Quantification of causal couplings via dynamical effects: a unifying perspective

Information transfers and flows in Markov chains as dynamical causal effects

A logical sequence of information-theoretic quantifiers of directional (causal) couplings in Markov chains is generated within the framework of dynamical causal effects (DCEs), starting from the simplest DCEs (in terms of localization of their functional elements) and proceeding step-by-step to more complex ones. Thereby, a system of 11 quantifiers is readily obtained, some of them coinciding with previously known causality measures widely used in time series analysis and often called "information transfers" or "flows" (transfer entropy, Ay-Polani information flow, Liang-Kleeman information flow, information response, etc.,) By construction, this step-by-step generation reveals logical relationships between all these quantifiers as specific DCEs. As a further concretization, diverse quantitative relationships between the transfer entropy and the Liang-Kleeman information flow are found both rigorously and numerically for coupled two-state Markov chains.

Contributions to surface air temperature trends estimated from climate time series: Medium-term causalities

Contributions of various natural and anthropogenic factors to trends of surface air temperatures at different latitudes of the Northern and Southern hemispheres on various temporal horizons are estimated from climate data since the 19th century in empirical autoregressive models. Along with anthropogenic forcing, we assess the impact of several natural climate modes including Atlantic Multidecadal Oscillation, El-Nino/Southern Oscillation, Interdecadal Pacific Oscillation, Pacific Decadal Oscillation, and Antarctic Oscillation. On relatively short intervals of the length of two or three decades, contributions of climate variability modes are considerable and comparable to the contributions of greenhouse gases and even exceed the latter. On longer intervals of about half a century and greater, the contributions of greenhouse gases dominate at all latitudinal belts including polar, middle, and tropical ones.

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.

Generative formalism of causality quantifiers for processes

The concept of dynamical causal effect (DCE) is generalized and equipped with a formalism which allows one to formulate in a unified manner and interrelate a variety of causality quantifiers used in time series analysis. An elementary DCE from a subsystem Y to a subsystem X is defined within the stochastic dynamical systems framework as a response of a future X state to an appropriate variation of an initial (X,Y)-state distribution or a certain parameter of Y or of the coupling element Y→X; this response is quantified in a probabilistic sense via a certain distinction functional; elementary DCEs are assembled over a set of initial variations via an assemblage functional. To include all those aspects, a "triple brackets formula" for the general DCE is suggested and serves as a first principle to produce specific causality quantifiers as realizations of the general DCE. As an application, transfer entropy and Liang-Kleeman information flow are related surprisingly as opposite limit cases in a family of DCEs; it is shown that their "nats per time unit" may differ drastically. The suggested DCE viewpoint links any formal causality quantifier to "intervention-effect" experiments, i.e., future responses to initial variations, and so provides its dynamical interpretation, opening a way to its further physical interpretations in studies of physical systems.

Inferring directionality of coupled dynamical systems using Gaussian process priors: Application on neurovascular systems

Dynamical system theory has recently shown promise for uncovering causality and directionality in complex systems, particularly using the method of convergent cross mapping (CCM). In spite of its success in the literature, the presence of process noise raises concern about CCM's ability to uncover coupling direction. Furthermore, CCM's capacity to detect indirect causal links may be challenged in simulated unidrectionally coupled Rossler-Lorenz systems. To overcome these limitations, we propose a method that places a Gaussian process prior on a cross mapping function (named GP-CCM) to impose constraints on local state space neighborhood comparisons. Bayesian posterior likelihood and evidence ratio tests, as well as surrogate data analyses are performed to obtain a robust statistic for dynamical coupling directionality. We demonstrate GP-CCM's performance with respect to CCM in synthetic data simulation as well as in empirical electroencephelography (EEG) and functional near infrared spectroscopy (fNIRS) activity data. Our findings show that GP-CCM provides a statistic that consistently reports indirect causal structures in non-separable unidirectional system interactions; GP-CCM also provides coupling direction estimates in noisy physiological signals, showing that EEG likely causes, i.e., drives, fNIRS dynamics.

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.

Noise robust approach to reconstruction of van der Pol-like oscillators and its application to Granger causality

Van der Pol oscillators and their generalizations are known to be a fundamental model in the theory of oscillations and their applications. Many objects of a different nature can be described using van der Pol-like equations under some circumstances; therefore, methods of reconstruction of such equations from experimental data can be of significant importance for tasks of model verification, indirect parameter estimation, coupling analysis, system classification, etc. The previously reported techniques were not applicable to time series with large measurement noise, which is usual in biological, climatological, and many other experiments. Here, we present a new approach based on the use of numerical integration instead of the differentiation and implicit approximation of a nonlinear dissipation function. We show that this new technique can work for noise levels up to 30% by standard deviation from the signal for different types of autonomous van der Pol-like systems and for ensembles of such systems, providing a new approach to the realization of the Granger-causality idea.

Phase-dynamic causalities within dynamical effects framework

This work investigates numerics of several widely known phase-dynamic quantifiers of directional (causal) couplings between oscillatory systems: transfer entropy (TE), differential quantifier, and squared-coefficients quantifier based on an evolution map. The study is performed on the system of two stochastic Kuramoto oscillators within the framework of dynamical causal effects. The quantifiers are related to each other and to an asymptotic effect of the coupling on phase diffusion. Several novel findings are listed as follows: (i) for a non-synchronous regime and high enough noise levels, the TE rate multiplied by a certain characteristic time (called here reduced TE) equals twice an asymptotic effect of a directional coupling on phase diffusion; (ii) "information flow" expressed by the TE rate unboundedly rises with the coupling coefficient even in the domain of effective synchronization; (iii) in any effective synchronization regime, the reduced TE is equal to 1/8 n.u. in each direction for equal coupling coefficients and equal noise intensities, and it is in general a simple function of the ratio of noise intensities and the ratio of coupling coefficients.

Transfer entropies within dynamical effects framework

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

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

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

Models of communication and control for brain networks: distinctions, convergence, and future outlook

Recent advances in computational models of signal propagation and routing in the human brain have underscored the critical role of white-matter structure. A complementary approach has utilized the framework of network control theory to better understand how white matter constrains the manner in which a region or set of regions can direct or control the activity of other regions. Despite the potential for both of these approaches to enhance our understanding of the role of network structure in brain function, little work has sought to understand the relations between them. Here, we seek to explicitly bridge computational models of communication and principles of network control in a conceptual review of the current literature. By drawing comparisons between communication and control models in terms of the level of abstraction, the dynamical complexity, the dependence on network attributes, and the interplay of multiple spatiotemporal scales, we highlight the convergence of and distinctions between the two frameworks. Based on the understanding of the intertwined nature of communication and control in human brain networks, this work provides an integrative perspective for the field and outlines exciting directions for future work.

Detecting and quantifying causal associations in large nonlinear time series datasets

Identifying causal relationships and quantifying their strength from observational time series data are key problems in disciplines dealing with complex dynamical systems such as the Earth system or the human body. Data-driven causal inference in such systems is challenging since datasets are often high dimensional and nonlinear with limited sample sizes. Here, we introduce a novel method that flexibly combines linear or nonlinear conditional independence tests with a causal discovery algorithm to estimate causal networks from large-scale time series datasets. We validate the method on time series of well-understood physical mechanisms in the climate system and the human heart and using large-scale synthetic datasets mimicking the typical properties of real-world data. The experiments demonstrate that our method outperforms state-of-the-art techniques in detection power, which opens up entirely new possibilities to discover and quantify causal networks from time series across a range of research fields.

Transient and equilibrium causal effects in coupled oscillators

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

Distinguishing between direct and indirect directional couplings in large oscillator networks: Partial or non-partial phase analyses?

