The Stockwell Transform in Studying the Dynamics of Brain Functions

  • Cheng Liu
  • William Gaetz
  • Hongmei Zhu
Part of the Operator Theory: Advances and Applications book series (OT, volume 205)


The dynamics of brain functional activities make time-frequency analysis a powerful tool in revealing its neuronal mechanisms. In this paper, we extend the definition of several widely used measures in spectral analysis, including the power spectral density function, coherence function and phaselocking value, from the classic Fourier domain to the time-frequency plane using the Stockwell transform. The comparisons between the Stockwell-based measures and the Morlet wavelet-based measures are addressed from both theoretical and numerical perspectives. The Stockwell approach has advantages over the Morlet wavelet approach in terms of easy interpretation and fast computation. A magnetoencephalography study using the Stockwell analysis reveals interesting temporal interaction between contralateral and ipsilateral motor cortices under the multi-source interference task.


Stockwell transforms Morlet wavelet transforms power density function coherence function phase-locking value dynamics of brain functions 

Mathematics Subject Classification (2000)

Primary 62M15 65R10 92C55 Secondary 42C40 47G10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. S. Bendal and A. G. Piersol, Random Data: Analysis and Measurement Procedures, 2nd Edition, John Wiley, 1986.Google Scholar
  2. [2]
    G. Bush and L. M. Shin, The multi-source interference task: an fMRI task that reliably activates the cingulo-frontal-parietal cognitive/attention network, Nat. Protoc. 1 (1) (2006), 308–313.CrossRefGoogle Scholar
  3. [3]
    G. Bush, T. J. Spencer, J. Holmes, L. M. Shin, E. M. Valera, L. J. Seidman, N. Makris, C. Surman, M. Aleardi, E. Mick and J. Biederman, Functional magnetic resonance imaging of methylphenidate and placebo in attention-deficit/hyperactivity disorder during the multi-source interference task, Arch Gen Psychigtry 65 (1) (2008) 102–114.CrossRefGoogle Scholar
  4. [4]
    G. Buzsaki, Rhythms of the Brain, Oxford University Press, 2006.Google Scholar
  5. [5]
    D. Cheyne, A. C. Bostan, W. Gaetz and E. W. Pang, Event-related beamforming: a robust method for presurgical functional mapping using MEG, Clin. Neurophysiol. 118 (8) (2007), 1691–1704.CrossRefGoogle Scholar
  6. [6]
    L. Cohen, Time-Frequency Analysis, Prentice-Hall, 1995.Google Scholar
  7. [7]
    I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992.Google Scholar
  8. [8]
    M. Ding, S. L. Bressler, W. Yang, H. Liang, Short-window spectral analysis of cortical event-related potentials by adaptive multivariate autoregressive modeling: data preprocessing, model validation, and variability assessment, Biol. Cybern. 83 (2000), 35–45.zbMATHCrossRefGoogle Scholar
  9. [9]
    B. Efron, The Jackknife, the Bootstrap, and Other Resampling Plans, Sixth Printing, SIAM, 1994.Google Scholar
  10. [10]
    P. C. Gibson, M. P. Lamoureux and G. F. Margrave, Stockwell and wavelet transforms, J. Fourier Anal. and Appl. 12 (2006), 713–721.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    B. G. Goodyear, H. Zhu, R. A. Brown and J. R. Mitchell, Removal of phase artifacts from fMRI data using a Stockwell transform filter improves brain activity detection, Magn. Reson. Med. 51 (2004), 16–21.CrossRefGoogle Scholar
  12. [12]
    A. Grossmann, J. Morlet and T. Paul, Transforms associated to square integrable representations: general results, J. Math. Phys. 26 (1985), 2473–2479.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, 2001.Google Scholar
  14. [14]
    Q. Guo, S. Molahajloo and M. W. Wong, Modified Stockwell transforms and time-frequency analysis in New Developments in Pseudo-Differential Operators, Birkhäuser, 2009, 275–285.Google Scholar
  15. [15]
    Q. Guo and M. W. Wong, Modified Stockwell Transforms, Memorie della Accademia delle Scienze di Torino, Classe di Scienze, Fische, Matemiche e Naturali, Serie V 32, 2008.Google Scholar
  16. [16]
    K. Keissar, L. R. Davrath and S. Akselrod, Time-Frequency Wavelet Transform Coherence of cardio-respiratory signals during exercise, Computers in Cardiology 33 (2006), 733–736.Google Scholar
  17. [17]
    J. P. Lachaux, A. Lutz, D. Rudrauf, D. Cosmelli, M. Le Van Quyen, J. Martinerie and F. Varela, Estimating the time-course of coherence between single-trial brain signals: an introduction to wavelet coherence, Neurophysiol. Clin 62 (3) (2002), 157–174.CrossRefGoogle Scholar
  18. [18]
    J. P. Lachaux, E. Rodriguez, M. Le Van Quyen, A. Lutz, J. Martinerie and F. J. Varela, Studying single-trials of phase synchronous activity in the brain, Int. J. Bifur. Chaos 10 (2000), 2429–2439.Google Scholar
  19. [19]
    J. P. Lachaux, E. Rodriguez, J. Martinerie and F. J. Varela, Measuring phase synchrony in brain signals, Hum. Brain Map. 8 (4) (1999), 194–208.CrossRefGoogle Scholar
  20. [20]
    L. Lee, M. H. Lee and A. Mechelli, The functional brain connectivity workshop: report and commentary, Comput. Neural Syst. 14 (2003), R5–R15.CrossRefGoogle Scholar
  21. [21]
    M. Le Van Quyen, J. Foucher, J. Lachaux, E. Rodriguez, A. Lutz, J. Martinerie and F. J. Varela, Comparison of Hilbert transform and wavelet methods for the analysis of neuronal synchrony, J. Neuroscience Methods 111 (2), (2001), 83–98.CrossRefGoogle Scholar
  22. [22]
    X. Li, X. Yao, J. Fox and J. G. Jefferys, Interaction dynamics of neuronal oscillations analysed using wavelet transforms, J. Neuroscience Methods 160 (1) (2007), 178–185.CrossRefGoogle Scholar
  23. [23]
    S. Marple, Digital Spectra Analysis with Applications, Prentice-Hall, 1987.Google Scholar
  24. [24]
    C. R. Pinnegar, Polalization analysis and polarization filtering of three component signals with the time-frequency S-transform, in Geophys. J. Int. 165 (2) (2006), 596–606.CrossRefGoogle Scholar
  25. [25]
    B. Polloka and et al., The cerebral oscillatory network associated with auditorily paced finger movements, Neuro Image 24 (3) (2005), 646–655.Google Scholar
  26. [26]
    P. Sauseng and W. Klimesch, What does phase information of oscillatory brain activity tell us about cognitive processes?, Neuroscience Biobehavioral Rev. 32 (5) (2008), 1001–1013.CrossRefGoogle Scholar
  27. [27]
    R. G. Stockwell, L. Mansinha and R. P. Lowe, Localization of the complex spectrum: the S transform, IEEE Trans Signal Proc. 44 (4) (1996), 998–1001.CrossRefGoogle Scholar
  28. [28]
    H. Zhu, B. G. Goodyear, M. L. Lauzon, R. A. Brown, G. S. Mayer, A. G. Law, L. Mansinha and J. R. Mitchell, A new local multiscale Fourier analysis for MRI, in Med. Phys. 30 (2003), 1134–1141.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Cheng Liu
    • 1
  • William Gaetz
    • 2
  • Hongmei Zhu
    • 3
  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada
  2. 2.Department of Medical ImagingUniversity of TorontoTorontoCanada
  3. 3.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations