Skip to main content
SpringerLink
Log in

Mathematical Methods of Signal Processing in Neuroscience

  • Chapter
  • First Online:
Wavelets in Neuroscience

Part of the book series: Springer Series in Synergetics ((SSSYN))

Abstract

This chapter offers a brief introduction to the novel advanced mathematical methods of analysis and processing of neurophysiological data. First, we give the rationale for the development of specific mathematical approaches for decoding information from non-stationary neurophysiological processes with time-varying features. Second, we focus on the development of mathematical methods for automatic processing and analysis of neurophysiological signals, more specifically, in the development of brain–computer interfaces (BCIs). Finally, we give an overview of the main applications of wavelet analysis in neuroscience, from the microlevel (the dynamics of individual cells or intracellular processes) to the macrolevel (dynamics of large-scale neuronal networks in the brain as a whole, ascertained by analyzing electro- and magnetoencephalograms).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. G. Buzsaki, A. Draguhn, Neuronal oscillations in cortical networks. Science 304, 1926 (2004)

    Article  ADS  Google Scholar 

  2. H.D. Abarbanel, M.I. Rabinovich, A. Selverston, M.V. Bazhenov, R. Huerta, M.M. Sushchik, L.L. Rubchinskii, Synchronisation in neural networks. Phys.–Usp. 39(4), 337 (1996)

    Google Scholar 

  3. V.I. Nekorkin, Nonlinear oscillations and waves in neurodynamics. Phys.–Usp. 51(3), 295 (2008)

    Google Scholar 

  4. B.P. Bezruchko, V.I. Ponomarenko, M.D. Prokhorov, D.A. Smirnov, P.A. Tass, Modeling nonlinear oscillatory systems and diagnostics of coupling between them using chaotic time series analysis: applications in neurophysiology. Phys.–Usp. 51(3), 304 (2008)

    Google Scholar 

  5. M.I. Rabinovich, M.K. Muezzinoglu, Nonlinear dynamics of the brain: emotion and cognition. Phys.–Usp. 53(4), 357 (2010)

    Google Scholar 

  6. M.I. Rabinovich, K.J. Friston, P. Varona (eds.), Principles of Brain Dynamics: Global State Interactions (MIT, Cambridge, 2012)

    Google Scholar 

  7. H.C. Tuckwell, Introduction to Theoretical Neurobiology (Cambridge University Press, Cambridge, 1988)

    Book  Google Scholar 

  8. W.J. Freeman, Mass Action in the Nervous System (Academic, New York, 1975)

    Google Scholar 

  9. L.S. da Fernando, Neural mechanisms underlying brain waves: from neural membranes to networks. Electroencephalogr. Clin. Neurophysiol. 79, 81 (1991)

    Article  Google Scholar 

  10. G.R. Ivanitskii, A.B. Medvinskii, M.A. Tsyganov, From the dynamics of population autowaves generated by living cells to neuroinformatics. Physics-Uspekhi 37(10), 961 (1994)

    Google Scholar 

  11. W.J. Freeman, Mesoscopic neurodynamics: from neuron to brain. J. Physiol. (France) 94, 303 (2000)

    Google Scholar 

  12. G.N. Borisyuk, R.M. Borisyuk, Y.B. Kazanovich, G.R. Ivanitskii, Models of neural dynamics in brain information processing—the developments of ‘the decade’. Phys.–Usp. 45(10), 1073 (2002)

    Google Scholar 

  13. A. Villacorta-Atienza, M.G. Velarde, V.A. Makarov, Compact internal representation of dynamic situations: neural network implementing the causality principle. Biol. Cybern. 103, 285 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. V.A. Makarov, A. Villacorta-Atienza, in Recurrent Neural Networks for Temporal Data Processing (INTECH’2011, Shanghai), ed. by H. Cardot (2011), pp. 81–102

    Google Scholar 

  15. N.P. Castellanos, V.A. Makarov, Recovering EEG brain signals: artifact suppression with wavelet enhanced independent component analysis. J. Neurosci. Methods 158, 300 (2006)

    Article  Google Scholar 

  16. A. Villacorta-Atienza, M.G. Velarde, V.A. Makarov, Compact internal representation of dynamic situations: neural network implementing the causality principle. Biol. Cybern. 103, 285 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. A.N. Pavlov, A.E. Hramov, A.A. Koronovskii, Y.E. Sitnikova, V.A. Makarov, A.A. Ovchinnikov, Wavelet analysis in neurodynamics. Phys.–Usp. 55(9), 845 (2012)

