Detectability of Granger causality for subsampled continuous-time neurophysiological processes

Sex-dependent nonlinear Granger connectivity patterns of brain aging in healthy population

BACKGROUND

Brain aging is a complex process that involves functional alterations in multiple subnetworks and brain regions. However, most previous studies investigating aging-related functional connectivity (FC) changes using resting-state functional magnetic resonance images (rs-fMRIs) have primarily focused on the linear correlation between brain subnetworks, ignoring the nonlinear casual properties of fMRI signals.

METHODS

We introduced the neural Granger causality technique to investigate the sex-dependent nonlinear Granger connectivity (NGC) during aging on a publicly available dataset of 227 healthy participants acquired cross-sectionally in Leipzig, Germany.

RESULTS

Our findings indicate that brain aging may cause widespread declines in NGC at both regional and subnetwork scales. These findings exhibit high reproducibility across different network sparsities, demonstrating the efficacy of static and dynamic analysis strategies. Females exhibit greater heterogeneity and reduced stability in NGC compared to males during aging, especially the NGC between the visual network and other subnetworks. Besides, NGC strengths can well reflect the individual cognitive function, which may therefore work as a sensitive metric in cognition-related experiments for individual-scale or group-scale mechanism understanding.

CONCLUSION

These findings indicate that NGC analysis is a potent tool for identifying sex-dependent brain aging patterns. Our results offer valuable perspectives that could substantially enhance the understanding of sex differences in neurological diseases in the future, especially in degenerative disorders.

Dynamical independence: Discovering emergent macroscopic processes in complex dynamical systems

We introduce a notion of emergence for macroscopic variables associated with highly multivariate microscopic dynamical processes. Dynamical independence instantiates the intuition of an emergent macroscopic process as one possessing the characteristics of a dynamical system "in its own right," with its own dynamical laws distinct from those of the underlying microscopic dynamics. We quantify (departure from) dynamical independence by a transformation-invariant Shannon information-based measure of dynamical dependence. We emphasize the data-driven discovery of dynamically independent macroscopic variables, and introduce the idea of a multiscale "emergence portrait" for complex systems. We show how dynamical dependence may be computed explicitly for linear systems in both time and frequency domains, facilitating discovery of emergent phenomena across spatiotemporal scales, and outline application of the linear operationalization to inference of emergence portraits for neural systems from neurophysiological time-series data. We discuss dynamical independence for discrete- and continuous-time deterministic dynamics, with potential application to Hamiltonian mechanics and classical complex systems such as flocking and cellular automata.

Granger causality analysis for calcium transients in neuronal networks, challenges and improvements

One challenge in neuroscience is to understand how information flows between neurons in vivo to trigger specific behaviors. Granger causality (GC) has been proposed as a simple and effective measure for identifying dynamical interactions. At single-cell resolution however, GC analysis is rarely used compared to directionless correlation analysis. Here, we study the applicability of GC analysis for calcium imaging data in diverse contexts. We first show that despite underlying linearity assumptions, GC analysis successfully retrieves non-linear interactions in a synthetic network simulating intracellular calcium fluctuations of spiking neurons. We highlight the potential pitfalls of applying GC analysis on real in vivo calcium signals, and offer solutions regarding the choice of GC analysis parameters. We took advantage of calcium imaging datasets from motoneurons in embryonic zebrafish to show how the improved GC can retrieve true underlying information flow. Applied to the network of brainstem neurons of larval zebrafish, our pipeline reveals strong driver neurons in the locus of the mesencephalic locomotor region (MLR), driving target neurons matching expectations from anatomical and physiological studies. Altogether, this practical toolbox can be applied on in vivo population calcium signals to increase the selectivity of GC to infer flow of information across neurons.

