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.Comment: accepted for publications, Journal of Computational Neuroscienc

Causal conditioning and instantaneous coupling in causality graphs

The paper investigates the link between Granger causality graphs recently formalized by Eichler and directed information theory developed by Massey and Kramer. We particularly insist on the implication of two notions of causality that may occur in physical systems. It is well accepted that dynamical causality is assessed by the conditional transfer entropy, a measure appearing naturally as a part of directed information. Surprisingly the notion of instantaneous causality is often overlooked, even if it was clearly understood in early works. In the bivariate case, instantaneous coupling is measured adequately by the instantaneous information exchange, a measure that supplements the transfer entropy in the decomposition of directed information. In this paper, the focus is put on the multivariate case and conditional graph modeling issues. In this framework, we show that the decomposition of directed information into the sum of transfer entropy and information exchange does not hold anymore. Nevertheless, the discussion allows to put forward the two measures as pillars for the inference of causality graphs. We illustrate this on two synthetic examples which allow us to discuss not only the theoretical concepts, but also the practical estimation issues.Comment: submitte

Algorithms of causal inference for the analysis of effective connectivity among brain regions

In recent years, powerful general algorithms of causal inference have been developed. In particular, in the framework of Pearl’s causality, algorithms of inductive causation (IC and IC*) provide a procedure to determine which causal connections among nodes in a network can be inferred from empirical observations even in the presence of latent variables, indicating the limits of what can be learned without active manipulation of the system. These algorithms can in principle become important complements to established techniques such as Granger causality and Dynamic Causal Modeling (DCM) to analyze causal influences (effective connectivity) among brain regions. However, their application to dynamic processes has not been yet examined. Here we study how to apply these algorithms to time-varying signals such as electrophysiological or neuroimaging signals. We propose a new algorithm which combines the basic principles of the previous algorithms with Granger causality to obtain a representation of the causal relations suited to dynamic processes. Furthermore, we use graphical criteria to predict dynamic statistical dependencies between the signals from the causal structure. We show how some problems for causal inference from neural signals (e.g., measurement noise, hemodynamic responses, and time aggregation) can be understood in a general graphical approach. Focusing on the effect of spatial aggregation, we show that when causal inference is performed at a coarser scale than the one at which the neural sources interact, results strongly depend on the degree of integration of the neural sources aggregated in the signals, and thus characterize more the intra-areal properties than the interactions among regions. We finally discuss how the explicit consideration of latent processes contributes to understand Granger causality and DCM as well as to distinguish functional and effective connectivity

Hierarchy of neural organization in the embryonic spinal cord: Granger-causality graph analysis of in vivo calcium imaging data

The recent development of genetically encoded calcium indicators enables monitoring in vivo the activity of neuronal populations. Most analysis of these calcium transients relies on linear regression analysis based on the sensory stimulus applied or the behavior observed. To estimate the basic properties of the functional neural circuitry, we propose a network-based approach based on calcium imaging recorded at single cell resolution. Differently from previous analysis based on cross-correlation, we used Granger-causality estimates to infer activity propagation between the activities of different neurons. The resulting functional networks were then modeled as directed graphs and characterized in terms of connectivity and node centralities. We applied our approach to calcium transients recorded at low frequency (4 Hz) in ventral neurons of the zebrafish spinal cord at the embryonic stage when spontaneous coiling of the tail occurs. Our analysis on population calcium imaging data revealed a strong ipsilateral connectivity and a characteristic hierarchical organization of the network hubs that supported established propagation of activity from rostral to caudal spinal cord. Our method could be used for detecting functional defects in neuronal circuitry during development and pathological conditions

Looking behind Granger causality

Granger causality as a popular concept in time series analysis is widely applied in empirical research. The interpretation of Granger causality tests in a cause-effect context is, however, often unclear or even controversial, so that the causality label has faded away. Textbooks carefully warn that Granger causality does not imply true causality and preferably refer the Granger causality test to a forecasting technique. Applying theory of inferred causation, we develop in this paper a method to uncover causal structures behind Granger causality. In this way we re-substantialize the causal attribution in Granger causality through providing an causal explanation to the conditional dependence manifested in Granger causality.Granger Causality; Time Series Causal Model; Graphical Model

Causal connectivity of evolved neural networks during behavior

To show how causal interactions in neural dynamics are modulated by behavior, it is valuable to analyze these interactions without perturbing or lesioning the neural mechanism. This paper proposes a method, based on a graph-theoretic extension of vector autoregressive modeling and 'Granger causality,' for characterizing causal interactions generated within intact neural mechanisms. This method, called 'causal connectivity analysis' is illustrated via model neural networks optimized for controlling target fixation in a simulated head-eye system, in which the structure of the environment can be experimentally varied. Causal connectivity analysis of this model yields novel insights into neural mechanisms underlying sensorimotor coordination. In contrast to networks supporting comparatively simple behavior, networks supporting rich adaptive behavior show a higher density of causal interactions, as well as a stronger causal flow from sensory inputs to motor outputs. They also show different arrangements of 'causal sources' and 'causal sinks': nodes that differentially affect, or are affected by, the remainder of the network. Finally, analysis of causal connectivity can predict the functional consequences of network lesions. These results suggest that causal connectivity analysis may have useful applications in the analysis of neural dynamics

Graphical modelling of multivariate time series

We introduce graphical time series models for the analysis of dynamic relationships among variables in multivariate time series. The modelling approach is based on the notion of strong Granger causality and can be applied to time series with non-linear dependencies. The models are derived from ordinary time series models by imposing constraints that are encoded by mixed graphs. In these graphs each component series is represented by a single vertex and directed edges indicate possible Granger-causal relationships between variables while undirected edges are used to map the contemporaneous dependence structure. We introduce various notions of Granger-causal Markov properties and discuss the relationships among them and to other Markov properties that can be applied in this context.Comment: 33 pages, 7 figures, to appear in Probability Theory and Related Field