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Chinese Science Bulletin 2006 Vol. 51 No. 24 3059—3064 DOI: 10.1007/s11434-006-2213-y

A new measure to character-ize multifractality of sleep electroencephalogram MA Qianli1,2, NING Xinbao1, WANG Jun2 & BIAN Chunhua1

1. State Key Laboratory of Modern Acoustics, Department of Electronics Science and Engineering, Institute for Biomedical Electronic Engi-neering, Nanjing University, Nanjing 210093, China;

2. Province Key Laboratory of Image Processing & Image Communica-tion, College of Telecommunications and Information Engineering, Nanjing University of Posts & Telecommunications, Nanjing 210003, China

Correspondence should be addressed to Ning Xinbao (email: xbning@ nju.edu.cn) Received January 23, 2006; accepted April 7, 2006

Abstract Traditional methods for nonlinear dy-namic analysis, such as correlation dimension, Lyapunov exponent, approximate entropy, detrended fluctuation analysis, using a single parameter, cannot fully describe the extremely sophisticated behavior of electroencephalogram (EEG). The multifractal for-malism reveals more “hidden” information of EEG by using singularity spectrum to characterize its nonlin-ear dynamics. In this paper, the zero-crossing time intervals of sleep EEG were studied using multifractal analysis. A new multifractal measure Δasα was pro-posed to describe the asymmetry of singularity spec-trum, and compared with the singularity strength range Δα that was normally used as a degree indi-cator of multifractality. One-way analysis of variance and multiple comparison tests showed that the new measure we proposed gave better discrimination of sleep stages, especially in the discrimination be-tween sleep and awake, and between sleep stages 3 and 4.

Keywords: sleep, electroencephalogram, multifractality.

The research of nonlinear dynamics in electroen-cephalogram (EEG) has made much headway in recent years. Nonlinear analysis methods have been success-fully applied to the studies of brain functions and pathological changes in EEG[1―3]. These studies also proved that EEG exhibited at least partly chaotic char-acteristics. The detrended fluctuation analysis (DFA)[4], which has been widely used recently, revealed the long-

range power-law correlation in EEG, indicating time scale invariant and fractal structure[5,6]. However, EEG is also rather noisy, displaying short-term decorrelation like white noise, and consequently, the EEG has been traditionally considered as a linear stationary random process. The paradoxical combination of short-term decorrelation and long-range correlation, stochastic and deterministic suggests that a single nonlinear parameter, such as largest Lyapunov exponent, correlation dimen-sion, fractal dimension, scaling exponent, etc., may not be able to fully characterize the “stochastic chaos” (as named by Freeman[7]) nature of EEG.

The long time behavior of chaotic, nonlinear dy-namic systems can often be characterized by (mono) fractal or multifractal measures. Monofractals are ho-mogeneous in the sense that they have the same scaling properties, characterized by a single singularity expo-nent throughout the entire signal. In contrast, multi-fractals can be decomposed into many (possibly infinite) sub-sets characterized by different exponents. Multi-fractal signals are intrinsically more complex and in-homogeneous than monofractals. Multifractal models have been used to account for scale invariance proper-ties of various objects in very different domains rang-ing from the energy dissipation or the velocity field in turbulent flows[8] to underlying hierarchical structure in proteins[9]. Physiologic signals generated by complex self-regulating systems, such as heartbeat interval, electrocardiogram, gait etc. have been proven to be multifractal, and the degree of multifractality often re-lates to pathological state or natural aging process[10–12]. The multifractal formalism has been applied to EEG analysis recently[13,14]. EEGs taken from healthy sub-jects and epilepsy patients, during imaginary and real visual-motor tracking tasks, and from different re-cording sites have been studied, and the results suggest that the multifractal formalism might be a good method for characterizing EEG dynamics.

In most of the present studies, nonlinear analysis was applied directly to the EEG amplitude fluctuation time series acquired by discrete-time sampling. However, EEG is a typical signal which is particularly susceptible to biological and experimental artifacts that manifest themselves as large amplitude deviations, and this leads to a questioning of the validity of techniques based on generalizations of geometric box counting, such as various approaches of the multifractal analysis. In this paper, we used the zero-crossing time intervals (CTI),

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which were derived from zero-crossings of EEG time series, to study the multifractality of EEG.

1 Zero-crossing time interval series of EEG Zero-crossing series is a frequency-based series, de-

fined as the set of zero-crossing events. Zero-crossings are robust to amplitude artifacts. And in addition, be-cause it contains only frequency information, it is use-ful to distinguish the contributions from frequency and/ or amplitude separately and to understand whether the frequency profile alone has fractal characteristics, in-dependent of amplitude changes. Zero-crossings have been used historically to automate EEG analysis over long time-periods, because they can clearly distinguish low frequency and high-frequency activity by measur-ing the time between each crossing with respect to the frequency band of interest[15].

However, despite the long research history, using zero-crossings to study nonlinearity in EEG dynamics appears to be scarce. Watters and Martins[6] used DFA to analyze the “EEG walk” constructed from the zero-crossings of EEG. They observed a mean scaling exponent across all subjects and sites of α = 0.67, which indicates time scale invariance and fractal struc-ture of EEG zero-crossings. In the present study, to adapt to multifractal analysis, we used another series which was also constructed from zero-crossings.

Let the EEG be x(t). The zero-crossing time is the level set {ti, x(ti) = 0} where the index i registers the order of the zero crossing event. In practice, {ti} is first determined by linear interpolation and then used to de-fine the set of crossing time intervals (CTI) C = {τi = ti+1 − ti}. Fig. 1 shows the amplitude fluctuations, zero-crossings and CTI series of a segment of EEG signal. Frequency activities are evidently exhibited in

the graph of zero-crossings (Fig. 1(b)) and CTI series (Fig. 1(c)).

The CTI of the fractional Brownian motion has been proven to follow a power law distribution[16]: p(τ) ~ τ-v, where p(τ) is the probability density function (PDF). Fig. 2 shows the PDF estimate of EEG CTI. It can be seen from the figure that the PDF also follows a power law distribution, suggesting fractal structure in EEG CTI.

2 Multifractal singularity spectrum The multifractal singularity spectrum was first pro-

posed by Halsey et al. [17] to describe normalized dis- tributions (measures) lying upon possibly fractal sets, for example those arising from dynamical systems the- ory. They focus upon the scaling properties of such measures, by considering their singularities, which are characterized by two indices: α (Lipschitz-Holder ex-ponent), which determines the strength of their singu-larities; and f, which describes how densely they are distributed. The spectrum of singularities is described by giving the possible range of α values and the func-tion f(α). The measures mentioned above correspond to varied physical quantities according to the specific problem under study, such as CTI (time intervals) and amplitude fluctuations (voltage) of EEG in this paper.

Suppose that we cover the support of the measure with boxes of size L, and the number of the boxes is N. Then N~L−1. Define Pi (L) = Mi(L)/MT as the probabil-ity in the ith box, where Mi(L) is the integrated measure in the ith box and MT is the total amount of the meas-ures. The singularity strength α is defined as ( ) ~ i

iP L Lα . (1) The smaller the exponent α, the more singular the

Fig. 1. Amplitude fluctuations (a), corresponding zero-crossings (b) and CTI series (c) of a segment of EEG signal.

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Fig. 2. Log-log plot of estimated (○) p(τ) of EEG CTI series. The solid line shows least square fit to the PDF. measure, and the “stronger” the singularity. It is a characterization of the scaling in the ith region or spa- tial location. If we count the number of the boxes N(α) where the probability Pi has singularity strength be-tween α and α+dα, then f (α) can be loosely defined as the fractal dimension (box dimension or Hausdorff di-mension, to describe the density of the distribution) of the set of boxes with singularity strength α by )(~)( αα fLN − . (2)

Large errors will be induced if we estimate α and f directly from eqs. (1) and (2) or by Legendre transform from the generalized dimensions Dq. Chhabra and Jen-sen[8] proposed an algorithm for determining the f(α) directly from experimental data. We briefly introduce their algorithm in the following. Please refer to ref. [8] for more details.

