Biological Cybernetics

, Volume 103, Issue 6, pp 463–469 | Cite as

Information theoretic interpretation of frequency domain connectivity measures

  • Daniel Y. TakahashiEmail author
  • Luiz A. Baccalá
  • Koichi Sameshima
Original Paper


In order to provide adequate multivariate measures of information flow between neural structures, modified expressions of partial directed coherence (PDC) and directed transfer function (DTF), two popular multivariate connectivity measures employed in neuroscience, are introduced and their formal relationship to mutual information rates are proved.


Information flow Partial directed coherence Directed transfer function Mutual information rate Granger causality 


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  1. Astolfi L, Cincotti F, Mattia D, Marciani MG, Baccalá LA, Fallani FDV, Salinari S, Ursino M, Zavaglia M, Ding L, Edgar JC, Miller GA, He B, Babiloni F (2007) Comparison of different cortical connectivity estimators for high-resolution EEG recordings. Hum Brain Mapp 28: 143–157CrossRefPubMedGoogle Scholar
  2. Baccalá LA, Sameshima K (2001a) Overcoming the limitations of correlation analysis for many simultaneously processed neural structures. In: Nicolelis MAL (ed) Progress in brain research, vol 130. Advances in Neural Population Coding. Elsevier, Amsterdam, pp 33–47Google Scholar
  3. Baccalá LA, Sameshima K (2001b) Partial directed coherence: a new concept in neural structure determination. Biol Cybern 84(6): 463–474CrossRefPubMedGoogle Scholar
  4. Baccalá LA, Sameshima K, Ballester G, Valle AC, Timo-Iaria C (1999) Studying the interaction between brain structures via directed coherence and Granger causality. Appl Signal Process 5: 40–48CrossRefGoogle Scholar
  5. Baccalá LA, Takahashi DY, Sameshima K (2006) Computer intensive testing for the influence between time-series. In: Schelter B, Winterhalder M, Timmer J (eds) Handbook of time series analysis. Weinheim, Wiley-VCH, pp 411–435CrossRefGoogle Scholar
  6. Baccalá LA, Takahashi DY, Sameshima K (2007) Generalized partial directed coherence. In: Cardiff proceedings of the 2007 15th international conference on digital signal processing (DSP 2007), pp 162–166Google Scholar
  7. Blinowska K, Kus R, Kaminski M, Janiszewska J (2010) Transmission of brain activity during cognitive task. Brain Topogr 23(2): 205–213CrossRefPubMedGoogle Scholar
  8. Gelfand IM, Yaglom AM (1959) Calculation of amount of information about a random function contained in another such function. Am Math Soc Transl Ser 2: 3–52Google Scholar
  9. Geweke JF (1982) Measurement of linear dependence and feedback between multiple time series. J Am Stat Assoc 77: 304–313CrossRefGoogle Scholar
  10. Geweke JF (1984) Measures of conditional linear dependence and feedback between time series. J Am Stat Assoc 79: 907–915CrossRefGoogle Scholar
  11. Granger CWJ (1969) Investigating causal relation by econometric models and cross-spectral methods. Econometrica 37: 424–438CrossRefGoogle Scholar
  12. Hannan EJ (1970) Multiple time series. John Wiley & Sons Inc., New YorkCrossRefGoogle Scholar
  13. Hosoya Y (1991) The decomposition and measurement of the interdependency between second-order stationary processes. Probab Theory Relat Fields 88: 429–444CrossRefGoogle Scholar
  14. Hosoya Y (2001) Elimination of third-series effect and defining partial measures of causality. J Time Ser Anal 22: 537–554CrossRefGoogle Scholar
  15. Kaminski M, Blinowska K (1991) A new method of the description of the information flow in the brain structures. Biol Cybern 65: 203–210CrossRefPubMedGoogle Scholar
  16. Lütkepohl H (1993) Introduction to multiple time series analysis. Springer-Verlag, BerlinGoogle Scholar
  17. Priestley MB (1981) Spectral analysis and time series. Academic Press, LondonGoogle Scholar
  18. Schelter B, Timmer J, Eichler M (2009) Assessing the strength of directed influences among neural signals using renormalized partial directed coherence. J Neurosci Methods 179(1): 121–130CrossRefPubMedGoogle Scholar
  19. Sommer FT, Wichert A (2003) Exploratory analysis and data modeling in functional neuroimaging. MIT Press, CambridgeGoogle Scholar
  20. Takahashi DY (2009) Medidas de Fluxo de Informação com Aplicação em Neurociência. PhD Thesis, Programa Interunidades de Pós-graduação em Bioinformática da Universidade de São Paulo [K. Sameshima and L. A. Baccalá, Advisors]Google Scholar
  21. Takahashi DY, Baccalá LA, Sameshima K (2010) Frequency domain connectivity: an information theoretic perspective. In: Proceedings of the 32nd annual international conference of the IEEE engineering in medicine and biology society, Buenos Aires. IEEE Press, New York, pp 1726–1729Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Daniel Y. Takahashi
    • 1
    Email author
  • Luiz A. Baccalá
    • 2
  • Koichi Sameshima
    • 3
  1. 1.Mathematics and Statistics InstituteUniversity of São PauloSão PauloBrazil
  2. 2.Telecommunications and Control Engineering Department of Escola PolitécnicaUniversity of São PauloSão PauloBrazil
  3. 3.Department of Radiology and Oncology, Faculdade de MedicinaUniversity of São PauloSão PauloBrazil

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