Altered nonlinear Granger causality interactions in the large-scale brain networks of patients with schizophrenia

Objective.It has been demonstrated that schizophrenia (SZ) is characterized by functional dysconnectivity involving extensive brain networks. However, the majority of previous studies utilizing resting-state functional magnetic resonance imaging (fMRI) to infer abnormal functional connectivity (FC) in patients with SZ have focused on the linear correlation that one brain region may influence another, ignoring the inherently nonlinear properties of fMRI signals.Approach. In this paper, we present a neural Granger causality (NGC) technique for examining the changes in SZ's nonlinear causal couplings. We develop static and dynamic NGC-based analyses of large-scale brain networks at several network levels, estimating complicated temporal and causal relationships in SZ patients.Main results. We find that the NGC-based FC matrices can detect large and significant differences between the SZ and healthy control groups at both the regional and subnetwork scales. These differences are persistent and significantly overlapped at various network sparsities regardless of whether the brain networks were built using static or dynamic techniques. In addition, compared to controls, patients with SZ exhibited extensive NGC confusion patterns throughout the entire brain.Significance. These findings imply that the NGC-based FCs may be a useful method for quantifying the abnormalities in the causal influences of patients with SZ, hence shedding fresh light on the pathophysiology of this disorder.

The Consensus Problem in Polities of Agents with Dissimilar Cognitive Architectures

Agents interacting with their environments, machine or otherwise, arrive at decisions based on their incomplete access to data and their particular cognitive architecture, including data sampling frequency and memory storage limitations. In particular, the same data streams, sampled and stored differently, may cause agents to arrive at different conclusions and to take different actions. This phenomenon has a drastic impact on polities-populations of agents predicated on the sharing of information. We show that, even under ideal conditions, polities consisting of epistemic agents with heterogeneous cognitive architectures might not achieve consensus concerning what conclusions to draw from datastreams. Transfer entropy applied to a toy model of a polity is analyzed to showcase this effect when the dynamics of the environment is known. As an illustration where the dynamics is not known, we examine empirical data streams relevant to climate and show the consensus problem manifest.

Consistent Changes in Cortico-Subthalamic Directed Connectivity Are Associated With the Induction of Parkinsonism in a Chronically Recorded Non-human Primate Model

Parkinson’s disease is a neurological disease with cardinal motor signs including bradykinesia and tremor. Although beta-band hypersynchrony in the cortico-basal ganglia network is thought to contribute to disease manifestation, the resulting effects on network connectivity are unclear. We examined local field potentials from a non-human primate across the naïve, mild, and moderate disease states (model was asymmetric, left-hemispheric dominant) and probed power spectral density as well as cortico-cortical and cortico-subthalamic connectivity using both coherence and Granger causality, which measure undirected and directed effective connectivity, respectively. Our network included the left subthalamic nucleus (L-STN), bilateral primary motor cortices (L-M1, R-M1), and bilateral premotor cortices (L-PMC, R-PMC). Results showed two distinct peaks (Peak A at 5–20 Hz, Peak B at 25–45 Hz) across all analyses. Power and coherence analyses showed widespread increases in power and connectivity in both the Peak A and Peak B bands with disease progression. For Granger causality, increases in Peak B connectivity and decreases in Peak A connectivity were associated with the disease. Induction of mild disease was associated with several changes in connectivity: (1) the cortico-subthalamic connectivity in the descending direction (L-PMC to L-STN) decreased in the Peak A range while the reciprocal, ascending connectivity (L-STN to L-PMC) increased in the Peak B range; this may play a role in generating beta-band hypersynchrony in the cortex, (2) both L-M1 to L-PMC and R-M1 to R-PMC causalities increased, which may either be compensatory or a pathologic effect of disease, and (3) a decrease in connectivity occurred from the R-PMC to R-M1. The only significant change seen between mild and moderate disease was increased right cortical connectivity, which may reflect compensation for the left-hemispheric dominant moderate disease state.