First, the entropy S of the process from which the measures arise is given by ∑−=

iii PPS log , (3)

and the Hausdorff dimension dh of the measure theo-retical support of the measure can be related to the en-tropy by a theorem by Billingsley which gives

1

1lim loglog

N

h i iN id P P

N→∞ =

= − ∑ . (4)

We can use these results to evaluate f (α) for a mul-tifractal measure P(x). This is done by first constructing a one-parameter family of normalized measures μ(q) where the probabilities in the boxes of size L are ( , ) [ ( )] [ ( )]q q

i jj

q L P L P Lμ = ∑ . (5)

The parameter q provides a microscope for exploring different regions of the singular measure. For q>1, μ(q)

amplifies the more singular region of P, while for q<1 it accentuates the less singular regions, and for q=1 the measure μ(1) replicates the original measure. Then the Hausdorff dimension of the measure theoretical support of μ(q), i.e. f , is given by

1( ) lim ( , ) log[ ( , )]log i iiN

f q q L q LN

μ μ→∞

= − ∑

0

( , ) log[ ( , )]lim .

logi ii

L

q L q LL

μ μ→

= ∑ (6)

In addition, we can compute the average value of the singularity strength αi=logPi/logL with respect to μ(q) by evaluating

1( ) lim ( , ) log ( )log i iiN

q q L P LN

α μ→∞

= − ∑

0

( , ) log ( )lim .

logi ii

L

q L P LL

μ→

= ∑ (7)

Eqs. (6) and (7) provide a relationship between a Hausdorff dimension f and an average singularity strength α as implicit functions of the parameter q.

The range of the singularity strength is defined as max min ,α α αΔ = − (8) where α max = α (q → −∞) and α min = α (q → +∞). Δα is a measure of the deviation from monofractal, indi-cating the degree of multifractality. Smaller Δα indi-cates that the measure tends to be monofractal, and contrarily, larger Δα indicates multifractality[12].

3 Application to sleep EEG analysis The experimental data used in this study come from

PhysioBank “The Sleep-EDF Database”[18,19], which is a collection of sleep recordings from 8 healthy subjects. The recordings were obtained from Caucasian males and females (21―35 years old) without any medication. Each record contains horizontal EOG, FpzCz and PzOz EEG, sampled at 100 Hz, and is accompanied by a hypnogram. Hypnograms are sleep staging results manually scored for every 30 s epoch according to Rechtschaffen & Kales.

We applied multifractal analysis to CTI series of sleep EEG. Fig. 3 shows an example of the linear fit to Σμilogμi vs. log L with varied q from the bottom in the order of q = 1, 3 and 5. It can be seen from the figure that there is no ambiguity in determining the slopes, and different q correspond to different slopes. This con-firms multifractality in EEG CTI series.

Then the multifractal singularity spectra were esti-mated for every sleep stage. For convenience, we will

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Fig. 3. Σμilogμi vs. logL with q = 1 (◊), 3 (□) and 5 (○). The solid lines show linear square fits correspondingly. use “Wake” for denoting the state of wakefulness, “S1”―“S4” for sleep stages 1―4 and REM for rapid eye movement sleep in the following description and figures. Fig. 4 shows the singularity spectra of varied sleep stages. It can be observed from the figure that the spectra are asymmetric, and there are differences in the spectra’s asymmetry among sleep stages. In the state of wakefulness, the region of q>0 (corresponding to smaller α) is larger than the region of q<0 (corre-sponding to larger α), i.e. the singularity spectrum is more concentrated on the “stronger” region of singular-ity. And with the transition into deeper sleep, the region of q>0 decreases, and contrarily the region of q<0 in-creases, i.e. the singularity spectrum moves towards the “weaker” region of singularity. The asymmetry of REM is between S1 and S2.

Fig. 4. Multifractal singularity spectra (f(α) vs. α) of varied sleep stages.

We introduce a measure Δasα to describe the

asymmetry of EEG multifractal singularity spectrum and its variation with sleep state transition which is

defined as +− Δ−Δ≡Δ αααas , (9)

where Δα− = αmax –α0 is the range of the singularity strength with q<0, Δα+ = α0 –αmin is the range of the singularity strength with q>0, and α0 = α (q = 0) is the singularity strength where the singularity spectrum ob-tains its maximum.

