Scale invariance of human electroencephalogram signals in sleep

Scale invariance of human electroencephalogram signals in sleep

Shi-Min Cai,¹Zhao-Hui Jiang,¹ Tao Zhou,^2,3,

*

1Department of Electronic Science and Technology, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China

2Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China

3Department of Physics, University of Fribourg, Chemin du Muse 3, CH-1700 Fribourg, Switzerland

4Institute of Complex Adaptive Systems, Shanghai Academy of System Science, Shanghai 200093, People’s Republic of China

In this paper, we investigate the dynamical properties of electroencephalogram共EEG兲signals of humans in sleep. By using a modified random walk method, we demonstrate that scale-invariance is embedded in EEG signals after a detrending procedure is applied. Furthermore, we study the dynamical evolution of the prob- ability density function共PDF兲of the detrended EEG signals by nonextensive statistical modeling. It displays a scale-independent property, which is markedly different from the usual scale-dependent PDF evolution and cannot be described by the Fokker-Planck equation.

PACS number共s兲: 87.19.Nn, 05.40.⫺a, 89.75.Da

I. INTRODUCTION

The analysis of electroencephalogram 共EEG兲 signals at- tracts extensive attention from various research fields, since it cannot only help us to understand the dynamical mecha- nism of human brain activities, but also be potentially useful in clinics in diagnosing some neural diseases. Some previous work has been done on human EEG signals in sleep and other physiological states. In Refs.关1–3兴the correlation di- mension and Lyapunov exponent were calculated to charac- terize and discriminate the sleep stages. Leeet al.provided evidence for the long-range power-law correlations embed- ded in EEG signals 关4兴. The mean scaling exponents were distinguished according to rapid eye movement共REM兲, non- REM, and awake stages, and gradually increased from stage 1 to stages 2, 3, and 4 in non-REM sleep. Hwaet al.found a variable scaling behavior in two regions, and described the topology plot of the scaling exponents in these two regions, which reveals the spatial structure of the nonlinear electronic activity 关5兴. Random matrix theory is used to demonstrate the existence of generic and subject-independent features of the ensemble of correlation matrices extracted from human EEG signals 关6兴. Yuanet al. found similar long-range tem- poral correlations and power-law distribution of the incre- ments of EEG signals after filtering out the␣ ^and␤^waves 关7兴.

Furthermore, some very recent work关8,9兴pointed out that the sleep-wake transitions exhibit a scale-invariant pattern and embed a self-organized criticality共see also Ref.关10兴for the concept of self-organized criticality兲. In the present pa- per, the Tsallis entropy is used to analyze a series of human EEG signals in sleep. Robust scale invariance was discov- ered for the EEG signals of brains in sleep, which does, to some extent, indicate that the human brain activity in sleep may be related to a self-organized critical system.

We use the Massachusetts Institute of Technology共MIT兲– Beth Israel Hospital 共BIH兲 polysomnographic database,

which is a collection of recordings of multiple physiological signals during sleep. Subjects were monitored in Boston’s Beth Israel Hospital Sleep Laboratory for evaluation of chronic obstructive sleep apnea syndrome, and to test the effects of constant positive airway pressure, a standard thera- peutic intervention that usually prevents or substantially re- duces airway obstruction in these subjects. The database con- tains four-, six-, and seven-channel polysomnographic recordings, each with an electrocardiogram共ECG兲signal an- notated beat by beat, and with an EEG signal annotated with respect to sleep stages关12兴. The records were digitized at a sampled interval of 250 Hz and 12 bits precision. The poly- somnographic wave forms were displayed on CRT display and edited by using a program called WAVE 共wave-form analysis, viewer, and editor兲, which was developed at Mas- sachusetts Institute of Technology. The sleep stage was an- notated at 30 s intervals according to the criteria of Rech- schaffen and Kales, denoted by six discrete levels—1, 2, 3, 4, REM, and awake 共stages 1, 2, 3, and 4 belong to non- REM sleep兲 关13兴. More details of the MIT-BIH polysomno- graphic database collection can be found in Ref.关14兴. In this paper, we chose the experimental data with the criterion that the subject record contain at least five states, with the persis- tent length of the state larger than 10⁵. Under this criterion, we chose ten subjects, and in total 40 samples: nine samples for the awake state, eight samples for the REM state, five samples for stage 1, nine samples for stage 2, six samples for stage 3, and three samples for stage 4. The testers in the experimental procedure were patients with diseases like ob- structive apnea with arousal, hypopnoea with arousal, and obstructive apnea. However, the disease status could only be observed in a short time period during the transition between states. The chosen experimental data were required to cover sufficiently long time periods in which the testers did not detect the disease. The average length of records in each stage was larger than 10⁵. The smallest average length was 1.21⫻10⁵ for stage 3 共corresponding to 7 min兲, while the largest contained 4.525⫻10⁵ samples for the awake stage 共corresponding to 28 min兲. A representative example is shown in Fig.1. In addition to Fig.1, all the experimental

