Milan Paluš

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.

Interactions between cardiac, respiratory and EEG-δ oscillations in rats during anaesthesia

We hypothesized that, associated with the state of anaesthesia, characteristic changes exist in both cardio‐respiratory and cerebral oscillator parameters and couplings, perhaps varying with depth of anaesthesia. Electrocardiograms (ECGs), respiration and electroencephalograms (EEGs) were recorded from two groups of 10 rats during the entire course of anaesthesia following the administration of a single bolus of ketamine–xylazine (KX group) or pentobarbital (PB group). The phase dynamics approach was then used to extract the instantaneous frequencies of heart beat, respiration and slow δ‐waves (within 0.5–3.5 Hz). The amplitudes of δ‐ and θ‐waves were analysed by use of a time–frequency representation of the EEG signal within 0.5–7.5 Hz obtained by wavelet transformation, using the Morlet mother wavelet. For the KX group, where slow δ‐waves constituted the dominant spectral component, the Hilbert transform was applied to obtain the instantaneous δ‐frequency. The θ‐activity was spread over too wide a spectral range for its phase to be meaningfully defined. For both agents, we observed two distinct phases of anaesthesia, with a marked increase in θ‐wave activity occurring on passage from a deeper phase of anaesthesia to a shallower one. In other respects, the effects of the two anaesthetics were very different. For KX anaesthesia, the two phases were separated by a marked change in all three instantaneous frequencies: stable, deep, anaesthesia with small frequency variability was followed by a sharp transition to shallow anaesthesia with large frequency variability, lasting until the animal awoke. The transition occurred 16–76 min after injection of the anaesthetic, with simultaneous reduction in the δ‐wave amplitude. For PB anaesthesia, the two epochs were separated by the return of a positive response to the pinch test at 53–94 min, following which it took a further period of 45–70 min for the animal to awaken. δ‐Waves were not apparent at any stage of PB anaesthesia. We applied non‐linear dynamics and information theory to seek evidence of causal relationships between the cardiac, respiratory and slow δ‐oscillations. We demonstrate that, for both groups, respiration drives the cardiac oscillator during deep anaesthesia. During shallow KX anaesthesia the direction either reverses, or the cardio‐respiratory interaction becomes insignificant; in the deep phase, there is a unidirectional deterministic interaction of respiration with slow δ‐oscillations. For PB anaesthesia, the cardio‐respiratory interaction weakens during the second phase but, otherwise, there is no observable change in the interactions. We conclude that non‐linear dynamics and information theory can be used to identify different stages of anaesthesia and the effects of different anaesthetics.

EEG phase synchronization in patients with paranoid schizophrenia

Recent findings suggest that specific deficits in neural synchrony and binding may underlie cognitive disturbances in schizophrenia and that key aspects of schizophrenia pathology involve discoordination and disconnection of distributed processes in multiple cortical areas associated with cognitive deficits. In the present study we aimed to investigate the underlying cortical mechanism of disturbed frontal-temporal-central-parietal connectivity in schizophrenia by examination of the synchronization patterns using wavelet phase synchronization index and coherence between all defined couples of 8 EEG signals recorded at different cortical sites in its relationship to positive and negative symptoms of schizophrenia. 31 adult schizophrenic outpatients with diagnosis of paranoid schizophrenia (mean age 27.4) were assessed in the study. The obtained results present the first quantitative evidence indicating direct relationship between wavelet phase synchronization and coherence in pairs of EEG signals recorded from frontal, temporal, central and parietal brain areas and positive and negative symptoms of schizophrenia. The performed analysis demonstrates that the level of phase synchronization and coherence in some pairs of EEG signals is inversely related to positive symptoms, negative symptoms and general psychopathology in temporal scales (frequency ranges) given by wavelet frequencies (WFs) equal to or higher than 7.56 Hz, and positively related to negative symptoms in wavelet frequencies equal to or lower than 5.35 Hz. This finding suggests that higher and lower frequencies may play a specific role in binding and connectivity and may be related to decreased or increased synchrony with specific manifestation in cognitive deficits of schizophrenia.

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 Earths 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.

Reliability of Inference of Directed Climate Networks Using Conditional Mutual Information

Across geosciences, many investigated phenomena relate to specific complex systems consisting of intricately intertwined interacting subsystems. Such dynamical complex systems can be represented by a directed graph, where each link denotes an existence of a causal relation, or information exchange between the nodes. For geophysical systems such as global climate, these relations are commonly not theoretically known but estimated from recorded data using causality analysis methods. These include bivariate nonlinear methods based on information theory and their linear counterpart. The trade-off between the valuable sensitivity of nonlinear methods to more general interactions and the potentially higher numerical reliability of linear methods may affect inference regarding structure and variability of climate networks. We investigate the reliability of directed climate networks detected by selected methods and parameter settings, using a stationarized model of dimensionality-reduced surface air temperature data from reanalysis of 60-year global climate records. Overall, all studied bivariate causality methods provided reproducible estimates of climate causality networks, with the linear approximation showing

From nonlinearity to causality : statistical testing and inference of physical mechanisms underlying complex dynamics

Principles and applications of statistical testing as a tool for inference of underlying mechanisms from experimental time series are discussed. The computational realizations of the test null hypothesis known as the surrogate data are introduced within the context of discerning nonlinear dynamics from noise, and discussed in examples of testing for nonlinearity in atmospheric dynamics, solar cycle and brain signals. The concept is further generalized for detection of directional interactions, or causality in bivariate time series.

