Estimating the decomposition of predictive information in multivariate systems

Characterization of cardiorespiratory coupling via a variability-based multi-method approach: Application to postural orthostatic tachycardia syndrome

There are several mechanisms responsible for the dynamical link between heart period (HP) and respiration (R), usually referred to as cardiorespiratory coupling (CRC). Historically, diverse signal processing techniques have been employed to study CRC from the spontaneous fluctuations of HP and respiration (R). The proposed tools differ in terms of rationale and implementation, capturing diverse aspects of CRC. In this review, we classify the existing methods and stress differences with the aim of proposing a variability-based multi-method approach to CRC evaluation. Ten methodologies for CRC estimation, namely, power spectral decomposition, traditional and causal squared coherence,\;information transfer, cross-conditional entropy, mixed prediction, Shannon entropy of the latency between heartbeat and inspiratory/expiratory onset, conditional entropy of the phase dynamics, synchrogram-based analysis, pulse-respiration quotient, and joint symbolic dynamics, are considered. The ability of these techniques was exemplified over recordings acquired from patients suffering from postural orthostatic tachycardia syndrome (POTS) and healthy controls. Analyses were performed at rest in the supine position (REST) and during head-up tilt (HUT). Although most of the methods indicated that at REST, the CRC was lower in POTS patients and decreased more evidently during HUT in POTS, peculiar differences stressed the complementary value of the approaches. The multiple perspectives provided by the variability-based multi-method approach to CRC evaluation help the characterization of a pathological state and/or the quantification of the effect of a postural challenge. The present work stresses the need for the application of multiple methods to derive a more complete evaluation of the CRC in humans.

Response inhibition in premotor cortex corresponds to a complex reshuffle of the mesoscopic information network

Recent studies have explored functional and effective neural networks in animal models; however, the dynamics of information propagation among functional modules under cognitive control remain largely unknown. Here, we addressed the issue using transfer entropy and graph theory methods on mesoscopic neural activities recorded in the dorsal premotor cortex of rhesus monkeys. We focused our study on the decision time of a Stop-signal task, looking for patterns in the network configuration that could influence motor plan maturation when the Stop signal is provided. When comparing trials with successful inhibition to those with generated movement, the nodes of the network resulted organized into four clusters, hierarchically arranged, and distinctly involved in information transfer. Interestingly, the hierarchies and the strength of information transmission between clusters varied throughout the task, distinguishing between generated movements and canceled ones and corresponding to measurable levels of network complexity. Our results suggest a putative mechanism for motor inhibition in premotor cortex: a topological reshuffle of the information exchanged among ensembles of neurons.

A method to assess linear self-predictability of physiologic processes in the frequency domain: application to beat-to-beat variability of arterial compliance

The concept of self-predictability plays a key role for the analysis of the self-driven dynamics of physiological processes displaying richness of oscillatory rhythms. While time domain measures of self-predictability, as well as time-varying and local extensions, have already been proposed and largely applied in different contexts, they still lack a clear spectral description, which would be significantly useful for the interpretation of the frequency-specific content of the investigated processes. Herein, we propose a novel approach to characterize the linear self-predictability (LSP) of Gaussian processes in the frequency domain. The LSP spectral functions are related to the peaks of the power spectral density (PSD) of the investigated process, which is represented as the sum of different oscillatory components with specific frequency through the method of spectral decomposition. Remarkably, each of the LSP profiles is linked to a specific oscillation of the process, and it returns frequency-specific measures when integrated along spectral bands of physiological interest, as well as a time domain self-predictability measure with a clear meaning in the field of information theory, corresponding to the well-known information storage, when integrated along the whole frequency axis. The proposed measure is first illustrated in a theoretical simulation, showing that it clearly reflects the degree and frequency-specific location of predictability patterns of the analyzed process in both time and frequency domains. Then, it is applied to beat-to-beat time series of arterial compliance obtained in young healthy subjects. The results evidence that the spectral decomposition strategy applied to both the PSD and the spectral LSP of compliance identifies physiological responses to postural stress of low and high frequency oscillations of the process which cannot be traced in the time domain only, highlighting the importance of computing frequency-specific measures of self-predictability in any oscillatory physiologic process.

NLDyn - An open source MATLAB toolbox for the univariate and multivariate nonlinear dynamical analysis of physiological data

BACKGROUND AND OBJECTIVE

We present NLDyn, an open-source MATLAB toolbox tailored for in-depth analysis of nonlinear dynamics in biomedical signals. Our objective is to offer a user-friendly yet comprehensive platform for researchers to explore the intricacies of time series data.

METHODS

NLDyn integrates approximately 80 distinct methods, encompassing both univariate and multivariate nonlinear dynamics, setting it apart from existing solutions. This toolbox combines state-of-the-art nonlinear dynamical techniques with advanced multivariate entropy methods, providing users with powerful analytical capabilities. NLDyn enables analyses with or without a sliding window, and users can easily access and customize default parameters.

RESULTS

NLDyn generates results that are both exportable and visually informative, facilitating seamless integration into research and presentations. Its ongoing development ensures it remains at the forefront of nonlinear dynamics analysis.

CONCLUSIONS

NLDyn is a valuable resource for researchers in the biomedical field, offering an intuitive interface and a wide array of nonlinear analysis tools. Its integration of advanced techniques empowers users to gain deeper insights from their data. As we continually refine and expand NLDyn's capabilities, we envision it becoming an indispensable tool for the exploration of complex dynamics in biomedical signals.

Information-Theoretic Analysis of Cardio-Respiratory Interactions in Heart Failure Patients: Effects of Arrhythmias and Cardiac Resynchronization Therapy

The properties of cardio-respiratory coupling (CRC) are affected by various pathological conditions related to the cardiovascular and/or respiratory systems. In heart failure, one of the most common cardiac pathological conditions, the degree of CRC changes primarily depend on the type of heart-rhythm alterations. In this work, we investigated CRC in heart-failure patients, applying measures from information theory, i.e., Granger Causality (GC), Transfer Entropy (TE) and Cross Entropy (CE), to quantify the directed coupling and causality between cardiac (RR interval) and respiratory (Resp) time series. Patients were divided into three groups depending on their heart rhythm (sinus rhythm and presence of low/high number of ventricular extrasystoles) and were studied also after cardiac resynchronization therapy (CRT), distinguishing responders and non-responders to the therapy. The information-theoretic analysis of bidirectional cardio-respiratory interactions in HF patients revealed the strong effect of nonlinear components in the RR (high number of ventricular extrasystoles) and in the Resp time series (respiratory sinus arrhythmia) as well as in their causal interactions. We showed that GC as a linear model measure is not sensitive to both nonlinear components and only model free measures as TE and CE may quantify them. CRT responders mainly exhibit unchanged asymmetry in the TE values, with statistically significant dominance of the information flow from Resp to RR over the opposite flow from RR to Resp, before and after CRT. In non-responders this asymmetry was statistically significant only after CRT. Our results indicate that the success of CRT is related to corresponding information transfer between the cardiac and respiratory signal quantified at baseline measurements, which could contribute to a better selection of patients for this type of therapy.

Complex Brain-Heart Mapping in Mental and Physical Stress

OBJECTIVE

The central and autonomic nervous systems are deemed complex dynamic systems, wherein each system as a whole shows features that the individual system sub-components do not. They also continuously interact to maintain body homeostasis and appropriate react to endogenous and exogenous stimuli. Such interactions are comprehensively referred to functional brain-heart interplay (BHI). Nevertheless, it remains uncertain whether this interaction also exhibits complex characteristics, that is, whether the dynamics of the entire nervous system inherently demonstrate complex behavior, or if such complexity is solely a trait of the central and autonomic systems. Here, we performed complexity mapping of the BHI dynamics under mental and physical stress conditions.

METHODS AND PROCEDURES

Electroencephalographic and heart rate variability series were obtained from 56 healthy individuals performing mental arithmetic or cold-pressure tasks, and physiological series were properly combined to derive directional BHI series, whose complexity was quantified through fuzzy entropy.

RESULTS

The experimental results showed that BHI complexity is mainly modulated in the efferent functional direction from the brain to the heart, and mainly targets vagal oscillations during mental stress and sympathovagal oscillations during physical stress.

CONCLUSION

We conclude that the complexity of BHI mapping may provide insightful information on the dynamics of both central and autonomic activity, as well as on their continuous interaction.

CLINICAL IMPACT

This research enhances our comprehension of the reciprocal interactions between central and autonomic systems, potentially paving the way for more accurate diagnoses and targeted treatments of cardiovascular, neurological, and psychiatric disorders.

Feasibility of Ultra-Short-Term Analysis of Heart Rate and Systolic Arterial Pressure Variability at Rest and during Stress via Time-Domain and Entropy-Based Measures

Heart Rate Variability (HRV) and Blood Pressure Variability (BPV) are widely employed tools for characterizing the complex behavior of cardiovascular dynamics. Usually, HRV and BPV analyses are carried out through short-term (ST) measurements, which exploit ~five-minute-long recordings. Recent research efforts are focused on reducing the time series length, assessing whether and to what extent Ultra-Short-Term (UST) analysis is capable of extracting information about cardiovascular variability from very short recordings. In this work, we compare ST and UST measures computed on electrocardiographic R-R intervals and systolic arterial pressure time series obtained at rest and during both postural and mental stress. Standard time-domain indices are computed, together with entropy-based measures able to assess the regularity and complexity of cardiovascular dynamics, on time series lasting down to 60 samples, employing either a faster linear parametric estimator or a more reliable but time-consuming model-free method based on nearest neighbor estimates. Our results are evidence that shorter time series down to 120 samples still exhibit an acceptable agreement with the ST reference and can also be exploited to discriminate between stress and rest. Moreover, despite neglecting nonlinearities inherent to short-term cardiovascular dynamics, the faster linear estimator is still capable of detecting differences among the conditions, thus resulting in its suitability to be implemented on wearable devices.

Pairwise and higher-order measures of brain-heart interactions in children with temporal lobe epilepsy

OBJECTIVE

While it is well-known that epilepsy has a clear impact on the activity of both the central nervous system (CNS) and the autonomic nervous system (ANS), its role on the complex interplay between CNS and ANS has not been fully elucidated yet. In this work, pairwise and higher-order predictability measures based on the concepts of Granger causality (GC) and Partial Information Decomposition (PID) were applied on time series of electroencephalographic (EEG) brain wave amplitude and heart rate variability (HRV) in order to investigate directed brain-heart interactions associated with the occurrence of focal epilepsy.

APPROACH

HRV and the envelopes of δ and α EEG activity recorded from ipsilateral (ipsi-EEG) and contralateral (contra-EEG) scalp regions were analyzed in 18 children suffering from temporal lobe epilepsy monitored during pre-ictal, ictal and post-ictal periods. After linear parametric model identification, we compared pairwise GC measures computed between HRV and a single EEG component with PID measures quantifying the unique, redundant and synergistic information transferred from ipsi-EEG and contra-EEG to HRV.

MAIN RESULTS

The analysis of GC revealed a dominance of the information transfer from EEG to HRV and negligible transfer from HRV to EEG, suggesting that CNS activities drive the ANS modulation of the heart rhythm, but did not evidence clear differences between δ and α rhythms, ipsi-EEG and contra-EEG, or pre- and post-ictal periods. On the contrary, PID revealed that epileptic seizures induce a reorganization of the interactions from brain to heart, as the unique predictability of HRV originated from the ipsi-EEG for the δ waves and from the contra-EEG for the α waves in the pre-ictal phase, while these patterns were reversed after the seizure.

SIGNIFICANCE

These results highlight the importance of considering higher-order interactions elicited by PID for the study of the neuro-autonomic effects of focal epilepsy, and may have neurophysiological and clinical implications.

Causal Inference in Time Series in Terms of Rényi Transfer Entropy

Uncovering causal interdependencies from observational data is one of the great challenges of a nonlinear time series analysis. In this paper, we discuss this topic with the help of an information-theoretic concept known as Rényi’s information measure. In particular, we tackle the directional information flow between bivariate time series in terms of Rényi’s transfer entropy. We show that by choosing Rényi’s parameter α, we can appropriately control information that is transferred only between selected parts of the underlying distributions. This, in turn, is a particularly potent tool for quantifying causal interdependencies in time series, where the knowledge of “black swan” events, such as spikes or sudden jumps, are of key importance. In this connection, we first prove that for Gaussian variables, Granger causality and Rényi transfer entropy are entirely equivalent. Moreover, we also partially extend these results to heavy-tailed α-Gaussian variables. These results allow establishing a connection between autoregressive and Rényi entropy-based information-theoretic approaches to data-driven causal inference. To aid our intuition, we employed the Leonenko et al. entropy estimator and analyzed Rényi’s information flow between bivariate time series generated from two unidirectionally coupled Rössler systems. Notably, we find that Rényi’s transfer entropy not only allows us to detect a threshold of synchronization but it also provides non-trivial insight into the structure of a transient regime that exists between the region of chaotic correlations and synchronization threshold. In addition, from Rényi’s transfer entropy, we could reliably infer the direction of coupling and, hence, causality, only for coupling strengths smaller than the onset value of the transient regime, i.e., when two Rössler systems are coupled but have not yet entered synchronization.