*zhutou@ustc.edu

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Published in "Physical Review E 76: 061903, 2007"

which should be cited to refer to this work.

results shown in this paper were obtained by averaging over the ten chosen subjects.

II. SCALE INVARIANCE OF DETRENDED EEG SIGNALS Consider an EEG series, denoted by兵x_i其 共i= 1 , 2 , . . . ,N兲, whose scaling characteristics are detected through the fol- lowing procedure.

Step 1. We construct the profile series Y_j=兺i=1 j x_i, j

= 1 , 2 , . . . ,N, and considerY_jas thewalk displacementof the resulting random walk.

Step 2.We divide the profile series into nonoverlapping segments with equal length and fit each segment with a second-order polynomial function. We regard the fitting re- sults as trends; a stationary series can be obtained by elimi- nating the trends from the profile series.

Step 3.After the detrending procedure, we define the in- crement of this modified profile series at a scales as ⌬sYj

=Y^ⴱ_j+s−Y^ⴱ_j, whereY^ⴱ_j is the deviation from the polynomial fit.

Step 4.Scale invariance共self-similarity兲in the stationary series implies that the probability distribution function共PDF兲 satisfies

P共x,s兲= 1

␴s

P

冉

冊

where␴sdenotes the standard deviation at time scales. Ob- viously,P共0 ,s兲=P共0兲1/␴s.

Changing the time scales from 2¹ to 2¹⁰, the normalized PDFs of⌬sY exhibit scale-invariant共self-similar兲 behaviors as presented in Fig. 2. That is to say, those PDFs can be rescaled into a single master curve, as shown in Fig.3. The scale invariance of the detrended EEG signals suggests that a quasistationary property is embedded in the distributions of time scales. Therefore, it helps us to search for stable distri- butions to mimic them.

III. NONEXTENSIVE STATISTICAL MODELING OF DETRENDED EEG SIGNALS

From the results sketched in the preceding section, here we use the Tsallis entropy to model the PDFs. The Tsallis entropy was introduced by Tsallis through generalizing the standard Boltzmann-Gibbs theory关15兴, and is given by

Sq=k

1 −

冕

q− 1

冉 冕

冊

In the limitq→1,S_q degenerates to the Boltzmann-Gibbs- Shannon entropy as

FIG. 1. A set of representative records of EEG signals in differ- ent stages. Each entire experimental data set includes more than 10⁵ data points, while only a small fraction are plotted.

FIG. 2. P共0 ,s兲 as a function of the time sampling scale s. A power-law scaling behavior is observed for about three orders of magnitude. The data points for awake, REM, and stages 1, 2, 3, and 4 are obtained by averaging nine, eight, five, nine, six, and three samples, respectively.

