution of the scalp fields and suppress unwanted interferences, the EEG data have been spatially filtered. Further, event related synchronization and desynchronization phenomena and phase synchronization analysis have been employed for the study of functionally active brain areas and their connectivity during real and imagery finger tapping. Our results show a fronto-parietal co-activation during both real and imagined movements and similar connectivity patterns among contralateral brain areas. The results support the hypothesis that functional connectivity over the contralateral hemisphere during finger tapping is preserved in imagery. The approach and results can be regarded as indicative evidences of a new strategy for recognizing imagined movements in EEG-based brain computer interface research.

Estimating nonlinear interdependences in dynamical systems using cellular nonlinear networks

We propose a method for estimating nonlinear interdependences between time series using cellular nonlinear networks. Our approach is based on the nonlinear dynamics of interacting nonlinear elements. We apply it to time series of coupled nonlinear model systems and to electroencephalographic time series from an epilepsy patient, and we show that an accurate approximation of symmetric and asymmetric realizations of a nonlinear interdependence measure can be achieved, thus allowing one to detect the strength and direction of couplings.

Estimation of the direction of the coupling by conditional probabilities of recurrence

We introduce a method to detect and quantify the asymmetry of the coupling between two interacting systems based on their recurrence properties. This method can detect the direction of the coupling in weakly as well as strongly coupled systems. It even allows detecting the asymmetry of the coupling in the more challenging case of structurally different systems and it is very robust against noise. We also address the problem of detecting the asymmetry of the coupling in passive experiments, i.e., when the strength of the coupling cannot be systematically changed, which is of great relevance for the analysis of experimental time series.

Relative performance of mutual information estimation methods for quantifying the dependence among short and noisy data

Commonly used dependence measures, such as linear correlation, cross-correlogram, or Kendall's tau , cannot capture the complete dependence structure in data unless the structure is restricted to linear, periodic, or monotonic. Mutual information (MI) has been frequently utilized for capturing the complete dependence structure including nonlinear dependence. Recently, several methods have been proposed for the MI estimation, such as kernel density estimators (KDEs), k -nearest neighbors (KNNs), Edgeworth approximation of differential entropy, and adaptive partitioning of the XY plane. However, outstanding gaps in the current literature have precluded the ability to effectively automate these methods, which, in turn, have caused limited adoptions by the application communities. This study attempts to address a key gap in the literature-specifically, the evaluation of the above methods to choose the best method, particularly in terms of their robustness for short and noisy data, based on comparisons with the theoretical MI estimates, which can be computed analytically, as well with linear correlation and Kendall's tau . Here we consider smaller data sizes, such as 50, 100, and 1000, and within this study we characterize 50 and 100 data points as very short and 1000 as short. We consider a broader class of functions, specifically linear, quadratic, periodic, and chaotic, contaminated with artificial noise with varying noise-to-signal ratios. Our results indicate KDEs as the best choice for very short data at relatively high noise-to-signal levels whereas the performance of KNNs is the best for very short data at relatively low noise levels as well as for short data consistently across noise levels. In addition, the optimal smoothing parameter of a Gaussian kernel appears to be the best choice for KDEs while three nearest neighbors appear optimal for KNNs. Thus, in situations where the approximate data sizes are known in advance and exploratory data analysis and/or domain knowledge can be used to provide a priori insights into the noise-to-signal ratios, the results in the paper point to a way forward for automating the process of MI estimation.

Directionality of coupling from bivariate time series: how to avoid false causalities and missed connections

We discuss some problems encountered in inference of directionality of coupling, or, in the case of two interacting systems, in inference of causality from bivariate time series. We identify factors and influences that can lead to either decreased test sensitivity or false detections and propose ways to cope with them in order to perform tests with high sensitivity and a low rate of false positive results.

Revealing direction of coupling between neuronal oscillators from time series: phase dynamics modeling versus partial directed coherence

The problem of determining directional coupling between neuronal oscillators from their time series is addressed. We compare performance of the two well-established approaches: partial directed coherence and phase dynamics modeling. They represent linear and nonlinear time series analysis techniques, respectively. In numerical experiments, we found each of them to be applicable and superior under appropriate conditions: The latter technique is superior if the observed behavior is "closer" to limit-cycle dynamics, the former is better in cases that are closer to linear stochastic processes.