    Google Scholar 

  18. P.L. Nunez, K. Srinivasan, Electric Fields of the Brain: The Neurophysics of EEG (Oxford University Press, New York, 1981)

    Google Scholar 

  19. E. Niedermeyer, F.L. da Silva, Electroencephalography: Basic Principles, Clinical Applications, and Related Fields (Lippincot Williams & Wilkins, Philadelphia, 2004)

    Google Scholar 

  20. A. Berenyi, M. Belluscio, D. Mao, G. Buzsaki, Closed-loop control of epilepsy by transcranial electrical stimulation. Science 337(6095), 735 (2012)

    Google Scholar 

  21. A. Luttjohann, E.L.M. van Luijtelaar, The dynamics of cortico-thalamo-cortical interactions at the transition from pre-ictal to ictal LFPs in absence epilepsy. Neurobiol. Dis. 47, 49 (2012)

    Article  Google Scholar 

  22. A. Luttjohann, J.M. Schoffelen, E.L.M. van Luijtelaar, Peri-ictal network dynamics of spike–wave discharges: phase and spectral characteristics. Exp. Neurol. 239, 235 (2013)

    Article  Google Scholar 

  23. M. Hämäläinen, R. Hari, R.J. Ilmoniemi, J. Knuutila, O.V. Lounasmaa, Magnetoencephalography: theory, instrumentation, and applications to noninvasive studies of the working human brain. Rev. Mod. Phys. 65, 413 (1993)

    Article  ADS  Google Scholar 

  24. P. Hansen, M. Kringelbach, R. Salmelin (eds.), MEG: An Introduction to Methods (Oxford University Press, New York, 2010)

    Google Scholar 

  25. I. Westmijse, P. Ossenblok, B. Gunning, E.L.M. van Luijtelaar, Onset and propagation of spike and slow wave discharges in human absence epilepsy: a MEG study. Epilepsia 50, 2538 (2009)

    Article  Google Scholar 

  26. E. Ahissar, P.M. Knutsen, Object localization with whiskers. Biol. Cybern. 98, 449 (2008)

    Article  MATH  Google Scholar 

  27. V.A. Makarov, A.N. Pavlov, A.N. Tupitsyn, F. Panetsos, A. Moreno, Stability of neural firing in the trigeminal nuclei under mechanical whisker stimulation. Comput. Intell. Neurosci. 2010, 340541 (2010)

    Article  Google Scholar 

  28. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, 1992)

    Book  MATH  Google Scholar 

  29. A. Aldroubi, M. Unser, Wavelets in Medicine and Biology (CRC, Boca Raton, 1996)

    MATH  Google Scholar 

  30. J.C. Van den Berg (eds.), Wavelets in Physics (Cambridge University Press, Cambridge, 2004)

    MATH  Google Scholar 

  31. N.E. Huang, Z. Shen, S.R. Long, A new view of nonlinear water waves: the Hilbert spectrum. Ann. Rev. Fluid Mech. 31, 417 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  32. N.E. Huang, Z. Shen, S.R. Long, M.C. Wu, H.H. Shih, Q. Zheng, N.C. Yen, C.C. Tung, H.H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. A: Math. Phys. Eng. Sci. 454, 903 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. N.E. Huang, Z. Shen, Hilbert–Huang Transform and Its Applications (World Scientific, Singapore, 2005)

    MATH  Google Scholar 

  34. B. Boashash, in Advances in Spectrum Analysis and Array Processing, ed. by S. Haykin (Prentice Hall, Englewood Cliffs, 1990), pp. 418–517

    Google Scholar 

  35. S. Qian, D. Chen, Joint Time–Frequency Analysis (Prentice Hall, Upper Saddle River, 1996)

    Google Scholar 

  36. W. Mecklenbrauker, F. Hlawatsch, The Wigner Distribution: Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997)

    Google Scholar 

  37. A. Grossman, J. Morlet, Decomposition of Hardy function into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15(4), 273 (1984)

    Google Scholar 

  38. M.B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S.G. Mallat, Y. Meyer, L. Raphael, Wavelets and Their Applications and Data Analysis (Jones and Bartlett, Boston, 1992)