Statistical Perspective on Functional and Causal Neural Connectomics: A Comparative Study

Representation of brain network interactions is fundamental to the translation of neural structure to brain function. As such, methodologies for mapping neural interactions into structural models, i.e., inference of functional connectome from neural recordings, are key for the study of brain networks. While multiple approaches have been proposed for functional connectomics based on statistical associations between neural activity, association does not necessarily incorporate causation. Additional approaches have been proposed to incorporate aspects of causality to turn functional connectomes into causal functional connectomes, however, these methodologies typically focus on specific aspects of causality. This warrants a systematic statistical framework for causal functional connectomics that defines the foundations of common aspects of causality. Such a framework can assist in contrasting existing approaches and to guide development of further causal methodologies. In this work, we develop such a statistical guide. In particular, we consolidate the notions of associations and representations of neural interaction, i.e., types of neural connectomics, and then describe causal modeling in the statistics literature. We particularly focus on the introduction of directed Markov graphical models as a framework through which we define the Directed Markov Property—an essential criterion for examining the causality of proposed functional connectomes. We demonstrate how based on these notions, a comparative study of several existing approaches for finding causal functional connectivity from neural activity can be conducted. We proceed by providing an outlook ahead regarding the additional properties that future approaches could include to thoroughly address causality.

Detection of Cause-Effect Relations Based on Information Granulation and Transfer Entropy

Causality inference is a process to infer Cause-Effect relations between variables in, typically, complex systems, and it is commonly used for root cause analysis in large-scale process industries. Transfer entropy (TE), as a non-parametric causality inference method, is an effective method to detect Cause-Effect relations in both linear and nonlinear processes. However, a major drawback of transfer entropy lies in the high computational complexity, which hinders its real application, especially in systems that have high requirements for real-time estimation. Motivated by such a problem, this study proposes an improved method for causality inference based on transfer entropy and information granulation. The calculation of transfer entropy is improved with a new framework that integrates the information granulation as a critical preceding step; moreover, a window-length determination method is proposed based on delay estimation, so as to conduct appropriate data compression using information granulation. The effectiveness of the proposed method is demonstrated by both a numerical example and an industrial case, with a two-tank simulation model. As shown by the results, the proposed method can reduce the computational complexity significantly while holding a strong capability for accurate casuality detection.

Inferring network properties from time series using transfer entropy and mutual information: Validation of multivariate versus bivariate approaches

Functional and effective networks inferred from time series are at the core of network neuroscience. Interpreting properties of these networks requires inferred network models to reflect key underlying structural features. However, even a few spurious links can severely distort network measures, posing a challenge for functional connectomes. We study the extent to which micro- and macroscopic properties of underlying networks can be inferred by algorithms based on mutual information and bivariate/multivariate transfer entropy. The validation is performed on two macaque connectomes and on synthetic networks with various topologies (regular lattice, small-world, random, scale-free, modular). Simulations are based on a neural mass model and on autoregressive dynamics (employing Gaussian estimators for direct comparison to functional connectivity and Granger causality). We find that multivariate transfer entropy captures key properties of all network structures for longer time series. Bivariate methods can achieve higher recall (sensitivity) for shorter time series but are unable to control false positives (lower specificity) as available data increases. This leads to overestimated clustering, small-world, and rich-club coefficients, underestimated shortest path lengths and hub centrality, and fattened degree distribution tails. Caution should therefore be used when interpreting network properties of functional connectomes obtained via correlation or pairwise statistical dependence measures, rather than more holistic (yet data-hungry) multivariate models.

We compare bivariate and multivariate methods for inferring networks from time series, which are generated using a neural mass model and autoregressive dynamics. We assess their ability to reproduce key properties of the underlying structural network. Validation is performed on two macaque connectomes and on synthetic networks with various topologies (regular lattice, small-world, random, scale-free, modular). Even a few spurious links can severely bias key network properties. Multivariate transfer entropy performs best on all topologies for longer time series.