Measuring the Rate of Information Transfer in Point-Process Data: Application to Cardiovascular Interactions

We present the implementation to cardiovascular variability of a method for the information-theoretic estimation of the directed interactions between event-based data. The method allows to compute the transfer entropy rate (TER) from a source to a target point process in continuous time, thus overcoming the severe limitations associated with time discretization of event-based processes. In this work, the method is evaluated on coupled cardiovascular point processes representing the heartbeat dynamics and the related peripheral pulsation, first using a physiologically-based simulation model and then studying real point-process data from healthy subjects monitored at rest and during postural stress. Our results document the ability of TER to detect direction and strength of the interactions between cardiovascular processes, also highlighting physiologically plausible interaction mechanisms.

Entropy and Multifractal-Multiscale Indices of Heart Rate Time Series to Evaluate Intricate Cognitive-Autonomic Interactions

Recent research has clarified the existence of a networked system involving a cortical and subcortical circuitry regulating both cognition and cardiac autonomic control, which is dynamically organized as a function of cognitive demand. The main interactions span multiple temporal and spatial scales and are extensively governed by nonlinear processes. Hence, entropy and (multi)fractality in heart period time series are suitable to capture emergent behavior of the cognitive-autonomic network coordination. This study investigated how entropy and multifractal-multiscale analyses could depict specific cognitive-autonomic architectures reflected in the heart rate dynamics when students performed selective inhibition tasks. The participants (N=37) completed cognitive interference (Stroop color and word task), action cancellation (stop-signal) and action restraint (go/no-go) tasks, compared to watching a neutral movie as baseline. Entropy and fractal markers (respectively, the refined composite multiscale entropy and multifractal-multiscale detrended fluctuation analysis) outperformed other time-domain and frequency-domain markers of the heart rate variability in distinguishing cognitive tasks. Crucially, the entropy increased selectively during cognitive interference and the multifractality increased during action cancellation. An interpretative hypothesis is that cognitive interference elicited a greater richness in interactive processes that form the central autonomic network while action cancellation, which is achieved via biasing a sensorimotor network, could lead to a scale-specific heightening of multifractal behavior.

Neural Estimator of Information for Time-Series Data with Dependency

Novel approaches to estimate information measures using neural networks are well-celebrated in recent years both in the information theory and machine learning communities. These neural-based estimators are shown to converge to the true values when estimating mutual information and conditional mutual information using independent samples. However, if the samples in the dataset are not independent, the consistency of these estimators requires further investigation. This is of particular interest for a more complex measure such as the directed information, which is pivotal in characterizing causality and is meaningful over time-dependent variables. The extension of the convergence proof for such cases is not trivial and demands further assumptions on the data. In this paper, we show that our neural estimator for conditional mutual information is consistent when the dataset is generated with samples of a stationary and ergodic source. In other words, we show that our information estimator using neural networks converges asymptotically to the true value with probability one. Besides universal functional approximation of neural networks, a core lemma to show the convergence is Birkhoff’s ergodic theorem. Additionally, we use the technique to estimate directed information and demonstrate the effectiveness of our approach in simulations.

An Information-Theoretic Framework to Measure the Dynamic Interaction Between Neural Spike Trains

OBJECTIVE

While understanding the interaction patterns among simultaneous recordings of spike trains from multiple neuronal units is a key topic in neuroscience, existing methods either do not consider the inherent point-process nature of spike trains or are based on parametric assumptions. This work presents an information-theoretic framework for the model-free, continuous-time estimation of both undirected (symmetric) and directed (Granger-causal) interactions between spike trains.

METHODS

The framework computes the mutual information rate (MIR) and the transfer entropy rate (TER) for two point processes X and Y, showing that the MIR between X and Y can be decomposed as the sum of the TER along the directions X → Y and Y → X. We present theoretical expressions and introduce strategies to estimate efficiently the two measures through nearest neighbor statistics.

RESULTS

Using simulations of independent and coupled point processes, we show the accuracy of MIR and TER to assess interactions even for weakly coupled and short realizations, and demonstrate the superiority of continuous-time estimation over the standard discrete-time approach. We also apply the MIR and TER to real-world data, specifically, recordings from in-vitro preparations of spontaneously-growing cultures of cortical neurons. Using this dataset, we demonstrate the ability of MIR and TER to describe how the functional networks between recording units emerge over the course of the maturation of the neuronal cultures.

CONCLUSION AND SIGNIFICANCE

the proposed framework provides principled measures to assess undirected and directed spike train interactions with more efficiency and flexibility than previous discrete-time or parametric approaches, opening new perspectives for the analysis of point-process data in neuroscience and many other fields.

Toward brain-heart computer interfaces: a study on the classification of upper limb movements using multisystem directional estimates

OBJECTIVE

Brain-computer interfaces (BCI) exploit computational features from brain signals to perform a given task. Despite recent neurophysiology and clinical findings indicating the crucial role of functional interplay between brain and cardiovascular dynamics in locomotion, heartbeat information remains to be included in common BCI systems. In this study, we exploit the multidimensional features of directional and functional interplay between electroencephalographic and heartbeat spectra to classify upper limb movements into three classes.

APPROACH

We gathered data from 26 healthy volunteers that performed 90 movements; the data were processed using a recently proposed framework for brain-heart interplay (BHI) assessment based on synthetic physiological data generation. Extracted BHI features were employed to classify, through sequential forward selection scheme and k-nearest neighbors algorithm, among resting state and three classes of movements according to the kind of interaction with objects.

MAIN RESULTS

The results demonstrated that the proposed brain-heart computer interface (BHCI) system could distinguish between rest and movement classes automatically with an average 90% of accuracy.

SIGNIFICANCE

Further, this study provides neurophysiology insights indicating the crucial role of functional interplay originating at the cortical level onto the heart in the upper limb neural control. The inclusion of functional BHI insights might substantially improve the neuroscientific knowledge about motor control, and this may lead to advanced BHCI systems performances.

