Causality Measures Research Articles

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.

“The Time of That Other Interpretation”: Gesture, Symptom, and Rilke’sThe Notebooks of Malte Laurids Brigge

“The Time of That Other Interpretation”: Gesture, Symptom, and Rilke’s<i>The Notebooks of Malte Laurids Brigge</i>

Feedback density and causal complexity of simulation model structure

ABSTRACT Measures of simulation model complexity generally focus on outputs; we propose measuring the complexity of a model’s causal structure to gain insight into its fundamental character. This article introduces tools for measuring causal complexity. First, we introduce a method for developing a model’s causal structure diagram, which characterises the causal interactions present in the code. Causal structure diagrams facilitate comparison of simulation models, including those from different paradigms. Next, we develop metrics for evaluating a model’s causal complexity using its causal structure diagram. We discuss cyclomatic complexity as a measure of the intricacy of causal structure and introduce two new metrics that incorporate the concept of feedback, a fundamental component of causal structure. The first new metric introduced here is feedback density, a measure of the cycle-based interconnectedness of causal structure. The second metric combines cyclomatic complexity and feedback density into a comprehensive causal complexity measure. Finally, we demonstrate these complexity metrics on simulation models from multiple paradigms and discuss potential uses and interpretations. These tools enable direct comparison of models across paradigms and provide a mechanism for measuring and discussing complexity based on a model’s fundamental assumptions and design.

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

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 of undirected instantaneous causality and (ii) a novel measure of GC including instantaneous effects. An effective procedure to speed up the optimization of parameters in this frame is also presented. After illustrating the proposed formalism in a theoretical example, we apply it to two datasets of cardiovascular and respiratory time series and compare the values obtained within the frequency bands of physiological interest by the proposed total measure of causality with those derived from the standard GC analysis. We find that the inclusion of instantaneous causality allows us to correctly disentangle the baroreflex mechanism from the effects related to cardiorespiratory interactions. Moreover, studying how controlling the respiratory rhythm acts on cardiovascular interactions, we document an increase of the direct (non-baroreflex mediated) influence of respiration on the heart rate in the respiratory frequency band when switching from spontaneous to paced breathing. This article is part of the theme issue ‘Advanced computation in cardiovascular physiology: new challenges and opportunities’.

Evaluation of Directed Causality Measures and Lag Estimations in Multivariate Time-Series.

The detection of causal effects among simultaneous observations provides knowledge about the underlying network, and is a topic of interests in many scientific areas. Over the years different causality measures have been developed, each with their own advantages and disadvantages. However, an extensive evaluation study is missing. In this work we consider some of the best-known causality measures i.e., cross-correlation, (conditional) Granger causality index (CGCI), partial directed coherence (PDC), directed transfer function (DTF), and partial mutual information on mixed embedding (PMIME). To correct for noise-related spurious connections, each measure (except PMIME) is tested for statistical significance based on surrogate data. The performance of the causality metrics is evaluated on a set of simulation models with distinct characteristics, to assess how well they work in- as well as outside of their “comfort zone.” PDC and DTF perform best on systems with frequency-specific connections, while PMIME is the only one able to detect non-linear interactions. The varying performance depending on the system characteristics warrants the use of multiple measures and comparing their results to avoid errors. Furthermore, lags between coupled variables are inherent to real-world systems and could hold essential information on the network dynamics. They are however often not taken into account and we lack proper tools to estimate them. We propose three new methods for lag estimation in multivariate time series, based on autoregressive modelling and information theory. One of the autoregressive methods and the one based on information theory were able to reliably identify the correct lag value in different simulated systems. However, only the latter was able to maintain its performance in the case of non-linear interactions. As a clinical application, the same methods are also applied on an intracranial recording of an epileptic seizure. The combined knowledge from the causality measures and insights from the simulations, on how these measures perform under different circumstances and when to use which one, allow us to recreate a plausible network of the seizure propagation that supports previous observations of desynchronisation and synchronisation during seizure progression. The lag estimation results show absence of a relationship between connectivity strength and estimated lag values, which contradicts the line of thinking in connectivity shaped by the neuron doctrine.

Stratified Restricted Mean Survival Time Model for Marginal Causal Effect in Observational Survival Data

Time to event outcomes is commonly encountered in epidemiologic research. Multiple papers have discussed the inadequacy of using the hazard ratio as a causal effect measure due to its noncollapsibility and the time-varying nature. In this paper, we further clarified that the hazard ratio might be used as a conditional causal effect measure, but it is generally not a valid marginal effect measure, even under randomized design. We proposed to use the restricted mean survival time (RMST) difference as a causal effect measure, since it essentially measures the mean difference over a specified time horizon and has a simple interpretation as the area under survival curves. For observational studies, propensity score adjustment can be implemented with RMST estimation to remove observed confounding bias. We proposed a propensity score stratified RMST estimation strategy, which performs well in our simulation evaluation and is relatively easy to implement for epidemiologists in practice. Our stratified RMST estimation includes two different versions of implementation, depending on whether researchers want to involve regression modeling adjustment, which provides a powerful tool to examine the marginal causal effect with observational survival data.

Stimulus evoked causality estimation in stereo-EEG

Objective. Stereo-electroencephalography (SEEG) has recently gained importance in analyzing brain functions. Its high temporal resolution and spatial specificity make it a powerful tool to investigate the strength, direction, and spectral content of brain networks interactions, especially when these connections are stimulus-evoked. However, choosing the best approach to evaluate the flow of information is not trivial, due to the lack of validated methods explicitly designed for SEEG. Approach. We propose a novel non-parametric statistical test for event-related causality (ERC) assessment on SEEG recordings. Here, we refer to the ERC as the causality evoked by a particular part of the stimulus (a response window (RW)). We also present a data surrogation method to evaluate the performance of a causality estimation algorithm. We finally validated our pipeline using surrogate SEEG data derived from an experimentally collected dataset, and compared the most used and successful measures to estimate effective connectivity, belonging to the Geweke–Granger causality framework. Main results. Here we show that our workflow correctly identified all the directed connections in the RW of the surrogate data and prove the robustness of the procedure against synthetic noise with amplitude exceeding physiological-plausible values. Among the causality measures tested, partial directed coherence performed best. Significance. This is the first non-parametric statistical test for ERC estimation explicitly designed for SEEG datasets. The pipeline, in principle, can also be applied to the analysis of any type of time-varying estimator, if there exists a clearly defined RW.

Dynamical ergodicity DDA reveals causal structure in time series.

Determining synchronization, causality, and dynamical similarity in highly complex nonlinear systems like brains is challenging. Although distinct, these measures are related by the unknown deterministic structure of the underlying dynamical system. For two systems that are not independent on each other, either because they result from a common process or they are already synchronized, causality measures typically fail. Here, we introduce dynamical ergodicity to assess dynamical similarity between time series and then combine this new measure with cross-dynamical delay differential analysis to estimate causal interactions between time series. We first tested this approach on simulated data from coupled Rössler systems where ground truth was known. We then applied it to intracranial electroencephalographic data from patients with epilepsy and found distinct dynamical states that were highly predictive of epileptic seizures.

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.

Granger-Causality Inference of the Existence of Unobserved Important Components in Network Analysis.

Detecting causal interrelationships in multivariate systems, in terms of the Granger-causality concept, is of major interest for applications in many fields. Analyzing all the relevant components of a system is almost impossible, which contrasts with the concept of Granger causality. Not observing some components might, in turn, lead to misleading results, particularly if the missing components are the most influential and important in the system under investigation. In networks, the importance of a node depends on the number of nodes connected to this node. The degree of centrality is the most commonly used measure to identify important nodes in networks. There are two kinds of degree centrality, which are in-degree and out-degree. This manuscrpt is concerned with finding the highest out-degree among nodes to identify the most influential nodes. Inferring the existence of unobserved important components is critical in many multivariate interacting systems. The implications of such a situation are discussed in the Granger-causality framework. To this end, two of the most recent Granger-causality techniques, renormalized partial directed coherence and directed partial correlation, were employed. They were then compared in terms of their performance according to the extent to which they can infer the existence of unobserved important components. Sub-network analysis was conducted to aid these two techniques in inferring the existence of unobserved important components, which is evidenced in the results. By comparing the results of the two conducted techniques, it can be asserted that renormalized partial coherence outperforms directed partial correlation in the inference of existing unobserved important components that have not been included in the analysis. This measure of Granger causality and sub-network analysis emphasizes their ubiquitous successful applicability in such cases of the existence of hidden unobserved important components.

Bidirectional Connectivity Between Broca's Area and Wernicke's Area During Interactive Verbal Communication.

Aim: This investigation aims to advance the understanding of neural dynamics that underlies live and natural interactions during spoken dialogue between two individuals. Introduction: The underlying hypothesis is that functional connectivity between canonical speech areas in the human brain will be modulated by social interaction. Methods: Granger causality was applied to compare directional connectivity across Broca's and Wernicke's areas during verbal conditions consisting of interactive and noninteractive communication. Thirty-three pairs of healthy adult participants alternately talked and listened to each other while performing an object naming and description task that was either interactive or not during hyperscanning with functional near-infrared spectroscopy (fNIRS). In the noninteractive condition, the speaker named and described a picture-object without reference to the partner's description. In the interactive condition, the speaker performed the same task but included an interactive response about the preceding comments of the partner. Causality measures of hemodynamic responses from Broca's and Wernicke's areas were compared between real, surrogate, and shuffled trials within dyads. Results: The interactive communication was characterized by bidirectional connectivity between Wernicke's and Broca's areas of the listener's brain. Whereas this connectivity was unidirectional in the speaker's brain. In the case of the noninteractive condition, both speaker's and listener's brains showed unidirectional top-down (Broca's area to Wernicke's area) connectivity. Conclusion: Together, directional connectivity as determined by Granger analysis reveals bidirectional flow of neuronal information during dynamic two-person verbal interaction for processes that are active during listening (reception) and not during talking (production). Findings are consistent with prior contrast findings (general linear model) showing neural modulation of the receptive language system associated with Wernicke's area during a two-person live interaction. Impact statement The neural dynamics that underlies real-life social interactions is an emergent topic of interest. Dynamically coupled cross-brain neural mechanisms between interacting partners during verbal dialogue have been shown within Wernicke's area. However, it is not known how within-brain long-range neural mechanisms operate during these live social functions. Using Granger causality analysis, we show bidirectional neural activity between Broca's and Wernicke's areas during interactive dialogue compared with a noninteractive control task showing only unidirectional activity. Findings are consistent with an Interactive Brain Model where long-range neural mechanisms process interactive processes associated with rapid and spontaneous spoken social cues.

Disrupted communication of the temporoparietal junction in patients with major depressive disorder.

Patients with major depressive disorder (MDD) suffer impairment in the transmission and integration of internal and external information sources. Accumulating evidence suggests that the temporoparietal junction (TPJ) is important for multiple cognitive and social functions and may act as a key node for the integration of internal and external information. Therefore, the TPJ's aberrant interaction mechanism may underpin MDD psychopathology. To answer this question, we conducted a comprehensive study using resting-state functional magnetic imaging data recorded from 74 patients with MDD and 69 normal controls. First, we examined whether TPJ was the most prominent region with altered functional/effective connectivity with multiple depression-related regions/networks, based on either zero-lag correlations or temporal mutual information (total interdependence and Granger causality) measurements. Accordingly, we derived a network model that depicts alterations of TPJ-connectivity in patients with MDD. Lastly, we performed a cross-approach comparison demonstrating more conducive indicators in delineating the network alteration model. Functional/effective connectivity between the TPJ and major functional networks that govern internal and external-driven information resources was attenuated in patients with MDD. TPJ acts like a key node for information-inflow and integration of multiple information streams. Therefore, dysfunctional connectivity indicators may serve as effective biomarkers for MDD. MDD is associated with the breakdown of the TPJ interaction model and its connections with the default mode network and the task-positive network.

Depression Diagnosis Modeling With Advanced Computational Methods: Frequency-Domain eMVAR and Deep Learning.

Electroencephalogram (EEG)-based automated depression diagnosis systems have been suggested for early and accurate detection of mood disorders. EEG signals are highly irregular, nonlinear, and nonstationary in nature and are traditionally studied from a linear viewpoint by means of statistical and frequency features. Since, linear metrics present certain limitations and nonlinear methods have proven to be an efficient tool in understanding the complexities of the brain in the identification of underlying behavior of biological signals, such as electrocardiogram, EEG and magnetoencephalogram and thus, can be applied to all nonstationary signals. Various nonlinear algorithms can be used in the analysis of EEG signals. In this research paper, we aim to develop a novel methodology for EEG-based depression diagnosis utilizing 2 advanced computational techniques: frequency-domain extended multivariate autoregressive (eMVAR) and deep learning (DL). We proposed a hybrid method comprising a pretrained ResNet-50 and long-short term memory (LSTM) to capture depression-specific information and compared with a strong conventional machine learning (ML) framework having eMVAR connectivity features. The following 8 causality measures, which interpret the interaction mechanisms among spectrally decomposed oscillations, were used to extract features from multivariate EEG time series: directed coherence (DC), directed transfer function (DTF), partial DC (PDC), generalized PDC (gPDC), extended DC (eDC), delayed DC (dDC), extended PDC (ePDC), and delayed PDC (dPDC). The classification accuracies were 84% with DC, 85% with DTF, 95.3% with PDC, 95.1% with gPDC, 84.8% with eDC, 84.6% with dDC, 84.2% with ePDC, and 95.9% with dPDC for the eMVAR framework. Through a DL framework (ResNet-50 + LSTM), the classification accuracy was achieved as 90.22%. The results demonstrate that our DL methodology is a competitive alternative to the strong feature extraction-based ML methods in depression classification.