Granger Causality on forward and Reversed Time Series

In this study, the information flow time arrow is investigated for stochastic data defined by vector autoregressive models. The time series are analyzed forward and backward by different Granger causality detection methods. Besides the normal distribution, which is usually required for the validity of Granger causality analysis, several other distributions of predictive errors are considered. A clear effect of a change in the order of cause and effect on the time-reversed series of unidirectionally connected variables was detected with standard Granger causality test (GC), when the product of the connection strength and the ratio of the predictive errors of the driver and the recipient was below a certain level, otherwise bidirectional causal connection was detected. On the other hand, opposite causal link was detected unconditionally by the methods based on the time reversal testing, but they were not able to detect correct bidirectional connection. The usefulness of the backward analysis is manifested in cases where falsely detected unidirectional connections can be rejected by applying the result obtained after the time reversal, and in cases of uncorrelated causally independent variables, where the absence of a causal link detected by GC on the original series should be confirmed on the time-reversed series.

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 boundaries of state-space Granger causality analysis applied to BOLD simulated data: A comparative modelling and simulation approach

BACKGROUND

The analysis of connectivity has become a fundamental tool in human neuroscience. Granger Causality Mapping is a data-driven method that uses Granger Causality (GC) to assess the existence and direction of influence between signals, based on temporal precedence of information. More recently, a theory of Granger causality has been developed for state-space (SS-GC) processes, but little is known about its statistical validation and application on functional magnetic resonance imaging (fMRI) data.

NEW METHOD

We explored different multivariate computational frameworks to define the optimal combination for GC estimation. We hypothesized a new heuristic, combining SS-GC with a distinct statistical validation technique, Time Reversed Testing, validating it on synthetic data. We test its performance with a number of experimental parameters, including block structure, sampling frequency, noise and system mean pairwise correlation, using a statistical framework of binary classification.

RESULTS

We found that SS-GC with time reversed testing outperforms other frameworks. The results validate the application of SS-GC to generative models. When estimating reliable causal relations, SS-GC returns promising results, especially when considering synthetic data with a high impact of noise and sampling rate.

CONCLUSIONS

In this study, we empirically explored the boundaries of SS-GC with time reversed testing, a data-driven causality analysis framework with potential applicability to fMRI data.

Decoding Task-Specific Cognitive States with Slow, Directed Functional Networks in the Human Brain

Flexible functional interactions among brain regions mediate critical cognitive functions. Such interactions can be measured using functional magnetic resonance imaging (fMRI) data either with instantaneous (zero-lag) or lag-based (time-lagged) functional connectivity. Because the fMRI hemodynamic response is slow, and is sampled at a timescale (seconds) several orders of magnitude slower than the underlying neural dynamics (milliseconds), simulation studies have shown that lag-based fMRI functional connectivity, measured with approaches like Granger–Geweke causality (GC), provides spurious and unreliable estimates of underlying neural interactions. Experimental verification of this claim is challenging because neural ground truth connectivity is often unavailable concurrently with fMRI recordings. Here we demonstrate that, despite these widely held caveats, GC networks estimated from fMRI recordings contain useful information for classifying task-specific cognitive states. We estimated instantaneous and lag-based GC functional connectivity networks using fMRI data from 1000 participants (Human Connectome Project database). A linear classifier, trained on either instantaneous or lag-based GC, reliably discriminated among seven different task and resting brain states, with >80% cross-validation accuracy. With network simulations, we demonstrate that instantaneous and lag-based GC exploited interactions at fast and slow timescales, respectively, to achieve robust classification. With human fMRI data, instantaneous and lag-based GC identified complementary, task–core networks. Finally, variations in GC connectivity explained inter-individual variations in a variety of cognitive scores. Our findings show that instantaneous and lag-based methods reveal complementary aspects of functional connectivity in the brain, and suggest that slow, directed functional interactions, estimated with fMRI, may provide useful markers of behaviorally relevant cognitive states.

