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NEW METHODS OF NONLlNEAR AND SYMBOLIC DYNAMICS IN SLEEP EEG-SIGNAL ANALYSIS
W.Klonowski'*, E. Olejarczykl, R. Stepienl and W. Szelenberge.-z J Institute ofBiocybernetics and Biomedical Engineering, Polish Academy ofSciences, 4 Trojdena St., 02-109 Warsaw. Poland, *
[email protected]. waw.pl 1 Department ofPsychiatry, Medical University of Warsaw, 27 Nowowiejska St., 00-665 Warsaw. Poland. wald@psych. wawpl
Abstract: We demonstrate that the methods of time series analysis supplied by Nonlinear Dynamics and by Symbolic Dynamics are applicable to sleep EEG-data analysis and may in future be used for computerized construction of hypnograms that are important tools in sleep disorder diagnosis. Keywords: Biomedical systems, Chaos theory, Fractals, Nonlinear analysis, Signal processing.
I. INTRODUCTION
these polysornnogram were collected according to standard 10-20 system from 16 channels, filtered with a bandpass filter 0.5 - 70.0 Hz and sampled with 128 kHz.
There exist serious needs for new methods in digital signal and image processing sufficiently simple that Medical Doctors may find these methods appropriate for biomedical applications. To satisfy these needs Medicine may benefit from methods developed in Nonlinear and Symbolic Dynamics (cf. Klonowski et aI., 2001; Klonowski, 2002a; Klonowski, 2002b; Wessel et aI., 2002; Zebrowski, 2002), in particular the methods analyzing time series in time domain.
2. HIGUCHI'S FRACTAL DIMENSION Higuchi's fractal dimension, Or , is calculated directly from the time series, without embedding the data in a phase space. It is, in fact, fractal dimension of the curve representing the signal under consideration in time domain, and so it is always between I and 2, since a simple curve has, of course, dimension equal 1 and a plane has dimension equal 2. The fractional part of Dr is a measure of the curve (and so of the signal it represents) complexity (cf. Higuchi, 1988; Accardo et aI., 1997; Blaszczyk and Klonowski, 2001; Klonowski et al.. 2001b; Klonowski et al.. 2002, Klonowski, 2002b). Dr should not be misled with fractal dimension of an attractor in the system's phase space.
We consider here two such methods - calculating Higuchi's [ractal dimension of the signal and an entropic method applied to symbolically encoded signal derivative. We analyze sleep EEG-signal and demonstrate how these two methods show the subjects passing quasi-periodically during normal sleep through different states - awake, stages 1, 2, 3, 4, and REM (Rapid Eye Movement). Data for analysis were provided by the 1st Dept. of Psychiatry, Medical University of Warsaw (Chair Prof. W.Szelenberger). Polysomnograms were collected using data acquisition system made in Poland by P.I.M. ELMIKO, Warsaw. EEGsignals in
EEG-signal complexity measured by Dr depends on the state of brain and so it changes due to different brain pathologies, e.g. epileptic seizures, as well as to physiological shifts, e.g. due to the stage of sleep.
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2.0
r---------------------------------,
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+--------------------------------1
1.1
~~~~~"""".;::,...._--------------------------I
ID
+------------_--
--' stage
Fig. 1. Higuchi's fractal dimension for 15 insomniacs. Abscissa: 0 - awake; 5 - REM corresponding sleep stages. Ordinate: Dr from I to 2 (every 0.2).
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sleep; 1,2,3,4
~-++I"----l-II---++---__+#____rlJ_----_+__HH*__jr_-_I,;__1 ....-___1
1.0
.... c '.4
time [hours]
Fig.2. Higuchi's fractal dimension shows quasi-periodicity of a healthy person's sleep: one channel whole night EEG-signal (abscissa - ca. 6.5 hours) was transformed into Dt
O+----.~,__---.-
-50
o
50
100
150
200
250
300
350
400
t[min] Fig. 4. Distribution of the symbol "0" (corresponding to 8-bit sequence [00000000) in the binary series representing signal derivative) in the whole symbolic series encoding the same all-night EEG-signal (about 6.5 hours) Klonowski, W. (2002b). Chaotic dynamics applied to signal complexity in phase space and in time domain. Chaos. Solitons and Fractals. 14, 1379-1387.
4. CONCLUSIONS Application of methods borrowed from Nonlinear Dynamics and from Symbolic Dynamics to EEGsignal analysis, aimed at more efficient and more user-friendly computer-assisted diagnosis of sleep disorders, seems very promising. The simplest method - algorithm calculating Higuchi's fractal dimension of EEG-signal in time domain - has already been implemented in the data acquisition / data analysis system DigiTrack™ made commercially by P.I.M. ELMIKO.
Klonowski, W., E.Olejarczyk and R Stepien (2002). Complexity of EEG-signal in Time Domain - Possible Biomedical Application. In: Experimental Chaos. AlP Conference Proceedings (Boccaletti, S., BJ. Gluckman, 1. Kurths, L.M. Pecora and M.L. Spano, Eds.), Vol. 622, pp. 155-160,. Melville, New York. Klonowski, W., E. Olejarczyk and R Stepien (2001a). Nonlinear dynamics. From conforrnons to human brain. Technology and Health Care 9,87-89; also http://www.ibib.waw.pl/-lbaf
ACKNOWLEDGEMENTS This work was partially supported by the State Committee for Scientific Research (KBN) grant NO.4 T11 F 01922 and IBBE PAS project St/24/03.
Klonowski, W., E.Olejarczyk and R Stepien (2001 b). Chaocity and Dimensional Complexity of EEG-Signal. In: Proceedings of 2001 International Symposium on Nonlinear Theory and its Applications (NOLTA 2001), Miyagi, Japan, vol. 2, pp. 399-402; also http://www.ibib.waw.pl/-lbaj
REFERENCES Accardo, A., M. Affinito, M. Carrozzi and F.Bouquet (1997). Use of the fractal dimension for the analysis of electroencephalographic time series. Biol.Cybern.77, 339-350.
Wessel, N, U. Schwarz, PJ. Saparin and J. Kurths (2002). Symbolic Dynamics for Medical Data Analysis. In: Attractors. Signals. and Synergetics. (W. Klonowski, Ed.). Frontiers on Nonlinear Dynamics Vol. I, pp. 45-61. Pabst Science Publishers, Lengerich, Berlin.
Blaszczyk J.W. and W. Klonowski (2001). Postural stability and fractal dynamics. Acta Neurobiol. Exp. 61, 105-112; Erratum ibid. p. 327. Higuchi, T. (1988). Approach to an irregular time series on the basis of the fractal theory. Physica D, 31, 277-283.
Zebrowski, JJ. (2002). Types of Dynamical Order and some Physiological Processes. In: Attractors. Signals. and Synergetics. (W. Klonowski, Ed.). Frontiers on Nonlinear Dynamics Vol. 1, pp. 243-255. Pabst Science Publishers, Lengerich, Berlin.
Klonowski, W. (2002a). Nonlinear Dynamics From Micro- to Macro-Cosmos. In: Attractors. Signals. and Synergetics. (W.Klonowski, Ed.). Frontiers on Nonlinear Dynamics Vol. I, 1115. PabstScience Publishers, Lengerich, Berlin.
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