    Google Scholar 

  39. Y. Meyer, Wavelets: Algorithms and Applications (SIAM, Philadelphia, 1993)

    MATH  Google Scholar 

  40. Y. Meyer, Wavelets and Operators (Cambridge University Press, Cambridge, 1992)

    MATH  Google Scholar 

  41. J.J. Benedetto, A.I. Zayed, Sampling, Wavelets, and Tomography (Birkháuser, Boston, 2004)

    Book  MATH  Google Scholar 

  42. J.C. Letelier, P.P. Weber, Spike sorting based on discrete wavelet transform coefficients. J. Neurosci. Methods 101, 93 (2000)

    Article  Google Scholar 

  43. E. Hulata, R. Segev, E. Ben-Jacob, A method for spike sorting and detection based on wavelet packets and Shannon’s mutual information. J. Neurosci. Methods 117, 1 (2002)

    Article  Google Scholar 

  44. Q.R. Quiroga, Z. Nadasdy, Y. Ben-Shaul, Unsupervised spike detection and sorting with wavelets and superparamagnetic clustering. Neural Comput. 16, 1661 (2004)

    Article  MATH  Google Scholar 

  45. R.Q. Quiroga, A. Kraskov, T. Kreuz, P. Grassberger, Perfomance of different synchronization measures in real data: a case study on electroencephalographic signals. Phys. Rev. E 65, 041903 (2002)

    Article  ADS  Google Scholar 

  46. J. Morlet, G. Arens, E. Fourgeau, D. Glard, Wave propagation and sampling theory. Part I. Complex signal and scattering in multilayered media. Geophysics 47(2), 203 (1982)

    Google Scholar 

  47. J. Morlet, G. Arens, E. Fourgeau, D. Giard, Wave propagation and sampling theory. Part II. Sampling theory and complex waves. Geophysics 47(2), 222 (1982)

    Google Scholar 

  48. P.S. Addison, The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science Engineering, Medicine and Finance (Institute of Physics Publishing, Bristol, 2002)

    Book  Google Scholar 

  49. G. Kaiser, A Friendly Guide to Wavelets (Springer/Birkhauser, Boston, 1994)

    MATH  Google Scholar 

  50. S.G. Mallat, A Wavelet Tour of Signal Processing (Academic, New York, 1998)

    MATH  Google Scholar 

  51. N.M. Astaf’eva, Wavelet analysis: basic theory and some applications. Phys.–Usp. 39(11), 1085 (1996)

    Google Scholar 

  52. I.M. Dremin, O.V. Ivanov, V.A. Nechitailo, Wavelets and their uses. Phys.–Usp. 44(5), 447 (2001)

    Google Scholar 

  53. A.A. Koronovskii, A.E. Hramov, Continuous Wavelet Analysis and Its Applications (Fizmatlit, Moscow, 2003)

    Google Scholar 

  54. V.G. Anfinogentov, A.A. Koronovskii, A.E. Hramov, Wavelet analysis and its applications for examination of behaviour of nonlinear dynamical systems of different nature. BRAS: Phys. 64(12), 2383 (2000)

    Google Scholar 

  55. B. Torresani, Continuous Wavelet Transform (Savoire, Paris, 1995)

    Google Scholar 

  56. M. Vetterli, J. Kovacevic, Wavelets and Subband Coding (Prentice Hall, Englewood Cliffs, 1995)

    MATH  Google Scholar 

  57. S.T. Ali, J.P. Antoine, J.P. Gazeau, Coherent States, Wavelets and Their Generalizations (Springer, New York, 1999)

    Google Scholar 

  58. Y. Meyer, R. Coifman, Calderon–Zygmund and Multilinear Operators (Cambridge University Press, Cambridge, 1997)

    MATH  Google Scholar 

  59. D.E. Postnov, Evaluation of a continuous wavelet transform by solving the Cauchy problem for a system of partial differential equations. Comput. Math. Math. Phys. 46(1), 73 (2006)

    Google Scholar 

  60. J.J. Benedetto, M. Frazier, Wavelets: Mathematics and Applications (CRC, Boca Raton, 1994)

    MATH  Google Scholar 

  61. R. Gencay, F. Selcuk, B. Whitcher, An Introduction to Wavelets and Other Filtering Methods in Finance and Economics (Academic, San Diego, 2001)

