Nonlinear Analysis of Sleep Stages Using Detrended Fluctuation Analysis: Normal vs. Sleep Apnea

Transcription

1 onlinear Analysis of Sleep Stages Using Detrended Fluctuation Analysis: ormal vs. Sleep Apnea JOG-MI LEE, DAE-JI KIM, I-YOUG KIM, and SU I. KIM Department of Biomedical Engineering Hanyang University Sungdong P.O. Box 55, Seoul, KOREA Abstract: - The purpose of this paper is to compare the characteristics of EEGs, which typically exhibit non-stationarity and long-range correlations, by calculating its scaling exponents in between sleep apnea and the normal conditions. Detrended fluctuation analysis (DFA), which is suitable for non-stationary time series, is used to analyze the fluctuation of the EEG dynamics by calculating its scaling exponents. As we classified the stages into REM (R: Rapid Eye Movemen, on-rem (), and Waken (W), each mean scaling exponent was significantly (p<0.00) different from the other stages in both sleep apnea and normal conditions. Two groups show significant (p<0.00) differences in all the stages. While the sleep apnea group shows >R>W by order of the mean scaling exponent value, the normal group shows >W>R. The fact of the reversed order of R and W between two groups may be related to some characteristics of sleep apnea. It was also noted that monotonic increasing in the stage to 4 in both groups. We conclude that the scaling exponents could be used to classify the sleep stages automatically, and give some evidence for the non-linear dynamics of the sleep stages that varies with the condition of the patients. Key-words: - Detrended-Fluctuation-Analysis (DFA); Electroencephalogram (EEG); Sleep Apnea; Scaling Exponents; on-linear Dynamics Introduction The non-linear analysis has been considered as a useful method in the analysis of the biological time series, which exhibits typically complex dynamics. However, as the statistical characteristics of biological signals often change with time and are typically both highly irregular and non-stationary in many cases, such analysis is so complicated [][2]. For example, human brain activities normally fluctuate in a complex manner. These fluctuations may arise from a complex non-linear dynamical system rather than an epiphenomenona of environmental stimuli [3]. Since the difference in activities of the brain causes difference in characteristics of EEG, it has been attempted to investigate the brain activity through analyzing EEG. It has been believed that it is valid, as a first approximation, to conceive of local neural networks in the cortex as deterministic dynamical systems [4] and to interpret the correlation dimension as providing information about cortical dynamics. This allows us to use the mathematical theory of non-linear systems to analyze EEG. In recent years, long-range power-law correlations have been discovered in a remarkably wide variety of systems [5][6][7]. Such long-range power-law correlations are a physical fact that in turn gives rise to the increasingly appreciated fractal geometry of nature. The quantification of power-law correlations with a critical exponent may give useful information for understanding electroencephalogram (EEG) properties [8]. Many physiological time series have been known to exhibit long-range power-law correlations under healthy conditions []. Since it has been believed that the characteristics of EEG are changed according to the sleep stages, many studies have reported on the automatic sleep staging using EEG. Some group tried to use non-linear dynamics method, such as correlation