Spatio-Temporal Clustering of Epileptic ECOG

The spatio-temporal mechanisms underlying the generation of epileptic seizures is not yet clearly understood. In this study, we attempt to quantify the spatio-temporal interactions of an epileptic brain by using a previously proposed SOM-based Similarity Index (SI) measure. We further show that spectral clustering approach can be appropriately used to determine the average spatial mappings in the brain at different stages of a seizure, by interpreting the SOM-SI values as affinity matrices. Results involving two pairs of seizures of an epileptic patient suggest that there may not be a regular pattern associated with channels's spatio-temporal dynamics during the inter-ictal to prepost ictal transition.

A novel approach to the detection of synchronisation in EEG based on empirical mode decomposition

Transient neural assemblies mediated by synchrony in particular frequency ranges are thought to underlie cognition. We propose a new approach to their detection, using empirical mode decomposition (EMD), a data-driven approach removing the need for arbitrary bandpass filter cut-offs. Phase locking is sought between modes. We explore the features of EMD, including making a quantitative assessment of its ability to preserve phase content of signals, and proceed to develop a statistical framework with which to assess synchrony episodes. Furthermore, we propose a new approach to ensure signal decomposition using EMD. We adapt the Hilbert spectrum to a time-frequency representation of phase locking and are able to locate synchrony successfully in time and frequency between synthetic signals reminiscent of EEG. We compare our approach, which we call EMD phase locking analysis (EMDPL) with existing methods and show it to offer improved time-frequency localisation of synchrony.

Granger causality and cross recurrence plots in rheochaos

Our stress relaxation measurements on wormlike micelles using a Rheo-SALS (rheology + small angle light scattering) apparatus allow simultaneous measurements of the stress and the scattered depolarized intensity. The latter is sensitive to orientational ordering of the micelles. To determine the presence of causal influences between the stress and the depolarized intensity time series, we have used the technique of linear and nonlinear Granger causality. We find there exists a feedback mechanism between the two time series and that the orientational order has a stronger causal effect on the stress than vice versa. We have also studied the phase space dynamics of the stress and the depolarized intensity time series using the recently developed technique of cross recurrence plots (CRPs). The presence of diagonal line structures in the CRPs unambiguously proves that the two time series share similar phase space dynamics.

Mutual nonlinear prediction of cardiovascular variability series: comparison between exogenous and autoregressive exogenous models

A model-based approach to perform mutual nonlinear prediction of short cardiovascular variability series is presented. The approach is based on identifying exogenous (X) and autoregressive exogenous (ARX) models by K-nearest neighbors local linear approximation, and estimates the predictability of a series given the other as the squared correlation between original and predicted values of the series. The method was first tested on simulations reproducing different types of interaction between non-identical Henon maps, and then applied to heart rate (HR) and blood pressure (BP) variability series measured in healthy subjects at rest and after head-up tilt. Simulations showed that different coupling conditions were always detected by the X model but not by the ARX model. The comparison between X and ARX models suggested the presence of oscillatory sources determining the regularity of HR and BP dynamics independently of their closed-loop mutual regulation. The transition from supine to upright position was associated with an enhancement of the HR and BP mutual regulation, compatible with the activation of the sympathetic nervous system induced by tilt.

Synchronization likelihood with explicit time-frequency priors

Cognitive processing requires integration of information processed simultaneously in spatially distinct areas of the brain. The influence that two brain areas exert on each others activity is usually governed by an unknown function, which is likely to have nonlinear terms. If the functional relationship between activities in different areas is dominated by the nonlinear terms, linear measures of correlation may not detect the statistical interdependency satisfactorily. Therefore, algorithms for detecting nonlinear dependencies may prove invaluable for characterizing the functional coupling in certain neuronal systems, conditions or pathologies. Synchronization likelihood (SL) is a method based on the concept of generalized synchronization and detects nonlinear and linear dependencies between two signals (Stam, C.J., van Dijk, B.W., 2002. Synchronization likelihood: An unbiased measure of generalized synchronization in multivariate data sets. Physica D, 163: 236-241.). SL relies on the detection of simultaneously occurring patterns, which can be complex and widely different in the two signals. Clinical studies applying SL to electro- or magnetoencephalography (EEG/MEG) signals have shown promising results. In previous implementations of the algorithm, however, a number of parameters have lacked a rigorous definition with respect to the time-frequency characteristics of the underlying physiological processes. Here we introduce a rationale for choosing these parameters as a function of the time-frequency content of the patterns of interest. The number of parameters that can be arbitrarily chosen by the user of the SL algorithm is thereby decreased from six to two. Empirical evidence for the advantages of our proposal is given by an application to EEG data of an epileptic seizure and simulations of two unidirectionally coupled Hénon systems.

