Lead/Lag directionality is not generally equivalent to causality in nonlinear systems: Comparison of phase slope index and conditional mutual information.

Applications of causal techniques to neural time series have increased extensively over last decades, including a wide and diverse family of methods focusing on electroencephalogram (EEG) analysis. Besides connectivity inferred in defined frequency bands, there is a growing interest in the analysis of cross-frequency interactions, in particular phase and amplitude coupling and directionality. Some studies show contradicting results of coupling directionality from high frequency to low frequency signal components, in spite of generally considered modulation of a high-frequency amplitude by a low-frequency phase. We have compared two widely used methods to estimate the directionality in cross frequency coupling: conditional mutual information (CMI) and phase slope index (PSI). The latter, applied to infer cross-frequency phase-amplitude directionality from animal intracranial recordings, gives opposite results when comparing to CMI. Both metrics were tested in a numerically simulated example of unidirectionally coupled Rössler systems, which helped to find the explanation of the contradictory results: PSI correctly estimates the lead/lag relationship which, however, is not generally equivalent to causality in the sense of directionality of coupling in nonlinear systems, correctly inferred by using CMI with surrogate data testing.

Compression complexity with ordinal patterns for robust causal inference in irregularly sampled time series.

Distinguishing cause from effect is a scientific challenge resisting solutions from mathematics, statistics, information theory and computer science. Compression-Complexity Causality (CCC) is a recently proposed interventional measure of causality, inspired by Wiener-Granger's idea. It estimates causality based on change in dynamical compression-complexity (or compressibility) of the effect variable, given the cause variable. CCC works with minimal assumptions on given data and is robust to irregular-sampling, missing-data and finite-length effects. However, it only works for one-dimensional time series. We propose an ordinal pattern symbolization scheme to encode multidimensional patterns into one-dimensional symbolic sequences, and thus introduce the Permutation CCC (PCCC). We demonstrate that PCCC retains all advantages of the original CCC and can be applied to data from multidimensional systems with potentially unobserved variables which can be reconstructed using the embedding theorem. PCCC is tested on numerical simulations and applied to paleoclimate data characterized by irregular and uncertain sampling and limited numbers of samples.

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.

Cross-Scale Causality and Information Transfer in Simulated Epileptic Seizures.

An information-theoretic approach for detecting causality and information transfer was applied to phases and amplitudes of oscillatory components related to different time scales and obtained using the wavelet transform from a time series generated by the Epileptor model. Three main time scales and their causal interactions were identified in the simulated epileptic seizures, in agreement with the interactions of the model variables. An approach consisting of wavelet transform, conditional mutual information estimation, and surrogate data testing applied to a single time series generated by the model was demonstrated to be successful in the identification of all directional (causal) interactions between the three different time scales described in the model. Thus, the methodology was prepared for the identification of causal cross-frequency phase-phase and phase-amplitude interactions in experimental and clinical neural data.

Causality and Information Transfer Between the Solar Wind and the Magnetosphere-Ionosphere System.

An information-theoretic approach for detecting causality and information transfer is used to identify interactions of solar activity and interplanetary medium conditions with the Earth's magnetosphere-ionosphere systems. A causal information transfer from the solar wind parameters to geomagnetic indices is detected. The vertical component of the interplanetary magnetic field (Bz) influences the auroral electrojet (AE) index with an information transfer delay of 10 min and the geomagnetic disturbances at mid-latitudes measured by the symmetric field in the H component (SYM-H) index with a delay of about 30 min. Using a properly conditioned causality measure, no causal link between AE and SYM-H, or between magnetospheric substorms and magnetic storms can be detected. The observed causal relations can be described as linear time-delayed information transfer.

Coupling in complex systems as information transfer across time scales.

