Estimating the decomposition of predictive information in multivariate systems

Two step transfer entropy - An estimator of delayed directional couplings between multivariate EEG time series

Quantifying delayed directional couplings between electroencephalographic (EEG) time series requires an efficient method of causal network inference. This is especially due to the limited knowledge about the underlying dynamics of the brain activity. Recent methods based on information theoretic measures such as Transfer Entropy (TE) made significant progress on this issue by providing a model-free framework for causality detection. However, TE estimation from observed data is not a trivial task, especially when the number of variables is large which is the case in a highly complex system like human brain. Here we propose a computationally efficient procedure for TE estimation based on using sets of the Most Informative Variables that effectively contribute to resolving the uncertainty of the destination. In the first step of this method, some conditioning sets are determined through a nonlinear state space reconstruction; then in the second step, optimal estimation of TE is done based on these sets. Validation of the proposed method using synthetic data and neurophysiological signals demonstrates computational efficiency in quantifying delayed directional couplings compared with the common TE analysis.

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.

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

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

Are Nonlinear Model-Free Conditional Entropy Approaches for the Assessment of Cardiac Control Complexity Superior to the Linear Model-Based One?

OBJECTIVE

We test the hypothesis that the linear model-based (MB) approach for the estimation of conditional entropy (CE) can be utilized to assess the complexity of the cardiac control in healthy individuals.

METHODS

An MB estimate of CE was tested in an experimental protocol (i.e., the graded head-up tilt) known to produce a gradual decrease of cardiac control complexity as a result of the progressive vagal withdrawal and concomitant sympathetic activation. The MB approach was compared with traditionally exploited nonlinear model-free (MF) techniques such as corrected approximate entropy, sample entropy, corrected CE, two k -nearest-neighbor CE procedures and permutation CE. Electrocardiogram was recorded in 17 healthy subjects at rest in supine position and during head-up tilt with table angles of 15°, 30°, 45°, 60°, and 75°. Heart period (HP) was derived as the temporal distance between two consecutive R-wave peaks and analysis was carried out over stationary sequences of 256 successive HPs.

RESULTS

The performance of the MB method in following the progressive decrease of HP complexity with tilt table angles was in line with those of MF approaches and the MB index was remarkably correlated with the MF ones.

CONCLUSION

The MB approach can be utilized to monitor the changes of the complexity of the cardiac control, thus speeding up dramatically the CE calculation.

SIGNIFICANCE

The remarkable performance of the MB approach challenges the notion, generally assumed in cardiac control complexity analysis based on CE, about the need of MF techniques and could allow real-time applications.

An Information-Theoretic Framework to Map the Spatiotemporal Dynamics of the Scalp Electroencephalogram

We present the first application of the emerging framework of information dynamics to the characterization of the electroencephalography (EEG) activity. The framework provides entropy-based measures of information storage (self entropy, SE) and information transfer (joint transfer entropy (TE) and partial TE), which are applied here to detect complex dynamics of individual EEG sensors and causal interactions between different sensors. The measures are implemented according to a model-free and fully multivariate formulation of the framework, allowing the detection of nonlinear dynamics and direct links. Moreover, to deal with the issue of volume conduction, a compensation for instantaneous effects is introduced in the computation of joint and partial TE. The framework is applied to resting state EEG measured from healthy subjects in the eyes open (EO) and eyes closed (EC) conditions, evidencing condition-dependent patterns indicative of how information is distributed in the EEG sensor space. The SE was uniformly low during EO and significantly higher in the posterior areas during EC. The joint and partial TE were saturated by instantaneous effects, documenting how volume conduction blurs the detection of information flow in the EEG. However, the use of compensated TE measures led us to evidence meaningful patterns like the presence of local sinks of information flow and propagation motifs, and the emergence of prevalent front-to-back EEG propagation during EC. These findings support the feasibility of our information-theoretic approach to assess the spatiotemporal dynamics of the scalp EEG in different conditions.

Predictability decomposition detects the impairment of brain-heart dynamical networks during sleep disorders and their recovery with treatment

This work introduces a framework to study the network formed by the autonomic component of heart rate variability (cardiac processη) and the amplitude of the different electroencephalographic waves (brain processes δ, θ, α, σ, β) during sleep. The framework exploits multivariate linear models to decompose the predictability of any given target process into measures of self-, causal and interaction predictability reflecting respectively the information retained in the process and related to its physiological complexity, the information transferred from the other source processes, and the information modified during the transfer according to redundant or synergistic interaction between the sources. The framework is here applied to theη,δ,θ,α,σ,βtime series measured from the sleep recordings of eight severe sleep apnoea-hypopnoea syndrome (SAHS) patients studied before and after long-term treatment with continuous positive airway pressure (CPAP) therapy, and 14 healthy controls. Results show that the full and self-predictability of η, δ and θ decreased significantly in SAHS compared with controls, and were restored with CPAP forδandθbut not forη The causal predictability of η and δ occurred through significantly redundant source interaction during healthy sleep, which was lost in SAHS and recovered after CPAP. These results indicate that predictability analysis is a viable tool to assess the modifications of complexity and causality of the cerebral and cardiac processes induced by sleep disorders, and to monitor the restoration of the neuroautonomic control of these processes during long-term treatment.

Quantifying information transfer and mediation along causal pathways in complex systems

Measures of information transfer have become a popular approach to analyze interactions in complex systems such as the Earth or the human brain from measured time series. Recent work has focused on causal definitions of information transfer aimed at decompositions of predictive information about a target variable, while excluding effects of common drivers and indirect influences. While common drivers clearly constitute a spurious causality, the aim of the present article is to develop measures quantifying different notions of the strength of information transfer along indirect causal paths, based on first reconstructing the multivariate causal network. Another class of novel measures quantifies to what extent different intermediate processes on causal paths contribute to an interaction mechanism to determine pathways of causal information transfer. The proposed framework complements predictive decomposition schemes by focusing more on the interaction mechanism between multiple processes. A rigorous mathematical framework allows for a clear information-theoretic interpretation that can also be related to the underlying dynamics as proven for certain classes of processes. Generally, however, estimates of information transfer remain hard to interpret for nonlinearly intertwined complex systems. But if experiments or mathematical models are not available, then measuring pathways of information transfer within the causal dependency structure allows at least for an abstraction of the dynamics. The measures are illustrated on a climatological example to disentangle pathways of atmospheric flow over Europe.