Mixed predictability and cross-validation to assess non-linear Granger causality in short cardiovascular variability series

A method to evaluate the direction and strength of causal interactions in bivariate cardiovascular and cardiorespiratory series is presented. The method is based on quantifying self and mixed predictability of the two series using nearest-neighbour local linear approximation. It returns two causal coupling indexes measuring the relative improvement in predictability along direct and reverse directions, and a directionality index indicating the preferential direction of interaction. The method was implemented through a cross-validation approach that allowed quantification of directionality without constraining the embedding of the series, and fully exploited the available data to maximise the prediction accuracy. Validation on short simulated bivariate time series demonstrated the ability of the method to capture different degrees of unidirectional and bidirectional interaction. Moreover, application to representative examples of heart rate, systolic arterial pressure and respiration series allowed the inference of causal relationships related to known physiological mechanisms and experimental conditions.

Quantitative evaluation of linear and nonlinear methods characterizing interdependencies between brain signals

Brain functional connectivity can be characterized by the temporal evolution of correlation between signals recorded from spatially-distributed regions. It is aimed at explaining how different brain areas interact within networks involved during normal (as in cognitive tasks) or pathological (as in epilepsy) situations. Numerous techniques were introduced for assessing this connectivity. Recently, some efforts were made to compare methods performances but mainly qualitatively and for a special application. In this paper, we go further and propose a comprehensive comparison of different classes of methods (linear and nonlinear regressions, phase synchronization, and generalized synchronization) based on various simulation models. For this purpose, quantitative criteria are used: in addition to mean square error under null hypothesis (independence between two signals) and mean variance computed over all values of coupling degree in each model, we provide a criterion for comparing performances. Results show that the performances of the compared methods are highly dependent on the hypothesis regarding the underlying model for the generation of the signals. Moreover, none of them outperforms the others in all cases and the performance hierarchy is model dependent.

Nonlinear parametric model for Granger causality of time series

The notion of Granger causality between two time series examines if the prediction of one series could be improved by incorporating information of the other. In particular, if the prediction error of the first time series is reduced by including measurements from the second time series, then the second time series is said to have a causal influence on the first one. We propose a radial basis function approach to nonlinear Granger causality. The proposed model is not constrained to be additive in variables from the two time series and can approximate any function of these variables, still being suitable to evaluate causality. Usefulness of this measure of causality is shown in two applications. In the first application, a physiological one, we consider time series of heart rate and blood pressure in congestive heart failure patients and patients affected by sepsis: we find that sepsis patients, unlike congestive heart failure patients, show symmetric causal relationships between the two time series. In the second application, we consider the feedback loop in a model of excitatory and inhibitory neurons: we find that in this system causality measures the combined influence of couplings and membrane time constants.

Measuring direction in the coupling of biological oscillators: a case study for electroreceptors of paddlefish

Recently developed methods for estimating directionality in the coupling between oscillators were tested on experimental time series data from electroreceptors of paddlefish, because each electroreceptor contains two distinct types of noisy oscillators. One type of oscillator is in the sensory epithelia, and another type is in the terminals of afferent neurons. Based on morphological organization and our previous work, we expected unidirectional coupling, whereby epithelial oscillations synaptically influence the spiking oscillators of afferent neurons. Using directionality analysis we confirmed unidirectional coupling of oscillators embedded in electroreceptors. We studied the performance of directionality algorithms for decreasing length of data. Also, we experimentally varied the strength of oscillator coupling, to test the effect of coupling strength on directionality algorithms.

