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In the study of complex systems from observed multivariate time series, insight into the evolution of one system may be under investigation, which can be explained by the information storage of the system and the information transfer from other interacting systems. We present a framework for the model-free estimation of information storage and information transfer computed as the terms composing the predictive information about the target of a multivariate dynamical process. The approach tackles the curse of dimensionality employing a nonuniform embedding scheme that selects progressively, among the past components of the multivariate process, only those that contribute most, in terms of conditional mutual information, to the present target process. Moreover, it computes all information-theoretic quantities using a nearest-neighbor technique designed to compensate the bias due to the different dimensionality of individual entropy terms. The resulting estimators of prediction entropy, storage entropy, transfer entropy, and partial transfer entropy are tested on simulations of coupled linear stochastic and nonlinear deterministic dynamic processes, demonstrating the superiority of the proposed approach over the traditional estimators based on uniform embedding. The framework is then applied to multivariate physiologic time series, resulting in physiologically well-interpretable information decompositions of cardiovascular and cardiorespiratory interactions during head-up tilt and of joint brain-heart dynamics during sleep.

This study proposes an integrated approach, framed in the emerging fields of network physiology and information dynamics, for the quantitative analysis of brain-heart interaction networks during sleep. With this approach, the time series of cardiac vagal autonomic activity and brain wave activities measured respectively as the normalized high frequency component of heart rate variability and the EEG power in the δ, θ, α, σ, and β bands, are considered as realizations of the stochastic processes describing the dynamics of the heart system and of different brain sub-systems. Entropy-based measures are exploited to quantify the predictive information carried by each (sub)system, and to dissect this information into a part actively stored in the system and a part transferred to it from the other connected systems. The application of this approach to polysomnographic recordings of ten healthy subjects led us to identify a structured network of sleep brain-brain and brain-heart interactions, with the node described by the β EEG power acting as a hub which conveys the largest amount of information flowing between the heart and brain nodes. This network was found to be sustained mostly by the transitions across different sleep stages, as the information transfer was weaker during specific stages than during the whole night, and vanished progressively when moving from light sleep to deep sleep and to REM sleep.

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In this study, the physiological networks underlying the joint modulation of the parasympathetic component of heart rate variability (HRV) and of the different electroencephalographic (EEG) rhythms during sleep were assessed using two popular measures of directed interaction in multivariate time series, namely Granger causality (GC) and transfer entropy (TE). Time series representative of cardiac and brain activities were obtained in 10 young healthy subjects as the normalized high frequency (HF) component of HRV and EEG power in the δ, θ, α, σ, and β bands, measured during the whole duration of sleep. The magnitude and statistical significance of GC and TE were evaluated between each pair of series, conditional on the remaining series, using respectively a linear model-based approach exploiting regression models, and a nonlinear model-free approach combining nearest-neighbor entropy estimation with a procedure for dimensionality reduction. The contribution of nonlinear dynamics to the TE was also assessed using surrogate data. GC and TE consistently detected structured networks of physiological interactions, with links directed predominantly from HRV to the EEG waves in the brain-heart network, and from the σ and β EEG waves to the δ, θ, and α waves in the brain-brain network. While these common patterns supported the suitability of a linear model-based analysis, we also found a significant contribution of nonlinear dynamics, particularly involving the information transferred out of the δ node in the two networks. This suggested the importance of nonparametric TE estimation for evidencing the fine structure of the physiological networks underlying the autonomic regulation of cardiac and brain functions during sleep.