    Google Scholar 

  62. T. Strutz, Bilddatenkompression. Grundlagen, codierung, JPEG, MPEG, wavelets (Vieweg Braunschweig, Wiesbaden, 2002)

    Google Scholar 

  63. J.S. Walker, A Primer on Wavelets and Their Scientific Applications (CRC, Boca Raton, 1999)

    Book  MATH  Google Scholar 

  64. C.L. da Fontoura, J.R.M. Cesar, Shape Analysis and Classification: Theory and Practice (CRC, Boca Raton, 2001)

    Google Scholar 

  65. S. Jaffard, Y. Meyer, R. Ryan, Wavelets: Tools for Science and Technology (SIAM, Philadelphia, 2001)

    Book  Google Scholar 

  66. M.V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software (A.K. Peters, Wellesley, 1994)

    MATH  Google Scholar 

  67. E.B. Postnikov, E.A. Lebedeva, Decomposition of strong nonlinear oscillations via modified continuous wavelet transform. Phys. Rev. E 82(5), 057201 (2010)

    Google Scholar 

  68. C. Guger, H. Ramoser, G. Pfurtscheller, Real-time EEG analysis for a brain–computer interface (BCI) with subject-specific spatial patterns. IEEE Trans. Neural Syst. Rehabil. Eng. 8(4), 447 (2000)

    Google Scholar 

  69. S.G. Mason, G.E. Birch, A general framework for brain–computer interface design. IEEE Trans. Neural Syst. Rehabil. Eng. 11(1), 70 (2003)

    Google Scholar 

  70. S. Makeig, S. Enghoff, T.P. Jung, T.J. Sejnowski, A natural basis for efficient brain-actuated control. IEEE Trans. Neural Syst. Rehabil. Eng. 8, 208 (2000)

    Article  Google Scholar 

  71. N.E. Sviderskaya, T.N. Dashinskaja, G.V. Taratunova, Spatial organization of EEG activation during the creative processes. J. High. Nerv. Act. 51, 393 (2001)

    Google Scholar 

  72. N.E. Sviderskaya, Spatial Organization of Electroencephalogram (VGMA Press, Voronezh, 2008)

    Google Scholar 

  73. N.E. Sviderskaya, A.G. Antonov, Effect of individual psychological characteristics on the spatial organization of EEG during nonverbally-divergent mindset. Hum. Physiol. 34(5), 34 (2008)

    Google Scholar 

  74. L.A. Farwell, E. Donchin, Talking off the top of your head: toward a mental prosthesis utilizing event related brain potentials. EEG Clin. Neurophysiol. 70, 510 (1988)

    Article  Google Scholar 

  75. J.N. Mak, Y. Arbel, J.W. Minett, L.M. McCane, B. Yuksel, D. Ryan, D. Thompson, L. Bianchi, D. Erdogmus, Optimizing the P300-based brain–computer interface: current status, limitations and future directions. J. Neural Eng. 8, 025003 (2011)

    Article  ADS  Google Scholar 

  76. M. Huang, P. Wu, Y. Liu, L. Bi, H. Chen, Application and contrast in brain–computer interface between Hilbert–Huang transform and wavelet transform, in The 9th International Conference for Young Computer Scientists (ICYCS’08), Zhang Jia Jie, 18–21 Nov 2008, pp. 1706–1710

    Google Scholar 

  77. M.R. Kousarrizi, A.A. Ghanbari, M. Teshnehlab, M. Aliyari, A. Gharaviri, Feature extraction and classification of EEG signals using wavelet transform, SVM and artificial neural networks for brain–computer interfaces, in International Joint Conference on Bioinformatics, Systems Biology and Intelligent Computing, 2009 (IJCBS’09), Shanghai, 3–5 Aug 2009, pp. 352–355

    Google Scholar 

  78. T. Bassani, J.C. Nievola, Pattern recognition for brain–computer interface on disabled subjects using a wavelet transformation, in IEEE Symposium on Computational Intelligence in Bioinformatics and Computational Biology (CIBCB’08), Sun Valley Idaho, 15–17 Sept 2008, pp. 180–186

    Google Scholar 

  79. V.M. Vaughan, Guest editorial brain–computer interface technology: a review of the second international meeting. IEEE Trans. Neural Syst. Rehabil. Eng. 11(2), 94 (2003)

    Google Scholar 

  80. U. Hoffmann, J.M. Vesin, T. Ebrahimi, K. Diserens, An efficient P300-based brain–computer interface for disabled subjects. J. Neurosci. Methods 167(1), 115 (2008)

