Synergy, redundancy and unnormalized Granger causality

We analyze by means of Granger causality the effect of synergy and redundancy in the inference (from time series data) of the information flow between subsystems of a complex network. Whilst fully conditioned Granger causality is not affected by synergy, the pairwise analysis fails to put in evidence synergetic effects. We show that maximization of the total Granger causality to a given target, over all the possible partitions of the set of driving variables, puts in evidence redundant multiplets of variables influencing the target, provided that an unnormalized definition of Granger causality is adopted. Along the same lines we also introduce a pairwise index of synergy (w.r.t. to information flow to a third variable) which is zero when two independent sources additively influence a common target; thus, this definition differs from previous definitions of synergy.

Cardiovascular and cardiorespiratory coupling analyses: a review

Recently, methods have been developed to analyse couplings in dynamic systems. In the field of medical analysis of complex cardiovascular and cardiorespiratory systems, there is growing interest in how insights may be gained into the interaction between regulatory mechanisms in healthy and diseased persons. The couplings within and between these systems can be linear or nonlinear. However, the complex mechanisms involved in cardiovascular and cardiorespiratory regulation very likely interact with each other in a nonlinear way. Recent advances in nonlinear dynamics and information theory have allowed the multivariate study of information transfer between time series. They therefore might be able to provide additional diagnostic and prognostic information in medicine and might, in particular, be able to complement traditional linear coupling analysis techniques. In this review, we describe the approaches (Granger causality, nonlinear prediction, entropy, symbolization, phase synchronization) most commonly applied to detect direct and indirect couplings between time series, especially focusing on nonlinear approaches. We will discuss their capacity to quantify direct and indirect couplings and the direction (driver-response relationship) of the considered interaction between different biological time series. We also give their basic theoretical background, their basic requirements for application, their main features and demonstrate their usefulness in different applications in the field of cardiovascular and cardiorespiratory coupling analyses.

Cross validation for selection of cortical interaction models from scalp EEG or MEG

A cross-validation (CV) method based on state-space framework is introduced for comparing the fidelity of different cortical interaction models to the measured scalp electroencephalogram (EEG) or magnetoencephalography (MEG) data being modeled. A state equation models the cortical interaction dynamics and an observation equation represents the scalp measurement of cortical activity and noise. The measured data are partitioned into training and test sets. The training set is used to estimate model parameters and the model quality is evaluated by computing test data innovations for the estimated model. Two CV metrics normalized mean square error and log-likelihood are estimated by averaging over different training/test partitions of the data. The effectiveness of this method of model selection is illustrated by comparing two linear modeling methods and two nonlinear modeling methods on simulated EEG data derived using both known dynamic systems and measured electrocorticography data from an epilepsy patient.

Causal relationships between frequency bands of extracellular signals in visual cortex revealed by an information theoretic analysis

Characterizing how different cortical rhythms interact and how their interaction changes with sensory stimulation is important to gather insights into how these rhythms are generated and what sensory function they may play. Concepts from information theory, such as Transfer Entropy (TE), offer principled ways to quantify the amount of causation between different frequency bands of the signal recorded from extracellular electrodes; yet these techniques are hard to apply to real data. To address the above issues, in this study we develop a method to compute fast and reliably the amount of TE from experimental time series of extracellular potentials. The method consisted in adapting efficiently the calculation of TE to analog signals and in providing appropriate sampling bias corrections. We then used this method to quantify the strength and significance of causal interaction between frequency bands of field potentials and spikes recorded from primary visual cortex of anaesthetized macaques, both during spontaneous activity and during binocular presentation of naturalistic color movies. Causal interactions between different frequency bands were prominent when considering the signals at a fine (ms) temporal resolution, and happened with a very short (ms-scale) delay. The interactions were much less prominent and significant at coarser temporal resolutions. At high temporal resolution, we found strong bidirectional causal interactions between gamma-band (40–100 Hz) and slower field potentials when considering signals recorded within a distance of 2 mm. The interactions involving gamma bands signals were stronger during movie presentation than in absence of stimuli, suggesting a strong role of the gamma cycle in processing naturalistic stimuli. Moreover, the phase of gamma oscillations was playing a stronger role than their amplitude in increasing causations with slower field potentials and spikes during stimulation. The dominant direction of causality was mainly found in the direction from MUA or gamma frequency band signals to lower frequency signals, suggesting that hierarchical correlations between lower and higher frequency cortical rhythms are originated by the faster rhythms.

Nonlinear connectivity by Granger causality

The communication among neuronal populations, reflected by transient synchronous activity, is the mechanism underlying the information processing in the brain. Although it is widely assumed that the interactions among those populations (i.e. functional connectivity) are highly nonlinear, the amount of nonlinear information transmission and its functional roles are not clear. The state of the art to understand the communication between brain systems are dynamic causal modeling (DCM) and Granger causality. While DCM models nonlinear couplings, Granger causality, which constitutes a major tool to reveal effective connectivity, and is widely used to analyze EEG/MEG data as well as fMRI signals, is usually applied in its linear version. In order to capture nonlinear interactions between even short and noisy time series, a few approaches have been proposed. We review them and focus on a recently proposed flexible approach has been recently proposed, consisting in the kernel version of Granger causality. We show the application of the proposed approach on EEG signals and fMRI data.

Kernel-Granger causality and the analysis of dynamical networks

We propose a method of analysis of dynamical networks based on a recent measure of Granger causality between time series, based on kernel methods. The generalization of kernel-Granger causality to the multivariate case, here presented, shares the following features with the bivariate measures: (i) the nonlinearity of the regression model can be controlled by choosing the kernel function and (ii) the problem of false causalities, arising as the complexity of the model increases, is addressed by a selection strategy of the eigenvectors of a reduced Gram matrix whose range represents the additional features due to the second time series. Moreover, there is no a priori assumption that the network must be a directed acyclic graph. We apply the proposed approach to a network of chaotic maps and to a simulated genetic regulatory network: it is shown that the underlying topology of the network can be reconstructed from time series of node's dynamics, provided that a sufficient number of samples is available. Considering a linear dynamical network, built by preferential attachment scheme, we show that for limited data use of the bivariate Granger causality is a better choice than methods using L1 minimization. Finally we consider real expression data from HeLa cells, 94 genes and 48 time points. The analysis of static correlations between genes reveals two modules corresponding to well-known transcription factors; Granger analysis puts in evidence 19 causal relationships, all involving genes related to tumor development.

Kernel method for nonlinear granger causality

Important information on the structure of complex systems can be obtained by measuring to what extent the individual components exchange information among each other. The linear Granger approach, to detect cause-effect relationships between time series, has emerged in recent years as a leading statistical technique to accomplish this task. Here we generalize Granger causality to the nonlinear case using the theory of reproducing kernel Hilbert spaces. Our method performs linear Granger causality in the feature space of suitable kernel functions, assuming arbitrary degree of nonlinearity. We develop a new strategy to cope with the problem of overfitting, based on the geometry of reproducing kernel Hilbert spaces. Applications to coupled chaotic maps and physiological data sets are presented.

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.