Redundancy and synergy in interactions among basic cardiovascular oscillations

2020

The cardiovascular control system comprises a complex network of various control mechanisms operating on many time scales resulting in complex and mutually interconnected output signals (e.g. heart rate, systolic and diastolic blood pressures). The analysis of these interconnections could noninvasively provide an information on the regulatory mechanisms involved in cardiovascular control and thus could be potentially applied to better characterize cardiovascular dysregulation in pathological conditions. Our study demonstrates that the strength of interactions among signals changes with the time scale and as a response to changed autonomic state (orthostasis compared to supine rest). Novel i…

Spectral decomposition of cerebrovascular and cardiovascular interactions in patients prone to postural syncope and healthy controls.

2022

We present a framework for the linear parametric analysis of pairwise interactions in bivariate time series in the time and frequency domains, which allows the evaluation of total, causal and instantaneous interactions and connects time- and frequency-domain measures. The framework is applied to physiological time series to investigate the cerebrovascular regulation from the variability of mean cerebral blood flow velocity (CBFV) and mean arterial pressure (MAP), and the cardiovascular regulation from the variability of heart period (HP) and systolic arterial pressure (SAP). We analyze time series acquired at rest and during the early and late phase of head-up tilt in subjects developing or…

Quantification of Different Regulatory Pathways Contributing to Heartbeat Dynamics during Multiple Stimuli: a Proof of the Concept.

2019

The dynamical interplay between brain and heart is mediated by several feedback mechanisms including the central autonomic network and baroreflex loop at a peripheral level, also for a short-term regulation. State of the art focused on the characterization of each regulatory pathway through a single stressor elicitation. However, no studies targeted the actual quantification of different mediating routes leading to the generation of heartbeat dynamics, particularly in case of combined exogenous stimuli. In this study, we propose a new approach based on computational modeling to quantify the contribution of multiple concurrent stimuli in modulating cardiovascular dynamics. In this prelimina…

Multivariate and Multiscale Complexity of Long-Range Correlated Cardiovascular and Respiratory Variability Series

2020

Assessing the dynamical complexity of biological time series represents an important topic with potential applications ranging from the characterization of physiological states and pathological conditions to the calculation of diagnostic parameters. In particular, cardiovascular time series exhibit a variability produced by different physiological control mechanisms coupled with each other, which take into account several variables and operate across multiple time scales that result in the coexistence of short term dynamics and long-range correlations. The most widely employed technique to evaluate the dynamical complexity of a time series at different time scales, the so-called multiscale …

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

2018

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…

Multiscale Information Storage of Linear Long-Range Correlated Stochastic Processes

2019

Information storage, reflecting the capability of a dynamical system to keep predictable information during its evolution over time, is a key element of intrinsic distributed computation, useful for the description of the dynamical complexity of several physical and biological processes. Here we introduce a parametric approach which allows one to compute information storage across multiple timescales in stochastic processes displaying both short-term dynamics and long-range correlations (LRC). Our analysis is performed in the popular framework of multiscale entropy, whereby a time series is first "coarse grained" at the chosen timescale through low-pass filtering and downsampling, and then …

Quantifying multidimensional control mechanisms of cardiovascular dynamics during multiple concurrent stressors

2021

Heartbeat regulation is achieved through different routes originating from central autonomic network sources, as well as peripheral control mechanisms. While previous studies successfully characterized cardiovascular regulatory mechanisms during a single stressor, to the best of our knowledge, a combination of multiple concurrent elicitations leading to the activation of different autonomic regulatory routes has not been investigated yet. Therefore, in this study, we propose a novel modeling framework for the quantification of heartbeat regulatory mechanisms driven by different neural routes. The framework is evaluated using two heartbeat datasets gathered from healthy subjects undergoing p…

Assessing Transfer Entropy in cardiovascular and respiratory time series: A VARFI approach

2021

In the study of complex biomedical systems represented by multivariate stochastic processes, such as the cardiovascular and respiratory systems, an issue of great relevance is the description of the system dynamics spanning multiple temporal scales. Recently, the quantification of multiscale complexity based on linear parametric models, incorporating autoregressive coefficients and fractional integration, encompassing short term dynamics and long-range correlations, was extended to multivariate time series. Within this Vector AutoRegressive Fractionally Integrated (VARFI) framework formalized for Gaussian processes, in this work we propose to estimate the Transfer Entropy, or equivalently G…