Quantification of the Direct Solar Impact on Some Components of the Hydro-Climatic System.

This study addresses the causal links between external factors and the main hydro-climatic variables by using a chain of methods to unravel the complexity of the direct sun–climate link. There is a gap in the literature on the description of a complete chain in addressing the structures of direct causal links of solar activity on terrestrial variables. This is why the present study uses the extensive facilities of the application of information theory in view of recent advances in different fields. Additionally, by other methods (e.g., neural networks) we first tested the existent non-linear links of solar–terrestrial influences on the hydro-climate system. The results related to the solar impact on terrestrial phenomena are promising, which is discriminant in the space-time domain. The implications prove robust for determining the causal measure of climate variables under direct solar impact, which makes it easier to consider solar activity in climate models by appropriate parametrizations. This study found that hydro-climatic variables are sensitive to solar impact only for certain frequencies (periods) and have a coherence with the Solar Flux only for some lags of the Solar Flux (in advance).

Sepsis-Pathophysiology and Therapeutic Concepts.

Sepsis is a life-threatening condition and a global disease burden. Today, the heterogeneous syndrome is defined as severe organ dysfunction caused by a dysregulated host response to infection, with renewed emphasis on immune pathophysiology. Despite all efforts of experimental and clinical research during the last three decades, the ability to positively influence course and outcome of the syndrome remains limited. Evidence-based therapy still consists of basic causal and supportive measures, while adjuvant interventions such as blood purification or targeted immunotherapy largely remain without proof of effectiveness so far. With this review, we aim to provide an overview of sepsis immune pathophysiology, to update the choice of therapeutic approaches targeting different immunological mechanisms in the course of sepsis and septic shock, and to call for a paradigm shift from the pathogen to the host response as a potentially more promising angle.

Comparison of frequency domain measures based on spectral decomposition for spontaneous baroreflex sensitivity assessment after Acute Myocardial Infarction

The objective of this study is to present a new method to assess in the frequency domain the directed interactions between the spontaneous variability of systolic arterial pressure (SAP) and heart period (HP) from their linear model representation, and to apply it for studying the baroreflex control of arterial pressure in healthy physiological states and after acute myocardial infarction (AMI). The method is based on pole decomposition of the model transfer function and on the following evaluation of causal measures of coupling and gain from the poles associated to low frequency (0.04−0.15 Hz) oscillatory components. It is compared with traditional non-causal approaches for the spectral analysis of the baroreflex gain, and with causal approaches based on the directed coherence, in a group of AMI patients and in Young and Old healthy controls studied at rest and during head-up tilt. Analysis of feedforward interactions from RR to SAP is also performed. Our results support the importance of using local causal approaches to quantify separately baroreflex and feedforward interactions between RR and SAP, allowing both to confirm known physiological behaviors (e.g., weaker baroreflex effectiveness and lower sensitivity after AMI) and to suggest novel findings (e.g., preserved low frequency baroreflex response to postural stress after AMI). We also find that the postural stress alters feedforward interactions selectively across groups, being related to decreased coupling only in Young and to increased gain mostly in AMI. These results have a significance for the clinical assessment of the baroreflex and the physiological evaluation of cardiovascular interactions.

Local and distant responses to single pulse electrical stimulation reflect different forms of connectivity

Measuring connectivity in the human brain involves innumerable approaches using both noninvasive (fMRI, EEG) and invasive (intracranial EEG or iEEG) recording modalities, including the use of external probing stimuli, such as direct electrical stimulation. To examine how different measures of connectivity correlate with one another, we compared ‘passive’ measures of connectivity during resting state conditions to the more ‘active’ probing measures of connectivity with single pulse electrical stimulation (SPES). We measured the network engagement and spread of the cortico-cortico evoked potential (CCEP) induced by SPES at 53 out of 104 total sites across the brain, including cortical and subcortical regions, in patients with intractable epilepsy (N=11) who were undergoing intracranial recordings as a part of their clinical care for identifying seizure onset zones. We compared the CCEP network to functional, effective, and structural measures of connectivity during a resting state in each patient. Functional and effective connectivity measures included correlation or Granger causality measures applied to stereoEEG (sEEGs) recordings. Structural connectivity was derived from diffusion tensor imaging (DTI) acquired before intracranial electrode implant and monitoring (N=8). The CCEP network was most similar to the resting state voltage correlation network in channels near to the stimulation location. In contrast, the distant CCEP network was most similar to the DTI network. Other connectivity measures were not as similar to the CCEP network. These results demonstrate that different connectivity measures, including those derived from active stimulation-based probing, measure different, complementary aspects of regional interrelationships in the brain.