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S₁= −

冕

= 1, as well as the constraint 具具x²典典q=␴², leads to the q-Gaussian distribution共q⬍3兲

Gq共x,s兲= 1

Z_q共s兲„1 −␤共s兲兵共1 −q兲关x−¯x共s兲兴²其…₊^1/1−q, 共4兲

where Z_q共s兲 is a normalization constant, ␤共s兲 is explicitly related to the variance of the distribution, and the subscript + indicates thatGq共x,s兲is non-negative关16兴.Gq→1共x,s兲recov- ers the usual Gaussian distribution. Theq-Gaussian PDF can describe a set of stable distributions from Gaussian to Lévy regimes 关17兴 by adjusting the value of q with appropriate time-dependent parameters␤共s兲andZ_q共s兲 关18兴. The distribu- tion falls into the Lévy regime in the interval 5/3⬍q⬍3, withq= 5/3 the critical value.

The results in Fig.4show that the PDF of the awake stage falls into the Lévy regime with q being equal to 1.94. It exhibits sharp kurtosis and a long-tail distribution, distin- guished from those of REM and non-REM stages. It should

be noted that we shift the distributions by dividing them by their standard deviations and plot only the cases of time scale s= 2,8,32,128,1024 to make the figure clear.

The specific values of␤共s兲for all scales are shown in Fig.

5. Interestingly, ␤共s兲 does not dissipate as the time scale s increases, unlike the behavior of ␤共s兲 recently reported in financial markets 共see Fig. 11 in Ref. 关19兴兲, in which ␤共s兲 decreases in a power-law form with time scales, indicating scale-dependent PDF evolution. In other words, it demon- strates that the dynamical evolution of EEG signals is not coincident with the diffusion process described by the Fokker-Planck equation关20兴.

Another significant equation of the nonextensive statisti- cal approach is theq-exponential function, which reads

eq共x,s兲= 1

Zq共s兲兵1 −␶共s兲关共1 −q兲兩x−¯x共s兲兩兴其₊^1/1−q, 共5兲 where the parameter␶共s兲is the relaxation rate of the distri- bution. Clearly, in the limitq→1,

e₁共x,s兲= 1

Zq共s兲exp关−␶共s兲兩x−¯x共s兲兩兴. 共6兲 Because the statistical distributions of the detrended incre- ments of EEG signals in the sleep stages exhibit an approxi- FIG. 3.共Color online兲Rescaled increment PDFs for six stages.

Obviously, curves with different time scales can well collapse onto a single master curve, demonstrating the existence of a quasistation- ary property. The different values of the time scalesare presented in the right panel, increasing as 2 , 4 , 8 , . . . , 1024. The data points for awake, REM, and stages 1, 2, 3, and 4 are obtained by averaging nine, eight, five, nine, six, and three samples, respectively.

FIG. 4. 共Color online兲 Rescaled increment PDFs of all stages with the approximate fit using nonextensive statistical modeling.

We use a q-Gaussian function to fit the awake stage, and a q-exponential function to fit other five stages. The awake stage falls into the Lévy regime with the best-fit parameter q= 1.94. In the REM stage, the values of q are slightly different; while in each non-REM stage, they are almost the same. All the data points for awake, REM, and stages 1, 2, 3, and 4 are obtained by averaging nine, eight, five, nine, six, and three samples, respectively.

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mately exponential form, we use theq-exponential model to quantify them, as shown in Fig. 4. The values of q for the REM and the non-REM stages are a little bit larger than 1.

This means that the fluctuation of human brain activities in the sleep stage converges to a normal exponential pattern. In particular, the EEG signals exhibit aq-invariant pattern for different time scales in all the four stages within non-REM sleep. The relaxation rates of the distributions are also ap- proximately invariant, as shown in Fig.5. However, in the REM stage, the values of q change slightly, and only the center part of the distribution can be well fitted by the present model. This irregularity of brain electrical activity in the REM stage may result from the acute neural activity关21兴.

The nonextensive statistical approach, modeling the de- trended increment’s PDF of EEG signals with an invariant parameterq, demonstrates the scale-independent property of the system. In order to further test the existence of this ob- served property, we randomize the empirical series of the awake stage by shuffling关22,23兴and show a fit for this arti- ficial distributions at different scales in Fig.6. Clearly, the parameterqwill approach the Gaussian regime共q= 1兲as the time scale increases. This result strongly illuminates that the scale-independent property of human brain activity in sleep is remarkably different from the turbulentlike scale- dependent evolution关24兴. Since the fluctuation in a system near a critical point is generally associated with scale invari- ance, the existence of a scale-dependent property of EEG signals indicates that the human brain activity in sleep may be related to a self-organized critical system, supporting a prior report about this issue关11兴.

IV. CONCLUSION

In this paper, several dynamical properties of human EEG signals in sleep are investigated. We first use a modified ran- dom walk method to construct the profile series including the information of the EEG signals. After a detrending proce- dure, we obtain a stationary series and define the increments

of the resulting random walk at multiple scales. In order to characterize the dynamical process of brain electronic activ- ity, we then study the P共0 ,s兲 of the PDF of normalized in- crements as a function ofs. With this choice we investigate the point of each probability distribution that is least affected by the noise introduced by the experimental data set. Scale invariance in both awake and sleep stages is obtained; thus one can rescale the distributions at different scales into a single master curve.

Aiming to investigate this property, we use the nonexten- sive statistical approach to model these processes. The dy- namical evolution of the detrended increments’ PDF in the awake stage can be well fitted by theq-Gaussian distribution with an invariant parameterq= 1.94. It demonstrates that the PDFs of the awake stage fall into the Lévy regime. In con- trast, aq-exponential distribution is used to mimic the PDFs of the sleep stages. In particular, the non-REM stage exhibits scale-independent distributions; while for the REM stage, the analysis suggests a complex distributional form with slightly different values of q. Note that, in many prior quantitative methods, like entropy and Lyapunov estimates, the REM and awake states are indistinguishable based on the entire time scale. Instead, here we analyze the EEG series at different time scales, and find a great difference of theq value in the awake state共i.e.,q= 1.94兲, which may be due to the extreme neural activity. However, the real biological reason is not clear thus far. We hope this sharply different q value can reveal some information that could be useful for a future and in-depth exploration.

In a recent work关24兴, Lin and Hughson proposed a tur- bulentlike cascade model, which describes a scale-dependent PDF evolution, to mimic the human heart rate; the validity of the model is, now, being challenged by the critical scaling invariance found in real human heart-rate processes关25,26兴.

In this paper, we demonstrate that the process of brain elec- tric activity is remarkably different from a turbulentlike cas- cade evolution, similar to what was found by Kiyonolet al.

关25,26兴. It is generally accepted that the complex dynamics of the heart rate is caused by an intricate balance between the two branches of the autonomic nervous system: the parasym- FIG. 5. 共Color online兲 ␤共s兲 and ␶共s兲 versus s for awake and

non-REM stages. The values of␤共s兲do not dissipate assincreases.

In particular,␶共s兲of non-REM sleep converges to an invariant pat- tern. All the data points for awake and stages 1, 2, 3, and 4 are obtained by averaging nine, five, nine, six, and three samples, respectively.

FIG. 6. 共Color online兲Increment PDF of randomized series of awake stage and fitting curves with different parameters q. The parameterqrapidly approaches the Gaussian regime共q= 1兲as the time scale increases. For clarity, we shifted the distributions by dividing them by their standard deviations.

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pathetic共PNS兲and the sympathetic 共SNS兲nervous systems, which respectively decrease and increase the heart rate. The autonomic nervous system is controlled by the central ner- vous system of the brain. Therefore, even though the electro- cardiograph and electroencephalogram are different, their similarities may not be a coincidence. Although the compari- son of the EEG and ECG in this paper could not present a convincing link between the scale-invariant properties of heart rate and EEG, the discussion of this aspect may en- lighten readers and can provide some insights into the under- lying dynamical mechanism of brain activity. In addition,

like the corresponding empirical studies on human ECG sig- nals, this work could provide some criteria for theoretical models of human EEG signals.

ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China under Grants No. 70571075, No.

