Sleep spindles detection from human sleep EEG signals using autoregressive (AR) model: a surrogate data approach

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1 J. Biomedical Science and Engineering, 29, 2, doi:.4236/bise Published Online Setember 29 (htt:// Slee sindles detection from human slee EEG signals using autoregressive (A) model: a surrogate data aroach Venatarishnan Perumalsamy, Sangeetha Sanaranarayanan 2, Suanesh aamony 3 Deartment of nformation Technology, Thiagaraar College of Engineering, Madurai, ndia; 2 Deartment of Electrical and Electronics Engineering, Sethu nstituite of Technology, Madurai, ndia; 3 Deartment of Electronics and Communication Engineering, Thiagaraar College of Engineering, Madurai, ndia. vit@tce.edu; viggee_tce2@yahoo.co.in; rshece@tce.edu eceived 5 May 29; revised 9 July 29; acceted 29 July 29. ABSTACT A new algorithm for the detection of slee sindles from human slee EEG with surrogate data aroach is resented. Surrogate data aroach is the state of the art technique for nonlinear sectral analysis. n this aer, by develoing autoregressive (A) models on short segment of the EEG is described as a suerosition of harmonic oscillating with daming and frequency in time. Slee sindle events are detected, whenever the daming of one or more frequencies falls below a redefined threshold. Based on a surrogate data, a method was roosed to test the hyothesis that the original data were generated by a linear Gaussian rocess. This method was tested on human slee EEG signal. The algorithm wor well for the detection of slee sindles and in addition the analysis reveals the alha and beta band activities in EEG. The rigorous statistical framewor roves essential in establishing these results. Keywords: A Model; LPC; Slee Sindles; Surrogate Data. NTODUCTON Oscillatory signal activities are ubiquitous in the biomedical signals []. Multielectrode recordings rovide the oortunity to study signal oscillations from a networ ersective. To assess signal interactions in the frequency domain, one often alies methods, such as ordinary coherence and Granger causality sectra [2] that are formulated within the frame wor of linear stochastic rocess. Electroencehalogram (EEG) is one of the most imortant electrohysiological techniques used in human clinical and basic slee research. n 979 Barlow roosed linear modeling system which has a longlasting history in EEG analysis [3]. The models are mainly considered as a mathematical descrition of the signal and less as a biohysical model of the underlying neuronal mechanisms. n 985, Frannaszczua et al. [4] roosed a model to interret linear models as damed harmonic oscillators generating EEG activity based on the equivalence between stochastically driven harmonic oscillators and autoregressive (A) models. There is a unique transformation between the A coefficients and the frequencies and daming coefficients of the corresonding oscillators. n articular at times when the EEG is dominated by a certain rhythmic activity e.g. in the case of slee sindles or alha activity, on might exect, that this activity will be reected by a ole with a corresonding frequency and low daming. This idea was the staring oint of our analysis [5]. The slee EEG is always not stationary. However, we demonstrated that the effects of non stationary become relevant only with scales longer than s [6]. Therefore, short segments with duration of around s are sufficiently described by linear models. The non stationary in longer time scales might be reected by the variation of the A-coefficients and thus by the corresonding frequencies and daming coefficients. Based on the above considerations we roose an easy way to define oscillatory events. They are detected, whenever the daming of one of the oles of a s A model is below a redefined threshold. The method of surrogate data is a tool to test whether data were generated by some class of model. n 992 the method of surrogate data roosed by Theiler et al. [7] is a general rocedure to test whether data are consistent with some class of models. n order to test the hyothesis that the data are consistent with being generated by a linear system, the Fourier Transform (FT) algorithm is alied. Based on a examle and the theory of linear stochastic systems we will show that this algorithm roduce correct Published Online Setember 29 in Scies. htt://