Partial phase synchronization for multivariate synchronizing systems

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

Variability of EEG synchronization prior to and during observation and execution of a sequential finger movement

The aim of this study was to test the validity of mirror neuron activity in humans through analysis of electroencephalographic (EEG) functional connectivity during an action not directed towards an object. We investigated changes in EEG interchannel synchronization prior to and during action execution and also prior to and during observation of the same action. Twelve participants observed a simple finger movement sequence. In a second testing session they physically executed the movement. EEGs were recorded from 19 active sites across the cortex. Activity was considered in four frequency bands (7-10 Hz, 10-13 Hz, 13-20 Hz, and 20-30 Hz) using a new measure: synchronization likelihood. This technique considers rapid changes in signal synchronization and spatiotemporal patterns of coherence. The results revealed no statistically significant difference in synchronization likelihood between the observation and execution data. We found an increase in synchronization over a broad frequency range during task processing and suggest that this may reflect interregional cortical coupling of intricately and hierarchically interconnected networks that are active in a similar way during both observation and execution of a movement. While EEG may be insensitive to differences present during the observation and execution of a movement, the results of the present study shed some light on the general mechanisms of cognitive integration.

Bivariate nonlinear prediction to quantify the strength of complex dynamical interactions in short-term cardiovascular variability

A nonlinear prediction method for investigating the dynamic interdependence between short length time series is presented. The method is a generalization to bivariate prediction of the univariate approach based on nearest neighbor local linear approximation. Given the input and output series x and y, the relationship between a pattern of samples of x and a synchronous sample of y was approximated with a linear polynomial whose coefficients were estimated from an equation system including the nearest neighbor patterns in x and the corresponding samples in y. To avoid overfitting and waste of data, the training and testing stages of the prediction were designed through a specific out-of-sample cross validation. The robustness of the method was assessed on short realizations of simulated processes interacting either linearly or nonlinearly. The predictor was then used to characterize the dynamical interaction between the short-term spontaneous fluctuations of heart period (RR interval) and systolic arterial pressure (SAP) in healthy young subjects. In the supine position, the predictability of RR given SAP was low and influenced by nonlinear dynamics. After head-up tilt the predictability increased significantly and was mostly due to linear dynamics. These findings were related to the larger involvement of the baroreflex regulation from SAP to RR in upright than in supine humans, and to the simplification of the RR-SAP coupling occurring with the tilt-induced alteration of the neural regulation of the cardiovascular rhythms.

A distributed computing system for multivariate time series analyses of multichannel neurophysiological data

We present a client-server application for the distributed multivariate analysis of time series using standard PCs. We here concentrate on analyses of multichannel EEG/MEG data, but our method can easily be adapted to other time series. Due to the rapid development of new analysis techniques, the focus in the design of our application was not only on computational performance, but also on high flexibility and expandability of both the client and the server programs. For this purpose, the communication between the server and the clients as well as the building of the computational tasks has been realized via the Extensible Markup Language (XML). Running our newly developed method in an asynchronous distributed environment with random availability of remote and heterogeneous resources, we tested the system's performance for a number of different univariate and bivariate analysis techniques. Results indicate that for most of the currently available analysis techniques, calculations can be performed in real time, which, in principle, allows on-line analyses at relatively low cost.

Enhanced synchrony in epileptiform activity? Local versus distant phase synchronization in generalized seizures

Synchronization is a fundamental characteristic of complex systems and a basic mechanism of self-organization. A traditional, accepted perspective on epileptiform activity holds that hypersynchrony covering large brain regions is a hallmark of generalized seizures. However, a few recent reports have described substantial fluctuations in synchrony before and during ictal events, thus raising questions as to the widespread synchronization notion. In this study, we used magnetoencephalographic recordings from epileptic patients with generalized seizures and normal control subjects to address the extent of the phase synchronization (phase locking) in local (neighboring) and distant cortical areas and to explore the ongoing temporal dynamics for particular ranges of frequencies at which synchrony occurs, during interictal and ictal activity. Synchronization patterns were found to differ somewhat depending on the epileptic syndrome, with primary generalized absence seizures displaying more long-range synchrony in all frequency bands studied (3-55 Hz) than generalized tonic motor seizures of secondary (symptomatic) generalized epilepsy or frontal lobe epilepsy. However, all seizures were characterized by enhanced local synchrony compared with distant synchrony. There were fluctuations in the synchrony between specific cortical areas that varied from seizure to seizure in the same patient, but in most of the seizures studied, regardless of semiology, there was a constant pattern in the dynamics of synchronization, indicating that seizures proceed by a recruitment of neighboring neuronal networks. Together, these data indicate that the concept of widespread "hypersynchronous" activity during generalized seizures may be misleading and valid only for very specific neuronal ensembles and circumstances.