Decreased directed functional connectivity in the psychedelic state

Neuroimaging studies of the psychedelic state offer a unique window onto the neural basis of conscious perception and selfhood. Despite well understood pharmacological mechanisms of action, the large-scale changes in neural dynamics induced by psychedelic compounds remain poorly understood. Using source-localised, steady-state MEG recordings, we describe changes in functional connectivity following the controlled administration of LSD, psilocybin and low-dose ketamine, as well as, for comparison, the (non-psychedelic) anticonvulsant drug tiagabine. We compare both undirected and directed measures of functional connectivity between placebo and drug conditions. We observe a general decrease in directed functional connectivity for all three psychedelics, as measured by Granger causality, throughout the brain. These data support the view that the psychedelic state involves a breakdown in patterns of functional organisation or information flow in the brain. In the case of LSD, the decrease in directed functional connectivity is coupled with an increase in undirected functional connectivity, which we measure using correlation and coherence. This surprising opposite movement of directed and undirected measures is of more general interest for functional connectivity analyses, which we interpret using analytical modelling. Overall, our results uncover the neural dynamics of information flow in the psychedelic state, and highlight the importance of comparing multiple measures of functional connectivity when analysing time-resolved neuroimaging data.

Restricted information in a two-step cascade

A cell must sense extracellular and intracellular fluctuations and respond appropriately to survive for optimal cellular functioning. Accordingly, a cell builds up biochemical networks which can transduce information of extracellular and intracellular fluctuations accurately. We consider a generic two-step cascade as a model gene regulatory network containing three regulatory proteins S, X, and Y connected as S→X→Y. The intermediate node X is a stochastic variable, acts as an obstacle, and impedes the information flow from S to Y. We quantify the information that is restricted by X using the tools of information theory and term this as restricted information. In this context, we further propose two measurable quantities, restricted efficiency and information transfer efficiency. The former determines how efficiently X restricts the upstream information coming from S, while the latter computes the efficiency of X to pass the upstream information toward Y. We also quantify the information that is being uniquely transferred from X to Y, which determines the extent of the ability of X to act as a source of information. Our analysis shows that when the signal strength (or mean population of S, 〈s〉) is low, the intermediate X can carry forward the upstream information reliably as well, as it acts as a better source of information, thereby increasing the fidelity of the network. But at the high signal strength, X restricts most of the upstream information, and its ability to act as a source of information gets reduced. This leads to a loss of fidelity of the network.

Transitions in information processing dynamics at the whole-brain network level are driven by alterations in neural gain

A key component of the flexibility and complexity of the brain is its ability to dynamically adapt its functional network structure between integrated and segregated brain states depending on the demands of different cognitive tasks. Integrated states are prevalent when performing tasks of high complexity, such as maintaining items in working memory, consistent with models of a global workspace architecture. Recent work has suggested that the balance between integration and segregation is under the control of ascending neuromodulatory systems, such as the noradrenergic system, via changes in neural gain (in terms of the amplification and non-linearity in stimulus-response transfer function of brain regions). In a previous large-scale nonlinear oscillator model of neuronal network dynamics, we showed that manipulating neural gain parameters led to a 'critical' transition in phase synchrony that was associated with a shift from segregated to integrated topology, thus confirming our original prediction. In this study, we advance these results by demonstrating that the gain-mediated phase transition is characterized by a shift in the underlying dynamics of neural information processing. Specifically, the dynamics of the subcritical (segregated) regime are dominated by information storage, whereas the supercritical (integrated) regime is associated with increased information transfer (measured via transfer entropy). Operating near to the critical regime with respect to modulating neural gain parameters would thus appear to provide computational advantages, offering flexibility in the information processing that can be performed with only subtle changes in gain control. Our results thus link studies of whole-brain network topology and the ascending arousal system with information processing dynamics, and suggest that the constraints imposed by the ascending arousal system constrain low-dimensional modes of information processing within the brain.

Identification of Hidden Sources by Estimating Instantaneous Causality in High-Dimensional Biomedical Time Series

The study of connectivity patterns of a system’s variables, such as multi-channel electroencephalograms (EEG), is of utmost importance towards a better understanding of its internal evolutionary mechanisms. Here, the problem of estimating the connectivity network from multivariate time series in the presence of prominent unobserved variables is addressed. The causality measure of partial mutual information from mixed embedding (PMIME), designed to estimate direct lag-causal effects in the presence of many observed variables, is adapted to estimate also zero-lag effects, the so-called instantaneous causality. We term the proposed advanced method, PMIME0. The estimation of instantaneous causality by PMIME0 is a signature of the presence of hidden source in the observed system, as demonstrated analytically in a toy model. It is further demonstrated that the PMIME0 identifies the true instantaneous with great accuracy in a variety of high-dimensional dynamical systems. The method is applied to EEG data with epileptiform discharges (EDs), and the results imply a strong impact of unobserved confounders during the EDs. This finding comes as a possible explanation for the increased levels of causality during epileptic seizures estimated by some measures affected by the presence of a common source.

On the Sensitivity of Granger Causality to Errors-In-Variables, Linear Transformations and Subsampling

This article studies the sensitivity of Granger causality to the addition of noise, the introduction of subsampling, and the application of causal invertible filters to weakly stationary processes. Using canonical spectral factors and Wold decompositions, we give general conditions under which additive noise or filtering distorts Granger-causal properties by inducing (spurious) Granger causality, as well as conditions under which it does not. For the errors-in-variables case, we give a continuity result, which implies that: a 'small' noise-to-signal ratio entails 'small' distortions in Granger causality. On filtering, we give general necessary and sufficient conditions under which 'spurious' causal relations between (vector) time series are not induced by linear transformations of the variables involved. This also yields transformations (or filters) which can eliminate Granger causality from one vector to another one. In a number of cases, we clarify results in the existing literature, with a number of calculations streamlining some existing approaches.

Characterizing information-theoretic storage and transfer in continuous time processes

The characterization of information processing is an important task in complex systems science. Information dynamics is a quantitative methodology for modeling the intrinsic information processing conducted by a process represented as a time series, but to date has only been formulated in discrete time. Building on previous work which demonstrated how to formulate transfer entropy in continuous time, we give a total account of information processing in this setting, incorporating information storage. We find that a convergent rate of predictive capacity, comprising the transfer entropy and active information storage, does not exist, arising through divergent rates of active information storage. We identify that active information storage can be decomposed into two separate quantities that characterize predictive capacity stored in a process: active memory utilization and instantaneous predictive capacity. The latter involves prediction related to path regularity and so solely inherits the divergent properties of the active information storage, while the former permits definitions of pathwise and rate quantities. We formulate measures of memory utilization for jump and neural spiking processes and illustrate measures of information processing in synthetic neural spiking models and coupled Ornstein-Uhlenbeck models. The application to synthetic neural spiking models demonstrates that active memory utilization for point processes consists of discontinuous jump contributions (at spikes) interrupting a continuously varying contribution (relating to waiting times between spikes), complementing the behavior previously demonstrated for transfer entropy in these processes.

Comparing brain connectivity metrics: a didactic tutorial with a toy model and experimental data

OBJECTIVE

The objective of this paper is to didactically compare resting state connectivity networks computed using two different methods called phase locking value (PLV) and convergent cross-mapping (CCM). PLV is a ubiquitous measure of connectivity in electrophysiological research but is less often applied to fMRI BOLD timeseries since this model-based metric assumes that oscillatory coupling is a sufficient condition for connectivity. Alternatively, CCM is a model-free method, which detects potentially nonlinear causal influences based on the ability to estimate one timeseries with another and does not assume an oscillatory structure.

APPROACH

We use a toy dataset to test the PLV and CCM algorithms under different known synchronization conditions. Additionally, experimental resting state EEG and fMRI datasets are used for comparison.

MAIN RESULTS

The results show that the resting state brain networks computed using both algorithms produce similar results for both resting state EEG and fMRI datasets. For both neuroimaging datasets, the network characteristics follow the same trends and the similarity between the computed networks, for both algorithms, is highly significant.

SIGNIFICANCE

CCM is able to identify low or one-way connection strengths better than PLV but takes exponentially longer to compute. Based on these results, PLV provides a good metric for on-line network identification because it is both computationally fast and an excellent approximation of the network computed with CCM.

Directed functional connectivity using dynamic graphical models

There are a growing number of neuroimaging methods that model spatio-temporal patterns of brain activity to allow more meaningful characterizations of brain networks. This paper proposes dynamic graphical models (DGMs) for dynamic, directed functional connectivity. DGMs are a multivariate graphical model with time-varying coefficients that describe instantaneous directed relationships between nodes. A further benefit of DGMs is that networks may contain loops and that large networks can be estimated. We use network simulations and human resting-state fMRI (N = 500) to investigate the validity and reliability of the estimated networks. We simulate systematic lags of the hemodynamic response at different brain regions to investigate how these lags potentially bias directionality estimates. In the presence of such lag confounds (0.4-0.8 s offset between connected nodes), our method has a sensitivity of 72%-77% to detect the true direction. Stronger lag confounds have reduced sensitivity, but do not increase false positives (i.e., directionality estimates of the opposite direction). In human resting-state fMRI, the default mode network has consistent influence on the cerebellar, the limbic and the auditory/temporal networks. We also show a consistent reciprocal relationship between the visual medial and visual lateral network. Finally, we apply the method in a small mouse fMRI sample and discover a highly plausible relationship between areas in the hippocampus feeding into the cingulate cortex. We provide a computationally efficient implementation of DGM as a free software package for R.

Time, frequency, and time-varying Granger-causality measures in neuroscience

This article proposes a systematic methodological review and an objective criticism of existing methods enabling the derivation of time, frequency, and time-varying Granger-causality statistics in neuroscience. The capacity to describe the causal links between signals recorded at different brain locations during a neuroscience experiment is indeed of primary interest for neuroscientists, who often have very precise prior hypotheses about the relationships between recorded brain signals. The increasing interest and the huge number of publications related to this topic calls for this systematic review, which describes the very complex methodological aspects underlying the derivation of these statistics. In this article, we first present a general framework that allows us to review and compare Granger-causality statistics in the time domain, and the link with transfer entropy. Then, the spectral and the time-varying extensions are exposed and discussed together with their estimation and distributional properties. Although not the focus of this article, partial and conditional Granger causality, dynamical causal modelling, directed transfer function, directed coherence, partial directed coherence, and their variant are also mentioned.

The influence of filtering and downsampling on the estimation of transfer entropy

Transfer entropy (TE) provides a generalized and model-free framework to study Wiener-Granger causality between brain regions. Because of its nonparametric character, TE can infer directed information flow also from nonlinear systems. Despite its increasing number of applications in neuroscience, not much is known regarding the influence of common electrophysiological preprocessing on its estimation. We test the influence of filtering and downsampling on a recently proposed nearest neighborhood based TE estimator. Different filter settings and downsampling factors were tested in a simulation framework using a model with a linear coupling function and two nonlinear models with sigmoid and logistic coupling functions. For nonlinear coupling and progressively lower low-pass filter cut-off frequencies up to 72% false negative direct connections and up to 26% false positive connections were identified. In contrast, for the linear model, a monotonic increase was only observed for missed indirect connections (up to 86%). High-pass filtering (1 Hz, 2 Hz) had no impact on TE estimation. After low-pass filtering interaction delays were significantly underestimated. Downsampling the data by a factor greater than the assumed interaction delay erased most of the transmitted information and thus led to a very high percentage (67–100%) of false negative direct connections. Low-pass filtering increases the number of missed connections depending on the filters cut-off frequency. Downsampling should only be done if the sampling factor is smaller than the smallest assumed interaction delay of the analyzed network.

Information-theoretic analysis of the directional influence between cellular processes

Inferring the directionality of interactions between cellular processes is a major challenge in systems biology. Time-lagged correlations allow to discriminate between alternative models, but they still rely on assumed underlying interactions. Here, we use the transfer entropy (TE), an information-theoretic quantity that quantifies the directional influence between fluctuating variables in a model-free way. We present a theoretical approach to compute the transfer entropy, even when the noise has an extrinsic component or in the presence of feedback. We re-analyze the experimental data from Kiviet et al. (2014) where fluctuations in gene expression of metabolic enzymes and growth rate have been measured in single cells of E. coli. We confirm the formerly detected modes between growth and gene expression, while prescribing more stringent conditions on the structure of noise sources. We furthermore point out practical requirements in terms of length of time series and sampling time which must be satisfied in order to infer optimally transfer entropy from times series of fluctuations.

Specific transfer entropy and other state-dependent transfer entropies for continuous-state input-output systems

Since its original formulation in 2000, transfer entropy has become an invaluable tool in the toolbox of nonlinear dynamicists working with empirical data. Transfer entropy and its generalizations provide a precise definition of uncertainty and information transfer that are central to the coupled systems studied in nonlinear science. However, a canonical definition of state-dependent transfer entropy has yet to be introduced. We introduce a candidate measure, the specific transfer entropy, and compare its properties to both total and local transfer entropy. Specific transfer entropy makes possible both state- and time-resolved analysis of the predictive impact of a candidate input system on a candidate output system. We also present principled methods for estimating total, local, and specific transfer entropies from empirical data. We demonstrate the utility of specific transfer entropy and our proposed estimation procedures with two model systems, and find that specific transfer entropy provides more, and more easily interpretable, information about an input-output system compared to currently existing methods.

The impact of hemodynamic variability and signal mixing on the identifiability of effective connectivity structures in BOLD fMRI

PURPOSE

Multiple computational studies have demonstrated that essentially all current analytical approaches to determine effective connectivity perform poorly when applied to synthetic functional Magnetic Resonance Imaging (fMRI) datasets. In this study, we take a theoretical approach to investigate the potential factors facilitating and hindering effective connectivity research in fMRI.

MATERIALS AND METHODS

In this work, we perform a simulation study with use of Dynamic Causal Modeling generative model in order to gain new insights on the influence of factors such as the slow hemodynamic response, mixed signals in the network and short time series, on the effective connectivity estimation in fMRI studies.

RESULTS

First, we perform a Linear Discriminant Analysis study and find that not the hemodynamics itself but mixed signals in the neuronal networks are detrimental to the signatures of distinct connectivity patterns. This result suggests that for statistical methods (which do not involve lagged signals), deconvolving the BOLD responses is not necessary, but at the same time, functional parcellation into Regions of Interest (ROIs) is essential. Second, we study the impact of hemodynamic variability on the inference with use of lagged methods. We find that the local hemodynamic variability provide with an upper bound on the success rate of the lagged methods. Furthermore, we demonstrate that upsampling the data to TRs lower than the TRs in state-of-the-art datasets does not influence the performance of the lagged methods.

CONCLUSIONS

Factors such as background scale-free noise and hemodynamic variability have a major impact on the performance of methods for effective connectivity research in functional Magnetic Resonance Imaging.

Transfer entropy in continuous time, with applications to jump and neural spiking processes

Transfer entropy has been used to quantify the directed flow of information between source and target variables in many complex systems. While transfer entropy was originally formulated in discrete time, in this paper we provide a framework for considering transfer entropy in continuous time systems, based on Radon-Nikodym derivatives between measures of complete path realizations. To describe the information dynamics of individual path realizations, we introduce the pathwise transfer entropy, the expectation of which is the transfer entropy accumulated over a finite time interval. We demonstrate that this formalism permits an instantaneous transfer entropy rate. These properties are analogous to the behavior of physical quantities defined along paths such as work and heat. We use this approach to produce an explicit form for the transfer entropy for pure jump processes, and highlight the simplified form in the specific case of point processes (frequently used in neuroscience to model neural spike trains). Finally, we present two synthetic spiking neuron model examples to exhibit the pertinent features of our formalism, namely, that the information flow for point processes consists of discontinuous jump contributions (at spikes in the target) interrupting a continuously varying contribution (relating to waiting times between target spikes). Numerical schemes based on our formalism promise significant benefits over existing strategies based on discrete time formalisms.