Biological Cybernetics

, Volume 104, Issue 3, pp 197–207 | Cite as

Block coherence: a method for measuring the interdependence between two blocks of neurobiological time series

  • Aatira G. Nedungadi
  • Mingzhou Ding
  • Govindan RangarajanEmail author
Original Paper


Multisensor recordings are becoming commonplace. When studying functional connectivity between different brain areas using such recordings, one defines regions of interest, and each region of interest is often characterized by a set (block) of time series. Presently, for two such regions, the interdependence is typically computed by estimating the ordinary coherence for each pair of individual time series and then summing or averaging the results over all such pairs of channels (one from block 1 and other from block 2). The aim of this paper is to generalize the concept of coherence so that it can be computed for two blocks of non-overlapping time series. This quantity, called block coherence, is first shown mathematically to have properties similar to that of ordinary coherence, and then applied to analyze local field potential recordings from a monkey performing a visuomotor task. It is found that an increase in block coherence between the channels from V4 region and the channels from prefrontal region in beta band leads to a decrease in response time.


Coherence Spectral density matrix Multivariate process Granger causality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Albo Z, Viana Di Prisco G, Chen Y, Rangarajan G, Truccolo W, Feng J, Vertes RP, Ding M (2004) Is partial coherence a viable technique for identifying generators of neural oscillations. Biol Cybern 90: 318–326PubMedCrossRefGoogle Scholar
  2. Baccala LA, Sameshima K (2001) Partial directed coherence: a new concept in neural structure determination. Biol Cybern 84: 463–474PubMedCrossRefGoogle Scholar
  3. Bernasconi C, Konig P (1999) On the directionality of cortical interactions studied by structural analysis of electrophysiological recordings. Biol Cybern 81: 199–210PubMedCrossRefGoogle Scholar
  4. Boudjellaba H, Dufour J, Roy R (1992) Testing causality between two vectors in multivariate autoregressive moving average models. J Am Stat Assoc 87: 1082–1090CrossRefGoogle Scholar
  5. Bressler SL, Coppola R, Nakamura R (1993) Episodic multiregional cortical coherence at multiple frequencies during visual task performance. Nature 366: 153–156PubMedCrossRefGoogle Scholar
  6. Brillinger D, Guha A (2007) Mutual information in the frequency domain. J Stat Plan Inference 137(3): 1076–1084CrossRefGoogle Scholar
  7. Brovelli A, Ding MZ, Ledberg A, Chen YH, 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: 9849–9854PubMedCrossRefGoogle Scholar
  8. Brown EN, Kass RE, Mitra PP (2004) Multiple neural spike train data analysis: state-of-the-art and future challenges. Nat Neurosci 7: 456–461PubMedCrossRefGoogle Scholar
  9. Chen YH, Rangarajan G, Feng JF, Ding MZ (2004) Analyzing multiple nonlinear time series with extended Granger causality. Phys Lett A 324: 26–35CrossRefGoogle Scholar
  10. Chen Y, Bressler SL, Knuth KH, Truccolo WA, Ding M (2006a) Stochastic modeling of neurobiological time series: power, coherence, Granger causality, and separation of evoked responses from ongoing activity. Chaos 16: 026113PubMedCrossRefGoogle Scholar
  11. Chen Y, Bressler SL, Ding M (2006b) Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data. J Neurosci Methods 150: 228–237PubMedCrossRefGoogle Scholar
  12. Dhamala M, Rangarajan G, Ding M (2008a) Estimating Granger causality from Fourier and wavelet transforms of time series data. Phys Rev Lett 100(1–4): 018701PubMedCrossRefGoogle Scholar
