On directed information theory and Granger causality graphs

Directed information theory deals with communication channels with feedback. When applied to networks, a natural extension based on causal conditioning is needed. We show here that measures built from directed information theory in networks can be used to assess Granger causality graphs of stochastic processes. We show that directed information theory includes measures such as the transfer entropy, and that it is the adequate information theoretic framework needed for neuroscience applications, such as connectivity inference problems.     

Hippocampal theta-driving cells revealed by Granger causality

The two-dipole model of theta generation in hippocampal CA1 suggests that the inhibitory perisomatic theta dipole is generated by local GABAergic interneurons. Various CA1 interneurons fire preferentially at different theta phases, raising the question of how these theta-locked interneurons contribute to the generation of theta oscillations. We here recorded interneurons in the hippocampal CA1 area of freely behaving mice, and identified a unique subset of theta-locked interneurons by using the Granger causality approach. These cells fired in an extremely reliable theta-burst pattern at high firing rates (～90 Hz) during exploration and always locked to ascending phases of the theta waves. Among theta-locked interneurons we recorded, only these cells generated strong Granger causal influences on local field potential (LFP) signals within the theta band (4-12 Hz), and the influences were persistent across behavioral states. Our results suggest that this unique type of theta-locked interneurons serve as the local inhibitory theta dipole control cells in shaping hippocampal theta oscillations.     

Granger causality analysis of steady-state electroencephalographic signals during propofol-induced anaesthesia

Changes in conscious level have been associated with changes in dynamical integration and segregation among distributed brain regions. Recent theoretical developments emphasize changes in directed functional (i.e., causal) connectivity as reflected in quantities such as 'integrated information' and 'causal density'. Here we develop and illustrate a rigorous methodology for assessing causal connectivity from electroencephalographic (EEG) signals using Granger causality (GC). Our method addresses the challenges of non-stationarity and bias by dividing data into short segments and applying permutation analysis. We apply the method to EEG data obtained from subjects undergoing propofol-induced anaesthesia, with signals source-localized to the anterior and posterior cingulate cortices. We found significant increases in bidirectional GC in most subjects during loss-of-consciousness, especially in the beta and gamma frequency ranges. Corroborating a previous analysis we also found increases in synchrony in these ranges; importantly, the Granger causality analysis showed higher inter-subject consistency than the synchrony analysis. Finally, we validate our method using simulated data generated from a model for which GC values can be analytically derived. In summary, our findings advance the methodology of Granger causality analysis of EEG data and carry implications for integrated information and causal density theories of consciousness.     

Evaluating the effective connectivity of resting state networks using conditional Granger causality

The human brain has been documented to be spatially organized in a finite set of specific coherent patterns, namely resting state networks (RSNs). The interactions among RSNs, being potentially dynamic and directional, may not be adequately captured by simple correlation or anticorrelation. In order to evaluate the possible effective connectivity within those RSNs, we applied a conditional Granger causality analysis (CGCA) to the RSNs retrieved by independent component analysis (ICA) from resting state functional magnetic resonance imaging (fMRI) data. Our analysis provided evidence for specific causal influences among the detected RSNs: default-mode, dorsal attention, core, central-executive, self-referential, somatosensory, visual, and auditory networks. In particular, we identified that self-referential and default-mode networks (DMNs) play distinct and crucial roles in the human brain functional architecture. Specifically, the former RSN exerted the strongest causal influence over the other RSNs, revealing a top-down modulation of self-referential mental activity (SRN) over sensory and cognitive processing. In quite contrast, the latter RSN was profoundly affected by the other RSNs, which may underlie an integration of information from primary function and higher level cognition networks, consistent with previous task-related studies. Overall, our results revealed the causal influences among these RSNs at different processing levels, and supplied information for a deeper understanding of the brain network dynamics.     

Granger causality vs. dynamic Bayesian network inference: a comparative study

Background  

In computational biology, one often faces the problem of deriving the causal relationship among different elements such as genes, proteins, metabolites, neurons and so on, based upon multi-dimensional temporal data. Currently, there are two common approaches used to explore the network structure among elements. One is the Granger causality approach, and the other is the dynamic Bayesian network inference approach. Both have at least a few thousand publications reported in the literature. A key issue is to choose which approach is used to tackle the data, in particular when they give rise to contradictory results.     

A Granger causality measure for point process models of ensemble neural spiking activity

The ability to identify directional interactions that occur among multiple neurons in the brain is crucial to an understanding of how groups of neurons cooperate in order to generate specific brain functions. However, an optimal method of assessing these interactions has not been established. Granger causality has proven to be an effective method for the analysis of the directional interactions between multiple sets of continuous-valued data, but cannot be applied to neural spike train recordings due to their discrete nature. This paper proposes a point process framework that enables Granger causality to be applied to point process data such as neural spike trains. The proposed framework uses the point process likelihood function to relate a neuron's spiking probability to possible covariates, such as its own spiking history and the concurrent activity of simultaneously recorded neurons. Granger causality is assessed based on the relative reduction of the point process likelihood of one neuron obtained excluding one of its covariates compared to the likelihood obtained using all of its covariates. The method was tested on simulated data, and then applied to neural activity recorded from the primary motor cortex (MI) of a Felis catus subject. The interactions present in the simulated data were predicted with a high degree of accuracy, and when applied to the real neural data, the proposed method identified causal relationships between many of the recorded neurons. This paper proposes a novel method that successfully applies Granger causality to point process data, and has the potential to provide unique physiological insights when applied to neural spike trains.     

Granger causality between multiple interdependent neurobiological time series: blockwise versus pairwise methods

Granger causality is becoming an important tool for determining causal relations between neurobiological time series. For multivariate data, there is often the need to examine causal relations between two blocks of time series, where each block could represent a brain region of interest. Two alternative methods are available. In the pairwise method, bivariate autoregressive models are fit to all pairwise combinations involving one time series from the first block and one from the second. The total Granger causality between the two blocks is then derived by summing pairwise causality values from each of these models. This approach is intuitive but computationally cumbersome. Theoretically, a more concise method can be derived, which we term the blockwise Granger causality method. In this method, a single multivariate model is fit to all the time series, and the causality between the two blocks is then computed from this model. We compare these two methods by applying them to cortical local field potential recordings from monkeys performing a sensorimotor task. The obtained results demonstrate consistency between the two methods and point to the significance potential of utilizing Granger causality analysis in understanding coupled neural systems.     

Mixed predictability and cross-validation to assess non-linear Granger causality in short cardiovascular variability series.

A method to evaluate the direction and strength of causal interactions in bivariate cardiovascular and cardiorespiratory series is presented. The method is based on quantifying self and mixed predictability of the two series using nearest-neighbour local linear approximation. It returns two causal coupling indexes measuring the relative improvement in predictability along direct and reverse directions, and a directionality index indicating the preferential direction of interaction. The method was implemented through a cross-validation approach that allowed quantification of directionality without constraining the embedding of the series, and fully exploited the available data to maximise the prediction accuracy. Validation on short simulated bivariate time series demonstrated the ability of the method to capture different degrees of unidirectional and bidirectional interaction. Moreover, application to representative examples of heart rate, systolic arterial pressure and respiration series allowed the inference of causal relationships related to known physiological mechanisms and experimental conditions.     

Granger causality mapping during joint actions reveals evidence for forward models that could overcome sensory-motor delays

Studies investigating joint actions have suggested a central role for the putative mirror neuron system (pMNS) because of the close link between perception and action provided by these brain regions [1], [2], [3]. In contrast, our previous functional magnetic resonance imaging (fMRI) experiment demonstrated that the BOLD response of the pMNS does not suggest that it directly integrates observed and executed actions during joint actions [4]. To test whether the pMNS might contribute indirectly to the integration process by sending information to brain areas responsible for this integration (integration network), here we used Granger causality mapping (GCM) [5]. We explored the directional information flow between the anterior sites of the pMNS and previously identified integrative brain regions. We found that the left BA44 sent more information than it received to both the integration network (left thalamus, right middle occipital gyrus and cerebellum) and more posterior nodes of the pMNS (BA2). Thus, during joint actions, two anatomically separate networks therefore seem effectively connected and the information flow is predominantly from anterior to posterior areas of the brain. These findings suggest that the pMNS is involved indirectly in joint actions by transforming observed and executed actions into a common code and is part of a generative model that could predict the future somatosensory and visual consequences of observed and executed actions in order to overcome otherwise inevitable neural delays.     

Altered connectivity pattern of hubs in default-mode network with Alzheimer's disease: an Granger causality modeling approach

Background

Evidences from normal subjects suggest that the default-mode network (DMN) has posterior cingulate cortex (PCC), medial prefrontal cortex (MPFC) and inferior parietal cortex (IPC) as its hubs; meanwhile, these DMN nodes are often found to be abnormally recruited in Alzheimer''s disease (AD) patients. The issues on how these hubs interact to each other, with the rest nodes of the DMN and the altered pattern of hubs with respect to AD, are still on going discussion for eventual final clarification.

Principal Findings

To address these issues, we investigated the causal influences between any pair of nodes within the DMN using Granger causality analysis and graph-theoretic methods on resting-state fMRI data of 12 young subjects, 16 old normal controls and 15 AD patients respectively. We found that: (1) PCC/MPFC/IPC, especially the PCC, showed the widest and distinctive causal effects on the DMN dynamics in young group; (2) the pattern of DMN hubs was abnormal in AD patients compared to old control: MPFC and IPC had obvious causal interaction disruption with other nodes; the PCC showed outstanding performance for it was the only region having causal relation with all other nodes significantly; (3) the altered relation between hubs and other DMN nodes held potential as a noninvasive biomarker of AD.

Conclusions

Our study, to the best of our knowledge, is the first to support the hub configuration of the DMN from the perspective of causal relationship, and reveal abnormal pattern of the DMN hubs in AD. Findings from young subjects provide additional evidence for the role of PCC/MPFC/IPC acting as hubs in the DMN. Compared to old control, MPFC and IPC lost their roles as hubs owing to the obvious causal interaction disruption, and PCC was preserved as the only hub showing significant causal relations with all other nodes.     

On the Inference of Functional Circadian Networks Using Granger Causality

Being able to infer one way direct connections in an oscillatory network such as the suprachiastmatic nucleus (SCN) of the mammalian brain using time series data is difficult but crucial to understanding network dynamics. Although techniques have been developed for inferring networks from time series data, there have been no attempts to adapt these techniques to infer directional connections in oscillatory time series, while accurately distinguishing between direct and indirect connections. In this paper an adaptation of Granger Causality is proposed that allows for inference of circadian networks and oscillatory networks in general called Adaptive Frequency Granger Causality (AFGC). Additionally, an extension of this method is proposed to infer networks with large numbers of cells called LASSO AFGC. The method was validated using simulated data from several different networks. For the smaller networks the method was able to identify all one way direct connections without identifying connections that were not present. For larger networks of up to twenty cells the method shows excellent performance in identifying true and false connections; this is quantified by an area-under-the-curve (AUC) 96.88%. We note that this method like other Granger Causality-based methods, is based on the detection of high frequency signals propagating between cell traces. Thus it requires a relatively high sampling rate and a network that can propagate high frequency signals.     

Recurrent activity in higher order, modality non-specific brain regions: a Granger causality analysis of autobiographic memory retrieval

It has been proposed that the workings of the brain are mainly intrinsically generated recurrent neuronal activity, with sensory inputs as modifiers of such activity in both sensory and higher order modality non-specific regions. This is supported by the demonstration of recurrent neuronal activity in the visual system as a response to visual stimulation. In contrast recurrent activity has never been demonstrated before in higher order modality non-specific regions. Using magneto-encephalography and Granger causality analysis, we tested in a paralimbic network the hypothesis that stimulation may enhance causal recurrent interaction between higher-order, modality non-specific regions. The network includes anterior cingulate/medial prefrontal and posterior cingulate/medial parietal cortices together with pulvinar thalami, a network known to be effective in autobiographic memory retrieval and self-awareness. Autobiographic memory retrieval of previous personal judgments of visually presented words was used as stimuli. It is demonstrated that the prestimulus condition is characterized by causal, recurrent oscillations which are maximal in the lower gamma range. When retrieving previous judgments of visually presented adjectives, this activity is dramatically increased during the stimulus task as ascertained by Granger causality analysis. Our results confirm the hypothesis that stimulation may enhance causal interaction between higher order, modality non-specific brain regions, exemplified in a network of autobiographical memory retrieval.     

Causality and persistence in ecological systems: a nonparametric spectral granger causality approach

Abstract Directionality in coupling, defined as the linkage relating causes to their effects at a later time, can be used to explain the core dynamics of ecological systems by untangling direct and feedback relationships between the different components of the systems. Inferring causality from measured ecological variables sampled through time remains a formidable challenge further made difficult by the action of periodic drivers overlapping the natural dynamics of the system. Periodicity in the drivers can often mask the self-sustained oscillations originating from the autonomous dynamics. While linear and direct causal relationships are commonly addressed in the time domain, using the well-established machinery of Granger causality (G-causality), the presence of periodic forcing requires frequency-based statistics (e.g., the Fourier transform), able to distinguish coupling induced by oscillations in external drivers from genuine endogenous interactions. Recent nonparametric spectral extensions of G-causality to the frequency domain pave the way for the scale-by-scale decomposition of causality, which can improve our ability to link oscillatory behaviors of ecological networks to causal mechanisms. The performance of both spectral G-causality and its conditional extension for multivariate systems is explored in quantifying causal interactions within ecological networks. Through two case studies involving synthetic and actual time series, it is demonstrated that conditional G-causality outperforms standard G-causality in identifying causal links and their concomitant timescales.     

On the formulation of oxygen transfer equations

Summary It is shown that although chemical potential differences are the driving force for diffusion, the rate of mass transfer depends on gradients of concentration.