Non-linear dependence and teleconnections in climate data: sources, relevance, nonstationarity

Quantification of relations between measured variables of interest by statistical measures of dependence is a common step in analysis of climate data. The choice of dependence measure is key for the results of the subsequent analysis and interpretation. The use of linear Pearson’s correlation coefficient is widespread and convenient. On the other side, as the climate is widely acknowledged to be a nonlinear system, nonlinear dependence quantification methods, such as those based on information-theoretical concepts, are increasingly used for this purpose. In this paper we outline an approach that enables well informed choice of dependence measure for a given type of data, improving the subsequent interpretation of the results. The presented multi-step approach includes statistical testing, quantification of the specific non-linear contribution to the interaction information, localization of areas with strongest nonlinear contribution and assessment of the role of specific temporal patterns, including signal nonstationarities. In detail we study the consequences of the choice of a general nonlinear dependence measure, namely mutual information, focusing on its relevance and potential alterations in the discovered dependence structure. We document the method by applying it to monthly mean temperature data from the NCEP/NCAR reanalysis dataset as well as the ERA dataset. We have been able to identify main sources of observed non-linearity in inter-node couplings. Detailed analysis suggested an important role of several sources of nonstationarity within the climate data. The quantitative role of genuine nonlinear coupling at monthly scale has proven to be almost negligible, providing quantitative support for the use of linear methods for monthly temperature data.

Small-world topology of functional connectivity in randomly connected dynamical systems.

Characterization of real-world complex systems increasingly involves the study of their topological structure using graph theory. Among global network properties, small-world property, consisting in existence of relatively short paths together with high clustering of the network, is one of the most discussed and studied. When dealing with coupled dynamical systems, links among units of the system are commonly quantified by a measure of pairwise statistical dependence of observed time series (functional connectivity). We argue that the functional connectivity approach leads to upwardly biased estimates of small-world characteristics (with respect to commonly used random graph models) due to partial transitivity of the accepted functional connectivity measures such as the correlation coefficient. In particular, this may lead to observation of small-world characteristics in connectivity graphs estimated from generic randomly connected dynamical systems. The ubiquity and robustness of the phenomenon are documented by an extensive parameter study of its manifestation in a multivariate linear autoregressive process, with discussion of the potential relevance for nonlinear processes and measures.

The role of nonlinearity in computing graph-theoretical properties of resting-state functional magnetic resonance imaging brain networks.

In recent years, there has been an increasing interest in the study of large-scale brain activity interaction structure from the perspective of complex networks, based on functional magnetic resonance imaging (fMRI) measurements. To assess the strength of interaction (functional connectivity, FC) between two brain regions, the linear (Pearson) correlation coefficient of the respective time series is most commonly used. Since a potential use of nonlinear FC measures has recently been discussed in this and other fields, the question arises whether particular nonlinear FC measures would be more informative for the graph analysis than linear ones. We present a comparison of network analysis results obtained from the brain connectivity graphs capturing either full (both linear and nonlinear) or only linear connectivity using 24 sessions of human resting-state fMRI. For each session, a matrix of full connectivity between 90 anatomical parcel time series is computed using mutual information. For comparison, connectivity matrices obtained for multivariate linear Gaussian surrogate data that preserve the correlations, but remove any nonlinearity are generated. Binarizing these matrices using multiple thresholds, we generate graphs corresponding to linear and full nonlinear interaction structures. The effect of neglecting nonlinearity is then assessed by comparing the values of a range of graph-theoretical measures evaluated for both types of graphs. Statistical comparisons suggest a potential effect of nonlinearity on the local measures-clustering coefficient and betweenness centrality. Nevertheless, subsequent quantitative comparison shows that the nonlinearity effect is practically negligible when compared to the intersubject variability of the graph measures. Further, on the group-average graph level, the nonlinearity effect is unnoticeable.

Detecting modes with nontrivial dynamics embedded in colored noise: enhanced Monte Carlo SSA and the case of climate oscillations

Abstract Singular spectrum analysis (SSA) is a useful tool for identification and extraction of oscillatory or other signals from a noisy background. Its basic form, however, is reliable when a signal is embedded in white noise, while the presence of “colored” noises could lead to spurious results. Recently, Monte Carlo SSA, based on a so-called surrogate data technique, has been introduced in order to increase the reliability of detecting signals embedded in colored noises, which are usually present in geophysical data. We propose to enhance the Monte Carlo SSA by evaluating and testing the regularity of dynamics (quantified by so-called coarse-grained entropy rates) of the SSA modes against the colored noise null hypothesis, in addition to the test based on variance (eigenvalues). We demonstrate that such an approach can improve the test reliability in detection of relatively more regular dynamical modes than those obtained by decomposition of colored noises, in particular, in the identification of irregular oscillations embedded in red noise. The method is illustrated in the detection of oscillations with a period of eight years in historical temperature records obtained from several European locations, as well as in the detection of approximately five-year cycles in the global temperature series.