ISSN 1320-0682
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ISSN 1320-0682 |
| Source: | http://www.complexity.org.au/ci/vol05/watters/watters.html | Received: | 19/01/1998 | ||
| Vol 5: | Copyright 1998 | Accepted for publication: | 06/05/1998 |
Many analyses of the EEG signal using frequency decomposition methods
rest on the assumption that the fundamental signal-generating process has
stochastic properties (i.e., no long-range temporal correlations, or temporal
self-similarity). In this paper, the detrended fluctuation analysis (DFA)
algorithm of Peng et al. (1992) was used to determine whether the locally-detrended
EEG contained long-range correlations, indicating fractal structure, or
whether it was best characterised as a stochastic random-walk process.
If the EEG was generated by a random walk process, then an estimated scaling
parameter was predicted to be 
=0.5, whereas
the scaling parameter for a fractal process would lie in the range 0.5<
1.5. In this study, highly significant
departures from =0.5 were observed from an
ensemble of 32 EEG signals, 
()=1.26,
=0.06, indicating that the EEG samples
could not have been generated by a random walk process. This result also
suggests that the EEG is generated by a fractal process which contains
long-range temporal correlations of the order 1/f1.52.
The view of the EEG as a largely stochastic system was challenged during the 1980's with the publication of both data analysis (Babloyantz & Destexhe 1986) and models (Freeman 1987) of the EEG which were inspired by so-called "chaos theory". Support for the "chaos in the brain" was drawn from a number of sources, including the statistical properties of fractal and power-law systems, and the dynamics of chaotic systems, to form an approach known broadly as the "dynamical systems" study of EEG (see Watters 1999 for a review). In this paradigm, the EEG is usually reconstructed using the method of delays (Packard et al. 1980) into a higher-dimensional state space, with the correlation dimension computed using the Grassberger and Procaccia (1983) algorithm (or a faster box-counting equivalent; Theiler 1987) as an empirical measure of signal complexity. These dimensions are then typically associated with particular qualitative brain states, such as sleep (e.g., Roeschke & Aldenhoff 1992), or with quantitative performance on cognitive or perceptual tasks. For example, Stam et al. (1995) found that the EEG correlation dimension increased during arithmetic performance compared to an eyes-closed control condition, whilst Rapp et al.(1989) argued that any cognitive processing should increase the complexity of EEG. Alternatively, Gregson et al. (1993) found that EEG correlation dimension was highly correlated with the complexity of load in a visual scanning task.
Although the individual hypotheses varied greatly across these studies, the general view put forward was that EEG dynamics could not be attributed solely or even largely to random processes, since analyses of similar experiments based on these kinds of assumptions often supported completely orthogonal theoretical positions (such as the synchronisation/desynchronisation paradox). However, the use of time delay embedding and associated dynamical systems techniques has also proved to be problematic for making direct comparisons to the power spectrum of unidimensional EEG time series. In addition, the finding of higher-order dynamics does not explain the well-established noisy and often non-stationary characteristics of the EEG signal. One possible alternative to the dynamical systems methodology, which facilitates direct comparisons with stochastic models such as random walks, is to examine the fractal scaling properties in the EEG signal. Evidence of temporal self-similarity could explain both noisy characteristics which do not arise from artefacts, and/or fractal structure as estimated from scaling properties. In general, such scaling properties can be expressed by examining the power spectrum:
(1)
where 
is spectral density, and
is frequency. Scaling properties, especially of the strictly 1/
kind, are thought to be important in nature, from understanding the spatial
dynamics of vegetative ecosystems, to precisely measuring the length of
coastlines. An excellent review of this material is found in (Hastings
& Sugihara 1993). Although fractals in these examples were originally
conceived spatially, the theory of fractals has been successfully generalised
to measure temporal self-similarity.
Fractal and random-walk processes have much in common, such as the distribution of changes in the amplitude of a measured time series being gaussian. However, random walk processes have a single feature which distinguishes them from fractal processes - they exhibit time-proportional variance. Determining whether a time series exhibits time proportional variance can therefore be used as a criterion for deciding whether an EEG sample is generated by a random walk or fractal process.
However, there are significant empirical obstacles to distinguishing fractal from stochastic properties in the EEG, especially with respect to discriminating noise resulting from biological and recording artefact, and noise which forms an intrinsic part of the brain’s dynamics (Rapp 1993). Assumptions of stationarity required for the correct use of many algorithms are often ignored, much to the chagrin of statisticians (although the same criticism applies to the use of linear-stochastic methods). In addition, the use of surrogate data tests (Theiler 1995), where signal frequencies are period or phase-randomised to remove temporal structure, such as epileptic spike-and-wave sets, have been shown to produce similar correlation dimensions to the original signal under certain experimental conditions. This finding has been used as evidence for the EEG being generated by a stochastic process, however, it should be pointed out that Theiler only considered epileptic and not normal EEG.
Improvements in experimental verification of the scaling properties of EEG signals has recently been made possible by the development of an algorithm (Peng et al. 1994) which is far less sensitive to non-stationarity in time than other methods which are generalisations of geometric box-counting. It is largely on the grounds of the sensitivity of dimension estimating algorithms to noise that researchers who favour the stochastic view of the EEG have been able to dismiss many studies indicating power law properties and/or nonlinear dynamics (e.g., Rapp 1993). To illustrate the point, Figure 1 shows a sample recording from the central cortical site Cz. Cyclic activity is evident, as are local, apparently noisy deviations, as well as potential artefact from eye movement, or some other high-amplitude source (at approximately 9s). The whole question of correctly interpreting high-amplitude deviations as important signal features or as noise-related artefacts is therefore non-trivial.
(2)
where F(L) is the average fluctuation which usually increases linearly
with L. The scaling exponent can then be approximated
as the slope log of log(F(L)) against log(L). This gives the scaling relation:
(3)
The scaling exponent should therefore be
able to characterise any significant correlation properties of the EEG
signal. If the EEG signal was completely pointwise uncorrelated, like a
random walk not depending on past history, then the scaling exponent would
be strictly =0.5. Of course, there may be
some short-range correlations which would suggest a non-random walk process,
but if scaling exists over large box sizes, these spurious correlations
would disappear. Finding scaling exponents in the range
0.5< 1.0 would indicate long-range power-law correlations of the kind which are ubiquitous in nature. However, for the range 1.0<
1.5, the EEG signal would still have long-range correlations, but would approach the smoothness of Brownian noise, as characterised by the range 1.5<2.0 (Peng et al. 1992).
In terms of spectral density, the scaling exponent
is related to by the relation:
(4)
Thus, the special case of 1/f white noise, with
=1 and =1, is often viewed as a compromise
between the absence of long-range correlation, which characterises a random
walk, and the smoother pointwise correlation of Brownian noise. The DFA
method thus has two distinct advantages with respect to EEG analysis: firstly,
long series may be analysed without regard to the stationarity concerns
associated with traditional spectral methods, since the spurious detection
of long-range correlations in a series with superimposed non-stationary
trends is avoided. Secondly, it facilitates the reliable distinction of
different types of noise, based on scaling parameter ranges, which may
be important for understanding the origin of the EEG, if the process is
generated by stochastic-linear or non-linear-chaotic mechanisms.
If long-range correlations exist in the EEG, and if we control
for non-stationarity, an ensemble average over EEG samples should result
in significantly positive departures from
=0.5. This fact suggests the null hypothesis of this study, which is that
the EEG is a random walk with
0.5. In order to test the null hypothesis, EEG segments of 10 seconds duration
were recorded from eight subjects, from the frontal sites Fz, F3, and F4,
and the central site Cz (total N=32). Recordings were made in the range
3-100 Hz so that comparisons could be made to the range covered by typical
spectral bands of interest (e.g., alpha, 8-12 Hz), with a high sampling
rate of 500 Hz (i.e., 5000 data points per recording). These signals were
then analysed using DFA, and an ensemble average computed. The DFA was
computed using a C program supplied by Totts Gap Research, U.S.A and was
compiled on a SCO-UNIX workstation. The mean scaling exponent for the 32
samples was (
)=1.26, with standard deviation of 0.06. This average represents a significant
positive departure from =0.5, and implies
a mean spectral density of =1.52. Therefore,
on the basis of the samples presented, the null hypothesis can be rejected.
This rejection is made with some certainty, as the estimates computed in
the study possessed surprisingly little intra-group variability, given
the spatial distribution across scalp locus and individual differences.
In addition, the scaling exponent is in a range which is equivalent to
those found in other biological systems where 
phenomena are ubiquitous (e.g., long-range correlations in non-coding nucleotide
segments; Li & Kaneko 1992).
The results of this study indicate that, when controlling for non-stationarity,
there are non-trivial long-range correlations within the EEG signal, which
are incompatible with a random walk description. The fact that the EEG
contains fractal structure, whose characteristics can be distinguished
from a statistical model of the EEG as a random walk, has important implications
for the way that the EEG is analysed using spectral decomposition. However,
it should be noted that the EEG under eyes-closed rest, as examined here,
exhibited relative homeostasis, with long-range correlations of
=1.26, rather than =1. Further studies need
to determine whether, during other experimental conditions such as psychopharmacological
manipulation, strictly 1/f scaling would be observed. Certainly, the scaling
exponents computed here suggest that the EEG is more stable than the highly
chaotic activity observed at the single neurone level, and so the view
that Wright and Liley (1996) have advanced regarding the emergence of macroscopic
stability resulting microscopic nonlinearity might have some merit.
In addition, the finding of significant temporal self-similarity may explain some of the more robust results using experimental designs recently published in the "dynamical systems" approach to EEG analysis as outlined above. One example is the finding of a linear+quadratic dose-response relationship between caffeine and the correlation dimension of the EEG (Watters, Martin & Schreter 1998), where no consistent relationship between caffeine dosage and EEG power had previously been found using spectral methods in any of the major spectral bands. Across four different types of simple cognitive tasks, the correlation dimension model was able to explain an average 30% of the variance in the EEG data, compared to just 1% for a linear fit. When we consider that low-dimensional structure exists in state-space reconstructed EEG, in combination with the findings of fractal scaling presented in this study, the suggestion that the EEG could be generated by a random walk or similar stochastic process does not appear to be tenable. Of course, a more comprehensive study would require recordings to be made from more spatially-distant sites, perhaps over the entire scalp, to be able characterise potential differences in the spatial distribution of EEG complexity. However, as a pilot study, these results presented here are encouraging. Further studies are needed to determine whether the fractal characteristics observed are an invariant property of the signal and its dynamics, or subject to the same change over time as methods based on spectral decomposition.
A number of criticisms have been raised regarding the DFA method, particularly with respect to its original field of application in identifying long-range correlations in patchy nucleotide sequences (e.g., Voss 1992). However, the method has provided a successful way to overcome non-stationarity in the signal, and have obviously yielded useful results for this limited sample in line with our predictions. The DFA method is currently being improved by its authors, with Viswanathan et al. (1997), for example, introducing a two-parameter model, which involves accounting for variations not only over different spatial scales, but also sample size, which may improve upon the current single-parameter technique. This method will be applied to EEG analysis in a future paper.
Thus, the results of the present study indicate that the dynamics of
the EEG cannot be attributed largely to random processes, and that a persistent
process exhibiting temporal self-similarity must be accounted for. Of course,
this view must be reconciled with the many important empirical and theoretical
results associated with a stochastic view of EEG-generation, which is a
formidable task. This may prove possible with a more formal account of
the relationship between stochastic and fractal processes, but such an
account is beyond the scope of this present work. We feel that such a comparison
might be fruitfully achieved by examining a smaller bandwidth signal, perhaps
in the individual frequency ranges which are of interest to stochastic
researchers. In a future study, it is hoped to use the examine the relative
importance of frequency and amplitude information within the EEG signal
with respect to its fractal and/or stochastic characteristics, in the context
of competing claims from experimenters and clinicians over the importance
of each component respectively (for example, by the use of zero-crossing
segmentations to analyse frequency independently of signal amplitude).
The assistance of Professor James J. Wright and Dr. Frances Martin in
the preparation of this paper, and of Dr. Zoltan Schreter in data collection,
is gratefully acknowledged. The data reported in this study was collected
while the author was at the Department of Psychology, University of Tasmania,
with approval from the University of Tasmania Ethics Committee under the
terms of the Declaration of Helsinki, 1964. The author is now at the Department
of Computing, Macquarie University NSW 2109 AUSTRALIA.
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