Skip to main content
SpringerLink
Log in

On the spectral formulation of Granger causality

  • Original Paper
  • Published:
Biological Cybernetics Aims and scope Submit manuscript

Abstract

Spectral measures of causality are used to explore the role of different rhythms in the causal connectivity between brain regions. We study several spectral measures related to Granger causality, comprising the bivariate and conditional Geweke measures, the directed transfer function, and the partial directed coherence. We derive the formulation of dependence and causality in the spectral domain from the more general formulation in the information-theory framework. We argue that the transfer entropy, the most general measure derived from the concept of Granger causality, lacks a spectral representation in terms of only the processes associated with the recorded signals. For all the spectral measures we show how they are related to mutual information rates when explicitly considering the parametric autoregressive representation of the processes. In this way we express the conditional Geweke spectral measure in terms of a multiple coherence involving innovation variables inherent to the autoregressive representation. We also link partial directed coherence with Sims’ criterion of causality. Given our results, we discuss the causal interpretation of the spectral measures related to Granger causality and stress the necessity to explicitly consider their specific formulation based on modeling the signals as linear Gaussian stationary autoregressive processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Amblard PO, Michel O (2011) On directed information theory and Granger causality graphs. J Comput Neurosci 30(1): 7–16

    Article  PubMed  Google Scholar 

  • Baccalá L, Sameshima K (2001) Partial directed coherence: a new concept in neural structure determination. Biol Cybern 84(1): 463–474

    Article  PubMed  Google Scholar 

  • Baccalá L, Sameshima K, Ballester G, Do Valle A, Timo-Iaria C (1999) Studying the interaction between brain structures via directed coherence and Granger causality. Appl Signal Process 5: 40–48

    Article  Google Scholar 

  • Barnett L, Barrett AB, Seth AK (2009) Granger causality and transfer entropy are equivalent for Gaussian variables. Phys Rev Lett 103(23): 238701

    Article  PubMed  Google Scholar 

  • Bernasconi C, König P (1999) On the directionality of cortical interactions studied by structural analysis of electrophysiological recordings. Biol Cybern 81(3): 199–210

    Article  PubMed  CAS  Google Scholar 

  • Bernasconi C, von Stein A, Chiang C, König P (2000) Bi-directional interactions between visual areas in the awake behaving cat. Neuroreport 11(4): 689–692

    Article  PubMed  CAS  Google Scholar 

  • Besserve M, Schoelkopf B, Logothetis NK, Panzeri S (2010) Causal relationships between frequency bands of extracellular signals in visual cortex revealed by an information theoretic analysis. J Comput Neurosci 29(3): 547–566

    Article  PubMed  Google Scholar 

  • Bressler SL, Seth AK (2011) Wiener Granger causality: a well established methodology. Neuroimage 58(2): 323–329

    Article  PubMed  Google Scholar 

  • Bressler SL, Richter CG, Chen Y, Ding M (2007) Cortical functional network organization from autoregressive modeling of local field potential oscillations. Stat Med 26(21): 3875–3885

    Article  PubMed  Google Scholar 

  • Bressler SL, Tang W, Sylvester CM, Shulman GL, Corbetta M (2008) Top-down control of human visual cortex by frontal and parietal cortex in anticipatory visual spatial attention. J Neurosci 28(40): 10056–10061

    Article  PubMed  CAS  Google Scholar 

  • Brillinger D (1981) Time series. Data analysis and theory. Holden-Day, San Francisco

    Google Scholar 

  • Brovelli A, Ding M, Ledberg A, Chen Y, Nakamura R, Bressler SL (2004) Beta oscillations in a large-scale sensorimotor cortical network: Directional influences revealed by Granger causality. Proc Natl Acad Sci USA 101: 9849–9854

    Article  PubMed  CAS  Google Scholar 

  • Buzsáki G (2006) Rhythms of the brain. Oxford University Press, New York

    Book  Google Scholar 

  • Chamberlain G (1982) The general equivalence of Granger and Sims causality. Econometrica 50(3): 569–581

    Article  Google Scholar 

  • Chen Y, Bressler S, Ding M (2006) Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data. J Neurosci Methods 150(2): 228–237

    Article  PubMed  Google Scholar 

  • Cover TM, Thomas JA (2006) Elements of information theory, 2nd ed. Wiley, New York

    Google Scholar 

  • Dhamala M, Rangarajan G, Ding M (2008) Estimating Granger causality from fourier and wavelet transforms of time series data. Phys Rev Lett 100(1): 018701

    Article  PubMed  Google Scholar 

  • Ding M, Chen Y, Bressler SL (2006) Granger causality: basic theory and application to neuroscience. In: Schelter B, Winterhalder M, Timmer J (eds) Handbook of time series analysis: recent theoretical developments and applications. Weinheim, Wiley-VCH Verlag, pp 437–460

    Google Scholar 

  • Eichler M (2006) On the evaluation of information flow in multivariate systems by the directed transfer function. Biol Cybern 94(6): 469–482

    Article  PubMed  Google Scholar 

  • Florens J (2003) Some technical issues in defining causality. J Econ 112: 127–128

    Google Scholar 

  • Fries P (2005) A mechanism for cognitive dynamics: neuronal communication through neuronal coherence. Trends Cogn Sci 9(10): 474–480

    Article  PubMed  Google Scholar 

  • Friston KJ (1994) Functional and effective connectivity in neuroimaging: a synthesis. Hum Brain Mapp 2(5): 56–78

    Article  Google Scholar 

  • Gelfand I, Yaglom A (1959) Calculation of the amount of information about a random function contained in another such function. Am Math Soc Transl Ser 2(12): 199–246

    Google Scholar 

  • Geweke JF (1982) Measurement of linear dependence and feedback between multiple time series. J Am Stat Assoc 77(378): 304–313

    Article  Google Scholar 

  • Geweke JF (1984) Measures of conditional linear dependence and feedback between time series. J Am Stat Assoc 79(388): 907–915

    Article  Google Scholar 

  • Gourevitch B, Le Bouquin-Jeannes R, Faucon G (2006) Linear and nonlinear causality between signals: methods, examples and neurophysiological applications. Biol Cybern 95(4): 349–369

    Article  PubMed  Google Scholar 

  • Gourieroux C, Monfort A, Renault E (1987) Kullback causality measures. Ann Econ Stat 6/7:369–410

    Google Scholar 

  • Granger CWJ (1963) Economic processes involving feedback. Inf Control 6: 28–48

    Article  Google Scholar 

  • Granger CWJ (1980) Testing for causality: a personal viewpoint. J Econ Dyn Control 2(1): 329–352

    Article  Google Scholar 

  • Guo S, Seth AK, Kendrick KM, Zhou C, Feng J (2008a) Partial Granger causality-eliminating exogenous inputs and latent variables. J Neurosci Methods 172(1): 79–93

    Article  PubMed  Google Scholar 

  • Guo S, Wu J, Ding M, Feng J (2008) Uncovering interactions in the frequency domain. PLoS Comput Biol 4(5): e1000087

    Article  PubMed  Google Scholar 

  • Kaminski M, Blinowska K (1991) A new method of the description of the information flow in the brain structures. Biol Cybern 65(3): 203–210

    Article  PubMed  CAS  Google Scholar 

  • Kaminski M, Ding M, Truccolo W, Bressler S (2001) Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance. Biol Cybern 85(2): 145–157

    Article  PubMed  CAS  Google Scholar 

  • Kolmogorov A (1939) Sur l’interpolation et extrapolation des suites stationnaires. Comp Rend Acad Sci Paris 208: 2043–2045

    Google Scholar 

  • Kuersteiner G (2008) Granger-Sims causality. The new palgrave dictionary of economics, 2nd ed. MacMillan, Bedford

    Google Scholar 

  • Ladroue C, Guo S, Kendrick K, Feng J (2009) Beyond element-wise interactions: identifying complex interactions in biological processes. PLoS ONE 4(9): e6899

    Article  PubMed  Google Scholar 

  • Marko H (1973) Bidirectional communication theory–generalization of information-theory. IEEE Trans Commun 12: 1345–1351

    Article  Google Scholar 

  • Nedungadi AG, Rangarajan G, Jain N, Ding M (2009) Analyzing multiple spike trains with nonparametric Granger causality. J Comput Neurosci 27(1): 55–64

    Article  PubMed  Google Scholar 

  • Pereda E, Quian Quiroga R, Bhattacharya J (2005) Nonlinear multivariate analysis of neurophysiological signals. Prog Neurobiol 77: 1–37

    Article  PubMed  Google Scholar 

  • Priestley M (1981) Spectral analysis and time series. Academic Press Inc., San Diego

    Google Scholar 

  • Rissanen J, Wax M (1987) Measures of mutual information and causal dependence between 2 time-series. IEEE Trans Inf Theory 33(4): 598–601

    Article  Google Scholar 

  • Rozanov YA (1967) Stationary random processes. Holden-Day, San Francisco

    Google Scholar 

  • Schelter B, Winterhalder M, Eichler M, Peifer M, Hellwig B, Guschlbauer B, Lucking C, Dahlhaus R, Timmer J (2006) Testing for directed influences among neural signals using partial directed coherence. J Neurosci Methods 152(1-2): 210–219

    Article  PubMed  Google Scholar 

  • Schreiber T (2000) Measuring information transfer. Phys Rev Lett 85: 461–464

    Article  PubMed  CAS  Google Scholar 

  • Sims C (1972) Money, income, and causality. Am Econ Rev 62(4): 540–552

    Google Scholar 

  • Solo V (2008) On causality and mutual information. In: Proceedings of the 47th IEEE conference on decision and control, pp 4939–4944

  • Takahashi DY, Baccala LA, Sameshima K (2010) Information theoretic interpretation of frequency domain connectivity measures. Biol Cybern 103(6): 463–469

    Article  PubMed  Google Scholar 

  • Wiener N (1956) The theory of prediction. In: Beckenbach EF (eds) Modern mathematics for engineers. McGraw-Hill, New York

    Google Scholar 

  • Winterhalder M, Schelter B, Hesse W, Schwab K, Leistritz L, Klan D, Bauer R, Timmer J, Witte H (2005) Comparison directed of linear signal processing techniques to infer interactions in multivariate neural systems. Signal Process 85(11): 2137–2160

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018, Barcelona, Spain

    D. Chicharro

Authors
  1. D. Chicharro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Chicharro.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chicharro, D. On the spectral formulation of Granger causality. Biol Cybern 105, 331–347 (2011). https://doi.org/10.1007/s00422-011-0469-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00422-011-0469-z

Keywords

Search

Navigation