Dynamic task-linked switching between brain networks – A tri-network perspective

The highly influential tri-network model proposed by Menon integrates 3 key intrinsic brain networks — the central executive network (CEN), the salience network (SN), and the default mode network (DMN), into a single cohesive model underlying normal behaviour and cognition. A large body of evidence suggests that abnormal intra- and inter- network connectivity between these three networks underlies the various behavioural and cognitive dysfunctions observed in patients with neuropsychiatric conditions such as PTSD and depression. An important prediction of the tri-network model is that the DMN and CEN networks are anti-correlated under the control of the SN, such that if a task engages one of the two, the SN inhibits the activation of the other. To date most of the evidence surrounding the functions of these three core networks comes from either resting state analyses or in the context of a single task with respect to rest. Few studies have investigated multiple tasks simultaneously or characterized the dynamics of task switching. Hence, a careful investigation of the temporal dynamics of network activity during task switching is warranted. To accomplish this we collected fMRI data from 14 participants that dynamically switched between a 2-back working memory task and an autobiographical memory retrieval task, designed to activate the CEN, DMN and the SN. The fMRI data were used to 1. identify nodes and sub-networks within the three major networks involved in task-linked dynamic network switching, 2. characterize the temporal pattern of activation of these nodes and sub-networks, and finally 3. investigate the causal influence that these nodes and sub-networks exerted on each other. Using a combination of multivariate neuroimaging analyses, timecourse analyses and multivariate Granger causality measures to study the tri-network dynamics, the current study found that the SN co-activates with the task-relevant network, providing a mechanistic insight into SN-mediated network selection in the context of explicit tasks. Our findings also indicate active involvement of the posterior insula and some medial temporal nodes in task-linked functions of the SN and DMN, warranting their inclusion as network nodes in future studies of the tri-network model. These results add to the growing body of evidence showing the complex interplay of CEN, DMN and SN nodes and sub-networks required for adequate task-switching, characterizing a normative pattern of task-linked network dynamics within the context of Menon’s tri-network model.

Causality and Information Transfer Between the Solar Wind and the Magnetosphere–Ionosphere System

An information-theoretic approach for detecting causality and information transfer is used to identify interactions of solar activity and interplanetary medium conditions with the Earth’s magnetosphere–ionosphere systems. A causal information transfer from the solar wind parameters to geomagnetic indices is detected. The vertical component of the interplanetary magnetic field () influences the auroral electrojet (AE) index with an information transfer delay of 10 min and the geomagnetic disturbances at mid-latitudes measured by the symmetric field in the H component (SYM-H) index with a delay of about 30 min. Using a properly conditioned causality measure, no causal link between AE and SYM-H, or between magnetospheric substorms and magnetic storms can be detected. The observed causal relations can be described as linear time-delayed information transfer.

Time-Reversibility, Causality and Compression-Complexity.

Detection of the temporal reversibility of a given process is an interesting time series analysis scheme that enables the useful characterisation of processes and offers an insight into the underlying processes generating the time series. Reversibility detection measures have been widely employed in the study of ecological, epidemiological and physiological time series. Further, the time reversal of given data provides a promising tool for analysis of causality measures as well as studying the causal properties of processes. In this work, the recently proposed Compression-Complexity Causality (CCC) measure (by the authors) is shown to be free of the assumption that the "cause precedes the effect", making it a promising tool for causal analysis of reversible processes. CCC is a data-driven interventional measure of causality (second rung on the Ladder of Causation) that is based on Effort-to-Compress (ETC), a well-established robust method to characterize the complexity of time series for analysis and classification. For the detection of the temporal reversibility of processes, we propose a novel measure called the Compressive Potential based Asymmetry Measure. This asymmetry measure compares the probability of the occurrence of patterns at different scales between the forward-time and time-reversed process using ETC. We test the performance of the measure on a number of simulated processes and demonstrate its effectiveness in determining the asymmetry of real-world time series of sunspot numbers, digits of the transcedental number and heart interbeat interval variability.