2 dimension and Lyapunov exponents [9][0], but all the classical non-linear dynamics methods require the stationary conditions, which can rarely be obtained in the biological signals. We applied a detrended fluctuation analysis (DFA), which is known for its robustness against non-stationarity, to the EEG for quantifying long-range correlation property. With this method that permits the detection of long-range correlations embedded in a seemingly non-stationary time series, we could effectively distinguish each stage from sleep EEG [3]. The purpose of this study is to apply a new method on discriminating sleep stages using EEG, to quantify non-linear characteristics of the sleep dynamics, and to find out some characteristic differences between healthy and sleep apnea. 2 Problem Formulation 2. Materials In this study, we used two groups of data; the sleep apnea that are from MIT-BIH polysomnography database and the control data that are from the two healthy young subject s polysomnography. MIT-BIH polysomnography database, which consists of four-, six-, and seven-channel polysomnographic recordings, each with an ECG signal annotated beat-by-beat, and an EEG signal annotated with respect to sleep stages []. Records vary in duration from.25 to 6.5 hours and have been sampled at 250 samples per second. Sleep stage was annotated at 30-second intervals according to the criteria of Rechtschaffen and Kales with six discrete levels;, 2, 3, 4, REM (Rapid Eye Movemen and Wake. The control group data were acquired from two digital polysomnographic recordings of two healthy young male subjects in the Seoul ational University Hospital. Each subject s data were recorded for about 8 hours at the Division of Sleep Studies of Seoul ational University Hospital using the IPSS (Intelligent PolySomnographic System, Korea). Each data set contained around 950 epochs. After recording, three human experts scored these data sets individually according to Rechtschaffen and Kales sle ep staging criteria. 2.2 Scaling Exponents and Detrended Fluctuation Analysis (DFA) There are prominent fluctuations on the exponent characterizing long-range correlations in the finite-length sequences [2]. This scaling exponent is important since we may find correlation properties of the time series. When the statistical properties of time series are invariant to the rescaled series of its subset, it is said to be self-similar and defined as d α t y( a y( ) a () d, where represents that the statistical properties of both time series are identical. In detail, a time series, y(, has same probability distribution function as a same-sized and rescaled series, a α y( t a ). This exponent is called a self-similarity parameter or a scaling exponent, which is important since we can find correlation properties of the time series from it. Mathematically scaling exponent is calculated as ln M y ln s2 ln s α = = ln M x ln n2 ln n (2), where M y is magnification factor to the y-axis and M x is to the x-axis, n is a window size of the series and s is a standard deviation of each signal respectively. And is simply the slope of the line that connects (n, s ), (n 2, s 2 ) on the log-log plot. In practice, we calculate average scaling exponent of the time series by selecting adequate number of (n, s) and fitting a line from these points. ow let us consider the case of = 0 and this represents that two time series have the identical standard deviations. But it may happen when the given time series is purely random in spite of its size, e.g. white noise. Therefore, we need to distinguish meaningful time series from random data by other methods. Effective solution of this problem is to study the integration of the original time series [3]. In this paper, we calculate the fluctuations of EEG after the integration using DFA. Since many kinds of biological signal such like EEG are highly non-stationary, if we simply integrate the time series without careful consideration, non-stationarity of the time series may be increased. Therefore, we apply DFA to the analysis of biological data, which calculates the root-mean-square fluctuation of integrated and detrended time series, permits the detection of intrinsic self-similarity embedded in a non-stationary time series, and also avoids the spurious detection of apparent self-similarity [3]. The detail procedures are as follows. First, each of time series with the number of samples is integrated as

3 y( k) = k { X ( X ave } t= (3), where X( is the sequence at time t, and X ave is the average of entire time series. ext y(k), integrated time series, is divided into sub sequences of equal length, n. In each box, the y-coordinate of a least-square line which fits to the data is denoted by y n (k). Finally, the average fluctuation as a function of box size n, is given by 2 F( n) = [ y( k) y n ( k)] t= (4) At the log-log plot of F(n) vs. n, the slope is a scaling exponent, which is the characteristic of fluctuations [3]. It is known that white noise, where the value at one instant is not correlated with any previous value, has uncorrelated integrated series y(k) such like a random walk and its scaling exponent is equal to 0.5 [4]. The case of = is a special one; the time series corresponds to /f noise [5]. And =.5 indicates Brownian noise which is the integration of white noise [8]. When 0 < <0.5, power-law anti-correlations are present such that large values are more likely to be followed by small values and vice versa [3]. If is greater than 0.5 and less than or equal to.0, then there is a long-range power-law correlations in that we are interested [3]. We used three kinds of time series to verify the DFA algorithm that we developed: White noise, /f noise, which is computer-generated fractional Gaussian noise with long-range correlations with the scaling exponent value as 0.8 approximately, and Brownian noise. 2.3 ECG Cancellation in EEG In general, EEG data contains many artifacts such as ECG, muscle artifact, eye movement etc. Practical use of EEG is extremely limited due to such inevitable contamination by a large amplitude ECG, which brings on the erroneous interpretation of the normal record. Therefore, we rejected the ECG artifact from the EEG and used this filtered EEG in the DFA analysis. To do this, we subtracted the predetermined averaged ECG from the subsequent ECG-contaminated EEG channel (AKAMURA et al., 987). The raw EEG data, y( is assumed to be the summation of the real EEG, x( and the ECG artifacts, z( as y ( = x( + z( (5) The peak of ECG R-wave is determined from ECG-file in polysomnography data. During 60msec before and after the R-wave peak point, we calculate the difference of maximum and minimum value of the ECG and the EEG respectively. ECG channel of polysomnography data is divided by mean difference, and rescaled to the EEG. By subtracting the estimate of the artifacts, z est ( from the raw EEG, y(, the processed EEG data is obtained as follows xest( = y( zest( (6) 2.4 Statistical Analysis We used one-way analysis of variance (AOVA) to determine if there are any differences among the means of the stages of each groups and between the means of two groups, and did multiple comparison procedures to detect specifically which pairs of means are significantly different. All the statistical procedures were performed under the MATLAB 5.3 (The Mathworks, Inc. MA, USA). EEG Scaling Exponent 0.2 (a) (b) x x 0 4 Figure. Plot of typical EEG divided by each stage in (a). Data is captured from to in SLP0A.Each stages are labeled below EEG signal. Corresponding DFA results is plotted in (b). We grouped with the same stages of EEG, and applied to statistical methods. 3 Problem Solution We computed DFA on each consecutive 30-second (7500 samples) EEG, and obtained 0,004 scaling exponents for sleep apnea and,807 for control. We varied window size from 0 to 000 samples in computing DFA, and made a linear regression to estimate the scaling exponent. The DFA-computing program was written in MATLAB code, tested white

4 noise, Brownian noise and /f noise signal, and verified that it s estimating values approach to the theoretical ones. After computing DFA, we grouped the resulting values according to the staging annotation as Fig. 3., and test whether the mean of each group is significantly different or not. Fig. shows that the typical EEG pattern of each stage and its corresponding DFA computation results. ote that the scaling exponents of EEG increases as brain goes to deep sleep, which means EEG patterns are more likely to Brownian noise. Scailing Exponents ( a) Control Vs. Apnea Control Apnea dynamics of sleep EEG is likely to that of Brownian noise more and more, which tells that the dynamics of brain become less activated as sleep stage goes to deep. To apply DFA to clinical EEG for its staging, we need an additional algorithm in order to more accurately discriminate stage and REM, for example. We could find that the dynamics of EEG has long-range correlations, that is, the characteristics of /f noise rather than those of random noise. This is of interest since it motivates a new modeling approach to understanding for the brain function mechanism. Since we have no need to find the specific pattern of each stage in EEG, such as K-complex, spindles delta waves that are used in the conventional method, our suggested method using non-linear characteristics of the time series seems more suitable for automatic discrimination of sleep stages. In the future study, we will use DFA method to reveal the dynamics of sleep and the mechanism of sleep disorders, and will apply this method to much more clinical sleep data REM Waken State of Subjects Figure 2. The comparison between two groups for each stage. Each value was represented by meanstandard Error. Fig. 2 shows the means and standard deviations of stage, 2, 3, 4, REM and waken. We observed that scaling exponents become greater significantly as sleep stage goes to deep sleep. All the means between every two stages is significantly different except between stage and REM. It is noted that the waken stage shows relatively large standard deviation compared with others, while slow wave sleep (stage 3, 4) shows relatively small standard deviation. This fact reveals that the dynamics of brain is more consistent in the slow wave sleep, and more divergent in the waken stage. We tested the difference of the mean between on-rem and REM stage in Fig. 3, which shows significant difference. 4 Conclusion We proposed a new approach for discriminating the characteristics of EEG and analyzed its properties by experiment. We applied a detrended fluctuation analysis, DFA, to the non-stationary EEG time series. Our results showed that scaling exponents of sleep and waken EEG were different from each other. The deeper the sleep stage, the larger the scaling exponents. These larger scaling exponents show that according to stage the Scailing Exponents ( a) Control Vs. Apnea REM REM Waken State of Subjects Control Apnea Figure 3. The comparison between two groups for each stage. In this case, we have only three stages (,R,W). Each value was represented by meantandard error. References: [] VISWAATHA, G. M., PEG, C.-K., STALEY, H. E., and GOLDBERGER, A. L., Deviations from uniform power law scaling in nonstationary time series, Phys. Rev. E, 55, 997, pp [2] IVAOV, P. C., ROSEBLUM, M. G., PEG, C.-K., MIETUS, J., HAVLI, S., STALEY, H. E., and GOLDBERGER, A. L., Scaling behavior of heartbeat intervals obtained by wavelet-based time-series analysis, ature, 383, 996, pp

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