20
Views
17
CrossRef citations to date
0
Altmetric
Original Article

Nonlinear Dynamical Aspects of the Human Sleep EEG

&
Pages 109-129 | Received 18 Oct 1993, Published online: 07 Jul 2009

References

  • Achermann P., Hartmann R., Gunzinger A. Dimensional analysis of all night sleep EEG data. APSS annual meeting. 1993, abstract 47
  • Babloyantz A., Nicolis C., Salazar U. Evidence of chaotic dynamics. Physics Letters A 1985; 111: 152–156
  • Babloyantz A., Destexhe A. Low-dimensional chaos in an instance of epilepsy. Proceedings of the National Academy of Sciences USA 1986; 83: 3513–3517
  • Basar E. EEG-brain dynamics. Relation between EEG and brain evoked potentials. Elsevier/North-Holland., Amsterdam 1980
  • Chaos in brain function, E. Basar. Springer, Berlin 1990
  • Bingham S., Kot M. Multidimensional trees, range searching, and a correlation dimension algorithm of reduced complexity. Physics Letters A 1989; 140: 327
  • Braun S. Berechnung der Korrelationsdimension aus EEG-Daten. Mainz. 1993, M.Sc. thesis
  • Dvorak I. Taken versus multichannel reconstruction in EEG correlation exponent estimates. Physics Letters A 1990; 151: 225–233
  • Eckmann J. P., Kamphorst S. O., Ruelle D., Ciliberto S. Lyapunov exponents from time series. Physical Review A 1986; 34: 4971–4979
  • Ehlers C. L., Havstad J. W., Garfinkel A., Kupfer D. J. Nonlinear analysis of EEG sleep states. Neuropsychopharmacology 1991; 5: 167–176
  • Farmer J. D., Ott E., Yorke J. A. Dimension of chaotic attractors. Physica D 1983; 7: 153–180
  • Feigenbaum M. J. Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics 1978; 19: 25–52
  • Fell J., Röschke J., Beckmann P. Deterministic chaos and the first positive Lyapunov-exponent: A nonlinear analysis of the human electrocephalogram during sleep. Biological Cybernetics 1993; 69: 139–146
  • Frank G. W., Lookman T., Nerenberg M. A. H., Essex C., Lemieux J., Blume W. Chaotic time series analyses of epileptic seizures. Physica D 1990; 46: 427–438
  • Gallez D., Babloyantz A. Predictability of human EEG: a dynamical approach. Biological Cybernetics 1991; 64: 381–391
  • Goldbeter A., Lefever R. Dissipative structures for an allosteric model. Application to glycolytic oscillators. Biophysical Journal 1972; 12: 1302–1315
  • Graf K. E., Elbert T. Dimensional analysis of the waking EEG. Brain dynamics. Progress and perspectives, E. Basar, T. Bullock. Springer, Berlin 1989; 174–191
  • Grassberger P., Proccacia I. Characterization of strange attractors. Physical Review Letters 1983; 50: 346
  • Grassberger P. An optimized box-assisted algorithm for fractal dimensions. Physics Letters A 1990; 148: 63
  • Grassberger P., Schreiber T., Schaffrath C. Nonlinear time sequence analysis. International Journal of Bifurcation and Chaos 1991; 1: 521–547
  • Haken H. At least one Lyapunov-exponent vanishes, if the trajectory of an attractor does not contain a fixed point. Physics Letters A 1983; 94: 71–72
  • Iasemidis L. D., Sackellares J. C. The evolution with time of the spatial distribution of the largest Lyapunov-exponent on the human epileptic cortex. Measuring chaos in the human brain, W. S. Pritchard, D. W. Duke. World Scientific, Singapore 1991; 49–82
  • Kaplan J. L., York J. A. Chaotic behavior of multidimensional difference equations. Lecture Notes in Mathematics. Springer, New York 1979
  • Krystal A. D., Weiner R. D. The largest Lyapunov-exponent of the EEG in ECT seizures, W. S. Pritchard, D. W. Duke. World Scientific, Singapore 1991; 113–127
  • Lutzenberger W., Birbaumer N., Flor H., Rockstroh B., Elbert T. Dimensional analysis of the human EEG and intelligence. Neuroscience Letters 1992; 143: 10–14
  • Lorenz E. N. Deterministic nonperiodic flow. Journal of Atmosphere Sciences 1963; 20: 130
  • Mandelbrot B. The fractal geometry of nature. Freeman, New York 1982
  • Manneville P., Pomeau Y. Intermittency and the Lorenz model. Physics Letters A 1979; 75: 1
  • Mayer-Kress G., Holzfuß J. Analysis of the human electroencephalogram with methods from nonlinear dynamics. Temporal disorder in human oscillatory systems, L. Rensing, U. van der Heiden, M. C. Mackey. Springer, Berlin 1987; 57–68
  • Nan X., Jinghua X. The fractal dimension of EEG as a physical measure of conscious human brain activation. Bulletin of Mathematics and Biology 1988; 50: 559–565
  • Newhouse S., Ruelle D., Takens F. Occurrence of strange axiom-a attractors near quasiperiodic flow on Tm, m ≤ 3. Communications in Mathematics and Physics 1978; 64: 35
  • Osborne A. R., Provencale A. Finite correlation dimension for stochastic systems with power-law spectra. Physica D 1989; 35: 357–381
  • Parker T. S., Chua C. O. Practical Numerical Algorithms for Chaotic Systems. Springer, New York, Berlin 1989
  • Parlitz U. Identification of true and spurious Lyapunov exponents from time series. International Journal of Bifurcation and Chaos 1991; 2: 155–165
  • Pesin Y. B. Characteristic Lyapunov-exponents and smooth ergodic theory. Russian Mathematical Surveys 1977; 32: 55
  • Pijn J. P., Van Neerven J., Noest A., Lopes da Silva F. H. Chaos or noise in EEG signals; dependence on state and brain site. Electroencephalography and clinical Neurophysiology 1991; 79: 371–381
  • Poincaré H. La science et l'hypothese. Gauthier-Villars, Paris 1903
  • Principe J. C., Lo P. C. Towards the determination of the largest Lyapunov-exponent of EEG segments. Measuring chaos in the human brain, W. S. Pritchard, D. W. Duke. World Scientific, Singapore 1991; 156–166
  • Pritchard W. S., Duke D. W. Dimensional analysis of no-task human EEG using the Grass-berger-Procaccia method. Psychophysiology 1992; 29: 182–192
  • Rapp P. E., Albano A. M., Mees A. I. Calculation of correlation dimensions from experimental data. Progress and Problems. Dynamic patterns in complex systems, J. A. S. Kelso, A. J. Mandell, M. F. Schlesinger. World Scientific, Singapore 1988; 191–205
  • Ray W. J., Wells R., Elbert T., Lutzenberger W., Birbaumer N. EEG and chaos: dimension estimation of sensory and hypnotic processes. Measuring chaos in the human brain, W. S. Pritchard, D. W. Duke. World Scientific, Singapore 1991; 199–215
  • Rechtschaffen A., Kales A. A manual of standardized terminology, technics and scoring system for sleep stages of human subjects. US Government Printing Office, Washington, D.C. 1968, Public Health Service, NIH Publication No. 204
  • Renyi A. Probability Theory. North Holland., Amsterdam 1971
  • Röschke J., Basar E. Is the EEG a simple noise or a “strange attractor”?. Pflügers Archives 1985; 405: R45
  • Röschke J., Aldenhoff J. B. The dimensionality of human's electroencephalogram during sleep. Biological Cybernetics 1991; 64: 307–313
  • Röschke J. Strange attractors, chaotic behaviour and informational aspects of sleep-EEG data. Neuropsychobiology 1992; 25: 172–176
  • Röschke J., Fell J., Beckmann P. The calculation of the first positive Lyapunov-exponent in sleep EEG data. Electroencephalography and clinical Neurophysiology 1993; 86: 348–352
  • Rössler O. E. The chaotic hierarchy. Zeitschrft für Naturforschung 1979; 389: 788–801
  • Skarda C., Freeman W. Chaos and the new science of the brain. Concepts in Neuroscience 1990; 1: 275–285
  • Soong A. C. K., Stuart C. I. J.M. Evidence of chaotic dynamics underlying the human alpha-rhythm electroencephalogram. Biological cybernetics 1989; 62: 55–62
  • Takens F. Detecting strange attractors in turbulence. Lecture notes in mathematics (898). Springer, Berlin 1981
  • Theiler J. Spurious dimensions from correlation algorithms applied to limited time-series data. Physical Review A 1986; 34: 2427
  • Vastano J. A., Kostelich E. J. Comparison of algorithms for determining Lyapunov exponents from experimental data. Dimensions and Entropies in Chaotic Systems, G. Mayer-Kress. Springer, Berlin 1986; 100–107
  • Wolf A., Swift J. B., Swinney H. L., Vastano J. A. Determining Lyapunov exponents from a time series. Physica D 1985; 16: 285
  • Wolf A., Bessoir T. Diagnosing chaos in the space circle. Physica D 1991; 50: 239–258

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.