Multivariate Correlation Measures Reveal Structure and Strength of Brain-Body Physiological Networks at Rest and During Mental Stress

In this work, we extend to the multivariate case the classical correlation analysis used in the field of network physiology to probe dynamic interactions between organ systems in the human body. To this end, we define different correlation-based measures of the multivariate interaction (MI) within and between the brain and body subnetworks of the human physiological network, represented, respectively, by the time series of δ, θ, α, and β electroencephalographic (EEG) wave amplitudes, and of heart rate, respiration amplitude, and pulse arrival time (PAT) variability (η, ρ, π). MI is computed: (i) considering all variables in the two subnetworks to evaluate overall brain-body interactions; (ii) focusing on a single target variable and dissecting its global interaction with all other variables into contributions arising from the same subnetwork and from the other subnetwork; and (iii) considering two variables conditioned to all the others to infer the network topology. The framework is applied to the time series measured from the EEG, electrocardiographic (ECG), respiration, and blood volume pulse (BVP) signals recorded synchronously via wearable sensors in a group of healthy subjects monitored at rest and during mental arithmetic and sustained attention tasks. We find that the human physiological network is highly connected, with predominance of the links internal of each subnetwork (mainly η-ρ and δ-θ, θ-α, α-β), but also statistically significant interactions between the two subnetworks (mainly η-β and η-δ). MI values are often spatially heterogeneous across the scalp and are modulated by the physiological state, as indicated by the decrease of cardiorespiratory interactions during sustained attention and by the increase of brain-heart interactions and of brain-brain interactions at the frontal scalp regions during mental arithmetic. These findings illustrate the complex and multi-faceted structure of interactions manifested within and between different physiological systems and subsystems across different levels of mental stress.

Measuring the Rate of Information Exchange in Point-Process Data With Application to Cardiovascular Variability

The amount of information exchanged per unit of time between two dynamic processes is an important concept for the analysis of complex systems. Theoretical formulations and data-efficient estimators have been recently introduced for this quantity, known as the mutual information rate (MIR), allowing its continuous-time computation for event-based data sets measured as realizations of coupled point processes. This work presents the implementation of MIR for point process applications in Network Physiology and cardiovascular variability, which typically feature short and noisy experimental time series. We assess the bias of MIR estimated for uncoupled point processes in the frame of surrogate data, and we compensate it by introducing a corrected MIR (cMIR) measure designed to return zero values when the two processes do not exchange information. The method is first tested extensively in synthetic point processes including a physiologically-based model of the heartbeat dynamics and the blood pressure propagation times, where we show the ability of cMIR to compensate the negative bias of MIR and return statistically significant values even for weakly coupled processes. The method is then assessed in real point-process data measured from healthy subjects during different physiological conditions, showing that cMIR between heartbeat and pressure propagation times increases significantly during postural stress, though not during mental stress. These results document that cMIR reflects physiological mechanisms of cardiovascular variability related to the joint neural autonomic modulation of heart rate and arterial compliance.

Transfer entropies within dynamical effects framework

Transfer entropy (TE) is widely used in time-series analysis to detect causal couplings between temporally evolving objects. As a coupling strength quantifier, the TE alone often seems insufficient, raising the question of its further interpretations. Here the TE is related to dynamical causal effects (DCEs) which quantify long-term responses of a coupling recipient to variations in a coupling source or in a coupling itself: Detailed relationships are established for a paradigmatic stochastic dynamical system of bidirectionally coupled linear overdamped oscillators, their practical applications and possible extensions are discussed. It is shown that two widely used versions of the TE (original and infinite-history) can become qualitatively distinct, diverging to different long-term DCEs.

The Human Organism as an Integrated Interaction Network: Recent Conceptual and Methodological Challenges

The field of Network Physiology aims to advance our understanding of how physiological systems and sub-systems interact to generate a variety of behaviors and distinct physiological states, to optimize the organism's functioning, and to maintain health. Within this framework, which considers the human organism as an integrated network, vertices are associated with organs while edges represent time-varying interactions between vertices. Likewise, vertices may represent networks on smaller spatial scales leading to a complex mixture of interacting homogeneous and inhomogeneous networks of networks. Lacking adequate analytic tools and a theoretical framework to probe interactions within and among diverse physiological systems, current approaches focus on inferring properties of time-varying interactions-namely strength, direction, and functional form-from time-locked recordings of physiological observables. To this end, a variety of bivariate or, in general, multivariate time-series-analysis techniques, which are derived from diverse mathematical and physical concepts, are employed and the resulting time-dependent networks can then be further characterized with methods from network theory. Despite the many promising new developments, there are still problems that evade from a satisfactory solution. Here we address several important challenges that could aid in finding new perspectives and inspire the development of theoretic and analytical concepts to deal with these challenges and in studying the complex interactions between physiological systems.

Effective brain connectivity for fNIRS data analysis based on multi-delays symbolic phase transfer entropy

OBJECTIVE

Recently, effective connectivity (EC) calculation methods for functional near-infrared spectroscopy (fNIRS) data mainly face two problems: the first problem is that noise can seriously affect the EC calculation and even lead to false connectivity; the second problem is that it ignores the various real neurotransmission delays between the brain region, and instead uses a fixed delay coefficient for calculation.

APPROACH

To overcome these two issues, a delay symbolic phase transfer entropy (dSPTE) is proposed by developing traditional transfer entropy (TE) to estimate EC for fNIRS. Firstly, the phase time sequence was obtained from the original sequence by the Hilbert transform and state-space reconstruction was realized using a uniform embedding scheme. Then, a symbolization technique was applied based on a neural-gas algorithm to improve its noise robustness. Finally, the EC was calculated on multiple time delay scales to match different inter-region neurotransmission delays.

MAIN RESULTS

A linear AR model, a nonlinear model and a multivariate hybrid model were introduced to simulate the performance of dSPTE, and the results showed that the accuracy of dSPTE was the highest, up to 74.27%, and specificity was 100% which means no false connectivity. The results confirmed that the dSPTE method realized better noise robustness, higher accuracy, and correct identification even if there was a long delay between series. Finally, we applied dSPTE to fNIRS dataset to analyse the EC during the finger-tapping task, the results showed that EC strength of task state significantly increased compared with the resting state.

SIGNIFICANCE

The proposed dSPTE method is a promising way to measure the EC for fNIRS. It incorporates the phase information TE with a symbolic process for fNIRS analysis for the first time. It has been confirmed to be noise robust and suitable for the complex network with different coupling delays.

Estimating Conditional Transfer Entropy in Time Series Using Mutual Information and Nonlinear Prediction

We propose a new estimator to measure directed dependencies in time series. The dimensionality of data is first reduced using a new non-uniform embedding technique, where the variables are ranked according to a weighted sum of the amount of new information and improvement of the prediction accuracy provided by the variables. Then, using a greedy approach, the most informative subsets are selected in an iterative way. The algorithm terminates, when the highest ranked variable is not able to significantly improve the accuracy of the prediction as compared to that obtained using the existing selected subsets. In a simulation study, we compare our estimator to existing state-of-the-art methods at different data lengths and directed dependencies strengths. It is demonstrated that the proposed estimator has a significantly higher accuracy than that of existing methods, especially for the difficult case, where the data are highly correlated and coupled. Moreover, we show its false detection of directed dependencies due to instantaneous couplings effect is lower than that of existing measures. We also show applicability of the proposed estimator on real intracranial electroencephalography data.