  13. Dhamala M, Rangarajan G, Ding M (2008b) Analyzing information flow in brain networks with nonparametric Granger causality. NeuroImage 41: 354–362PubMedCrossRefGoogle Scholar
  14. Ding M, Bressler SL, Yang W, Liang H (2000) Short-window spectral analysis of cortical event-related potentials by adaptive multivariate autoregressive modeling: data preprocessing, model validation, and variability assessment. Biol Cybern 83: 35–45PubMedCrossRefGoogle Scholar
  15. Ding M, Chen Y, Bressler SL (2006) Granger causality: basic theory and applications to neuroscience. In: Schelter B, Winterhalder M, Timmer J (eds) Handbook of time series analysis. Wiley-VCH Verlag, pp 437–460Google Scholar
  16. Efron B, Tibshirani RJ (1993) An introduction to the bootstrap. Chapman and Hall/CRC, LondonGoogle Scholar
  17. Friedland S (1975) Generalised Hadamard inequality and its application. Linear Multilinear Algebra 2: 327–333CrossRefGoogle Scholar
  18. Friston KJ, Harrison L, Penny W (2003) Dynamic causal modeling. NeuroImage 19: 1273–1302PubMedCrossRefGoogle Scholar
  19. Gel’fand IM, Yaglom AM (1959) Calculation of the amount of information about a random function contained in another such function. Am Math Soc Transl Series 2 12: 199–246Google Scholar
  20. Geweke J (1982) Measurement of linear dependence and feedback between multiple time series. J Am Stat Assoc 77: 304–324CrossRefGoogle Scholar
  21. Geweke J (1984) Measures of conditional linear dependence and feedback between time series. J Am Stat Assoc 79: 907–915CrossRefGoogle Scholar
  22. Granger C (1969) Measures of conditional linear dependence and feedback between time series. Econometrica 37: 424–438CrossRefGoogle Scholar
  23. Harrison L, Penny WD, Friston KJ (2003) Multivariate autoregressive modeling of fMRI time series. NeuroImage 19: 1477–1491PubMedCrossRefGoogle Scholar
  24. Hesse W, Moller E, Arnold M, Schack B (2003) The use of time-variant EEG Granger causality for inspecting directed interdependencies of neural assemblies. J Neurosci Methods 124: 27–44PubMedCrossRefGoogle Scholar
  25. Hoffman KM, Kunze R (1971) Linear algebra. Prentice Hall, Englewood Cliffs, NJ, USAGoogle Scholar
  26. Horn RA, Johnson CR (1990) Matrix analysis. Cambridge University Press, LondonGoogle Scholar
  27. Jain N, Qi H-X, Kaas JH (2001) Longterm chronic multichannel recordings from sensorimotor cortex and thalamus of primates. Prog Brain Res 130: 63–72CrossRefGoogle Scholar
  28. Jarvis MR, Mitra PP (2001) Sampling properties of the spectrum and coherency of sequences of action potentials. Neural Comput 13: 717–749PubMedCrossRefGoogle Scholar
  29. Kaminski MJ, Blinowska KJ (1991) A new method of the description of the information flow in the brain structures by a modified directed transfer function (dDTF). Biol Cybern 65: 203–210PubMedCrossRefGoogle Scholar
  30. Kaminski M, Ding M, Truccolo WA, Bressler SL (2001) Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance. Biol Cybern 85: 145–157PubMedCrossRefGoogle Scholar
  31. Korzeniewska A, Manczak M, Kaminski M, Blinowska KJ, Kasicki S (2003) Determination of information flow direction among brain structures by a modified directed transfer function (dDTF)method. J Neurosci Methods 125: 195–207PubMedCrossRefGoogle Scholar
  32. Kus R, Kaminski M, Blinowska KJ (2004) Determination of EEG activity propagation: pairwise versus multichannel estimate. IEEE Trans Bio-Med Eng 51: 1501–1510CrossRefGoogle Scholar
  33. Ladroue C, Guo S, Kendrick K, Feng J (2009) Beyond element-wise interactions: identifying complex interactions in biological processes. PLoS ONE 4(9): e6899PubMedCrossRefGoogle Scholar