Complex systems such as the human brain or the Earth's climate consist of many subsystems interacting in intricate, nonlinear ways. Moreover, variability of such systems extends over broad ranges of spatial and temporal scales and dynamical phenomena on different scales also influence each other. In order to explain how to detect cross-scale causal interactions, we review information-theoretic formulation of the Granger causality in combination with computational statistics (surrogate data method) and demonstrate how this method can be used to infer driver-response relations from amplitudes and phases of coupled nonlinear dynamical systems. Considering complex systems evolving on multiple time scales, the reviewed methodology starts with a wavelet decomposition of a multi-scale signal into quasi-oscillatory modes of a limited bandwidth, described using their instantaneous phases and amplitudes. Then statistical associations, in particular, causality relations between phases or between phases and amplitudes on different time scales are tested using the conditional mutual information. As an application, we present the analysis of cross-scale interactions and information transfer in the dynamics of the El Niño Southern Oscillation. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Loss of neuronal network resilience precedes seizures and determines the ictogenic nature of interictal synaptic perturbations.

The mechanism of seizure emergence and the role of brief interictal epileptiform discharges (IEDs) in seizure generation are two of the most important unresolved issues in modern epilepsy research. We found that the transition to seizure is not a sudden phenomenon, but is instead a slow process that is characterized by the progressive loss of neuronal network resilience. From a dynamical perspective, the slow transition is governed by the principles of critical slowing, a robust natural phenomenon that is observable in systems characterized by transitions between dynamical regimes. In epilepsy, this process is modulated by synchronous synaptic input from IEDs. IEDs are external perturbations that produce phasic changes in the slow transition process and exert opposing effects on the dynamics of a seizure-generating network, causing either anti-seizure or pro-seizure effects. We found that the multifaceted nature of IEDs is defined by the dynamical state of the network at the moment of the discharge occurrence.

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.

Comparison of six methods for the detection of causality in a bivariate time series.

In this comparative study, six causality detection methods were compared, namely, the Granger vector autoregressive test, the extended Granger test, the kernel version of the Granger test, the conditional mutual information (transfer entropy), the evaluation of cross mappings between state spaces, and an assessment of predictability improvement due to the use of mixed predictions. Seven test data sets were analyzed: linear coupling of autoregressive models, a unidirectional connection of two Hénon systems, a unidirectional connection of chaotic systems of Rössler and Lorenz type and of two different Rössler systems, an example of bidirectionally connected two-species systems, a fishery model as an example of two correlated observables without a causal relationship, and an example of mediated causality. We tested not only 20000 points long clean time series but also noisy and short variants of the data. The standard and the extended Granger tests worked only for the autoregressive models. The remaining methods were more successful with the more complex test examples, although they differed considerably in their capability to reveal the presence and the direction of coupling and to distinguish causality from mere correlation.

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.

Smooth information flow in temperature climate network reflects mass transport.

A directed climate network is constructed by Granger causality analysis of air temperature time series from a regular grid covering the whole Earth. Using winner-takes-all network thresholding approach, a structure of a smooth information flow is revealed, hidden to previous studies. The relevance of this observation is confirmed by comparison with the air mass transfer defined by the wind field. Their close relation illustrates that although the information transferred due to the causal influence is not a physical quantity, the information transfer is tied to the transfer of mass and energy.

Small-world bias of correlation networks: From brain to climate.

Complex systems are commonly characterized by the properties of their graph representation. Dynamical complex systems are then typically represented by a graph of temporal dependencies between time series of state variables of their subunits. It has been shown recently that graphs constructed in this way tend to have relatively clustered structure, potentially leading to spurious detection of small-world properties even in the case of systems with no or randomly distributed true interactions. However, the strength of this bias depends heavily on a range of parameters and its relevance for real-world data has not yet been established. In this work, we assess the relevance of the bias using two examples of multivariate time series recorded in natural complex systems. The first is the time series of local brain activity as measured by functional magnetic resonance imaging in resting healthy human subjects, and the second is the time series of average monthly surface air temperature coming from a large reanalysis of climatological data over the period 1948-2012. In both cases, the clustering in the thresholded correlation graph is substantially higher compared with a realization of a density-matched random graph, while the shortest paths are relatively short, showing thus distinguishing features of small-world structure. However, comparable or even stronger small-world properties were reproduced in correlation graphs of model processes with randomly scrambled interconnections. This suggests that the small-world properties of the correlation matrices of these real-world systems indeed do not reflect genuinely the properties of the underlying interaction structure, but rather result from the inherent properties of correlation matrix.

Identifying causal gateways and mediators in complex spatio-temporal systems.

Identifying regions important for spreading and mediating perturbations is crucial to assess the susceptibilities of spatio-temporal complex systems such as the Earth's climate to volcanic eruptions, extreme events or geoengineering. Here a data-driven approach is introduced based on a dimension reduction, causal reconstruction, and novel network measures based on causal effect theory that go beyond standard complex network tools by distinguishing direct from indirect pathways. Applied to a data set of atmospheric dynamics, the method identifies several strongly uplifting regions acting as major gateways of perturbations spreading in the atmosphere. Additionally, the method provides a stricter statistical approach to pathways of atmospheric teleconnections, yielding insights into the Pacific-Indian Ocean interaction relevant for monsoonal dynamics. Also for neuroscience or power grids, the novel causal interaction perspective provides a complementary approach to simulations or experiments for understanding the functioning of complex spatio-temporal systems with potential applications in increasing their resilience to shocks or extreme events.

Multiscale atmospheric dynamics: cross-frequency phase-amplitude coupling in the air temperature.

Interactions between dynamics on different temporal scales of about a century long record of data of the daily mean surface air temperature from various European locations have been detected using a form of the conditional mutual information, statistically tested using the Fourier-transform and multifractal surrogate data methods. An information transfer from larger to smaller time scales has been observed as the influence of the phase of slow oscillatory phenomena with the periods around 6-11 yr on the amplitudes of the variability characterized by the smaller temporal scales from a few months to 4-5 yr. The overall effect of the slow oscillations on the interannual temperature variability within the range 1-2 ° C has been observed in large areas of Europe.

Comparison of coherence and phase synchronization of the human sleep electroencephalogram.

OBJECTIVE: Potential differences between coherence and phase synchronization analyses of human sleep electroencephalogram (EEG) are assessed and occurrences of phase vs. complete synchronization between EEG signals from different locations during different sleep stages are investigated. METHODS: Linear spectral coherence, mean phase coherence (MPC) z-score and Pearson's correlation coefficient of analytic amplitudes were evaluated for different spectral bands of whole-night EEG recordings from 25 healthy subjects. RESULTS: Coherence and MPC z-score demonstrated practically the same statistical differences between vigilance stages, confirming the findings of previous coherence-based studies. MPC z-score and amplitude correlations were most correlated (>0.5) between homologous interhemispheric positions and least correlated between nonhomologous interhemispheric positions and between fronto-occipital positions. CONCLUSIONS: Coherence and phase synchronization provided essentially the same information. Complete synchronization was manifested by highly coherent phases and correlated amplitudes, as well as by correlated changes of phase synchronization, coherence and amplitude correlations between vigilance states. In cases of weaker coupling, phase synchronization and coherence change in agreement, while behaviour of amplitude correlations differs. SIGNIFICANCE: Phase synchronization analysis is not superior to coherence analysis, although the coupling between EEG signals is dominated by phase synchronization which turns into complete synchronization in the most strongly coupled EEG signals.

Functional connectivity in resting-state fMRI: is linear correlation sufficient?