70571074, and No. 10635040. B.H.W. acknowledges 973 Project 2006CB705500.

关1兴J. Fell and J. Röoschke, Int. J. Neurosci. 76, 109共1994兲. 关2兴J. Fell, J. Röschke, K. Mann, and C. Schaffner, Clin. Neuro-

physiol. 98, 401共1996兲.

关3兴T. Kobayashi, S. Madokoro, T. Ota, H. Ihara, Y. Umezawa, J.

Murayama, H. Kosada, K. Misaki, and H. Nakagawa, Psychia- try Clin. Neurosci. 54, 278共2000兲.

关4兴J. M. Lee, D. J. Kim, I. Y. Kim, K. S. Park, and S. I. Kim, Comput. Biol. Med. 32, 37共2002兲.

关5兴R. C. Hwa and T. C. Ferree, Phys. Rev. E 66, 021901共2002兲. 关6兴P. Šeba, Phys. Rev. Lett. 91, 198104共2003兲.

关7兴J. W. Yuan, B. Zheng, C. P. Pan, Y. Z. Wu, and S. Trimper, Physica A 364, 315共2006兲.

关8兴C. C. Lo, T. Chou, T. Penzel, T. E. Scammell, R. E. Strecker, H. E. Staney, and P. C. Ivanov, Proc. Natl. Acad. Sci. U.S.A.

101, 17545共2004兲.

关9兴J. C. Comte, P. Ravassard, and P. A. Salin, Phys. Rev. E 73, 056127共2006兲.

关10兴P. Bak, How Nature Works: The Science of Self-Organized Criticality共Springer, New York, 1996兲.

关11兴L. de Arcangelis, C. Perrone-Capano, and H. J. Herrmann, Phys. Rev. Lett. 96, 028107共2006兲.

关12兴A. L. Goldberger, L. A. N. Amaral, L. Glass, J. M. Hausdorff, P. C. Ivanov, R. G. Mark, J. E. Mietus, G. B. Moody, C. K.

Peng, and H. E. Stanley, Circulation 101, e215共1999兲. 关13兴A. Rechschaffen and A. Kales,A Manual of Standardized Ter-

minology, Techniques and Scoring System for Sleep Stage of

Human Subjects 共U.S. Dept. of Health, Education, and Wel- fare, Bethesda, MD, 1968兲.

关14兴Y. Ichimaru and G. B. Moody, Psychiatry Clin. Neurosci. 53, 175共1999兲.

关15兴C. Tsallis, J. Stat. Phys. 52, 479共1988兲.

关16兴S. Abe and Y. Okamoto,Nonextensive Statistical Mechanics and Its Application共Springer, Berlin, 2001兲.

关17兴P. Lévy, Théorie de lÁddition des Variables Aléatories 共Gauthier-Villars, Paris, 1927兲.

关18兴C. Vignat and A. Plastino, Phys. Rev. E 74, 051124共2006兲. 关19兴A. A. G. Cortines and R. Riera, Physica A 377, 181共2007兲. 关20兴The Fokker-Planck equation is used to describe a scale-

dependent evolution process with asymptotic power-law rela- tion␤共s兲⬃s^␣.

关21兴E. R. Kandel, J. H. Schwartz, and T. M. Jessell,Principles of Neural Science共McGraw-Hill, New York, 2000兲.

关22兴J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D.

Farmer, Physica D 58, 77共1992兲.

关23兴J. Theiler and D. Prichard, Physica D 94, 221共1996兲. 关24兴D. C. Lin and R. L. Hughson, Phys. Rev. Lett. 86, 1650

共2001兲.

关25兴K. Kiyono, Z. R. Struzik, N. Aoyagi, S. Sakata, J. Hayano, and Y. Yamamoto, Phys. Rev. Lett. 93, 178103共2004兲.

关26兴K. Kiyono, Z. R. Struzik, N. Aoyagi, F. Togo, and Y. Yama- moto, Phys. Rev. Lett. 95, 058101共2005兲.

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