  34. Liang H, Bressler SL, Ding M, Truccolo WA, Nakamura R (2002) Synchronized activity in prefrontal cortex during anticipation of visuomotor processing. Neuroreport 13: 2011–2015PubMedCrossRefGoogle Scholar
  35. Lungarella M, Sporns O (2006) Mapping information flow in sensorimotor networks. PLoS Comput Biol 2: 1301–1312CrossRefGoogle Scholar
  36. Lutkepohl H (1991) Introduction to multiple timeseries analysis. Springer-Verlag, BerlinGoogle Scholar
  37. Mitrinovic DS, Pecaric JE, Fink AM (1993) Classical and new inequalities in analysis. Kluwer Academic Publishers, DordrechtGoogle Scholar
  38. Morf M et al (1978) Recursive multichannel maximum entropy spectral estimation. IEEE Trans GeoSci Elec, GE-16 (2), 85–94Google Scholar
  39. Pascual-Marqui RD (2007) Coherence and phase synchronisation: Generalisation to pairs of multivariate time-series and removal of zero led correlations. ArXiv:0706.1776v3Google Scholar
  40. Percival DB, Walden AT (1998) Analysis for physical applications. Cambridge University Press, London, UKGoogle Scholar
  41. Prichard D, Theiler J (1994) Generating surrogate data for time series with several simultaneously measured variables. Phys Rev Lett 73: 951–954PubMedCrossRefGoogle Scholar
  42. Roebroeck A, Formisano E, Goebel R (2005) Mapping directed influence over the brain using Granger causality and fMRI. Neuroimage 25: 230–242PubMedCrossRefGoogle Scholar
  43. Rosenberg JR, Halliday DM, Breeze P, Conway BA (1998) Identification of patterns of neuronal connectivity—partial spectra, partial coherence, and neuronal interactions. J Neurosci Methods 83: 57–72PubMedCrossRefGoogle Scholar
  44. Rozanov YA (1967) Stationary random process. Holden Day, San FranciscoGoogle Scholar
  45. Sayed AH, Kailath T (2001) A survey of spectral factorization methods. Numer Linear Algebra Appl 8: 467–496CrossRefGoogle Scholar
  46. Schelter B, Dahlhaus R, Leistritz L, Hesse W, Schack B, Kurths J, Timmer J, Witte H (2008) Multivariate time series analysis. In: Dahlhaus R, Kurths J, Maass P, Timmer J (eds) Mathematical methods in time series analysis and digital image processing. Springer, New York, pp 1–40CrossRefGoogle Scholar
  47. Tang A, Jackson D, Hobbs J, Chen W, Smith JL, Patel H, Prieto A, Petrusca D, Grivich MI, Sher A, Hottowy P, Dabrowski W, Litke AM, Beggs JM (2008) A maximum entropy model applied to spatial and temporal correlations from cortical networks in vitro. J Neurosci 28(2): 505–518PubMedCrossRefGoogle Scholar
  48. Wang X, Chen Y, Bressler SL, Ding M (2007) Granger causality between multiple interdependent neurobiological time series: blockwise versus pairwise methods. Int J Neural Syst 17: 71–78PubMedCrossRefGoogle Scholar
  49. Wu JH, Liu XG, Feng JF (2008) Detecting causality between different frequencies. J Neurosci Methods 167: 367–375PubMedCrossRefGoogle Scholar
  50. Zhang Y, Chen Y, Bressler SL, Ding M (2008) Response preparation and inhibition: the role of cortical sensorimotor beta rhythm. Neuroscience 156: 238–246PubMedCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Aatira G. Nedungadi
    • 1
    • 2
  • Mingzhou Ding
    • 3
  • Govindan Rangarajan
    • 4
    Email author
  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  2. 2.Computational Biology and Mathematical Modelling GroupCentre for Cellular and Molecular BiologyHyderabadIndia
  3. 3.J. Crayton Pruitt Family Department of Biomedical EnginerringUniversity of FloridaGainesvilleUSA
  4. 4.Department of Mathematics and Centre for NeuroscienceIndian Institute of ScienceBangaloreIndia

Personalised recommendations