Information Decomposition in Multivariate Systems: Definitions, Implementation and Application to Cardiovascular Networks

2016

The continuously growing framework of information dynamics encompasses a set of tools, rooted in information theory and statistical physics, which allow to quantify different aspects of the statistical structure of multivariate processes reflecting the temporal dynamics of complex networks. Building on the most recent developments in this field, this work designs a complete approach to dissect the information carried by the target of a network of multiple interacting systems into the new information produced by the system, the information stored in the system, and the information transferred to it from the other systems; information storage and transfer are then further decomposed into amou…

Multiscale partial information decomposition of dynamic processes with short and long-range correlations: theory and application to cardiovascular co…

2022

Abstract Objective. In this work, an analytical framework for the multiscale analysis of multivariate Gaussian processes is presented, whereby the computation of Partial Information Decomposition measures is achieved accounting for the simultaneous presence of short-term dynamics and long-range correlations. Approach. We consider physiological time series mapping the activity of the cardiac, vascular and respiratory systems in the field of Network Physiology. In this context, the multiscale representation of transfer entropy within the network of interactions among Systolic arterial pressure (S), respiration (R) and heart period (H), as well as the decomposition into unique, redundant and s…

Inclusion of Instantaneous Influences in the Spectral Decomposition of Causality: Application to the Control Mechanisms of Heart Rate Variability

2021

Heart rate variability is the result of several physiological regulation mechanisms, including cardiovascular and cardiorespiratory interactions. Since instantaneous influences occurring within the same cardiac beat are commonplace in this regulation, their inclusion is mandatory to get a realistic model of physiological causal interactions. Here we exploit a recently proposed framework for the spectral decomposition of causal influences between autoregressive processes [2] and generalize it by introducing instantaneous couplings in the vector autoregressive model (VAR). We show the effectiveness of the proposed approach on a toy model, and on real data consisting of heart period (RR), syst…

Information decomposition in the frequency domain: a new framework to study cardiovascular and cardiorespiratory oscillations

2021

While cross-spectral and information-theoretic approaches are widely used for the multivariate analysis of physiological time series, their combined utilization is far less developed in the literature. This study introduces a framework for the spectral decomposition of multivariate information measures, which provides frequency-specific quantifications of the information shared between a target and two source time series and of its expansion into amounts related to how the sources contribute to the target dynamics with unique, redundant and synergistic information. The framework is illustrated in simulations of linearly interacting stochastic processes, showing how it allows us to retrieve …

Information-theoretic assessment of cardiovascular variability during postural and mental stress

2016

This study was aimed at investigating the individual and combined effects of postural and mental stress on short-term cardiovascular regulation. To this end, we applied measures taken from the emerging framework of information dynamics on the beat-to-beat spontaneous variability of RR interval and systolic arterial pressure (SAP) measured from healthy subjects in the resting supine position and during the separate and simultaneous execution of experimental protocols performing head-up tilt (HUT) and mental arithmetics (MA). The information stored in RR interval variability, a measure inversely related to the complexity of the time series, increased significantly during HUT and HUT+MA compar…

Selection of blood pressure signal for baroreflex analysis

2020

This study aims to evaluate the strength of the causal coupling among systolic, mean and diastolic blood pressure (SBP, MBP and DBP) with heart period (RR interval) (evaluating cardiac chronotropic baroreflex arm) and peripheral vascular resistance (PVR) (evaluating vascular resistance baroreflex arm) in frequency domain using partial spectral decomposition method. We recorded beat-to-beat RR, SBP, MBP and DBP and PVR values in 39 volunteers during supine rest and head-up tilt. Our results showed that during supine rest the most dominant causal coupling was from DBP to RR in both low and high frequency bands and significantly decreased during orthostasis. The strength of spectral couplings …

Towards understanding the complexity of cardiovascular oscillations: Insights from information theory.

2018

Abstract Cardiovascular complexity is a feature of healthy physiological regulation, which stems from the simultaneous activity of several cardiovascular reflexes and other non-reflex physiological mechanisms. It is manifested in the rich dynamics characterizing the spontaneous heart rate and blood pressure variability (HRV and BPV). The present study faces the challenge of disclosing the origin of short-term HRV and BPV from the statistical perspective offered by information theory. To dissect the physiological mechanisms giving rise to cardiovascular complexity in different conditions, measures of predictive information, information storage, information transfer and information modificati…

Multiscale Information Decomposition Dissects Control Mechanisms of Heart Rate Variability at Rest and During Physiological Stress.

2019

Heart rate variability (HRV

Partial Information Decomposition in the Frequency Domain: Application to Control Mechanisms of Heart Rate Variability at Rest and During Postural St…

2020

We exploit a recently proposed framework for assessing causal influences in the frequency domain to construct the partial information decomposition (PID) for informational circuits of three variables, thus obtaining the spectral decomposition of redundancy, synergy and unique information. The approach is applied to heart period (HP), systolic pressure (SP) and respiration (RESP) variability series measured in healthy subjects in baseline and head up tilt conditions. Integrating the informational quantities in the respiratory band, the total influence from RESP to HP does not change in the two conditions. However, we find that in baseline RESP causes HP mostly through the direct pathway desc…

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

2017

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.…

Feasibility of Linear Parametric Estimation of Dynamic Information Measures to assess Physiological Stress from Short-Term Cardiovascular Variability

2021

Extensive efforts have been recently devoted to implement fast and reliable algorithms capable of assessing the physiological response of the organism to physiological stress. In this study, we propose the comparison between model-free and linear parametric methods as regards their ability to detect alterations in the dynamics and in the complexity of cardiovascular and respiratory variability evoked by postural and mental stress. Dynamic entropy (DE) and information storage (IS) measures were calculated on three physiological time-series, i.e. heart period, respiratory volume and systolic arterial pressure, on 61 healthy subjects monitored in resting conditions as well as during head-up ti…

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

2017

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 compar…

Efficient Computation of Multiscale Entropy over Short Biomedical Time Series Based on Linear State-Space Models

2017

The most common approach to assess the dynamical complexity of a time series across multiple temporal scales makes use of the multiscale entropy (MSE) and refined MSE (RMSE) measures. In spite of their popularity, MSE and RMSE lack an analytical framework allowing their calculation for known dynamic processes and cannot be reliably computed over short time series. To overcome these limitations, we propose a method to assess RMSE for autoregressive (AR) stochastic processes. The method makes use of linear state-space (SS) models to provide the multiscale parametric representation of an AR process observed at different time scales and exploits the SS parameters to quantify analytically the co…

Information transfer and information modification to identify the structure of cardiovascular and cardiorespiratory networks

2017

To fully elucidate the complex physiological mechanisms underlying the short-term autonomic regulation of heart period (H), systolic and diastolic arterial pressure (S, D) and respiratory (R) variability, the joint dynamics of these variables need to be explored using multivariate time series analysis. This study proposes the utilization of information-theoretic measures to measure causal interactions between nodes of the cardiovascular/cardiorespiratory network and to assess the nature (synergistic or redundant) of these directed interactions. Indexes of information transfer and information modification are extracted from the H, S, D and R series measured from healthy subjects in a resting…

Analysis of Cardiac Pulse Arrival Time Series at Rest and during Physiological Stress

2022

The study of cardiovascular dynamics is pivotal in the prevention and monitoring of cardiovascular diseases. Pulse Arrival Time (PAT) series contain information concerning not only the dynamics of the Autonomic Nervous System (ANS), but of all the systems involved in the regulation of cardiovascular homeostasis. This study aims to highlight how indexes extracted from PAT series in time-, frequency- and information-domain allow to discriminate among different physiological conditions. Analyses were carried out on 76 young healthy subjects, at rest and during orthostatic or mental stress. Our results show that PAT indexes vary according to the ANS condition, and may thus be useful parameters …

Assessing Transfer Entropy in cardiovascular and respiratory time series under long-range correlations.

2021

Heart Period (H) results from the activity of several coexisting control mechanisms, involving Systolic Arterial Pressure (S) and Respiration (R), which operate across multiple time scales encompassing not only short-term dynamics but also long-range correlations. In this work, multiscale representation of Transfer Entropy (TE) and of its decomposition in the network of these three interacting processes is obtained by extending the multivariate approach based on linear parametric VAR models to the Vector AutoRegressive Fractionally Integrated (VARFI) framework for Gaussian processes. This approach allows to dissect the different contributions to cardiac dynamics accounting for the simultane…

ANALYSIS OF RESPIRATORY SINUS ARRHYTHMIA MECHANISMS IN INFORMATION DOMAIN

2018

Vector Autoregressive Fractionally Integrated Models to Assess Multiscale Complexity in Cardiovascular and Respiratory Time Series

2020

Cardiovascular variability is the result of the activity of several physiological control mechanisms, which involve different variables and operate across multiple time scales encompassing short term dynamics and long range correlations. This study presents a new approach to assess the multiscale complexity of multivariate time series, based on linear parametric models incorporating autoregressive coefficients and fractional integration. The approach extends to the multivariate case recent works introducing a linear parametric representation of multiscale entropy, and is exploited to assess the complexity of cardiovascular and respiratory time series in healthy subjects studied during postu…

A new Frequency Domain Measure of Causality based on Partial Spectral Decomposition of Autoregressive Processes and its Application to Cardiovascular…

2019

We present a new method to quantify in the frequency domain the strength of directed interactions between linear stochastic processes. This issue is traditionally addressed by the directed coherence (DC), a popular causality measure derived from the spectral representation of vector autoregressive (AR) processes. Here, to overcome intrinsic limitations of the DC when it needs to be objectively quantified within specific frequency bands, we propose an approach based on spectral decomposition, which allows to isolate oscillatory components related to the pole representation of the vector AR process in the Z-domain. Relating the causal and non-causal power content of these components we obtain…

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

2019

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 applicabil…

Information domain analysis of respiratory sinus arrhythmia mechanisms.

2019

Ventilation related heart rate oscillations – respiratory sinus arrhythmia (RSA) – originate in human from several mechanisms. Two most important of them – the central mechanism (direct communication between respiratory and cardiomotor centers), and the peripheral mechanism (ventilation-associated blood pressure changes transferred to heart rate via baroreflex) have been described in previous studies. The major aim of this study was to compare the importance of these mechanisms in the generation of RSA non-invasively during various states by quantifying the strength of the directed interactions between heart rate, systolic blood pressure and respiratory volume signals. Seventy-eight healthy…

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

2019

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 …

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

2022

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…

Causal analysis of short-term cardiovascular variability: state-dependent contribution of feedback and feedforward mechanisms.

2016

Baroreflex function is usually assessed from spontaneous oscillations of blood pressure (BP) and cardiac RR interval assuming a unidirectional influence from BP to RR. However, the interaction of BP and RR is bidirectional—RR also influences BP. Novel methods based on the concept of Granger causality were recently developed for separate analysis of feedback (baroreflex) and feedforward (mechanical) interactions between RR and BP. We aimed at assessing the proportion of the two causal directions of the interactions between RR and systolic BP (SBP) oscillations during various conditions, and at comparing causality measures from SBP to RR with baroreflex gain indexes. Arterial BP and ECG sig…

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

2022

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 post…

Towards Disentangling the Contribution of Different Pathways for the Regulation of Cardiac Activity: A Pilot Study

2020

Heartbeat is dynamically regulated through mediating routes associated with central and peripheral feedback mechanisms. Previous studies focused on the quantification of these mechanisms in the presence of a single stressor. In this pilot study we propose a model aimed to quantify the contribution of different heartbeat regulatory routes while multiple stressors are administrated to the subject. The model is tested with Heart rate Variability (HRV) series from 26 subjects undergoing physical and affective stressors. Results show that the physical stressor prevalently (74%) contributes in mediating cardiac vagal control dynamics in case of concurrent affective elicitation. These results may …

Transfer Entropy Analysis of Pulse Arrival Time - Heart Period Interactions during Physiological Stress

2022

Although Heart Period (HP) variability is the most widely used measure to assess cardiovascular oscillations, its evaluation combined with that of Pulse Arrival Time (PAT) variability may provide additional information about cardiac dynamics and cardiovascular interactions. In this study, we computed the transfer entropy from PAT to HP in 76 subjects monitored at rest and during orthostatic and mental stress using both a model-free (k- Nearest Neighbors) and a linear parametric estimator. Our results show how the information flow between these two variables depends on the physiological condition and how the nonlinear measure captures more information than the linear one during orthostatic s…

Respiratory Sinus Arrhythmia Mechanisms in Young Obese Subjects

2020

Autonomic nervous system (ANS) activity and imbalance between its sympathetic and parasympathetic components are important factors contributing to the initiation and progression of many cardiovascular disorders related to obesity. The results on respiratory sinus arrhythmia (RSA) magnitude changes as a parasympathetic index were not straightforward in previous studies on young obese subjects. Considering the potentially unbalanced ANS regulation with impaired parasympathetic control in obese patients, the aim of this study was to compare the relative contribution of baroreflex and non-baroreflex (central) mechanisms to the origin of RSA in obese vs. control subjects. To this end, we applied…

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

2017

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)…

Extending the spectral decomposition of Granger causality to include instantaneous influences: application to the control mechanisms of heart rate va…

2021

Assessing Granger causality (GC) intended as the influence, in terms of reduction of variance of surprise, that a driver variable exerts on a given target, requires a suitable treatment of ‘instantaneous’ effects, i.e. influences due to interactions whose time scale is much faster than the time resolution of the measurements, due to unobserved confounders or insufficient sampling rate that cannot be increased because the mechanism of generation of the variable is inherently slow (e.g. the heartbeat). We exploit a recently proposed framework for the estimation of causal influences in the spectral domain and include instantaneous interactions in the modelling, thus obtaining (i) a novel index…

Spectral analysis of the beat-to-beat variability of arterial compliance

2022

Arterial compliance is an important parameter influencing ventricular-arterial coupling, depending on structural and functional mechanics of arteries. In this study, the spontaneous beat-to-beat variability of arterial compliance was investigated in time and frequency domains in thirty-nine young and healthy subjects monitored in the supine resting state and during head-up tilt. Spectral decomposition was applied to retrieve the spectral content of the time series associated to low (LF) and high frequency (HF) oscillatory components. Our results highlight: (i) a decrease of arterial compliance with tilt, in agreement with previous studies; (ii) an increase of the LF power content concurrent…

Vascular resistance arm of the baroreflex: methodology and comparison with the cardiac chronotropic arm.

2020

Baroreflex response consists of cardiac chronotropic (effect on heart rate), cardiac inotropic (on contractility), venous (on venous return) and vascular (on vascular resistance) arms. Because of its measurement simplicity, cardiac chronotropic arm is most often analysed. The aim was to introduce a method to assess vascular baroreflex arm, and to characterize its changes during stress. We evaluated the effect of orthostasis and mental arithmetics (MA) in 39 (22 female, median age: 18.7 yrs.) and 36 (21 female, 19.2 yrs.) healthy volunteers, respectively. We recorded systolic and mean blood pressure (SBP and MBP) by volume-clamp method and R-R interval (RR) by ECG. Cardiac output (CO) was re…

Feasibility of Ultra-short Term Complexity Analysis of Heart Rate Variability in Resting State and During Orthostatic Stress

2022

In this work, we study ultra-short term (UST) complexity of Heart Rate Variability (HRV) and its agreement with analysis of standard short-term (ST) HRV recordings obtained at rest and during orthostatic stress. Conditional Entropy (CE) measures have been computed using both a linear Gaussian approximation and a more accurate model-free approach based on nearest neighbors. The agreement between UST and ST indices has been compared via statistical tests and correlation analysis, suggesting the feasibility of exploiting faster algorithms and shorter time series for detecting changes in cardiovascular control during various states.

Assessing Transfer Entropy in cardiovascular and respiratory time series: A VARFI approach

2021

In the study of complex biomedical systems represented by multivariate stochastic processes, such as the cardiovascular and respiratory systems, an issue of great relevance is the description of the system dynamics spanning multiple temporal scales. Recently, the quantification of multiscale complexity based on linear parametric models, incorporating autoregressive coefficients and fractional integration, encompassing short term dynamics and long-range correlations, was extended to multivariate time series. Within this Vector AutoRegressive Fractionally Integrated (VARFI) framework formalized for Gaussian processes, in this work we propose to estimate the Transfer Entropy, or equivalently G…