Estimating Transfer Entropy in Continuous Time Between Neural Spike Trains or Other Event-Based Data

Measuring and understanding information storage and transfer in a simulated human gut microbiome

Considering biological systems as information processing entities and analyzing their organizational structure via information-theoretic measures has become an established approach in life sciences. We transfer this framework to a field of broad general interest, the human gut microbiome. We use BacArena, a software combining agent-based modelling and flux-balance analysis, to simulate a simplified human intestinal microbiome (SIHUMI). In a first step, we derive information theoretic measures from the simulated abundance data, and, in a second step, relate them to the metabolic processes underlying the abundance data. Our study provides further evidence on the role of active information storage as an indicator of unexpected structural change in the observed system. Besides, we show that information transfer reflects coherent behavior in the microbial community, both as a reaction to environmental changes and as a result of direct effective interaction. In this sense, purely abundance-based information theoretic measures can provide meaningful insight on metabolic interactions within bacterial communities. Furthermore, we shed light on the important however little noticed technical aspect of distinguishing immediate and delayed effects in the interpretation of local information theoretical measures.

Inference of monosynaptic connections from parallel spike trains: A review

This article presents a mini-review about the progress in inferring monosynaptic connections from spike trains of multiple neurons over the past twenty years. First, we explain a variety of meanings of "neuronal connectivity" in different research areas of neuroscience, such as structural connectivity, monosynaptic connectivity, and functional connectivity. Among these, we focus on the methods used to infer the monosynaptic connectivity from spike data. We then summarize the inference methods based on two main approaches, i.e., correlation-based and model-based approaches. Finally, we describe available source codes for connectivity inference and future challenges. Although inference will never be perfect, the accuracy of identifying the monosynaptic connections has improved dramatically in recent years due to continuous efforts.

A rigorous and versatile statistical test for correlations between stationary time series

In disciplines from biology to climate science, a routine task is to compute a correlation between a pair of time series and determine whether the correlation is statistically significant (i.e., unlikely under the null hypothesis that the time series are independent). This problem is challenging because time series typically exhibit autocorrelation and thus cannot be properly analyzed with the standard iid-oriented statistical tests. Although there are well-known parametric tests for time series, these are designed for linear correlation statistics and thus not suitable for the increasingly popular nonlinear correlation statistics. There are also nonparametric tests that can be used with any correlation statistic, but for these, the conditions that guarantee correct false positive rates are either restrictive or unclear. Here, we describe the truncated time-shift (TTS) test, a nonparametric procedure to test for dependence between 2 time series. We prove that this test correctly controls the false positive rate as long as one of the time series is stationary, a minimally restrictive requirement among current tests. The TTS test is versatile because it can be used with any correlation statistic. Using synthetic data, we demonstrate that this test performs correctly even while other tests suffer high false positive rates. In simulation examples, simple guidelines for parameter choices allow high statistical power to be achieved with sufficient data. We apply the test to datasets from climatology, animal behavior, and microbiome science, verifying previously discovered dependence relationships and detecting additional relationships.

Causal dynamics of sleep, circadian rhythm, and mood symptoms in patients with major depression and bipolar disorder: insights from longitudinal wearable device data

BACKGROUND

Sleep and circadian rhythm disruptions are common in patients with mood disorders. The intricate relationship between these disruptions and mood has been investigated, but their causal dynamics remain unknown.

METHODS

We analysed data from 139 patients (76 female, mean age = 23.5 ± 3.64 years) with mood disorders who participated in a prospective observational study in South Korea. The patients wore wearable devices to monitor sleep and engaged in smartphone-delivered ecological momentary assessment of mood symptoms. Using a mathematical model, we estimated their daily circadian phase based on sleep data. Subsequently, we obtained daily time series for sleep/circadian phase estimates and mood symptoms spanning >40,000 days. We analysed the causal relationship between the time series using transfer entropy, a non-linear causal inference method.

FINDINGS

The transfer entropy analysis suggested causality from circadian phase disturbance to mood symptoms in both patients with MDD (n = 45) and BD type I (n = 35), as 66.7% and 85.7% of the patients with a large dataset (>600 days) showed causality, but not in patients with BD type II (n = 59). Surprisingly, no causal relationship was suggested between sleep phase disturbances and mood symptoms.

INTERPRETATION

Our findings suggest that in patients with mood disorders, circadian phase disturbances directly precede mood symptoms. This underscores the potential of targeting circadian rhythms in digital medicine, such as sleep or light exposure interventions, to restore circadian phase and thereby manage mood disorders effectively.