Nonlinear Analysis of Cerebral Hemodynamic and Intracranial Pressure Signals for Characterization of Autoregulation

The objective of this study was to determine whether or not the underlying physiological systems that generates spontaneous arterial blood pressure (ABP), cerebral blood flow velocity (CBFV), and intracranial pressure signals could be adequately approximated as a linear stochastic process. Furthermore, a new measure (C) capable of capturing the degree of nonlinear dependency between two ABP and CBFV signals (including a time-varying situation) was proposed for quantifying the degree of cerebral blood flow autoregulation. A surrogate data test of fifteen ABP, CBFV, and intracranial pressure (ICP) segments was conducted for detecting whether there exists a statistically significant deviation from the null hypothesis of linear signals. The extension of the established block computation method of C measure to an adaptive one was achieved. This new algorithm was then applied to study the C evolution using brain injury patients data from a hyperventilation study and two propofol studies. Nonlinearity has not been detected for all the fifteen recordings, neither has nonlinear dependency between CBFV and ABP. However, their presences in some of the signal segments justified the adoption of a nonlinear measure of dependency capable of characterizing both linear and nonlinear correlations for inferring autoregulation status. C measure started to decrease with the introduction of hypocapnia state indicating that hyperventilation may reduce the dependency of CBFV on ABP fluctuations. On the other hand, complex patterns of C measure evolution were observed among 14 cases of propofol data indicating a nontrivial effect of propofol on the dependency of CBFV on ABP.

Lag synchrony measures dynamical processes underlying progression of seizure states

We investigate the dynamics of bursting behavior in an intact hippocampal preparation using causal entropy, an adaptive measure of lag synchrony. This analysis, together with a heuristic model of coupled bursting networks, separates experimentally observed bursting dynamics into two dynamical regimes, when bursting is driven by (1) the intranetwork dynamics of a single region, or (2) internetwork feedback between spatially disjoint neural populations. Our results suggest that the abrupt transition between these two states heralds the gradual desynchronization of bursting activity. These results illustrate how superficially homogeneous behavior across loosely coupled networks may harbor hidden, but robust, dynamical processes.

Approaches to Information-Theoretic Analysis of Neural Activity

Understanding how neurons represent, process, and manipulate information is one of the main goals of neuroscience. These issues are fundamentally abstract, and information theory plays a key role in formalizing and addressing them. However, application of information theory to experimental data is fraught with many challenges. Meeting these challenges has led to a variety of innovative analytical techniques, with complementary domains of applicability, assumptions, and goals.

Fronto-parietal coupling of brain rhythms in mild cognitive impairment: a multicentric EEG study

Electroencephalographic (EEG) data were recorded in 69 normal elderly (Nold), 88 mild cognitive impairment (MCI), and 109 mild Alzheimer's disease (AD) subjects at rest condition, to test whether the fronto-parietal coupling of EEG rhythms is in line with the hypothesis that MCI can be considered as a pre-clinical stage of the disease at group level. Functional coupling was estimated by synchronization likelihood of Laplacian-transformed EEG data at electrode pairs, which accounts for linear and non-linear components of that coupling. Cortical rhythms of interest were delta (2-4Hz), theta (4-8Hz), alpha 1 (8-10.5Hz), alpha 2 (10.5-13Hz), beta 1 (13-20Hz), beta 2 (20-30Hz), and gamma (30-40Hz). Compared to the Nold subjects, the AD patients presented a marked reduction of the synchronization likelihood (delta to gamma) at both fronto-parietal and inter-hemispherical (delta to beta 2) electrodes. As a main result, alpha 1 synchronization likelihood progressively decreased across Nold, MCI, and mild AD subjects at midline (Fz-Pz) and right (F4-P4) fronto-parietal electrodes. The same was true for the delta synchronization likelihood at right fronto-parietal electrodes (F4-P4). For these EEG bands, the synchronization likelihood correlated with global cognitive status as measured by the Mini Mental State Evaluation. The present results suggest that at group level, fronto-parietal coupling of the delta and alpha rhythms progressively becomes abnormal though MCI and mild AD. Future longitudinal research should evaluate whether the present EEG approach is able to predict the cognitive decline in individual MCI subjects.