2 V. Perumalsamy et al. / J. Biomedical Science and Engineering 2 (29) distribution of time series and therefore might not generally yield the correct distribution of the test statistics. The surrogate data method differs from a simle Monte Carlo imlementation of a hyothesis test in that it tests not against a single model, but a class of models, i.e. linear systems driven by Gaussian noise. The idea is to select single model from the class on the basis of the measured data x, and then do a Monte Carlo hyothesis test for the selected model and the original data. The statistical roerties of a time series generated by a linear rocess are secified by the autocovariance function (ACF) or equivalently by its Fourier Transform, the ower sectrum X (ω). The urose of this study is described as even without a rigorous mathematical foundation it is ossible to detect slee sindles as well as alha and beta activities from human slee EEG. The theoretical framewor of A model and the rocedure to generate the surrogate data are given in Section 2. The method is then tested on one simulation data set in Section 3. Section 4 describes slee sindles detection using A model with surrogate data aroach. esults are discussed and summarized in Section METHODS n this section, we start with A model of order and then roceed to outline the rocedure for generating surrogate data. 2.. A Model The Parametric descrition of the EEG signal by means of the A model maes ossible estimation of the transfer function of the system in the straight forward way. From the transfer function it is easy to find the differential equation describing the investigated rocess. Our detection algorithm is based on modeling s segments of the EEG time series using autoregressive (A) models of order. From the A ()-model a x n n () where a - Coefficients of the model ( a =) x n - The value of the samled signal at the moment n n - Zero mean uncorrelated white noise rocess. Alying the Z-transform to Eq. we obtain: where Scies Coyright 29 A( z) X ( z) E( z) (2) A( z) a z (3) X(z)-the Z-transform of the signal x E(z)-the Z-transform of the noise. f the system is stable, there exists A - (z) and we get X ( z) A ( z) E( z) (4) n the z domain this filter is exressed by the Eq.4 where A - (z) is the transfer function. Denoting it by H(z) and writing if exlicitly we obtain: H ( z) A ( Z) / a z (5) Multilying numerator and denominator H z we get ( z) z / a z (6) Factorizing the denominator gives the formula H ( z) z / ( z z ) (7) i where z r e Using the above formula to estimate the frequencies f /( 2) and daming coefficients lnr ( Denotes the samling interval). We assume that there are only single oles of H (Z) which can be written in the form H ( z) c ( z / z z ) (8) For the single ole coefficients c can be found according to the formula: ( z z ) H ( z) c limz z (9) z By means of the inverse transform z - from the Eq.8 the imulse resonse of the system: h (n) can be found. Since from the roerties of the z - transform we now that z ( z / z z )) ex( n. ln ) we obtain: z i ( H ( z)) h( n) z c ex( n. ln ) () z f the samling interval t was chosen according to the Nyquist theorem we can exress the imulse resonse as continuous function and write it in the form: t h( c ex(. lnz ) t c ex( a () where lnz t = t*n a t Lalace transform of the Eq. which corresonds to the transfer function H(s) of the continuous system is given by the formula: H ( s) c (2) s a The above exression can be obtained directly from Eq.8

3 296 V. Perumalsamy et al. / J. Biomedical Science and Engineering 2 (29) by means of the integral transform z. Eq.2 can be written as a ratio of two olynomials of the order - and b H ( s) c s s b s b c s c o o (3) The olynomial coefficients can be readily calculated from c and a. This form of the transfer function was found also by Freeman 975. t leads directly to the differential equations describing the system. The transfer function is the ratio of the Lalace transform of inut y( and outut x( functions: X ( s) L( x( ) H ( s) (4) Y ( s) L( y( ) d Since variable s corresonds to the oerator, from dt Eq.3 and 4 we get: c d dt d x( c x( cx( dt d d y( b y( b y( (5) dt dt n this way we have obtained the differential equation describing the system which is free of the arbitrary arameters. ts order is determined by the characteristic of the signal and may be found from the criteria based on the rincile of the maximum of entroy Surrogate Data Method Generally, a surrogate data testing method involves three ingredients: ) a null hyothesis; 2) a method to generate surrogate data; and 3) testing statistics for significance evaluation. The null hyothesis in the resent study is that investigated data from a linear Gaussian rocess. f the null hyothesis is reected, then we conclude that the data are either non-gaussian or come from nonlinear rocess. The surrogate data are generated in such a way that it is Gaussian distributed but has the same second order sectral roerties (in the bivariate, auto sectra) as the original data. The testing statistics the amlitude of the eridogram we will give the stes for generation the surrogate data and the theoretical rationale behind the stes [8]. Consider two zero mean stationary random rocesses x( and y(. These rocesses may or may not be linear Gaussian rocesses. Their one-sided auto sectra S and S can be estimated [9]. S S ( f ) 2lim T E[ X T 2 ( f, T ) ] 2 ( f ) 2limT E[ Y ( f, T ) ] (6) T where T is the duration of the data, the exectation is taen over multile realizations and X and Y are the Fourier Transform of x( and y(. Now we consider how to generate two zero-mean linear Gaussian rocesses x( and y( which have the same second order statistical roerties as x( and y (. Let exressed in real and imaginary arts, X, and Y, these Fourier Transforms are ' X ( f ) X ( f ). X ( f ) X and X, Y Y Y ( f ) Y ( f ). Y( f ) (7) Since x ( and y ( are normally distributed with zero mean values, X, X, Y and Y are also normally distributed with zero means [9]. The covariance matrix ( ) of X, X, Y and Y should be selected such that the auto-sectra and cross sectra of x( and y ( are the same as those of the original data, i.e, S S and S S n ractice, S and S estimated from the data. The question is how to draw Gaussian variables for X, X, Y and Y such that S S and S S. t can be shown that the relation between X, X, Y and Y and S and S are as follows []. E [ X X ] E[ YY] T E[ X X ] E[ Y Y ] S 4 T E[ Y Y ] E[ Y Y ] S 4 (8) According to these relations the covariance matrix ( ) for the real and imaginary arts of X and Y for each frequency is X T S E X x X X 4 S Choosing Y E y Y Y Y 4 S T S S S and S and S X, (9) X, Y and Y are then generated by samling from a Gaussian distribution with zero mean and covariance matrix. Once X, X, Y and Y are generated for each frequency, the surrogate data x ( and y ( are the inverse Fourier Transform of X ( f ) X ( f ). X ( f ) and X ( f ) X ( f ). X ( f ) Scies Coyright 29

4 V. Perumalsamy et al. / J. Biomedical Science and Engineering 2 (29) For easy imlementation, we summarize the earlier analysis as follows. Ste ) Estimate S and S for original data according to (6) Ste 2) Calculate Covariance matrix according to (9) Ste 3) Draw values (realizations) of X, X, Y and Y from the Gaussian rocesses with zero mean and covariance matrix ( ) for each frequency. This can be done for examle Matlab function based on the ziggurat method []. Ste 4) Tae the inverse Fourier Transform of X and Y to obtain the surrogate data x ( and y (. To ensure surrogate data are real valued, the negative frequency arts of X and Y are taen as the comlex conugate of the ositive frequency arts. Ste 5) eeat Stes 3) and 4) to generate multile realizations of the surrogate data. X and Y, could be drawn from Gaussian distribution with zero mean and covariance matrix x. This simlified method is similar to the method roosed by Timmer [2]. The robability density function for the extreme value distribution (tye ) with location arameter µ and scale arameter is σ [3]. x x y ex( ex( ) ) (2) The exact PDF is determined once µ and σ are nown. From this distribution, one can determine the threshold for any desired significance level (i.e. - value). 3. SMULATON EXAMPLE We have erformed simulation studies to test the effectiveness of the method roosed before. Matlab Signal rocessing and Sectral analysis toolbox is used in our analysis. 3.. Linear Bivariate A Model Driven by Gaussian White Noise The model is written as x(.8* x( t ).5x( t 2) ( y(.6x( t 5) ( (2) where ( and ( are uncorrelated Gaussian white noise with zero means and unit variances. The data set consists of M=5 realizations where each realization is of length K=52. For samling frequency of 28Hz, each realization has the duration of 4s. To erform the test P =5 surrogate data sets were generated following the rocedure in Section 2. The one sided ower sectra S and S for both original (solid curve) and one set of surrogate data (dotted curve) are shown in Figure. t is seen that the surrogate data s sectra nearly match well with that of the original data. This is an exected result. The maximum amlitude for x( original ower sectra is 3.89 [Figure (a)] and that for surrogate data x ( ower sectra is Similarly the maximum amlitude for y( original ower sectra is [Figure (b)] and that for surrogate data y ( ower sectra is Notice that the ower sectra amlitude is higher between to 3 Hz than other regions. Namely, the estimation variances are roortional to the ower sectral amlitudes as those frequencies. Considering that the surrogate data contour lots are comuted based on a randomly selected data set among P=5 available, these maximum value comarisons suggests the nown fact that there are no nonlinear or non-gaussian comonents in the original data. The histogram and Gaussian fit of inut data x( and y( and surrogate data x ( and y ( are generated by ziggurat algorithm shown in Figure 2. The PDFs are obtained from these arameters Figure. One sided ower sectra of a) x( and x ( b) y( and y (.The solid curve indicates the result from the original data and the dotted curve indicates the result from one of the 5 surrogate data sets. Scies Coyright 29

5 298 V. Perumalsamy et al. / J. Biomedical Science and Engineering 2 (29) according to Eq.2 and are lotted along with the histogram in blue color in Figure 3. Notice that the PDF here are multilied by the total number of surrogate data sets (5) to match the histogram. From the PDFs, the threshold for a significance level of <.5. By comaring the original data s maximum ower sectra values with the thresholds, it can be seen from the null hyothesis that the original data coming from a linear Gaussian rocesses cannot be reected, a theoretically exected result. (a) (b) Figure 2. Histogram and Gaussian fit of the a) original and b) surrogate data PDF G aussian Fit 3 PDF Gaussian Fit Figure 3. PDF of original ower sectra and the surrogate data ower sectra with Gaussian Fit. Scies Coyright 29

6 V. Perumalsamy et al. / J. Biomedical Science and Engineering 2 (29) Figure 4. µ and σ versus the number of surrogate data sets. We have also erformed a study to examine whether fitting an extreme value distribution to the emirical histogram is a viable aroach. The fitted model arameters µ and σ versus the number of surrogate data sets used are lotted in Figure 4. t can be seen after around 5 surrogate data sets, the estimation becomes linear. 4. SLEEP SPNDLES DETECTON A seven minutes recording of 9 channels of EEG (C3, A2, O, O2, C4, P3, P4, F3, and F4) was used in our test (6,27 trials) shown in Figure 5. Precise numbers of slee stages and standard characteristics derived form hynograms are in Table. Number of slee stages (7s records) are in the first column; data in the second column are the average values over 5 subects (C3A2, C4P4, F3C3, P3O, P4O2) related to total slee time, the beginning of slee is set as the first aearance of the slee Stage 2. Slee efficiencies is the ratio of time sent in Stages -4 or EM slee to the whole slee time, where also movement time and some awaenings during the night are included; in slee medicine this arameter discriminates some slee disorders. Surrogate data were generated following the rocedure in Section 2. The EEG derivation C3A2 and C4P4 was analyzed shown in Figure 6. Oscillatory events are detected, if the daming coefficients r falls above a re defined threshold and hence r, exceeds the corresonding threshold. n ractice, we use two thresholds, a lower one, r a, to detect candidate events scanning the EEG with non-overlaing -s segments. When r a is crossed we go bac to the revious segment and use a smaller ste size of /6s (overlaing -s segments). f r exceeds a second threshold r b > r a the beginning of an oscillatory events is detected. Figure 5. Human slee EEG signals (9 channels C3, A2, O, O2, C4, P3, P4, F3, and F4). Scies Coyright 29

7 3 V. Perumalsamy et al. / J. Biomedical Science and Engineering 2 (29) Table. Number of slee stages (first column) and slee efficiency and average ercentage of slee stages during the slee (mean ± standard deviation). Total 6,27 Slee Efficiency[%]: 92.6 ± 5.3 Waing 76 % Waing 8. ± 3.2 Stage 89 % Stage 6.9 ±.4 Stage % Stage 2 46 ± 7.3 Stage % Stage ± 4.2 Stage % Stage 4.9 ± 2.6 EM Slee 327 % EM Slee 7.9 ± 3.6 Movement time 7 % Movement time.7 ±.6 (a) Figure 6. Slee sindles detection for a) C3A2 and b) C4P4 channel (arrow mar reresentation). (b) The oscillatory events are terminated by the time r is lower than r b for the last time before it falls below r a. The frequency and time at the osition of the maximal value r max are considered as the frequency and occurrence of the event resectively. Sindles or Sigma waves are a oor indicator for slee onset. Some subects do not have discernable sigma waves. These sindles are more rominent in Stage 2 than Stage. n the resent analysis the order of the A model was set to =8 and the threshold \value set to r a =.75 and r b =.85 arameter were chosen in such a way, that clearly visible slee sindles were reliable detected by the algorithm. Slee stages were visually scored according to standard criteria in Figure 7. n Figure 7 shows the detected events from 2 recordings. The distributions show modes in four stages: in the delta (±:-2 Hz), in the fast delta (±2: 2-4 Hz), in the alha ( : 8-2 Hz) and sigma (slee sindles:.5-6 Hz) bands. These modes are also evident in the distribution of the events in the different slee stages (Figure 7). Alha waves redominate in waing, sindles in Stage 2 while the occurrence of delta waves decreases from Stage 2 to Stage 4 (deeening of slee). Delta waves are the revalent events in EM slee and Stage. Note that they also occur in Stage 2 with almost the same incidence. Events in the alha frequency range corresond to continuous alha activity during waing and to small amlitude, sometimes sindle-lie activity in NEM slee. However, in articular during NEM slee Stages 3 and 4, slow waves are often not detected as oscillatory events because they yield a relaxatory ole (frequency zero) in the A-model. Table 2 rovides detection of different frequency bands and their amlitude calculated from the Fast Fourier Transform. Table 2. Different frequency band detection and the corresonding true value determined from one sided Power sectra. S.No Wave Bandwidth (Hz) Time range (sec) Amlitude (µv) Delta, Theta Alha > 7 4 Beta > < 5 Slee Sindles (sigma) < 5 Scies Coyright 29

8 V. Perumalsamy et al. / J. Biomedical Science and Engineering 2 (29) Figure 7. Slee stages from Stage -Stage 4. The two most imortant characteristics of EEG elements are frequency and amlitude. Frequency is inverse value of duration of EEG segment. The frequency range is divided in to four bands: beta (2Hz and higher), alha (8-2Hz), theta (4-8Hz) and delta (, -4Hz). Amlitude of EEG is taen as the ea to ea value. After the magnitude of amlitude EEG signal is divided into low-voltage, middle and high- voltage EEG. For delta, theta and alha bands these values are following: -3 µv for low voltage, 3-7 µv for middle voltage and above 7 µv for high voltage EEG. For beta band, the values are lower: below µv low voltage, -25 µv middle voltage, and above 25 µv high voltage EEG. Beta activity is tyical during waefulness. Alha occurs in relaxed state with eyes closed. Theta waves are dominant in normal wae state in children; in adult eole theta activity aears only in small amount esecially in drowsiness. Delta waves are resent in dee slee; in healthy eole do not exist in wae state. Slee sindles are rhythmic activity in 2-4Hz frequency range and amlitude below 5 µv. The ower sectra for the original (solid line) and the surrogate (dotted line) data are shown in Figure 8. Three eas around -4Hz can be seen, indicating the resence of synchronized activities at these frequencies. According to the traditional classification first ea belongs to the delta band (-4Hz) and the second ea belongs to the alha band (8-2Hz) and third ea belongs to the beta band (>2Hz). Past research has examined whether such oscillatory neural activities self-coule and coule each other in a nonlinear fashion [7]. We study this issue with the new surrogate test method. Figure 8. One sided ower sectra estimated recording from P3O electrode on s eriod. The solid line is for the original data and dotted line is for surrogate data. Scies Coyright 29

9 32 V. Perumalsamy et al. / J. Biomedical Science and Engineering 2 (29) Table 3 rovides comarison between E. Olbrich et al. and our study. However, a few sindles are not detected by the algorithm with the chosen arameters, i.e. the maximum of r remains smaller than the threshold r b. Therefore, in our method roosed is using a surrogate data aroach needs to be further otimized for more sensitive sindle detection. 5. DSCUSSON AND SUMMAY n this study slee sindles were detected in s overlaed segments, wide.5-6hz band containing delta and theta was used for better eye movement searation. Alha activity was excluded as it would increase auto covariance. With consensus scoring we would exect the agreement to imrove [5]. The cost benefit of this laborious tas would have been low. There was no manual artifact handling and it is ossible that some misclassifications are the results of the artifacts undetected by the corresonding threshold r and were generated surrogate data with ziggurat method. n this algorithm there are two different thresholds but as it can be seen from Table 3, the amlitude criterion was most imortant (i.e. mostly amlitude in EM slee is less than 5 µv and NEM slee is greater than µv). Schwilden et al. [4] reorted that 9% of the EEG did not show any nonlinearity. They suggested that only under some athological conditions, such as eilesy, the brain signals manifest nonlinear effects. Other studies [5,6], have shown that non linear characteristics exist between theta and gamma EEG signals during short term memory rocessing. To settle these debates, one needs carefully constructed statistical tests. The method roosed here reresents our effort in this direction. We will contrast our method with other often alied techniques in this area. 5.. Comarison with other Surrogate Data Method Multile ways are currently in use to generate surrogate data sets to test the significance of the slee sindle detection. n one method, the Fourier Transform (FT) is estimated for each segment and the hase of the Fourier Transform is randomized without changing the Fourier amlitude. The result is then inverse Fourier Trans- formed to generate a surrogate time series [4,7]. While the hase randomization destroyed nonlinearity and leads to linear Gaussian rocess [8], the Fourier amlitudes are random variables for a stationary rocess as well. By eeing them constant, one loses an imortant degree of freedom [2]. n contrast, our method, in which both amlitudes and hases of the Fourier Transform are randomly generated, roduces surrogate data sets that are exlicitly Gaussian, linear and share the same second order statistics with the original data. Another method, called amlitude adusted Fourier Transform (AAFT), has been used in recent studies [9]. AAFT was designed to test the null hyothesis that the observed time series is a monotonic nonlinear transformation of a linear Gaussian rocess. This method has the same roblems as the revious hase randomization method. n addition, the generated surrogate data sets usually do not have the same ower sectrum as the original data, leading to false reections when the discriminating statistics are sensitive to second-order statistical roerties [2] Final emars We mae several additional remars regarding our method. First, generating the null hyothesis distribution for the test statistic is a very time-consuming rocess. The alication of the extreme value theory mitigates this roblem. Our examles show that the robability distribution function based on the extreme value theory fits the emirical distribution well. n agrees with that from the emirical histogram. Secondly, when the original data with length L need to be zero added to length N (N > L), the surrogate data we generated have length N. To mae sure that the surrogate data still have the same ower sectra with the original data, the surrogate data need to be normalized by L, instead of the real data length N. Third, the eriodogram method is used to estimate the ower sectra. Consequently, all of the limitations associated with the eriodogram method were inherent in our method, including oor frequency resolution for short data, fourth when we generated surrogate data set using ziggurat method does not generate random numbers effectively in tail regions because Matlab does not execute greater than 5 function calls []. Finally, Table 3. Comarison between A model and A model with surrogate data aroach. S. No. Name of the A-Model (E. Olbrich et al. 23) This study Electrode Bandwidth (Hz) Amlitude (µv) Bandwidth (Hz) Amlitude (µv) C3A C4P F3C3 elaxatory ole in the A model (zero frequency) P3O elaxatory ole in the A model (zero frequency) P4O2-6 <5 2-5 < Scies Coyright 29

