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Fractal dimension of electroencephalographic time series and underlying brain processes

Biological Cybernetics volume 73, pages 477–482 (1995)Cite this article

Abstract

Fractal dimension has been proposed as a useful measure for the characterization of electrophysiological time series. This paper investigates what the pointwise dimension of electroencephalographic (EEG) time series can reveal about underlying neuronal generators. The following theoretical assumptions concerning brain function were made (i) within the cortex, strongly coupled neural assemblies exist which oscillate at certain frequencies when they are active, (ii) several such assemblies can oscillate at a time, and (iii) activity flow between assemblies is minimal. If these assumptions are made, cortical activity can be considered as the weighted sum of a finite number of oscillations (plus noise). It is shown that the correlation dimension of finite time series generated by multiple oscillators increases monotonically with the number of oscillators. Furthermore, it is shown that a reliable estimate of the pointwise dimension of the raw EEG signal can be calculated from a time series as short as a few seconds. These results indicate that (i) The pointwise dimension of the EEG allows conclusions regarding the number of independently oscillating networks in the cortex, and (ii) a reliable estimate of the pointwise dimension of the EEG is possible on the basis of short raw signals.

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References

  • Achermann P, Hartmann R, Gunzinger A, Guggenbühl W, Borbely AA (1994) All-night sleep EEG and artificial stochastic control signals have similar correlation dimensions. Electroencephalogr Clin Neurophysiol 90:384–387

    Article  PubMed  Google Scholar 

  • Aertsen A, Arndt M (1993) Response synchronization in the visual cortex. Curr Opin Neurobiol 3:586–594

    Article  PubMed  Google Scholar 

  • Berge P, Pomeau Y, Vidal C (1986) Order within chaos. Wiley, New York

    Google Scholar 

  • Braitenberg V (1978) Cell assemblies in the cerebral cortex. In: Heim R, Palm G (eds) Theoretical approaches to complex systems. Lecture notes in biomathematics, vol 21. Springer, Berlin Heidelberg New York, pp 171–188

    Google Scholar 

  • Broomhead DS, King GP (1986) Extracting qualitative dynamics from experimental data. Physica D 20:217–236

    Article  Google Scholar 

  • Eckhorn R, Bauer R, Jordan W, et al. (1988) Coherent oscillations: a mechanism of feature linking in the visual cortex? Biol Cybern 60:121–130

    Article  PubMed  Google Scholar 

  • Eckmann JP, Ruelle D (1992) Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems. Physica D 56:185–187

    Article  Google Scholar 

  • Elbert T, Rockstroh B (1987) Threshold regulation: a key to the understanding of the combined dynamics of EEG and event related potentials. J Psychophysiol 4:317–333

    Google Scholar 

  • Elbert T, Ray WJ, Kowalik ZJ, Skinner JE, Graf KE, Birbaumer N (1994) Chaos and physiology: deterministic chaos in excitable cell assemblies. Phys Rev 74:1–47

    Google Scholar 

  • Farmer JD (1982) Information dimension and the probabilistic structure of chaos. Z Naturforsch 37a:1304–1325

    Google Scholar 

  • Farmer JD, Ott E, Yorke JA (1983) The dimension of chaotic attractors. Physica D 7:153–180

    Article  Google Scholar 

  • Gray CM, Singer W (1989) Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proc Natl Acad Sci USA 86:1698–1702

    PubMed  Google Scholar 

  • Gray, CM, König P, Engel AK, Singer W (1989) Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338:334–337

    Article  PubMed  Google Scholar 

  • Hebb DO (1949) The organization of behavior: a neuropsychological theory. Wiley, New York

    Google Scholar 

  • Lutzenberger W, Elbert T, Birbaumer N, Ray WJ, Schupp HT (1992) The scalp distribution of fractal dimension of the EEG and its variation with mental task. Brain Topogr 5:27–34

    Article  PubMed  Google Scholar 

  • Osborne AR, Provenzale A (1989) Finite correlation dimension for stochastic systems with power-law spectra. Physica D 35:357–381

    MathSciNet  Google Scholar 

  • Packard NH, Crutchfield JP, Farmer JD, Shaw RS (1980) Geometry from a time series. Phys Rev Lett 45:712–716

    Article  Google Scholar 

  • Palm G (1987) Comments on the paper ‘Threshold regulation: a key to the understanding of the combined dynamics of EEG and eventrelated potentials’ by T. Elbert and B. Rockstroh. J Psychophysiol 4:335–337

    Google Scholar 

  • Preissl H, Aertsen AMHJ (1992) Reconstruction and characterisation of neuronal dynamics: How attractive is chaos? In: Aertsen A, Braitenberg V (eds) Information processing in the cortex: experiments and theory. Springer, Berlin Heidelberg, New York, pp 285–297

    Google Scholar 

  • Procaccia I (1988) Complex or just complicated? Nature 333:489–499

    Google Scholar 

  • Pulvermüller F, Preissl H, Eulitz C, Pantev C, Lutzenberger W, Elbert T, Birbaumer N (1994) Brain rhythms, cell assemblies, and cognition: evidence from the processing of words and pseudowords. Psycoloquy 5 (48)

  • Röschke J, Aldenhoff J (1991) The dimensionality of human's electroencephalogram during sleep. Biol Cybern 64:307–313

    Article  PubMed  Google Scholar 

  • Röschke J, Aldenhoff JB (1992) A nonlinear approach to brain function: deterministic chaos and sleep EEG. Sleep 15:95–101

    PubMed  Google Scholar 

  • Singer W (1994) Putative functions of temporal correlations in neocortical processing. In: Koch C, Davis J (eds) Large scale neuronal theories of the brain. MIT Press, Boston, Mass

    Google Scholar 

  • Skarda CA, Freeman WJ (1987) How brains make chaos in order to make sense of the world. Behav Brain Sci 10:161–195

    Google Scholar 

  • Smith LA (1988) Intrinsic limits on dimension calculation. Phys Lett A 133:283–288

    Article  Google Scholar 

  • Theiler J (1986) Spurious dimensions from correlation algorithms applied to limited time series data. Phys Rev A 34:2427–2432

    Article  PubMed  Google Scholar 

  • Vastano JA, Kostelich EJ (1986) Comparison of algorithms for determining Lyapunov exponents from experimental data. In: Mayer-Kress G (ed) Dimensions and entropies in chaotic systems. Springer, Berlin Heidelberg, New York, pp 100–107

    Google Scholar 

  • Young LS (1982) Dimension, entropy and Lyapunov exponents. Ergod Th Dynam Sys 2:109–124

    Google Scholar 

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Affiliations

  1. Institut für Medizinische Psychologie und Verhaltensneurobiologie, Universität Tübingen, Gartenstrasse 29, D-72074, Tübingen, Germany

    Werner Lutzenberger, Hubert Preissl & Friedemann Pulvermüller

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  1. Werner Lutzenberger
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  2. Hubert Preissl
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  3. Friedemann Pulvermüller
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Lutzenberger, W., Preissl, H. & Pulvermüller, F. Fractal dimension of electroencephalographic time series and underlying brain processes. Biol. Cybern. 73, 477–482 (1995). https://doi.org/10.1007/BF00201482

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  • DOI: https://doi.org/10.1007/BF00201482

Keywords

  • Time Series
  • Fractal Dimension
  • Reliable Estimate
  • Finite Time
  • Correlation Dimension