A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis

The aim of this paper is to explain critical features of the human primary generalized epilepsies by investigating the dynamical bifurcations of a nonlinear model of the brain's mean field dynamics. The model treats the cortex as a medium for the propagation of waves of electrical activity, incorporating key physiological processes such as propagation delays, membrane physiology, and corticothalamic feedback. Previous analyses have demonstrated its descriptive validity in a wide range of healthy states and yielded specific predictions with regards to seizure phenomena. We show that mapping the structure of the nonlinear bifurcation set predicts a number of crucial dynamic processes, including the onset of periodic and chaotic dynamics as well as multistability. Quantitative study of electrophysiological data supports the validity of these predictions. Hence, we argue that the core electrophysiological and cognitive differences between tonic-clonic and absence seizures are predicted and interrelated by the global bifurcation diagram of the model's dynamics. The present study is the first to present a unifying explanation of these generalized seizures using the bifurcation analysis of a dynamical model of the brain.

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

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

Nonlinear multivariate analysis of neurophysiological signals

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

Assessing causality from multivariate time series

In this work we propose a general nonparametric test of causality for weakly dependent time series. More precisely, we study the problem of attribution, i.e., the proper comparison of the relative influence that two or more external dynamics trigger on a given system of interest. We illustrate the possible applications of the proposed methodology in very different fields like physiology and climate science.

Inferences about information flow and dispersal for spatially extended population systems using time-series data

This work explores an information-theoretic approach to drawing inferences about coupling of spatially extended ecological populations based solely on time-series of abundances. The efficacy of the approach, time-delayed mutual information, was explored using a spatially extended predator-prey model system in which populations at different patches were coupled via diffusive movement. The approach identified the relative magnitude and direction of information flow resulting from animal movement between populations, the change in information flow as a function of distance separating populations, and the diffusive nature of the information flow. In addition, when the diffusive movement was eliminated from the model, mutual information correctly provided no evidence of information flow, even when population synchrony was generated by a common environmental driving function. Thus, for this model system, time-delayed mutual information was useful in discriminating between the Moran effect and animal movement as causes of population synchrony, as well as in characterizing dispersal in terms of direction, relative speed and diffusive nature.

Quantitative analysis of chaotic synchronization by means of coherence

We use an index of chaotic synchronization based on the averaged coherence function for the quantitative analysis of the process of the complete synchronization loss in unidirectionally coupled oscillators and maps. We demonstrate that this value manifests different stages of the synchronization breaking. It is invariant to time delay and insensitive to small noise and distortions, which can influence the accessible signals at measurements. Peculiarities of the synchronization destruction in maps and oscillators are investigated.

Estimation of coupling between oscillators from short time series via phase dynamics modeling: limitations and application to EEG data

We demonstrate in numerical experiments that estimators of strength and directionality of coupling between oscillators based on modeling of their phase dynamics [D. A. Smirnov and B. P. Bezruchko, Phys. Rev. E 68, 046209 (2003)] are widely applicable. Namely, although the expressions for the estimators and their confidence bands are derived for linear uncoupled oscillators under the influence of independent sources of Gaussian white noise, they turn out to allow reliable characterization of coupling from relatively short time series for different properties of noise, significant phase nonlinearity of the oscillators, and nonvanishing coupling between them. We apply the estimators to analyze a two-channel human intracranial epileptic electroencephalogram (EEG) recording with the purpose of epileptic focus localization.