We investigate the relative merit of phase-based methods for inferring directional couplings in complex networks of weakly interacting dynamical systems from multivariate time-series data. We compare the evolution map approach and its partialized extension to each other with respect to their ability to correctly infer the network topology in the presence of indirect directional couplings for various simulated experimental situations using coupled model systems. In addition, we investigate whether the partialized approach allows for additional or complementary indications of directional interactions in evolving epileptic brain networks using intracranial electroencephalographic recordings from an epilepsy patient. For such networks, both direct and indirect directional couplings can be expected, given the brain's connection structure and effects that may arise from limitations inherent to the recording technique. Our findings indicate that particularly in larger networks (number of nodes ≫10), the partialized approach does not provide information about directional couplings extending the information gained with the evolution map approach.

Quantifying information transfer and mediation along causal pathways in complex systems

Measures of information transfer have become a popular approach to analyze interactions in complex systems such as the Earth or the human brain from measured time series. Recent work has focused on causal definitions of information transfer aimed at decompositions of predictive information about a target variable, while excluding effects of common drivers and indirect influences. While common drivers clearly constitute a spurious causality, the aim of the present article is to develop measures quantifying different notions of the strength of information transfer along indirect causal paths, based on first reconstructing the multivariate causal network. Another class of novel measures quantifies to what extent different intermediate processes on causal paths contribute to an interaction mechanism to determine pathways of causal information transfer. The proposed framework complements predictive decomposition schemes by focusing more on the interaction mechanism between multiple processes. A rigorous mathematical framework allows for a clear information-theoretic interpretation that can also be related to the underlying dynamics as proven for certain classes of processes. Generally, however, estimates of information transfer remain hard to interpret for nonlinearly intertwined complex systems. But if experiments or mathematical models are not available, then measuring pathways of information transfer within the causal dependency structure allows at least for an abstraction of the dynamics. The measures are illustrated on a climatological example to disentangle pathways of atmospheric flow over Europe.

Relating Granger causality to long-term causal effects

In estimation of causal couplings between observed processes, it is important to characterize coupling roles at various time scales. The widely used Granger causality reflects short-term effects: it shows how strongly perturbations of a current state of one process affect near future states of another process, and it quantifies that via prediction improvement (PI) in autoregressive models. However, it is often more important to evaluate the effects of coupling on long-term statistics, e.g., to find out how strongly the presence of coupling changes the variance of a driven process as compared to an uncoupled case. No general relationships between Granger causality and such long-term effects are known. Here, we pose the problem of relating these two types of coupling characteristics, and we solve it for a class of stochastic systems. Namely, for overdamped linear oscillators, we rigorously derive that the above long-term effect is proportional to the short-term effects, with the proportionality coefficient depending on the prediction interval and relaxation times. We reveal that this coefficient is typically considerably greater than unity so that small normalized PI values may well correspond to quite large long-term effects of coupling. The applicability of the derived relationship to wider classes of systems, its limitations, and its value for further research are discussed. To give a real-world example, we analyze couplings between large-scale climatic processes related to sea surface temperature variations in equatorial Pacific and North Atlantic regions.

Information thermodynamics of near-equilibrium computation

In studying fundamental physical limits and properties of computational processes, one is faced with the challenges of interpreting primitive information-processing functions through well-defined information-theoretic as well as thermodynamic quantities. In particular, transfer entropy, characterizing the function of computational transmission and its predictability, is known to peak near critical regimes. We focus on a thermodynamic interpretation of transfer entropy aiming to explain the underlying critical behavior by associating information flows intrinsic to computational transmission with particular physical fluxes. Specifically, in isothermal systems near thermodynamic equilibrium, the gradient of the average transfer entropy is shown to be dynamically related to Fisher information and the curvature of system's entropy. This relationship explicitly connects the predictability, sensitivity, and uncertainty of computational processes intrinsic to complex systems and allows us to consider thermodynamic interpretations of several important extreme cases and trade-offs.