Mutual Information Analysis of Brain-Body Interactions during different Levels of Mental stress

In this work, we analyze brain-heart interactions during different mental states computing mutual information (MI) between the dynamic activity of different physiological systems. In 18 healthy subjects monitored in a relaxed resting state and during a mental arithmetic and a serious game task, multichannel EEG, one lead ECG, respiration and blood volume pulse were collected via wireless non-invasive biosensors. From these signals, synchronous 300-second time series were extracted measuring brain activity via the δ, θ, α, and β EEG power, and activity of the body district via the ECG R-R interval η, the respiratory amplitude ρ and the pulse arrival time π. MI was computed using a linear estimator: (i) between {η,ρ,π} and {δ,θ,α,β}, to measure overall brain-body interactions; (ii) between each time series and the others of the same district, to measure information shared within a district; and (iii) between each time series of a district and all series of the other district, to evaluate individual contributions to the information shared between brain and body. Results document the existence of statistically significant brain-body interactions, with high MI values involving mainly the η body dynamics and the δ and β brain dynamics. State-dependent variations were mostly relevant to the MI of the brain system involving δ, θ, α during mental arithmetic, and α and β during serious game. Thus, MI can be useful to detect correlated activity within and between brain and body systems monitored simultaneously during different mental states.

A validity and reliability study of Conditional Entropy Measures of Pulse Rate Variability

In this work, we present the feasibility to use a simpler methodological approach for the assessment of the short-term complexity of Heart Rate Variability (HRV). Specifically, we propose to exploit Pulse Rate Variability (PRV) recorded through photoplethysmography in place of HRV measured from the ECG, and to compute complexity via a linear Gaussian approximation in place of the standard model-free methods (e.g., nearest neighbor entropy estimates) usually applied to HRV. Linear PRV-based and model-free HRV-based complexity measures were compared via statistical tests, correlation analysis and Bland-Altman plots, demonstrating an overall good agreement. These results support the applicability of the simpler proposed approach, which is faster and easier-to-implement, making our approach eligible for portable/wearable devices and thus broadening the out-of-lab accessibility of autonomic indexes.

Detecting Causality in Multivariate Time Series via Non-Uniform Embedding

Causal analysis based on non-uniform embedding schemes is an important way to detect the underlying interactions between dynamic systems. However, there are still some obstacles to estimating high-dimensional conditional mutual information and forming optimal mixed embedding vector in traditional non-uniform embedding schemes. In this study, we present a new non-uniform embedding method framed in information theory to detect causality for multivariate time series, named LM-PMIME, which integrates the low-dimensional approximation of conditional mutual information and the mixed search strategy for the construction of the mixed embedding vector. We apply the proposed method to simulations of linear stochastic, nonlinear stochastic, and chaotic systems, demonstrating its superiority over partial conditional mutual information from mixed embedding (PMIME) method. Moreover, the proposed method works well for multivariate time series with weak coupling strengths, especially for chaotic systems. In the actual application, we show its applicability to epilepsy multichannel electrocorticographic recordings.

Comparison of methods for the assessment of nonlinearity in short-term heart rate variability under different physiopathological states

Despite the widespread diffusion of nonlinear methods for heart rate variability (HRV) analysis, the presence and the extent to which nonlinear dynamics contribute to short-term HRV are still controversial. This work aims at testing the hypothesis that different types of nonlinearity can be observed in HRV depending on the method adopted and on the physiopathological state. Two entropy-based measures of time series complexity (normalized complexity index, NCI) and regularity (information storage, IS), and a measure quantifying deviations from linear correlations in a time series (Gaussian linear contrast, GLC), are applied to short HRV recordings obtained in young (Y) and old (O) healthy subjects and in myocardial infarction (MI) patients monitored in the resting supine position and in the upright position reached through head-up tilt. The method of surrogate data is employed to detect the presence and quantify the contribution of nonlinear dynamics to HRV. We find that the three measures differ both in their variations across groups and conditions and in the percentage and strength of nonlinear HRV dynamics. NCI and IS displayed opposite variations, suggesting more complex dynamics in O and MI compared to Y and less complex dynamics during tilt. The strength of nonlinear dynamics is reduced by tilt using all measures in Y, while only GLC detects a significant strengthening of such dynamics in MI. A large percentage of detected nonlinear dynamics is revealed only by the IS measure in the Y group at rest, with a decrease in O and MI and during T, while NCI and GLC detect lower percentages in all groups and conditions. While these results suggest that distinct dynamic structures may lie beneath short-term HRV in different physiological states and pathological conditions, the strong dependence on the measure adopted and on their implementation suggests that physiological interpretations should be provided with caution.

Reliability of Short-Term Heart Rate Variability Indexes Assessed through Photoplethysmography

The gold standard method to monitor heart rate variability (HRV) comprises measuring the time series of interbeat interval durations from electrocardiographic (ECG) recordings. However, due to the widespread use, simplicity and usability of photoplethysmographic (PPG) techniques, monitoring pulse rate variability (PRV) from pulse wave recordings has become a viable alternative to standard HRV analysis. The present study investigates the accuracy of PRV, measured as a surrogate of HRV, for the quantification of descriptive indexes computed in the time domain (mean, variance), frequency domain (low-to-high frequency power ratio LF/HF, HF band central frequency) and information domain (entropy, conditional entropy). We analyze short time series (300 intervals) of HRV measured from the ECG and of PRV acquired from Finometer device in 76 subjects monitored in the resting supine position (SU) and in the upright position during head-up tilt (HUT). Time, frequency and information domain indexes are computed for each HRV and PRV series and, for each index, the comparison between the two approaches is performed through statistical comparison of the distributions across subjects, robust linear regression, and Bland-Altman plots. Results of the comparison indicate an overall good agreement between PRV-based and HRV-based indexes, with an accuracy that is slightly lower during HUT than during SU, and for the band-power ratio and conditional entropy. These results suggest the feasibility of PRV-based assessment of HRV descriptive indexes, and suggest to further investigate the agreement in conditions of physiological stress.