Functional connectivity (FC) analysis is a prominent approach to analyzing fMRI data, especially acquired under the resting state condition. The commonly used linear correlation FC measure bears an implicit assumption of Gaussianity of the dependence structure. If only the marginals, but not all the bivariate distributions are Gaussian, linear correlation consistently underestimates the strength of the dependence. To assess the suitability of linear correlation and the general potential of nonlinear FC measures, we present a framework for testing and estimating the deviation from Gaussianity by means of comparing mutual information in the data and its Gaussianized counterpart. We apply this method to 24 sessions of human resting state fMRI. For each session, matrix of connectivities between 90 anatomical parcel time series is computed using mutual information and compared to results from its multivariate Gaussian surrogate that conserves the correlations but cancels any nonlinearity. While the group-level tests confirmed non-Gaussianity in the FC, the quantitative assessment revealed that the portion of mutual information neglected by linear correlation is relatively minor-on average only about 5% of the mutual information already captured by the linear correlation. The marginality of the non-Gaussianity was confirmed in comparisons using clustering of the parcels-the disagreement between clustering obtained from mutual information and linear correlation was attributable to random error. We conclude that for this type of data, practical relevance of nonlinear methods trying to improve over linear correlation might be limited by the fact that the data are indeed almost Gaussian.

Partitioning networks into clusters and residuals with average association.

We investigate the problem of detecting clusters exhibiting higher-than-average internal connectivity in networks of interacting systems. We show how the average association objective formulated in the context of spectral graph clustering leads naturally to a clustering strategy where each system is assigned to at most one cluster. A residual set is formed of the systems that are not members of any cluster. Maximization of the average association objective leads to a discrete optimization problem, which is difficult to solve, but a relaxed version can be solved using an eigendecomposition of the connectivity matrix. A simple approach to extracting clusters from a relaxed solution is described and developed by applying a variance maximizing solution to the relaxed solution, which leads to a method with increased accuracy and sensitivity. Numerical studies of theoretical connectivity models and of synchronization clusters in a lattice of coupled Lorenz oscillators are conducted to show the efficiency of the proposed approach. The method is applied to an experimentally obtained human resting state functional magnetic resonance imaging dataset and the results are discussed.

Identification of nonlinear oscillatory activity embedded in broadband neural signals.

Oscillatory phenomena in the brain activity and their synchronization are frequently studied using mathematical models and analytic tools derived from nonlinear dynamics. In many experimental situations, however, neural signals have a broadband character and if oscillatory activity is present, its dynamical origin is unknown. To cope with these problems, a framework for detecting nonlinear oscillatory activity in broadband time series is presented. First, a narrow-band oscillatory mode is extracted from a broadband background. Second, it is tested whether the extracted mode is significantly different from linearly filtered noise, modelled as a linear stochastic process possibly passed through a static nonlinear transformation. If a nonlinear oscillatory mode is positively detected, further analysis using nonlinear approaches such as the phase synchronization analysis can potentially bring new information. For linear processes, however, standard approaches such as the coherence analysis are more appropriate and provide sufficient description of underlying interactions with smaller computational effort. The method is illustrated in a numerical example and applied to analyze experimentally obtained human EEG time series from a sleeping subject.

High-frequency network activity, global increase in neuronal activity, and synchrony expansion precede epileptic seizures in vitro.

How seizures start is a major question in epilepsy research. Preictal EEG changes occur in both human patients and animal models, but their underlying mechanisms and relationship with seizure initiation remain unknown. Here we demonstrate the existence, in the hippocampal CA1 region, of a preictal state characterized by the progressive and global increase in neuronal activity associated with a widespread buildup of low-amplitude high-frequency activity (HFA) (>100 Hz) and reduction in system complexity. HFA is generated by the firing of neurons, mainly pyramidal cells, at much lower frequencies. Individual cycles of HFA are generated by the near-synchronous (within approximately 5 ms) firing of small numbers of pyramidal cells. The presence of HFA in the low-calcium model implicates nonsynaptic synchronization; the presence of very similar HFA in the high-potassium model shows that it does not depend on an absence of synaptic transmission. Immediately before seizure onset, CA1 is in a state of high sensitivity in which weak depolarizing or synchronizing perturbations can trigger seizures. Transition to seizure is characterized by a rapid expansion and fusion of the neuronal populations responsible for HFA, associated with a progressive slowing of HFA, leading to a single, massive, hypersynchronous cluster generating the high-amplitude low-frequency activity of the seizure.

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.