Testing dynamic correlations and nonlinearity in bivariate time series through information measures and surrogate data analysis.

The increasing availability of time series data depicting the evolution of physical system properties has prompted the development of methods focused on extracting insights into the system behavior over time, discerning whether it stems from deterministic or stochastic dynamical systems. Surrogate data testing plays a crucial role in this process by facilitating robust statistical assessments. This ensures that the observed results are not mere occurrences by chance, but genuinely reflect the inherent characteristics of the underlying system. The initial process involves formulating a null hypothesis, which is tested using surrogate data in cases where assumptions about the underlying distributions are absent. A discriminating statistic is then computed for both the original data and each surrogate data set. Significantly deviating values between the original data and the surrogate data ensemble lead to the rejection of the null hypothesis. In this work, we present various surrogate methods designed to assess specific statistical properties in random processes. Specifically, we introduce methods for evaluating the presence of autodependencies and nonlinear dynamics within individual processes, using Information Storage as a discriminating statistic. Additionally, methods are introduced for detecting coupling and nonlinearities in bivariate processes, employing the Mutual Information Rate for this purpose. The surrogate methods introduced are first tested through simulations involving univariate and bivariate processes exhibiting both linear and nonlinear dynamics. Then, they are applied to physiological time series of Heart Period (RR intervals) and respiratory flow (RESP) variability measured during spontaneous and paced breathing. Simulations demonstrated that the proposed methods effectively identify essential dynamical features of stochastic systems. The real data application showed that paced breathing, at low breathing rate, increases the predictability of the individual dynamics of RR and RESP and dampens nonlinearity in their coupled dynamics.

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.

Investigating Dynamic High-Order Interactions in Physiological Networks through Predictive Information Decomposition.

We present an approach to assess redundant and synergistic interactions in network systems via the information-theoretic analysis of multivariate physiological processes. The approach sets up a strategy to decompose the information shared between the present states of a group of random processes and their own past states into unique contributions arising from the past of subgroups of processes and redundant and synergistic contributions arising from the dynamic interaction among the subgroups. The method is illustrated in a theoretical example of linearly interacting Gaussian processes, showing that redundancy and synergy are related mostly to unidirectional coupling and to bidirectional coupling with internal dynamics. It is then applied to the network of short-term heart period, arterial pressure and respiratory variability probed in healthy subjects, showing that redundancy and synergy prevail respectively in cardiorespiratory interactions and in cardiovascular interactions in the resting state, and that postural stress increases the predictive information and the redundancy of physiological interactions.

Statistical Approaches to Characterize Functional Connectivity in Brain and Physiologic Networks on a Single-Subject Basis.

The trend toward personalized medicine necessitates drawing conclusions from descriptive indexes of physiopathological states estimated from individual recordings of biomedical signals, using statistical analyses that focus on subject-specific differences between experimental conditions. In this context, the present work introduces an approach to assess functional connectivity in brain and physiologic networks by pairwise information-theoretic measures of coupling between signals, whose significance and variations between conditions are statistically validated on a single-subject basis through the use of surrogate and bootstrap data analyses. The approach is illustrated on single-subject recordings of (i) resting-state functional magnetic resonance imaging (rest-fMRI) signals acquired in a pediatric patient with hepatic encephalography associated to a portosystemic shunt and undergoing liver vascular shunt correction, and of (ii) cardiovascular and cerebrovascular time series acquired at rest and during head-up tilt in a subject suffering from orthostatic intolerance.

Decomposing the Mutual Information Rate of Heart Period and Respiration Variability Series to Assess Cardiorespiratory Interactions.

Heart rate variability results from the coupled activity of the cardiovascular and cardiorespiratory systems, which have their own internal regulation mechanisms but also interact with each other and with the autonomic nervous system to maintain homeostasis. In this work, the assessment of these physiological mechanisms is carried out decomposing the Mutual Information Rate (MIR), an information-theoretic measure of the interdependence between coupled processes, into terms of entropy rate or conditional mutual information related respectively to complexity and causality measures. These measures are computed using a non-parametric approach based on nearest-neighbors. The proposed framework is first tested on simulated autoregressive processes and then applied to experimental data consisting of heart period and respiratory time series measured in healthy subjects monitored at rest and during head-up tilt. Our results evidence that MIR decomposition is able to highlight the interdependence of short-term physiological mechanisms of cardiorespiratory interactions during postural stress.