MAXENT3D_PID: An Estimator for the Maximum-Entropy Trivariate Partial Information Decomposition

Partial information decomposition (PID) separates the contributions of sources about a target into unique, redundant, and synergistic components of information. In essence, PID answers the question of “who knows what” of a system of random variables and hence has applications to a wide spectrum of fields ranging from social to biological sciences. The paper presents MaxEnt3D_Pid, an algorithm that computes the PID of three sources, based on a recently-proposed maximum entropy measure, using convex optimization (cone programming). We describe the algorithm and its associated software utilization and report the results of various experiments assessing its accuracy. Moreover, the paper shows that a hierarchy of bivariate and trivariate PID allows obtaining the finer quantities of the trivariate partial information measure.

Time, frequency and information domain analysis of short-term heart rate variability before and after focal and generalized seizures in epileptic children

OBJECTIVE

In this work we explore the potential of combining standard time and frequency domain indexes with novel information measures, to characterize pre- and post-ictal heart rate variability (HRV) in epileptic children, with the aim of differentiating focal and generalized epilepsy regarding the autonomic control mechanisms.

APPROACH

We analyze short-term HRV in 37 children suffering from generalized or focal epilepsy, monitored 10 s, 300 s, 600 s and 1800 s both before and after seizure episodes. Nine indexes are computed in time (mean, standard deviation of normal-to-normal intervals, root mean square of the successive differences (RMSSD)), frequency (low-to-high frequency power ratio LF/HF, normalized LF and HF power) and information (entropy, conditional entropy and self-entropy) domains. Focal and generalized epilepsy are compared through statistical analysis of the indexes and using linear discriminant analysis (LDA).

MAIN RESULTS

In children with focal epilepsy, early post-ictal phase is characterized by significant tachycardia, depressed HRV, increased LF power and LF/HF, and decreased complexity, progressively recovered across time windows after the episodes. Children with generalized seizures instead show significant tachycardia, lower RMSSD, higher LF power and LF/HF ratio before the seizure. These different behaviors are exploited by LDA analysis to separate focal and generalized epilepsy up to an accuracy of 75%. Results suggest a shift of the sympatho-vagal balance towards sympathetic dominance and vagal withdrawal, noticeable just after the termination of seizure episodes and then reverted in focal epilepsy, and persistent during inter-ictal and pre-ictal periods in generalized epilepsy.

SIGNIFICANCE

Our analysis helps in elucidating the pathophysiology of inter-ictal HRV autonomic control and the differential diagnosis of generalized and focal epilepsy. These findings may have clinical relevance since altered sympatho-vagal control can be related to a higher danger of morbidity and mortality, may reduce thresholds for life-threatening arrhythmias, and could be a biomarker of risk for sudden unexpected death in epilepsy.

Comparison of short-term heart rate variability indexes evaluated through electrocardiographic and continuous blood pressure monitoring

Heart rate variability (HRV) analysis represents an important tool for the characterization of complex cardiovascular control. HRV indexes are usually calculated from electrocardiographic (ECG) recordings after measuring the time duration between consecutive R peaks, and this is considered the gold standard. An alternative method consists of assessing the pulse rate variability (PRV) from signals acquired through photoplethysmography, a technique also employed for the continuous noninvasive monitoring of blood pressure. In this work, we carry out a thorough analysis and comparison of short-term variability indexes computed from HRV time series obtained from the ECG and from PRV time series obtained from continuous blood pressure (CBP) signals, in order to evaluate the reliability of using CBP-based recordings in place of standard ECG tracks. The analysis has been carried out on short time series (300 beats) of HRV and PRV in 76 subjects studied in different conditions: resting in the supine position, postural stress during 45° head-up tilt, and mental stress during computation of arithmetic test. Nine different indexes have been taken into account, computed in the time domain (mean, variance, root mean square of the successive differences), frequency domain (low-to-high frequency power ratio LF/HF, HF spectral power, and central frequency), and information domain (entropy, conditional entropy, self entropy). Thorough validation has been performed using comparison of the HRV and PRV distributions, robust linear regression, and Bland-Altman plots. Results demonstrate the feasibility of extracting HRV indexes from CBP-based data, showing an overall relatively good agreement of time-, frequency-, and information-domain measures. The agreement decreased during postural and mental arithmetic stress, especially with regard to band-power ratio, conditional, and self-entropy. This finding suggests to use caution in adopting PRV as a surrogate of HRV during stress conditions.

Interplay of synergy and redundancy in diamond motif

The formalism of partial information decomposition provides a number of independent components which altogether constitute the total information provided by the source variable(s) about the target variable(s). These non-overlapping terms are recognized as unique information, synergistic information, and redundant information. The metric of net synergy conceived as the difference between synergistic and redundant information is capable of detecting effective synergy, effective redundancy, and information independence among stochastic variables. The net synergy can be quantified using appropriate combinations of different Shannon mutual information terms. The utilization of the net synergy in network motifs with the nodes representing different biochemical species, involved in information sharing, uncovers rich store for exciting results. In the current study, we use this formalism to obtain a comprehensive understanding of the relative information processing mechanism in a diamond motif and two of its sub-motifs, namely, bifurcation and integration motif embedded within the diamond motif. The emerging patterns of effective synergy and effective redundancy and their contribution toward ensuring high fidelity information transmission are duly compared in the sub-motifs. Investigation on the metric of net synergy in independent bifurcation and integration motifs are also executed. In all of these computations, the crucial roles played by various systemic time scales, activation coefficients, and signal integration mechanisms at the output of the network topologies are especially emphasized. Following this plan of action, we become confident that the origin of effective synergy and effective redundancy can be architecturally justified by decomposing a diamond motif into bifurcation and integration motif. According to our conjecture, the presence of a common source of fluctuations creates effective redundancy. Our calculations reveal that effective redundancy empowers signal fidelity. Moreover, to achieve this, input signaling species avoids strong interaction with downstream intermediates. This strategy is capable of making the diamond motif noise-tolerant. Apart from the topological features, our study also puts forward the active contribution of additive and multiplicative signal integration mechanisms to nurture effective redundancy and effective synergy.

Cardiovascular and respiratory variability during orthostatic and mental stress: A comparison of entropy estimators

The aim of this study is to characterize cardiovascular and respiratory signals during orthostatic and mental stress as reflected in indices of entropy and complexity, providing a comparison between the performance of different estimators. To this end, the heart rate variability, systolic blood pressure, diastolic blood pressure and respiration time series were extracted from the recordings of 61 healthy volunteers undergoing a protocol consisting of supine rest, head-up tilt test and mental arithmetic task. The analysis was performed in the information domain using measures of entropy and conditional entropy, estimated through model-based (linear) and model-free (binning, nearest neighbor) approaches. Our results show that different types of stress elicited different responses in the employed indices. On one hand, entropy mainly reflected known changes in the variance of physiological time series. On the other hand, the information conveyed by conditional entropy allowed to characterize the complexity of the four time series during the two stress tasks: we found that cardiac and vascular dynamics underwent a reduction in complexity as a consequence of postural stress, while vascular and respiratory complexity increased as a result of mental stress. As for the performance of different estimators, we did not find substantial differences between model-based and model-free approaches, possibly indicating that significant non-linear dynamics did not appear in the studied conditions.