We studied 24-hour EEG records of four subjects (sc4002e0, sc4012e0, sc4102e0 and sc4112e0). The signals were divided into segments of 30 s for being consistent with the hypnograms. The total number of validated segments is 11314 (states distribute as, Wake: 7722; S1: 286; S2: 2036; S3: 289; S4: 240; REM: 741). Then CTI series was constructed for each segment, and was analyzed with multifractal formalism. Δasα and Δα were estimated for quantifying changes of multifractal singularity spectra. The hypnogram and the corre-sponding Δasα and Δα of subject sc4012e0 are plotted in Fig. 5 as an illustration of their variation with sleep state transition.

We grouped analysis results of all the four subjects according to sleep stages given in the database. Fig. 6 shows the means and standard deviations of the two measures at various sleep stages. We did one-way analysis of variance (ANOVA) to test the significance of the difference among the stages. To balance the group sizes, for each group of Wake, S2 and REM, 300 segments were randomly selected, and combined with all the available segments of S1, S3 and S4, and then tested with ANOVA. The test of Δasα came out with F(5, 1709) = 394.41, and Δα with F(5, 1709) = 287.96. To further study the discrimination of the states given by each measure, we also did multiple comparison test. The results show that, at a significance level of P<0.05, both Δasα and Δα cannot distinguish between the states of S1 and REM, and in addition, Δα also cannot distin-guish between the states of S3 and S4.

4 Discussion and conclusion In this paper, we studied the CTI series of EEG us-

ing multifractal formalism. We found evidence of mul-tifractal structures in EEG CTI series, suggesting mul-tifractality of EEG. Multifractal analysis was applied to EEG CTI series of varied sleep stages. We observed that the multifractal singularity spectrum of EEG CTI series was asymmetric, and there are differences in the spectra’s asymmetry among sleep stages. In the state of wakefulness, the singularity spectrum is more concen-

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Fig. 5. The hypnogram (a) and the corresponding Δasα (b) and Δα (c) of subject sc4012e0. The arrows denote short-term wakefulness in sleep.

Fig. 6. Means and standard deviations of Δasα (a) and Δα (b) of all the four subjects at various sleep stages.

trated on the “stronger” region of singularity. And with the transition into deeper sleep, the singularity spec-trum moves towards the “weaker” region of singularity. So we proposed a new measure Δasα to describe the asymmetry of EEG multifractal singularity spectrum. We compared the new measure with the singularity strength range Δα which was normally used as a degree indicator of multifractality. This was done by applying them to the discrimination of sleep stages.

The results of ANOVA and multiple comparison test indicates that, Δasα gives better discrimination of sleep stages overall, and also it gives good discrimination of the states S3 and S4 that Δα fails to do. However both of the two measures cannot distinguish between the states of S1 and REM, and it is difficult to automati-cally discriminate these two states at all times[20].

Carefully examining the curves of two measures in Fig. 5 and the statistic results in Fig. 6, we can find that the value’s fluctuation of Δasα in the state of wakeful-ness is smaller than that of Δα. And Δasα is obviously more sensitive to short-term wakefulness in sleep (de-noted as the arrows in Fig. 5).

In fact, both of the two measures can characterize the variation of EEG’s nonlinear properties with sleep state transition, and both have some limitation on that. Mul-tifractal singularity spectrum has much information on nonlinear properties of the system. It is worth to be further studied that how to find effective parameter from the spectrum to characterize brain functions and pathological changes in EEG.

Biological processes of human are extremely sophis-ticated. Sleep is not simply a state in which the brain is

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resting, but a dynamic, complicated condition during which the brain is quite active. The sleep stages can hardly be discriminated accurately by any single analy-sis method or any single measure at the present time. The new method and measure we proposed in this pa-per is not an exception that can distinguish sleep states accurately by itself. The R&K criteria which have been considered as a “golden standard” for sleep stage dis-crimination, also needs multiple bio-signals, and multi- parameters for manual scoring. Nevertheless, results of our method show that there are significant differences among sleep stages at the significant level of P<0.05. The purpose of our study is to find new measures by utilizing latest advances in nonlinear dynamics. We hope that by combining our new measure with linear/ nonlinear parameters that have been proven to be effec-tive, the automatic discrimination accuracy of sleep stages will be improved and meet the requirements of clinical practices finally.

Acknowledgements This work was supported by the Na-tional Natural Science Foundation of China (Grant No. 60501003).

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