Dimensional complexity and spectral properties of the human sleep EEG
Introduction
First attempts to apply methods from nonlinear time series analysis to electroencephalograms (EEG) were carried out in the framework of the chaos hypothesis, i.e. it was assumed that the EEG within a particular psycho-physiological state could be described by a deterministic chaotic system and therefore could be characterized by invariant measures such as the fractal dimension of the corresponding attractor or as the Lyapunov exponents. In the case of sleep EEG the sleep stages were considered as distinct psychophysiological states.
One of the first publications in this framework was by Babloyantz et al. (1985). They applied the Grassberger-Procaccia (GP) algorithm (Grassberger and Procaccia, 1983a, Grassberger and Procaccia, 1983b) for the determination of the correlation dimension to time series of sleep EEG segments. They reported decreasing correlation dimensions from REM sleep to sleep stage 2 to slow wave sleep. Since then numerous studies appeared confirming this result on larger samples and with different data sets (Ehlers et al., 1991, Röschke and Aldenhoff, 1991, Achermann et al., 1994b, Pradhan et al., 1995). In addition to low correlation dimensions also positive Lyapunov exponents have been reported (Fell et al., 1993), indicating that the EEG may result from a low-dimensional chaotic process.
Dimension analysis in the framework of the chaos hypothesis was not only applied to characterize the sleep EEG, but to all kinds of psychophysiological conditions in normal as well as pathological states.
Already from the beginning these results have been questioned for several reasons: From a general point of view it seems very unlikely that such a complex system as the brain should produce an activity which can be described by low-dimensional dynamics. Moreover, if the existence of a sufficiently low-dimensional attractor is assumed, then for a reliable estimate of the fractal dimension a time series has to fulfill requirements such as stationarity, a sufficiently large number of data points and a reasonable signal to noise ratio (see e.g. Kantz and Schreiber, 1997). It is very unlikely that these requirements are simultaneously met in the case of the EEG.
Moreover, the interpretation of low values of the correlation dimension as a sign of deterministic chaos has been challenged by Osborne and Provenzale (1989): they showed that correlated noise exhibiting power law spectra P(f)∝f−β, may result in low values of the correlation dimension. Furthermore, Rapp et al. (1993) applied a low-pass filter to white noise and showed that the resulting time series produced low correlation dimensions. These studies demonstrated that spurious low correlation dimensions may already result from linear stochastic processes. To avoid these spurious estimates, Theiler (1986) proposed a correction to the original GP algorithm, which reduces the effects of linear correlations. He showed that with this correction the scaling regions that lead to low dimensions for colored noise such as presented by Osborne and Provenzale (1989) disappeared. A first application of this correction to EEG data was reported by Theiler and Rapp (1996). They re-examined a set of human EEG data recorded during rest and while performing simple tasks which has previously been reported to exhibit low values of the correlation dimension. Using this correction, the scaling region was reduced or disappeared totally, leading the authors to conclude that no convincing evidence for low-dimensional behavior could be found. A similar result was obtained by Schreiber (1999) for a data set used for epileptic seizure prediction by Elger and Lehnertz (1998).
Many authors dropped the chaos hypothesis and referred to their dimension estimates not as an absolute measure of a fractal dimension of a strange attractor but as a relative measure of dimensional complexity (see e.g. Pritchard and Duke, 1992, Rey and Guillemant, 1997, and references therein). However, without referring to an attractor dimension the interpretation of the dimensional complexity turned out to be equivocal.
The present work focuses on two problems: First, to what extent does the dimensional complexity (DC) really measure nonlinear properties in sleep EEG? This question is related to the more general problem of nonlinearity in the sleep EEG. To answer it we compared DC of the original data with those of surrogate data, i.e. time series which resemble the linear properties and the amplitude distribution of the original time series, but are otherwise random (see Theiler et al., 1992, Schreiber and Schmitz, 2000, and references therein). Second, we studied the relationship between different spectral features and DC in order to identify those spectral features which have the largest impact on the DC.
Section snippets
EEG data
The sleep EEG of 4 healthy young right handed males (aged 23–25 years) was analyzed. The data are 8-h baseline recordings (23:00–07:00 h) of a previous study (Endo et al., 1998) performed in the sleep laboratory of the Institute of Pharmacology and Toxicology, University of Zurich. The EEG signals from the C3-A2 derivation were conditioned by the following analog filters: a high-pass filter (−3 dB at 0.16 Hz), a low-pass filter (−3 dB at 102 Hz, <−40 dB at 256 Hz), and a notch filter (50 Hz).
Results
Fig. 2 shows the sleep profile and the time courses of different measures for one all-night EEG. The averaged DC of the surrogate data is denoted by , while in the following the dimensional complexity of the original segment is labeled DC. The EEG recording was subdivided into consecutive, non-overlapping epochs of 1 min duration. The curves of all 4 EEG measures exhibit cyclic changes which coincide with the non-REM/REM sleep cycles. When Pδ and α are of highest values during stage 4,
Discussion
Our analysis showed that the variations of the dimensional complexity (DC) of the sleep EEG in the course of a night can be explained to a large extent by linear properties of the EEG time series. The surrogate data analysis revealed only weak differences between DC values of the original data and their surrogates. In sleep stage 2 only, we found a considerable number of segments with a nonlinear signature. We were able to identify two distinct processes which are responsible for these
Acknowledgements
We thank R. Badii and R. Füchslin for fruitful discussions and H.P. Landolt for comments on the manuscript. This work was supported by the Helmut Horten Stiftung, the Swiss National Science Foundation and the University of Zurich.
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