Characterizing information-theoretic storage and transfer in continuous time processes

The characterization of information processing is an important task in complex systems science. Information dynamics is a quantitative methodology for modeling the intrinsic information processing conducted by a process represented as a time series, but to date has only been formulated in discrete time. Building on previous work which demonstrated how to formulate transfer entropy in continuous time, we give a total account of information processing in this setting, incorporating information storage. We find that a convergent rate of predictive capacity, comprising the transfer entropy and active information storage, does not exist, arising through divergent rates of active information storage. We identify that active information storage can be decomposed into two separate quantities that characterize predictive capacity stored in a process: active memory utilization and instantaneous predictive capacity. The latter involves prediction related to path regularity and so solely inherits the divergent properties of the active information storage, while the former permits definitions of pathwise and rate quantities. We formulate measures of memory utilization for jump and neural spiking processes and illustrate measures of information processing in synthetic neural spiking models and coupled Ornstein-Uhlenbeck models. The application to synthetic neural spiking models demonstrates that active memory utilization for point processes consists of discontinuous jump contributions (at spikes) interrupting a continuously varying contribution (relating to waiting times between spikes), complementing the behavior previously demonstrated for transfer entropy in these processes.

Transient and equilibrium causal effects in coupled oscillators

Two quite different types of causal effects are given by (i) changes in near future states of a driven system under changes in a current state of a driving system and (ii) changes in statistical characteristics of a driven system dynamics under changes in coupling parameters, e.g., under switching the coupling off. The former can be called transient causal effects and can be estimated from a time series within the well established framework of the Wiener-Granger causality, while the latter represent equilibrium (or stationary) causal effects which are often most interesting but generally inaccessible to estimation from an observed time series recorded at fixed coupling parameters. In this work, relationships between the two kinds of causal effects are found for unidirectionally coupled stochastic linear oscillators depending on their frequencies and damping factors. Approximate closed-form expressions for these relationships are derived. Their limitations and possible extensions are discussed, and their practical applicability to extracting equilibrium causal effects from time series is argued.

Estimating Brain Connectivity With Varying-Length Time Lags Using a Recurrent Neural Network

OBJECTIVE

Computer-aided estimation of brain connectivity aims to reveal information propagation in brain automatically, which has great potential in clinical applications, e.g., epilepsy foci diagnosis. Granger causality is an effective tool for directional connection analysis in multivariate time series. However, most existing methods based on Granger causality assume fixed time lags in information transmission, while the propagation delay between brain signals is usually changing constantly.

METHODS

We propose a Granger causality estimator based on the recurrent neural network, called RNN-GC, to deal with the multivariate brain connectivity detection problem. Our model takes input of time-series signals with arbitrary length of transmission time lags and learns the information flow from the data using the gated RNN model, i.e., long short-term memory (LSTM). The LSTM model can sequentially update the gates in memory cells to determine how many preceding points should be considered for prediction. Therefore, the LSTM-based RNN-GC estimator works well on varying-length time lags and shows effectiveness even on very long transmission delays.

RESULTS

Experiments are carried out in comparison with other methods using both simulation data and epileptic electroencephalography signals. The RNN-GC estimator achieves superior performance in brain connectivity estimation and shows robustness in modeling multivariate connections with varying-length time lags.

CONCLUSION

The RNN-GC method is capable of modeling nonlinear and varying-length lagged information transmission and effective in directional brain connectivity estimation.

SIGNIFICANCE

The proposed method is promising to serve as a robust brain connection analysis tool in clinical applications.

Apparent remote synchronization of amplitudes: A demodulation and interference effect

A form of "remote synchronization" was recently described, wherein amplitude fluctuations across a ring of non-identical, non-linear electronic oscillators become entrained into spatially-structured patterns. According to linear models and mutual information, synchronization and causality dip at a certain distance, then recover before eventually fading. Here, the underlying mechanism is finally elucidated through novel experiments and simulations. The system non-linearity is found to have a dual role: it supports chaotic dynamics, and it enables the energy exchange between the lower and higher sidebands of a predominant frequency. This frequency acts as carrier signal in an arrangement resembling standard amplitude modulation, wherein the lower sideband and the demodulated baseband signals spectrally overlap. Due to a spatially-dependent phase relationship, at a certain distance near-complete destructive interference occurs between them, causing the observed dip. Methods suitable for detecting non-trivial entrainment, such as transfer entropy and the auxiliary system approach, nevertheless, reveal that synchronization and causality actually decrease with distance monotonically. Remoteness is, therefore, arguably only apparent, as also reflected in the propagation of external perturbations. These results demonstrate a complex mechanism of dynamical interdependence, and exemplify how it can lead to incorrectly inferring synchronization and causality.

Low-dimensional approximation searching strategy for transfer entropy from non-uniform embedding

Transfer entropy from non-uniform embedding is a popular tool for the inference of causal relationships among dynamical subsystems. In this study we present an approach that makes use of low-dimensional conditional mutual information quantities to decompose the original high-dimensional conditional mutual information in the searching procedure of non-uniform embedding for significant variables at different lags. We perform a series of simulation experiments to assess the sensitivity and specificity of our proposed method to demonstrate its advantage compared to previous algorithms. The results provide concrete evidence that low-dimensional approximations can help to improve the statistical accuracy of transfer entropy in multivariate causality analysis and yield a better performance over other methods. The proposed method is especially efficient as the data length grows.

Time, frequency, and time-varying Granger-causality measures in neuroscience

This article proposes a systematic methodological review and an objective criticism of existing methods enabling the derivation of time, frequency, and time-varying Granger-causality statistics in neuroscience. The capacity to describe the causal links between signals recorded at different brain locations during a neuroscience experiment is indeed of primary interest for neuroscientists, who often have very precise prior hypotheses about the relationships between recorded brain signals. The increasing interest and the huge number of publications related to this topic calls for this systematic review, which describes the very complex methodological aspects underlying the derivation of these statistics. In this article, we first present a general framework that allows us to review and compare Granger-causality statistics in the time domain, and the link with transfer entropy. Then, the spectral and the time-varying extensions are exposed and discussed together with their estimation and distributional properties. Although not the focus of this article, partial and conditional Granger causality, dynamical causal modelling, directed transfer function, directed coherence, partial directed coherence, and their variant are also mentioned.