FUNDING

Institute for Basic Science, the Human Frontiers Science Program Organization, the National Research Foundation of Korea, and the Ministry of Health & Welfare of South Korea.

Adaptive modeling and inference of higher-order coordination in neuronal assemblies: A dynamic greedy estimation approach

Central in the study of population codes, coordinated ensemble spiking activity is widely observable in neural recordings with hypothesized roles in robust stimulus representation, interareal communication, and learning and memory formation. Model-free measures of synchrony characterize coherent pairwise activity but not higher-order interactions, a limitation transcended by statistical models of ensemble spiking activity. However, existing model-based analyses often impose assumptions about the relevance of higher-order interactions and require repeated trials to characterize dynamics in the correlational structure of ensemble activity. To address these shortcomings, we propose an adaptive greedy filtering algorithm based on a discretized mark point-process model of ensemble spiking and a corresponding statistical inference framework to identify significant higher-order coordination. In the course of developing a precise statistical test, we show that confidence intervals can be constructed for greedily estimated parameters. We demonstrate the utility of our proposed methods on simulated neuronal assemblies. Applied to multi-electrode recordings from human and rat cortical assemblies, our proposed methods provide new insights into the dynamics underlying localized population activity during transitions between brain states.

Identifying Effective Connectivity between Stochastic Neurons with Variable-Length Memory Using a Transfer Entropy Rate Estimator

Information theory explains how systems encode and transmit information. This article examines the neuronal system, which processes information via neurons that react to stimuli and transmit electrical signals. Specifically, we focus on transfer entropy to measure the flow of information between sequences and explore its use in determining effective neuronal connectivity. We analyze the causal relationships between two discrete time series, X:=Xt:t∈Z and Y:=Yt:t∈Z, which take values in binary alphabets. When the bivariate process (X,Y) is a jointly stationary ergodic variable-length Markov chain with memory no larger than k, we demonstrate that the null hypothesis of the test-no causal influence-requires a zero transfer entropy rate. The plug-in estimator for this function is identified with the test statistic of the log-likelihood ratios. Since under the null hypothesis, this estimator follows an asymptotic chi-squared distribution, it facilitates the calculation of p-values when applied to empirical data. The efficacy of the hypothesis test is illustrated with data simulated from a neuronal network model, characterized by stochastic neurons with variable-length memory. The test results identify biologically relevant information, validating the underlying theory and highlighting the applicability of the method in understanding effective connectivity between neurons.

Ensemble learning and ground-truth validation of synaptic connectivity inferred from spike trains

Probing the architecture of neuronal circuits and the principles that underlie their functional organization remains an important challenge of modern neurosciences. This holds true, in particular, for the inference of neuronal connectivity from large-scale extracellular recordings. Despite the popularity of this approach and a number of elaborate methods to reconstruct networks, the degree to which synaptic connections can be reconstructed from spike-train recordings alone remains controversial. Here, we provide a framework to probe and compare connectivity inference algorithms, using a combination of synthetic ground-truth and in vitro data sets, where the connectivity labels were obtained from simultaneous high-density microelectrode array (HD-MEA) and patch-clamp recordings. We find that reconstruction performance critically depends on the regularity of the recorded spontaneous activity, i.e., their dynamical regime, the type of connectivity, and the amount of available spike-train data. We therefore introduce an ensemble artificial neural network (eANN) to improve connectivity inference. We train the eANN on the validated outputs of six established inference algorithms and show how it improves network reconstruction accuracy and robustness. Overall, the eANN demonstrated strong performance across different dynamical regimes, worked well on smaller datasets, and improved the detection of synaptic connectivity, especially inhibitory connections. Results indicated that the eANN also improved the topological characterization of neuronal networks. The presented methodology contributes to advancing the performance of inference algorithms and facilitates our understanding of how neuronal activity relates to synaptic connectivity.

Exploring the Short-Term Memory of Heart Rate Variability through Model-Free Information Measures

In this work, we perform a comparative analysis of discrete- and continuous-time estimators of information-theoretic measures quantifying the concept of memory utilization in short-term heart rate variability (HRV). Specifically, considering heartbeat intervals in discrete time we compute the measure of information storage (IS) and decompose it into immediate memory utilization (IMU) and longer memory utilization (MU) terms; considering the timings of heartbeats in continuous time we compute the measure of MU rate (MUR). All measures are computed through model-free approaches based on nearest neighbor entropy estimators applied to the HRV series of a group of 15 healthy subjects measured at rest and during postural stress. We find, moving from rest to stress, statistically significant increases of the IS and the IMU, as well as of the MUR. Our results suggest that both discrete-time and continuous-time approaches can detect the higher predictive capacity of HRV occurring with postural stress, and that such increased memory utilization is due to fast mechanisms likely related to sympathetic activation.

An information theoretic method to resolve millisecond-scale spike timing precision in a comprehensive motor program

Sensory inputs in nervous systems are often encoded at the millisecond scale in a precise spike timing code. There is now growing evidence in behaviors ranging from slow breathing to rapid flight for the prevalence of precise timing encoding in motor systems. Despite this, we largely do not know at what scale timing matters in these circuits due to the difficulty of recording a complete set of spike-resolved motor signals and assessing spike timing precision for encoding continuous motor signals. We also do not know if the precision scale varies depending on the functional role of different motor units. We introduce a method to estimate spike timing precision in motor circuits using continuous MI estimation at increasing levels of added uniform noise. This method can assess spike timing precision at fine scales for encoding rich motor output variation. We demonstrate the advantages of this approach compared to a previously established discrete information theoretic method of assessing spike timing precision. We use this method to analyze the precision in a nearly complete, spike resolved recording of the 10 primary wing muscles control flight in an agile hawk moth, Manduca sexta. Tethered moths visually tracked a robotic flower producing a range of turning (yaw) torques. We know that all 10 muscles in this motor program encode the majority of information about yaw torque in spike timings, but we do not know whether individual muscles encode motor information at different levels of precision. We demonstrate that the scale of temporal precision in all motor units in this insect flight circuit is at the sub-millisecond or millisecond-scale, with variation in precision scale present between muscle types. This method can be applied broadly to estimate spike timing precision in sensory and motor circuits in both invertebrates and vertebrates.

Information theoretic measures of causal influences during transient neural events

Introduction: Transient phenomena play a key role in coordinating brain activity at multiple scales, however their underlying mechanisms remain largely unknown. A key challenge for neural data science is thus to characterize the network interactions at play during these events. Methods: Using the formalism of Structural Causal Models and their graphical representation, we investigate the theoretical and empirical properties of Information Theory based causal strength measures in the context of recurring spontaneous transient events. Results: After showing the limitations of Transfer Entropy and Dynamic Causal Strength in this setting, we introduce a novel measure, relative Dynamic Causal Strength, and provide theoretical and empirical support for its benefits. Discussion: These methods are applied to simulated and experimentally recorded neural time series and provide results in agreement with our current understanding of the underlying brain circuits.

Network inference from short, noisy, low time-resolution, partial measurements: Application to C. elegans neuronal calcium dynamics

Network link inference from measured time series data of the behavior of dynamically interacting network nodes is an important problem with wide-ranging applications, e.g., estimating synaptic connectivity among neurons from measurements of their calcium fluorescence. Network inference methods typically begin by using the measured time series to assign to any given ordered pair of nodes a numerical score reflecting the likelihood of a directed link between those two nodes. In typical cases, the measured time series data may be subject to limitations, including limited duration, low sampling rate, observational noise, and partial nodal state measurement. However, it is unknown how the performance of link inference techniques on such datasets depends on these experimental limitations of data acquisition. Here, we utilize both synthetic data generated from coupled chaotic systems as well as experimental data obtained from Caenorhabditis elegans neural activity to systematically assess the influence of data limitations on the character of scores reflecting the likelihood of a directed link between a given node pair. We do this for three network inference techniques: Granger causality, transfer entropy, and, a machine learning-based method. Furthermore, we assess the ability of appropriate surrogate data to determine statistical confidence levels associated with the results of link-inference techniques.

A primer on entropy in neuroscience

Entropy is not just a property of a system - it is a property of a system and an observer. Specifically, entropy is a measure of the amount of hidden information in a system that arises due to an observer's limitations. Here we provide an account of entropy from first principles in statistical mechanics with the aid of toy models of neural systems. Specifically, we describe the distinction between micro and macrostates in the context of simplified binary-state neurons and the characteristics of entropy required to capture an associated measure of hidden information. We discuss the origin of the mathematical form of entropy via the indistinguishable re-arrangements of discrete-state neurons and show the way in which the arguments are extended into a phase space description for continuous large-scale neural systems. Finally, we show the ways in which limitations in neuroimaging resolution, as represented by coarse graining operations in phase space, lead to an increase in entropy in time as per the second law of thermodynamics. It is our hope that this primer will support the increasing number of studies that use entropy as a way of characterising neuroimaging timeseries and of making inferences about brain states.

Granger causality for compressively sensed sparse signals

Compressed sensing is a scheme that allows for sparse signals to be acquired, transmitted, and stored using far fewer measurements than done by conventional means employing the Nyquist sampling theorem. Since many naturally occurring signals are sparse (in some domain), compressed sensing has rapidly seen popularity in a number of applied physics and engineering applications, particularly in designing signal and image acquisition strategies, e.g., magnetic resonance imaging, quantum state tomography, scanning tunneling microscopy, and analog to digital conversion technologies. Contemporaneously, causal inference has become an important tool for the analysis and understanding of processes and their interactions in many disciplines of science, especially those dealing with complex systems. Direct causal analysis for compressively sensed data is required to avoid the task of reconstructing the compressed data. Also, for some sparse signals, such as for sparse temporal data, it may be difficult to discover causal relations directly using available data-driven or model-free causality estimation techniques. In this work, we provide a mathematical proof that structured compressed sensing matrices, specifically circulant and Toeplitz, preserve causal relationships in the compressed signal domain, as measured by Granger causality (GC). We then verify this theorem on a number of bivariate and multivariate coupled sparse signal simulations which are compressed using these matrices. We also demonstrate a real world application of network causal connectivity estimation from sparse neural spike train recordings from rat prefrontal cortex. In addition to demonstrating the effectiveness of structured matrices for GC estimation from sparse signals, we also show a computational time advantage of the proposed strategy for causal inference from compressed signals of both sparse and regular autoregressive processes as compared to standard GC estimation from original signals.

Data-driven causal analysis of observational biological time series

Complex systems are challenging to understand, especially when they defy manipulative experiments for practical or ethical reasons. Several fields have developed parallel approaches to infer causal relations from observational time series. Yet, these methods are easy to misunderstand and often controversial. Here, we provide an accessible and critical review of three statistical causal discovery approaches (pairwise correlation, Granger causality, and state space reconstruction), using examples inspired by ecological processes. For each approach, we ask what it tests for, what causal statement it might imply, and when it could lead us astray. We devise new ways of visualizing key concepts, describe some novel pathologies of existing methods, and point out how so-called 'model-free' causality tests are not assumption-free. We hope that our synthesis will facilitate thoughtful application of methods, promote communication across different fields, and encourage explicit statements of assumptions. A video walkthrough is available (Video 1 or https://youtu.be/AIV0ttQrjK8).

THPs: Topological Hawkes Processes for Learning Causal Structure on Event Sequences

Learning causal structure among event types on multitype event sequences is an important but challenging task. Existing methods, such as the Multivariate Hawkes processes, mostly assumed that each sequence is independent and identically distributed. However, in many real-world applications, it is commonplace to encounter a topological network behind the event sequences such that an event is excited or inhibited not only by its history but also by its topological neighbors. Consequently, the failure in describing the topological dependency among the event sequences leads to the error detection of the causal structure. By considering the Hawkes processes from the view of temporal convolution, we propose a topological Hawkes process (THP) to draw a connection between the graph convolution in the topology domain and the temporal convolution in time domains. We further propose a causal structure learning method on THP in a likelihood framework. The proposed method is featured with the graph convolution-based likelihood function of THP and a sparse optimization scheme with an Expectation-Maximization of the likelihood function. Theoretical analysis and experiments on both synthetic and real-world data demonstrate the effectiveness of the proposed method.

Early lock-in of structured and specialised information flows during neural development

The brains of many organisms are capable of complicated distributed computation underpinned by a highly advanced information processing capacity. Although substantial progress has been made towards characterising the information flow component of this capacity in mature brains, there is a distinct lack of work characterising its emergence during neural development. This lack of progress has been largely driven by the lack of effective estimators of information processing operations for spiking data. Here, we leverage recent advances in this estimation task in order to quantify the changes in transfer entropy during development. We do so by studying the changes in the intrinsic dynamics of the spontaneous activity of developing dissociated neural cell cultures. We find that the quantity of information flowing across these networks undergoes a dramatic increase across development. Moreover, the spatial structure of these flows exhibits a tendency to lock-in at the point when they arise. We also characterise the flow of information during the crucial periods of population bursts. We find that, during these bursts, nodes tend to undertake specialised computational roles as either transmitters, mediators, or receivers of information, with these roles tending to align with their average spike ordering. Further, we find that these roles are regularly locked-in when the information flows are established. Finally, we compare these results to information flows in a model network developing according to a spike-timing-dependent plasticity learning rule. Similar temporal patterns in the development of information flows were observed in these networks, hinting at the broader generality of these phenomena.

Assessing transfer entropy from biochemical data

We address the problem of evaluating the transfer entropy (TE) produced by biochemical reactions from experimentally measured data. Although these reactions are generally nonlinear and nonstationary processes making it challenging to achieve accurate modeling, Gaussian approximation can facilitate the TE assessment only by estimating covariance matrices using multiple data obtained from simultaneously measured time series representing the activation levels of biomolecules such as proteins. Nevertheless, the nonstationary nature of biochemical signals makes it difficult to theoretically assess the sampling distributions of TE, which are necessary for evaluating the statistical confidence and significance of the data-driven estimates. We resolve this difficulty by computationally assessing the sampling distributions using techniques from computational statistics. The computational methods are tested by using them in analyzing data generated from a theoretically tractable time-varying signal model, which leads to the development of a method to screen only statistically significant estimates. The usefulness of the developed method is examined by applying it to real biological data experimentally measured from the ERBB-RAS-MAPK system that superintends diverse cell fate decisions. A comparison between cells containing wild-type and mutant proteins exhibits a distinct difference in the time evolution of TE while any apparent difference is hardly found in average profiles of the raw signals. Such a comparison may help in unveiling important pathways of biochemical reactions.

Feedforward and feedback influences through distinct frequency bands between two spiking-neuron networks

Several studies on brain signals suggested that bottom-up and top-down influences are exerted through distinct frequency bands among visual cortical areas. It was recently shown that theta and gamma rhythms subserve feedforward, whereas the feedback influence is dominated by the alpha-beta rhythm in primates. A few theoretical models for reproducing these effects have been proposed so far. Here we show that a simple but biophysically plausible two-network motif composed of spiking-neuron models and chemical synapses can exhibit feedforward and feedback influences through distinct frequency bands. Different from previous studies, this kind of model allows us to study directed influences not only at the population level, by using a proxy for the local field potential, but also at the cellular level, by using the neuronal spiking series.

Toward practical causal epidemiology

Population attributable fraction (PAF), probability of causation, burden of disease, and related quantities derived from relative risk ratios are widely used in applied epidemiology and health risk analysis to quantify the extent to which reducing or eliminating exposures would reduce disease risks. This causal interpretation conflates association with causation. It has sometimes led to demonstrably mistaken predictions and ineffective risk management recommendations. Causal artificial intelligence (CAI) methods developed at the intersection of many scientific disciplines over the past century instead use quantitative high-level descriptions of networks of causal mechanisms (typically represented by conditional probability tables or structural equations) to predict the effects caused by interventions. We summarize these developments and discuss how CAI methods can be applied to realistically imperfect data and knowledge - e.g., with unobserved (latent) variables, missing data, measurement errors, interindividual heterogeneity in exposure-response functions, and model uncertainty. We recommend that CAI methods can help to improve the conceptual foundations and practical value of epidemiological calculations by replacing association-based attributions of risk to exposures or other risk factors with causal predictions of the changes in health effects caused by interventions.

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.

Causal Information Rate

Information processing is common in complex systems, and information geometric theory provides a useful tool to elucidate the characteristics of non-equilibrium processes, such as rare, extreme events, from the perspective of geometry. In particular, their time-evolutions can be viewed by the rate (information rate) at which new information is revealed (a new statistical state is accessed). In this paper, we extend this concept and develop a new information-geometric measure of causality by calculating the effect of one variable on the information rate of the other variable. We apply the proposed causal information rate to the Kramers equation and compare it with the entropy-based causality measure (information flow). Overall, the causal information rate is a sensitive method for identifying causal relations.

Relationship between mutual information and cross-correlation time scale of observability as measures of connectivity strength

The task of identifying and characterizing network structures out of experimentally observed time series is tackled by implementing different solutions, ranging from entropy-based techniques to the evaluation of the significance of observed correlation estimators. Among the metrics that belong to the first class, mutual information is of major importance due to the relative simplicity of implementation and its relying on the crucial concept of entropy. With regard to the second class, a method that allows us to assess the connectivity strength of a link in terms of a time scale of its observability via the significance estimate of measured cross correlation was recently shown to provide a reliable tool to study network structures. In this paper, we investigate the relationship between this last metric and mutual information by simultaneously assessing both metrics on large sets of data extracted from three experimental contexts, human brain magnetoencephalography, human brain electroencephalography, and surface wind measurements carried out on a small regional scale, as well as on simulated coupled, auto-regressive processes. We show that the relationship is well described by a power law and provide a theoretical explanation based on a simple noise and signal model. Besides further upholding the reliability of cross-correlation time scale of observability, the results show that the combined use of this metric and mutual information can be used as a valuable tool to identify and characterize connectivity links in a wide range of experimental contexts.

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.

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.