Inferring and quantifying causality in neuronal networks
© Chicharro et al; licensee BioMed Central Ltd. 2011
Published: 18 July 2011
The idea of inferring causal interactions from a variable X to another variable Y from the reduction of the prediction error of Y when including the past of X was formulated by Wiener  and formalized by Granger . While Granger provided a measure to study causal interactions for Gaussian linear processes, the underlying concept (Granger causality) can be used to derive a general criterion to test for causality . In particular, in the framework of information theory, transfer entropy  extends the measure proposed by Granger so that it is applicable to stationary and non-stationary finite-order Markov processes .
We here show that the concept of Granger causality can be used to reliably determine between which nodes in a network causal interactions exist, but in contrast to commonly held beliefs, it is not adequate to quantify the strength of these causal interactions. Instead we propose a two-step procedure to infer and quantify causality. First, a Granger causality based measure, like transfer entropy, is applied to build a directed graph indicating causal interactions between the nodes. Second, for nodes fulfilling some conditions we describe, an alternative measure is used to quantify the strength of the causal interactions.
This procedure is exemplified with bivariate stochastic processes for which the information theoretic measures can be calculated analytically. This avoids dealing with the problem of the identification of the underlying processes from the recorded time series, and the problem of estimating the probability distributions which further difficult the analysis of causal interactions from experimental data. These simple examples allow us to illustrate some drawbacks of using the measures based on the concept of Granger causality in the way they are commonly applied for the study of neural causal interactions [e. g. 6-8].
DC was supported by the grant 2010FI-B2 00079 of the 'Comissionat per a Universitats i Recerca del Departament d'Innovació, Universitats i Empresa de la Generalitat de Catalunya i el Fons Social Europeu'. RGA and AL acknowledge support from the Ramon y Cajal Program from the Spanish Government.
- Wiener N: The theory of prediction. Modern Mathematics for Engineers. Edited by: Beckenbach EF. 1957, New York: McGraw-HillGoogle Scholar
- Granger CWJ: Investigating causal relations by econometric models and cross-spectral methods. Econometrica. 1969, 37 (3): 424-438. 10.2307/1912791.View ArticleGoogle Scholar
- Granger CWJ: Testing for causality: A personal viewpoint. J Econ Dynamics and Control. 1980, 2 (1): 329-352. 10.1016/0165-1889(80)90069-X.View ArticleGoogle Scholar
- Schreiber T: Measuring information transfer. Phys Rev Lett. 2000, 85: 461-464. 10.1103/PhysRevLett.85.461.View ArticlePubMedGoogle Scholar
- Amblard PO, Michel O: On directed information theory and Granger causality graphs. J Comput Neurosci. 2010, doi:10.1007/s10827–010–0231–x 1–10Google Scholar
- Brovelli A, Ding M, Ledberg A, Chen Y, Nakamura R, Bressler SL: Beta oscillations in a large-scale sensorimotor cortical network: Directional influences revealed by Granger causality. P Natl Acad Sci USA. 2004, 101: 9849-9854. 10.1073/pnas.0308538101.View ArticleGoogle Scholar
- Roebroeck A, Formisano E, Goebel R: Mapping directed influence over the brain using Granger causality and fMRI. Neuroimage. 2005, 25 (1): 230-242. 10.1016/j.neuroimage.2004.11.017.View ArticlePubMedGoogle Scholar
- Vicente R, Wibral M, Lindner M, Pipa G: Transfer entropy: a model-free measure of effective connectivity for the neurosciences. J Comput Neurosci. 2010, doi:10.1007/s10827–010–0262–3 1–23Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.