Assessment of EEG synchronization based on state-space analysis

Cortical computation involves the formation of cooperative neuronal assemblies characterized by synchronous oscillatory activity. A traditional method for the identification of synchronous neuronal assemblies has been the coherence analysis of EEG signals. Here, we suggest a new method called S estimator, whereby cortical synchrony is defined from the embedding dimension in a state-space. We first validated the method on clusters of chaotic coupled oscillators and compared its performance to that of other methods for assessing synchronization. Then nine adult subjects were studied with high-density EEG recordings, while they viewed in the two hemifields (hence with separate hemispheres) identical sinusoidal gratings either arranged collinearly and moving together, or orthogonally oriented and moving at 90 degrees . The estimated synchronization increased with the collinear gratings over a cluster of occipital electrodes spanning both hemispheres, whereas over temporo-parietal regions of both hemispheres, it decreased with the same stimulus and it increased with the orthogonal gratings. Separate calculations for different EEG frequencies showed that the occipital clusters involved synchronization in the beta band and the temporal clusters in the alpha band. The gamma band appeared to be insensitive to stimulus diversity. Different stimulus configurations, therefore, appear to cause a complex rearrangement of synchronous neuronal assemblies distributed over the cortex, in particular over the visual cortex.

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

We compare two conceptually different approaches to the detection of weak directional couplings between two oscillatory systems from bivariate time series. The first approach is based on the analysis of the systems' phase dynamics, whereas the other one tests for interdependencies in the reconstructed state spaces of the systems. We analyze the sensitivity of both techniques to weak couplings in numerical experiments by considering couplings between almost identical as well as between significantly different nonlinear systems. We study different degrees of phase diffusion, test the robustness of the two techniques against observational noise, and investigate the influence of the time series length. Our results show that none of the two approaches is generally superior to the other, and we conclude that it is probably the combination of both techniques that would allow the most comprehensive and reliable characterization of coupled systems.

Time-related interdependence between low-frequency cortical electrical activity and respiratory activity in lizard,Gallotia galloti

Electroencephalograms of medial cortex and electromyograms of intercostal muscles (EMG-icm) were simultaneously recorded in the lizard, Gallotia galloti, during two daily time periods (at daytime, DTP: 1200-1600 h; by night, NTP: 0000-0400 h), to investigate whether a relationship exists between the respiratory and cortical electrical activity of reptiles, and, if so, how this relationship changes during the night rest period. Testing was carried out by studying interdependence between cortical electrical and respiratory activities, by means of linear and nonlinear signal analysis techniques. Both physiological activities were evaluated through simultaneous power signals, derived from the power of the low-frequency band of the electroencephalogram (pEEG-LF), and from the power of the EMG-icm (pEMG-icm), respectively. During both DTP and NTP, there was a significant coherence between both signals in the main frequency band of pEMG-icm. During both DTP and NTP, the nonlinear index N measured significant linear asymmetric interdependence between pEEG-LF and pEMG-icm. The N value obtained between pEEG-LF vs. pEMG-icm was greater than the one between pEMG-icm vs. pEEG-LF. This means that the system that generates the pEEG-LF is more complex than the one that generates the pEMG-icm, and suggests that the temporal variability of power in the low-frequency cortical electrical activity is driven by the power of the respiratory activity.

Assessing spatial coupling in complex population dynamics using mutual prediction and continuity statistics

A number of important questions in ecology involve the possibility of interactions or "coupling" among potential components of ecological systems. The basic question of whether two components are coupled (exhibit dynamical interdependence) is relevant to investigations of movement of animals over space, population regulation, food webs and trophic interactions, and is also useful in the design of monitoring programs. For example, in spatially extended systems, coupling among populations in different locations implies the existence of redundant information in the system and the possibility of exploiting this redundancy in the development of spatial sampling designs. One approach to the identification of coupling involves study of the purported mechanisms linking system components. Another approach is based on time series of two potential components of the same system and, in previous ecological work, has relied on linear cross-correlation analysis. Here we present two different attractor-based approaches, continuity and mutual prediction, for determining the degree to which two population time series (e.g., at different spatial locations) are coupled. Both approaches are demonstrated on a one-dimensional predator-prey model system exhibiting complex dynamics. Of particular interest is the spatial asymmetry introduced into the model as linearly declining resource for the prey over the domain of the spatial coordinate. Results from these approaches are then compared to the more standard cross-correlation analysis. In contrast to cross-correlation, both continuity and mutual prediction are clearly able to discern the asymmetry in the flow of information through this system.