Univariate and multivariate conditional entropy measures for the characterization of short-term cardiovascular complexity under physiological stress

OBJECTIVE

A defining feature of physiological systems under the neuroautonomic regulation is their dynamical complexity. The most common approach to assess physiological complexity from short-term recordings, i.e. to compute the rate of entropy generation of an individual system by means of measures of conditional entropy (CE), does not consider that complexity may change when the investigated system is part of a network of physiological interactions. This study aims at extending the concept of short-term complexity towards the perspective of network physiology, defining multivariate CE measures whereby multiple physiological processes are accounted for in the computation of entropy rates.

APPROACH

Univariate and multivariate CE measures are computed using state-of-the-art methods for entropy estimation and applied to time series of heart period (H), systolic (S) and diastolic (D) arterial pressure, and respiration (R) variability measured in healthy subjects monitored in a resting state and during conditions of postural and mental stress.

MAIN RESULTS

Compared with the traditional univariate metric of short-term complexity, multivariate measures provide additional information with plausible physiological interpretation, such as (i) the dampening of respiratory sinus arrhythmia and activation of the baroreflex control during postural stress; (ii) the increased complexity of heart period and blood pressure variability during mental stress, reflecting the effect of respiratory influences and upper cortical centers; (iii) the strong influence of D on S, mediated by left ventricular ejection fraction and vascular properties; (iv) the role of H in reducing the complexity of D, related to cardiac run-off effects; and (v) the unidirectional role of R in influencing cardiovascular variability.

SIGNIFICANCE

Our results document the importance of employing a network perspective in the evaluation of the short-term complexity of cardiovascular and respiratory dynamics across different physiological states.

Multiscale analysis of information dynamics for linear multivariate processes

In the study of complex physical and physiological systems represented by multivariate time series, an issue of great interest is the description of the system dynamics over a range of different temporal scales. While information-theoretic approaches to the multiscale analysis of complex dynamics are being increasingly used, the theoretical properties of the applied measures are poorly understood. This study introduces for the first time a framework for the analytical computation of information dynamics for linear multivariate stochastic processes explored at different time scales. After showing that the multiscale processing of a vector autoregressive (VAR) process introduces a moving average (MA) component, we describe how to represent the resulting VARMA process using statespace (SS) models and how to exploit the SS model parameters to compute analytical measures of information storage and information transfer for the original and rescaled processes. The framework is then used to quantify multiscale information dynamics for simulated unidirectionally and bidirectionally coupled VAR processes, showing that rescaling may lead to insightful patterns of information storage and transfer but also to potentially misleading behaviors.

Atypical transistor-based chaotic oscillators: Design, realization, and diversity

In this paper, we show that novel autonomous chaotic oscillators based on one or two bipolar junction transistors and a limited number of passive components can be obtained via random search with suitable heuristics. Chaos is a pervasive occurrence in these circuits, particularly after manual adjustment of a variable resistor placed in series with the supply voltage source. Following this approach, 49 unique circuits generating chaotic signals when physically realized were designed, representing the largest collection of circuits of this kind to date. These circuits are atypical as they do not trivially map onto known topologies or variations thereof. They feature diverse spectra and predominantly anti-persistent monofractal dynamics. Notably, we recurrently found a circuit comprising one resistor, one transistor, two inductors, and one capacitor, which generates a range of attractors depending on the parameter values. We also found a circuit yielding an irregular quantized spike-train resembling some aspects of neural discharge and another one generating a double-scroll attractor, which represent the smallest known transistor-based embodiments of these behaviors. Through three representative examples, we additionally show that diffusive coupling of heterogeneous oscillators of this kind may give rise to complex entrainment, such as lag synchronization with directed information transfer and generalized synchronization. The replicability and reproducibility of the experimental findings are good.

Entropy measures, entropy estimators, and their performance in quantifying complex dynamics: Effects of artifacts, nonstationarity, and long-range correlations

Entropy measures are widely applied to quantify the complexity of dynamical systems in diverse fields. However, the practical application of entropy methods is challenging, due to the variety of entropy measures and estimators and the complexity of real-world time series, including nonstationarities and long-range correlations (LRC). We conduct a systematic study on the performance, bias, and limitations of three basic measures (entropy, conditional entropy, information storage) and three traditionally used estimators (linear, kernel, nearest neighbor). We investigate the dependence of entropy measures on estimator- and process-specific parameters, and we show the effects of three types of nonstationarities due to artifacts (trends, spikes, local variance change) in simulations of stochastic autoregressive processes. We also analyze the impact of LRC on the theoretical and estimated values of entropy measures. Finally, we apply entropy methods on heart rate variability data from subjects in different physiological states and clinical conditions. We find that entropy measures can only differentiate changes of specific types in cardiac dynamics and that appropriate preprocessing is vital for correct estimation and interpretation. Demonstrating the limitations of entropy methods and shedding light on how to mitigate bias and provide correct interpretations of results, this work can serve as a comprehensive reference for the application of entropy methods and the evaluation of existing studies.

Basic cardiovascular variability signals: mutual directed interactions explored in the information domain

The study of short-term cardiovascular interactions is classically performed through the bivariate analysis of the interactions between the beat-to-beat variability of heart period (RR interval from the ECG) and systolic blood pressure (SBP). Recent progress in the development of multivariate time series analysis methods is making it possible to explore how directed interactions between two signals change in the context of networks including other coupled signals. Exploiting these advances, the present study aims at assessing directional cardiovascular interactions among the basic variability signals of RR, SBP and diastolic blood pressure (DBP), using an approach which allows direct comparison between bivariate and multivariate coupling measures. To this end, we compute information-theoretic measures of the strength and delay of causal interactions between RR, SBP and DBP using both bivariate and trivariate (conditioned) formulations in a group of healthy subjects in a resting state and during stress conditions induced by head-up tilt (HUT) and mental arithmetics (MA). We find that bivariate measures better quantify the overall (direct  +  indirect) information transferred between variables, while trivariate measures better reflect the existence and delay of directed interactions. The main physiological results are: (i) the detection during supine rest of strong interactions along the pathway RR  →  DBP  →  SBP, reflecting marked Windkessel and/or Frank-Starling effects; (ii) the finding of relatively weak baroreflex effects SBP  →  RR at rest; (iii) the invariance of cardiovascular interactions during MA, and the emergence of stronger and faster SBP  →  RR interactions, as well as of weaker RR  →  DBP interactions, during HUT. These findings support the importance of investigating cardiovascular interactions from a network perspective, and suggest the usefulness of directed information measures to assess physiological mechanisms and track their changes across different physiological states.