10 V. Perumalsamy et al. / J. Biomedical Science and Engineering 2 (29) the discrimination ower of our statistical test is unclear. This can be determined emirically by reeating the test many times on different realization for the data [2]. We will consider this issue as art of our future research. n summary, detection of slee sindles using the surrogate method roosed in this aer is shown to give accurate results when alied to test the significance of ower sectral amlitudes. t is based on solid statistical rinciles and overcomes some weanesses in revious methods for the same urose. t is exected to become a useful addition to the reertoire of nonlinear analysis methods for neuroscience and other biomedical signal rocessing alications. 6. ACKNOWLEDGEMENTS The authors wish to exress their ind thans to Ms. Kristina Susmaova for roviding EEG slee data and invaluable hel and discussions. EFEENCES [] Buzsai, G., (26) hythms of the brain, Oxford, U. K: Oxford University Press. [2] Chen, Y. H., Bressler, S. L., Knuth, K. H., Truccolo, W. A., and Ding, M. Z., (26) Stochastic modeling of neurobiological time series: Power, coherence, granger causality, and searation of evoed resonse from ongoing activity, Chaos, 6. [3] Barlow, J. S., (979) Comuterized clinical electroencehalograhy in ersective, EEE Transactions Biomed. Eng., 26 (7), [4] Franaszczu, P. J. and Blinowsa, K. J., (985) Linear model of brain electric activity EEG as a suerosition of damed oscillatory models, Biol. Cybern., 53, [5] Olbrich, E. and Achermann, P., (23) Oscillatory events in the human slee EEG detection and roerties, Elsevier Science March., 6. [6] Olbrich, E., Achermann, P., and Meier, P. F., (22) Dynamics of human slee EEG, Neuro comuting CNS. [7] Theiler, J., Euban, S., Longtin, A., Galdriian, B., and Farmer, J. D., (992) Testing for linearity in time series: The method of surrogate data, Phys. D., 58, 77 94,. [8] Wang, X., Chen, Y. H., and Ding, M. Z., (27) Testing for statistical significance in Bisectra: A surrogate data aroach and alication to neuroscience, EEE Transactions on Biomedical Engineering, 54(), [9] Wu, W. B., (25) Fourier transform of stationary rocesses, Proc. Amer. Math. Soc., 33, [] Bendat, J. and Persol, A., (986) andom data analysis and measurement rocedures, 2 nd ed. New Yor: Wiley. [] Marsagli, G. and Tsang, W. W., (2) The ziggurat method for generating random variables, J. Stat. Softw., 5, 6. [2] Timmer J., On Surrogate data testing for linearity based on the eriodogram, erints arxiv: com-gas/9593, 995. [3] Evans, M., Hastings, N., and Peacoc, B., (993) Statistical distributions, 2 nd ed, New Yor: Wiley. [4] Schwilden, H. and Jeleazcov, C., (22) Does the EEG during isoflurane/alfentanil anesthesia differ from linear random data, J. Clin. Monit, Comut., 7, [5] Schac, B., Vath, N., Petsche, H., Geissler, H. G., and Moller E., (22) Phase couling of theta-gamma EEG rhythms during short term memory rocessing, nt. J. Psychohys., 44, 43 63,. [6] Shils, J. L., Litt, M., Solnic, B. E., and Stecer, M. M., (996) Bisectral analysis of visual interactions in humans, Electroencehalograhy Clinical Neurohys., l98, [7] Schanze, T. and Echorn,., (997) Phase correlation among rhythms resent at different frequencies: Sectral methods, alication to microelectrode recording from visual cortex and functional imlications, nt. J. Psychohyus., 26, [8] Venema, V., Bachenr, S., ust, H. W., and Simmer, C., Statistical characteristics of surrogate data based on geohysical measurements, Non linear Processes Geohys., 3, [9] Chon, K. H., aghavan,., Chen, Y. M., aghavan, D. J., and Yi, K. P, (25) nteractions of TGF-deendent and myogeneic oscillations in tubular ressure, Amer. J. Physiol., 288, F298 FF37. [2] Schreiber, T. and Schmitz, A., (997) Discrimination ower of measures for nonlinearity in a time series, Phys, ev. E, 55, Scies Coyright 29