    Google Scholar 

  81. G. Pires, U. Nunes, M. Castelo-Branco, Comparison of a rowcolumn speller vs. a novel lateral single-character speller: assessment of BCI for severe motor disabled patients. Clin. Neurophysiol. 123, 1168 (2012)

    Google Scholar 

  82. N.V. Manyakov, N. Chumerin, A. Combaz, M.M. van Hulle, Comparison of classification methods for P300 brain–computer interface on disabled subjects. Comput. Intell. Neurosci. 2011, 519868 (2011)

    Article  Google Scholar 

  83. S. Lu, C. Guan, H. Zhang, Unsupervised brain–computer interface based on intersubject information and online adaptation. IEEE Trans. Neural Syst. Rehabil. Eng. 17, 135 (2009)

    Article  Google Scholar 

  84. P.R. Kennedy, R.A. Bakay, Restoration of neural output from a paralyzed patient by a direct brain connection. Neuroreport 9, 1707 (1998)

    Article  Google Scholar 

  85. J.R. Wolpaw, Brain–computer interfaces as new brain output pathways. J. Physiol. 579(Part 3), 613 (2007)

    Google Scholar 

  86. A.Y. Kaplan, S.L. Shishkin, I.P. Ganin, I.A. Basyul, A.Y. Zhigalov, Adapting the P300-based brain–computer interface for gaming: a review. IEEE Trans. Comput. Intell. AI Games (Special Issue on Brain/Neuronal–Computer Games Interfaces and Interaction) 5(2), 141 (2013)

    Google Scholar 

  87. I.P. Ganin, S.P. Shishkin, A.G. Kochetkova, Y.A. Kaplan, Brain–computer interface on the base of “wave P300”: study of the effect of stimulus number in the sequence of their presentation. Hum. Physiol. 38(2), 5 (2012)

    Google Scholar 

  88. G. Edlinger, C. Guger, Social environments, mixed communication and goal-oriented control application using a brain–computer interface, in Proceedings of the International Conference UAHCI 2011, Orlando. LNCS, vol. 6766 (2011), pp. 545–554

    Google Scholar 

  89. D.E. Newland, An Introduction to Random Vibrations, Spectral and Wavelet Analysis (Wiley, New York, 1993)

    Google Scholar 

  90. M. Holschneider, Wavelets: An Analysis Tool (Oxford University Press, Oxford, 1995)

    MATH  Google Scholar 

  91. C. Blatter, Wavelets: A Primer (A.K. Peters, Natick, 1998)

    MATH  Google Scholar 

  92. M. Farge, J.C. Hunt, J.C. Vassilicos, Wavelets, Fractals and Fourier Transforms (Oxford University Press, Oxford, 1995)

    Google Scholar 

  93. D.B. Percival, A.T. Walden, Wavelet Methods for Time Series Analysis (Cambridge University Press, Cambridge/New York, 2000)

    Book  MATH  Google Scholar 

  94. B.P. van Milligen, E. Sánchez, T. Estrada, C. Hidalgo, B. Brãnas, B. Carreras, L. Garsia, Wavelet bicoherence: a new turbulence analysis tool. Phys. Plasmas 2(8), 3017 (1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research and Education Center ‘Nonlinear Dynamics of Complex Systems’, Saratov State Technical University, Saratov, Russia

    Alexander E. Hramov, Alexey A. Koronovskii & Alexey N. Pavlov

  2. Department of Nonlinear Processes, Saratov State University, Saratov, Russia

    Alexander E. Hramov & Alexey A. Koronovskii

  3. Department of Applied Mathematics, Complutense University, Madrid, Spain

    Valeri A. Makarov

  4. Physics Department, Saratov State University, Saratov, Russia

    Alexey N. Pavlov

  5. Institute for Higher Nervous Activity and Neurophysiology, Russian Academy of Sciences, Moscow, Russia

    Evgenia Sitnikova

Authors
  1. Alexander E. Hramov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexey A. Koronovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Valeri A. Makarov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Alexey N. Pavlov
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Evgenia Sitnikova
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Hramov, A.E., Koronovskii, A.A., Makarov, V.A., Pavlov, A.N., Sitnikova, E. (2015). Mathematical Methods of Signal Processing in Neuroscience. In: Wavelets in Neuroscience. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43850-3_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-43850-3_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-43849-7

  • Online ISBN: 978-3-662-43850-3

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation