CHAOCITY AND DIMENSIONAL COMPLEXITY OF EEG-SIGNAL W.Klonowski, E.Olejarczyk, and R.Stepien Lab. of Biosignal Analysis Fundamentals, Institute of Biocybernetics and Biomedical Engineering, Polish Academy of Sciences, 4 Trojdena St., 02-109 Warsaw, Poland,
[email protected] Abstract: We tested for nonlinearity 16-channels EEG-data of 21 healthy human subjects by surrogate data method using S-map forecasting as a discriminant statistics, showing that in most cases one may not reject the null hypothesis that the signal was generated by a linear stochastic process. We also demonstrated that fractal dimension of EEG-signal in time domain works as a relative index of signal’s dimensional complexity and may be useful for doctors, e.g. in semi-automatic differentiation of sleep stages.
Introduction Human brain is an extremely complex system that generates complex electroencephalographic signals. An important question has been asked again and again – is brain chaotic? It cannot be fully stochastic (what in classical science was called “chaotic”) because then we would not be able e.g. to move our limbs purposely. It neither can be fully deterministic (in the classical mechanics’ sense) because then it could not be creative. Concept of deterministic chaos, as introduced in Nonlinear Dynamics, enables successful combination of the two notions that seems to contradict each other. Nonlinear Dynamics demonstrated that relatively simple systems can show extremely complicated dynamics, while very complex systems can in some circumstances show relatively simple dynamics. Human brain is (and it has to be) partially stochastic, partially chaotic system (cf. [1-3]). To be helpful for doctors the methods of signal analysis have to be simple and quick, so that they may be implemented on a typical PC and still work practically in real time on data that are noisy and highly non-stationary. While for a physicist words fractal dimension “ring the bell” of dimension in the system’s phase space, e.g. correlational dimension of the attractor, such methods are rather slow and too complicated for doctors, since they need embedding of EEGdata in a multidimensional phase space.
Embedding procedure needs establishing of at least two important parameters - embedding dimension and time delay - which may drastically differ from case to case; it also needs rather long (in theory infinite) stationary epochs of the considered signal. Meanwhile, the most interesting for doctors and neuro-scientists are just the points of nonstationarities. The method of dimensional complexity in time domain is quick, simple, and does not require setting up of parameters since data embedding in a phase space is not needed. At the same time it provides high data-compression, not in the sense that the original data could be retrieved from the transformed data, but in the sense that the transformed data act like bookmarks. Instead of skimming through hundreds and hundreds of screens, with 16 or more curves on each screen, doctors may scrutinize the transformed signal that is 80-100 times shorter to choose these fragments of the original signal that are “suspicious” and need further, more detailed consideration (cf. [4]). What is extremely important, dimensional complexity can be used as a relative quantitative index to characterize the signal, regardless of the nature of the underlying dynamics, i.e. even if the analyzed signal is not chaotic. Data acquisition Data for analysis were provided by the 1st Dept. of Psychiatry, Medical University of Warsaw. EEG-signals were collected according to international standard 10-20 from 16 channels [5] (Fig.2a) using DigiTrack™ system made in Poland by P.I.M. ELMIKO, Warsaw. The signals were filtered with a band-pass filter 0.5 - 70.0 Hz. Sampling frequency was 128 Hz. The subjects were healthy volunteers and persons suffering from insomnia.
Interpersonal differences in brain chaocity Recently we analyzed 16-channels EEG of 21 healthy human subjects under resting condition, performing the test for nonlinearity by surrogate data method [6] with the null hypothesis, H0 : “These EEG-data were generated by a linear stochastic process (LSP)” and with nonlinear prediction error (NPE) of either simple nonlinear prediction or of S-map forecasting (short for
Sequential Locally Weighted Global Linear Maps, SLWGLM [7]) as a discriminant statistic. The null hypothesis is rejected if the value of the discriminant statistic differs between the original data and the surrogate data (cf. [2,3]). What we observed is that in most cases the null hypothesis cannot be rejected, but the interpersonal differences in brain chaocity are really considerable (Fig.1).
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Categorized Histogram: SUBJECT x LSP Categorized by SUBJECT 16 14 12 10 8 6 4 2 0 16 14 12 10 8 6 4 2 0 16 14 12 10 8 6 4 2 0 16 14 12 10 8 6 4 2 0 16 14 12 10 8 6 4 2 0 Reject Accept
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Fig. 1. Verification of hypothesis H0 by S-map prediction for 21 healthy subjects. “Rejected” means that the EEG-signal observed on a certain channel is probably of nonlinear deterministic-chaotic nature. Subjects 16, 18, 19, 21 seem to be “more chaotic” than others. Dimensional complexity of sleep EEG Instead of using correlation dimension of a reconstructed system’s attractor, D2 , in the phase space, we prefer to use as a relative measure of complexity of EEG-signal fractal dimension calculated in time domain, Df , using e.g. the algorithm proposed by Higuchi (cf. [8,9]). Df characterizes complexity of the curve representing the signal on a plane, e.g. that of the line traced out on a moving paper tape by a pen of a classical signal recorder and so it always takes values between 1 and 2 – the greater is D (or, more precisely, the greater its fractional part is) the more complex is the signal under consideration. EEG-data are transformed into Df(t) curve, one local Df value for 100 data points, leading to ca.
100-fold compression of the original data. Df(t) correlates with power spectrum of the EEG-signal - for the intervals for which fractal dimension is below the average calculated for the whole signal, the power spectrum is moved towards lower frequencies and opposite [10]. The transformation is quick - the algorithm was implemented into our DigiTrack™ EEG-data viewer – original data as long as several hours are transformed into Df(t) (on a selected channel) in 30-50 sec. and the results are shown practically on one screen, on which the doctor can also see a short chosen fragment of the original signal, the signal’s power spectrum calculated for the same period, and the map of signal’s amplitude at a given moment.
We have demonstrated that Df can help to assess the effectiveness of therapy (cf. [4]) as well as to assess influence of magneto-stimulation on healthy subjects [11]. For EEG registered during sleep (together with other measured biosignals, like ECG, EOG, etc. called polysomnogram) Df(t) on a selected channel can be helpful in constructing so called hypnograms, [2]. The method is prone to signal artefacts [1].
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Results For computerized construction of hypnograms we propose to use fractal dimension of the current source density (CSD) EEG-signal (i.e. of the signal measured relatively to the average local value) on channel C3 (Fig.2b, cf. also [2,5]): C3CSD =[(C3-F3)+(C3-P3)+(C3-T3)+(C3-Cz)] /4
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Fig. 2. a. International 10-20 system viewed from vertex; b. Calculation of current source density – example of C3 CSD (after [5]). We calculated Higuchi’s fractal dimension of C3CSD signal of 15 healthy subjects [2] and 15 patients suffering with insomnia for each 4 secondlong epoch categorized by a specialist to one of five sleep stages or as waked. The results show that a relative index of signal’s dimensional complexity such as Df can be useful for doctors also in analysis of sleep EEG, e.g. in differentiation of sleep stages for a given person (Fig.3). Even a specialist cannot categorize stage 3 from stage 4 and stage 1 from REM based on EEG only. Conclusions We think that the difference between “chaotic” and “stochastic” is quantitative rather than
qualitative. A classical stochastic system would be one with an infinite number of degrees of freedom. If a system under consideration has many “irreducible” degrees of freedom it is still considered stochastic; if the number of degrees of freedom may be reduced to a few “really important” degrees of freedom, the system is considered to be deterministic-chaotic. In time domain, instead of speaking about degrees of freedom one should rather speak about subprocesses with different characteristic time scales. Nonlinear complexity quantifiers such as Higuchi’s fractal dimension can be used to characterize the signal regardless of the nature of the underlying dynamics, i.e. even if the signal is not chaotic.
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Fig. 3. EEG fractal dimension for 15 healthy subjects in wake state and different sleep stages References [1] W.Klonowski, E.Olejarczyk, and R.Stepien, “Nonlinear Dynamics - from Conformons to Human Brain”, Technology and Health Care 9, 87-90, 2001. [2] W.Klonowski, E.Olejarczyk, and R.Stepien, “Complexity of EEG-signal in Time Domain – Possible Biomedical Application”, in: Proceedings of Experimental Chaos 2001, the 6 th Experimental Chaos Conference, Potsdam, Germany, to be published. [3] R.Stepien, “Testing for Nonlinearity in EEG Signal of Healthy Subjects”, Acta Neurobiol. Exp. submitted. [4] W.Klonowski, E.Olejarczyk, and R.Stepien, “Nonlinear Quantifiers of EEE-signal Complexity”, in: Proceedings of 2000 International Symposium on Nonlinear Theory and its Application (NOLTA’2000) Dresden, Vol. 1, 261-264, 2000. [5] W.Szelenberger, Evoked Potentials (in Polish: "Potencjaly wywolane”), pp. 45-46, Wydawnictwo ELMIKO, Warsaw, 2001. [6] J.Theiler S.Eubk, A.Longin, B.Galdrikian and J.D.Farmer, “Testing for nonlinearity in time series: The method of surrogate data”, Physica D 58, 77, 1992.
[7] G.Sugihara, “Nonlinear forecasting for the classification of the natural time series”, Phil. Trans.R. Soc. Lond. A348, 477-495, 1994. [8] T.Higuchi, ”Approach to an irregular time series on the basis of the fractal theory”, Physica D, 31, 277-283, 1988. [9] A.Accardo, M.Affinito, M.Carrozzi, and F.Bouquet, ”Use of the fractal dimension for the analysis of electroencephalo-graphic time series”, Biol.Cybern. 77, 339-350, 1997. [10] W.Klonowski, J.Ciszewski, W.Jernajczyk, and K.Niedzielska, “Application of Chaos Theory and Fractal Analysis for EEGsignal Processing in Patients with Seasonal Affective Disorder”, in: Proceedings of 1999 International Symposium on Nonlinear Theory and its Applications (NOLTA'99), Hawaii, USA, Vol. 1, 339342, 1999. [11] W.Klonowski, E.Olejarczyk, and R.Stepien, “Nonlinear Dynamics of EEG-signal Reveals Influence of Magnetic Field on the Brain”, in: Proceedings Chicago2000 World Congress on Medical Physics and Biomed. Engineering, FR-A325-2, on CD (abstracts\4048-52180, short_papers\403852296); also http://www.ibib.waw.pl/~lbaf