A Method to Assess Granger Causality, Isolation and Autonomy in the Time and Frequency Domains: Theory and Application to Cerebrovascular Variability.

OBJECTIVE: Concepts of Granger causality (GC) and Granger autonomy (GA) are central to assess the dynamics of coupled physiologic processes. While causality measures have been already proposed and largely applied in time and frequency domains, measures quantifying self-dependencies are still limited to the time-domain formulation and lack of a clear spectral representation. METHODS: We embed into the classical linear parametric framework for computing GC from a driver random process X to a target process Y a measure of Granger Isolation (GI) quantifying the part of the dynamics of Y not originating from X, and a new spectral measure of GA assessing frequency-specific patterns of self-dependencies in Y. The framework is formulated in a way such that the full-frequency integration of the spectral GC, GI and GA measures returns the corresponding time-domain measures. The measures are illustrated in theoretical simulations and applied to time series of mean arterial pressure and cerebral blood flow velocity obtained in subjects prone to develop postural syncope and healthy controls. RESULTS: simulations show that GI is complementary to GC but not trivially related to it, while GA reflects the regularity of the internal dynamics of the analyzed target process. In the application to cerebrovascular interactions, spectral GA quantified the physiological response to postural stress of slow cerebral blood flow oscillations, while spectral GC and GI detected an altered response to postural stress in subjects prone to syncope, likely related to impaired cerebral autoregulation. CONCLUSION AND SIGNIFICANCE: The new spectral measures of GI and GA are useful complements to GC for the analysis of interacting oscillatory processes, and detect physiological and pathological responses to postural stress which cannot be traced in the time domain. The thorough assessment of causality, isolation and autonomy opens new perspectives for the analysis of coupled biological processes in both physiological and clinical investigations.

Time-varying information measures: an adaptive estimation of information storage with application to brain-heart interactions.

Network Physiology is a rapidly growing field of study that aims to understand how physiological systems interact to maintain health. Within the information theory framework the information storage (IS) allows to measure the regularity and predictability of a dynamic process under stationarity assumption. However, this assumption does not allow to track over time the transient pathways occurring in the dynamical activity of a physiological system. To address this limitation, we propose a time-varying approach based on the recursive least squares algorithm (RLS) for estimating IS at each time instant, in non-stationary conditions. We tested this approach in simulated time-varying dynamics and in the analysis of electroencephalographic (EEG) signals recorded from healthy volunteers and timed with the heartbeat to investigate brain-heart interactions. In simulations, we show that the proposed approach allows to track both abrupt and slow changes in the information stored in a physiological system. These changes are reflected in its evolution and variability over time. The analysis of brain-heart interactions reveals marked differences across the cardiac cycle phases of the variability of the time-varying IS. On the other hand, the average IS values exhibit a weak modulation over parieto-occiptal areas of the scalp. Our study highlights the importance of developing more advanced methods for measuring IS that account for non-stationarity in physiological systems. The proposed time-varying approach based on RLS represents a useful tool for identifying spatio-temporal dynamics within the neurocardiac system and can contribute to the understanding of brain-heart interactions.

Granger Causality Analysis of Transient Calcium Dynamics in the Honey Bee Antennal Lobe Network.

Odorant processing presents multiple parallels across animal species, and insects became relevant models for the study of olfactory coding because of the tractability of the underlying neural circuits. Within the insect brain, odorants are received by olfactory sensory neurons and processed by the antennal lobe network. Such a network comprises multiple nodes, named glomeruli, that receive sensory information and are interconnected by local interneurons participating in shaping the neural representation of an odorant. The study of functional connectivity between the nodes of a sensory network in vivo is a challenging task that requires simultaneous recording from multiple nodes at high temporal resolutions. Here, we followed the calcium dynamics of antennal lobe glomeruli and applied Granger causality analysis to assess the functional connectivity among network nodes in the presence and absence of an odorous stimulus. This approach revealed the existence of causal connectivity links between antennal lobe glomeruli in the absence of olfactory stimulation, while at odor arrival, the connectivity network's density increased and became stimulus-specific. Thus, such an analytical approach may provide a new tool for the investigation of neural network plasticity in vivo.

Comparison of discretization strategies for the model-free information-theoretic assessment of short-term physiological interactions.

This work presents a comparison between different approaches for the model-free estimation of information-theoretic measures of the dynamic coupling between short realizations of random processes. The measures considered are the mutual information rate (MIR) between two random processes X and Y and the terms of its decomposition evidencing either the individual entropy rates of X and Y and their joint entropy rate, or the transfer entropies from X to Y and from Y to X and the instantaneous information shared by X and Y. All measures are estimated through discretization of the random variables forming the processes, performed either via uniform quantization (binning approach) or rank ordering (permutation approach). The binning and permutation approaches are compared on simulations of two coupled non-identical Hènon systems and on three datasets, including short realizations of cardiorespiratory (CR, heart period and respiration flow), cardiovascular (CV, heart period and systolic arterial pressure), and cerebrovascular (CB, mean arterial pressure and cerebral blood flow velocity) measured in different physiological conditions, i.e., spontaneous vs paced breathing or supine vs upright positions. Our results show that, with careful selection of the estimation parameters (i.e., the embedding dimension and the number of quantization levels for the binning approach), meaningful patterns of the MIR and of its components can be achieved in the analyzed systems. On physiological time series, we found that paced breathing at slow breathing rates induces less complex and more coupled CR dynamics, while postural stress leads to unbalancing of CV interactions with prevalent baroreflex coupling and to less complex pressure dynamics with preserved CB interactions. These results are better highlighted by the permutation approach, thanks to its more parsimonious representation of the discretized dynamic patterns, which allows one to explore interactions with longer memory while limiting the curse of dimensionality.

Connectivity Analysis in EEG Data: A Tutorial Review of the State of the Art and Emerging Trends.

Understanding how different areas of the human brain communicate with each other is a crucial issue in neuroscience. The concepts of structural, functional and effective connectivity have been widely exploited to describe the human connectome, consisting of brain networks, their structural connections and functional interactions. Despite high-spatial-resolution imaging techniques such as functional magnetic resonance imaging (fMRI) being widely used to map this complex network of multiple interactions, electroencephalographic (EEG) recordings claim high temporal resolution and are thus perfectly suitable to describe either spatially distributed and temporally dynamic patterns of neural activation and connectivity. In this work, we provide a technical account and a categorization of the most-used data-driven approaches to assess brain-functional connectivity, intended as the study of the statistical dependencies between the recorded EEG signals. Different pairwise and multivariate, as well as directed and non-directed connectivity metrics are discussed with a pros-cons approach, in the time, frequency, and information-theoretic domains. The establishment of conceptual and mathematical relationships between metrics from these three frameworks, and the discussion of novel methodological approaches, will allow the reader to go deep into the problem of inferring functional connectivity in complex networks. Furthermore, emerging trends for the description of extended forms of connectivity (e.g., high-order interactions) are also discussed, along with graph-theory tools exploring the topological properties of the network of connections provided by the proposed metrics. Applications to EEG data are reviewed. In addition, the importance of source localization, and the impacts of signal acquisition and pre-processing techniques (e.g., filtering, source localization, and artifact rejection) on the connectivity estimates are recognized and discussed. By going through this review, the reader could delve deeply into the entire process of EEG pre-processing and analysis for the study of brain functional connectivity and learning, thereby exploiting novel methodologies and approaches to the problem of inferring connectivity within complex networks.

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

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 orthostatic syncope in response to prolonged postural stress, and in healthy controls. The spectral measures of total, causal and instantaneous coupling between HP and SAP, and between MAP and CBFV, are averaged in the low-frequency band of the spectrum to focus on specific rhythms, and over all frequencies to get time-domain measures. The analysis of cardiovascular interactions indicates that postural stress induces baroreflex involvement, and its prolongation induces baroreflex dysregulation in syncope subjects. The analysis of cerebrovascular interactions indicates that the postural stress enhances the total coupling between MAP and CBFV, and challenges cerebral autoregulation in syncope subjects, while the strong sympathetic activation elicited by prolonged postural stress in healthy controls may determine an increased coupling from CBFV to MAP during late tilt. These results document that the combination of time-domain and spectral measures allows us to obtain an integrated view of cardiovascular and cerebrovascular regulation in healthy and diseased subjects.

Resting State EEG Directed Functional Connectivity Unveils Changes in Motor Network Organization in Subacute Stroke Patients After Rehabilitation.

Brain plasticity and functional reorganization are mechanisms behind functional motor recovery of patients after an ischemic stroke. The study of resting-state motor network functional connectivity by means of EEG proved to be useful in investigating changes occurring in the information flow and find correlation with motor function recovery. In the literature, most studies applying EEG to post-stroke patients investigated the undirected functional connectivity of interacting brain regions. Quite recently, works started to investigate the directionality of the connections and many approaches or features have been proposed, each of them being more suitable to describe different aspects, e.g., direct or indirect information flow between network nodes, the coupling strength or its characteristic oscillation frequency. Each work chose one specific measure, despite in literature there is not an agreed consensus, and the selection of the most appropriate measure is still an open issue. In an attempt to shed light on this methodological aspect, we propose here to combine the information of direct and indirect coupling provided by two frequency-domain measures based on Granger's causality, i.e., the directed coherence (DC) and the generalized partial directed coherence (gPDC), to investigate the longitudinal changes of resting-state directed connectivity associated with sensorimotor rhythms α and ß, occurring in 18 sub-acute ischemic stroke patients who followed a rehabilitation treatment. Our results showed a relevant role of the information flow through the pre-motor regions in the reorganization of the motor network after the rehabilitation in the sub-acute stage. In particular, DC highlighted an increase in intra-hemispheric coupling strength between pre-motor and primary motor areas, especially in ipsi-lesional hemisphere in both α and ß frequency bands, whereas gPDC was more sensitive in the detection of those connection whose variation was mostly represented within the population. A decreased causal flow from contra-lesional premotor cortex towards supplementary motor area was detected in both α and ß frequency bands and a significant reinforced inter-hemispheric connection from ipsi to contra-lesional pre-motor cortex was observed in ß frequency. Interestingly, the connection from contra towards ipsilesional pre-motor area correlated with upper limb motor recovery in α band. The usage of two different measures of directed connectivity allowed a better comprehension of those coupling changes between brain motor regions, either direct or mediated, which mostly were influenced by the rehabilitation, revealing a particular involvement of the pre-motor areas in the cerebral functional reorganization.

A new Framework for the Spectral Information Decomposition of Multivariate Gaussian Processes.

Different information-theoretic measures are available in the literature for the study of pairwise and higher-order interactions in multivariate dynamical systems. While these measures operate in the time domain, several physiological and non-physiological systems exhibit a rich oscillatory content that is typically analyzed in the frequency domain through spectral and cross-spectral approaches. For Gaussian systems, the relation between information and spectral measures has been established considering coupling and causality measures, but not for higher-order interactions. To fill this gap, in this work we introduce an information-theoretic framework in the frequency domain to quantify the information shared between a target process and two sources, even multivariate, and to highlight the presence of redundancy and synergy in the analyzed dynamical system. Firstly, we simulate different linear interacting processes by showing the capability of the proposed framework to retrieve amounts of information shared by the processes in specific frequency bands which are not detectable by the related time-domain measures. Then, the framework is applied on EEG time series representative of the brain activity during a motor execution task in a group of healthy subjects.

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.

Preliminary development of a questionnaire to measure the extra-pulmonary symptoms of severe asthma.

BACKGROUND: Research into the effects of asthma treatments on the extra-pulmonary symptoms of severe asthma is limited by the absence of a suitable questionnaire. The aim was to create a questionnaire suitable for intervention studies by selecting symptoms that are statistically associated with asthma pathology and therefore may improve when pathology is reduced. METHODS: Patients attending a specialist asthma clinic completed the 65-item General Symptom Questionnaire (GSQ-65), a questionnaire validated for assessing symptoms of people with multiple medically unexplained symptoms. Lung function (FEV1%) and cumulative oral corticosteroids (OCS) calculated from maintenance dose plus exacerbations were obtained from clinic records. Pathology was represented by the two components of a principal component analysis (PCA) of FEV1% and OCS. LASSO regression was used to select symptoms that had high coefficients with these two principal components and occurred frequently in severe asthma. RESULTS: 100 patients provided data. PCA revealed two components, one where FEV1% and OCS were inversely related and another where they were directly related. LASSO regression revealed 39 symptoms with non-zero coefficients on one or more of the two principal components from which 16 symptoms were selected for the GSQ-A on the basis of magnitude of coefficient and frequency. Asthma symptoms measured by asthma control questionnaires were excluded. The GSQ-A correlated 0.33 and - 0.34 (p = 0.001) with the two principal components. CONCLUSION: The GSQ-A assesses the frequency of 16 heterogenous non-respiratory symptoms that are associated with asthma severity using the statistical combination of FEV1% and OCS.

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

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 amounts of information shared by the processes within specific frequency bands which are otherwise not detectable by time-domain information measures, as well as coupling features which are not detectable by spectral measures. Then, it is applied to the time series of heart period, systolic and diastolic arterial pressure and respiration variability measured in healthy subjects monitored in the resting supine position and during head-up tilt. We show that the spectral measures of unique, redundant and synergistic information shared by these variability series, integrated within specific frequency bands of physiological interest and reflect the mechanisms of short-term regulation of cardiovascular and cardiorespiratory oscillations and their alterations induced by the postural stress. This article is part of the theme issue 'Advanced computation in cardiovascular physiology: new challenges and opportunities'.

Estimation of Granger causality through Artificial Neural Networks: applications to physiological systems and chaotic electronic oscillators.

One of the most challenging problems in the study of complex dynamical systems is to find the statistical interdependencies among the system components. Granger causality (GC) represents one of the most employed approaches, based on modeling the system dynamics with a linear vector autoregressive (VAR) model and on evaluating the information flow between two processes in terms of prediction error variances. In its most advanced setting, GC analysis is performed through a state-space (SS) representation of the VAR model that allows to compute both conditional and unconditional forms of GC by solving only one regression problem. While this problem is typically solved through Ordinary Least Square (OLS) estimation, a viable alternative is to use Artificial Neural Networks (ANNs) implemented in a simple structure with one input and one output layer and trained in a way such that the weights matrix corresponds to the matrix of VAR parameters. In this work, we introduce an ANN combined with SS models for the computation of GC. The ANN is trained through the Stochastic Gradient Descent L1 (SGD-L1) algorithm, and a cumulative penalty inspired from penalized regression is applied to the network weights to encourage sparsity. Simulating networks of coupled Gaussian systems, we show how the combination of ANNs and SGD-L1 allows to mitigate the strong reduction in accuracy of OLS identification in settings of low ratio between number of time series points and of VAR parameters. We also report how the performances in GC estimation are influenced by the number of iterations of gradient descent and by the learning rate used for training the ANN. We recommend using some specific combinations for these parameters to optimize the performance of GC estimation. Then, the performances of ANN and OLS are compared in terms of GC magnitude and statistical significance to highlight the potential of the new approach to reconstruct causal coupling strength and network topology even in challenging conditions of data paucity. The results highlight the importance of of a proper selection of regularization parameter which determines the degree of sparsity in the estimated network. Furthermore, we apply the two approaches to real data scenarios, to study the physiological network of brain and peripheral interactions in humans under different conditions of rest and mental stress, and the effects of the newly emerged concept of remote synchronization on the information exchanged in a ring of electronic oscillators. The results highlight how ANNs provide a mesoscopic description of the information exchanged in networks of multiple interacting physiological systems, preserving the most active causal interactions between cardiovascular, respiratory and brain systems. Moreover, ANNs can reconstruct the flow of directed information in a ring of oscillators whose statistical properties can be related to those of physiological networks.

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.

Local Granger causality.

Granger causality (GC) is a statistical notion of causal influence based on prediction via linear vector autoregression. For Gaussian variables it is equivalent to transfer entropy, an information-theoretic measure of time-directed information transfer between jointly dependent processes. We exploit such equivalence and calculate exactly the local Granger causality, i.e., the profile of the information transferred from the driver to the target process at each discrete time point; in this frame, GC is the average of its local version. We show that the variability of the local GC around its mean relates to the interplay between driver and innovation (autoregressive noise) processes, and it may reveal transient instances of information transfer not detectable from its average values. Our approach offers a robust and computationally fast method to follow the information transfer along the time history of linear stochastic processes, as well as of nonlinear complex systems studied in the Gaussian approximation.

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.