Annals of Biomedical Engineering, Vol. 37, No. 1, January 2009 ( 2008) pp. 192–200 DOI: 10.1007/s10439-008-9579-8
Comparison of Power Spectrum Predictors in Computing Coherence Functions for Intracortical EEG Signals SERAP AYDıN Faculty of Engineering, Electrical and Electronics Engineering Department, Ondokuz Mayıs University, Kurupelit Campus, 55139 Kurupelit, Samsun, Turkey (Received 12 February 2008; accepted 29 September 2008; published online 22 October 2008)
Synchronic oscillations between two specified EEG channels are highly affected by both certain pathologic and physiologic conditions.3 In literature, frequency domain cross spectrum, the mutual information and phase synchronization where the phase of time series are extracted by using either Hilbert Transform or Wavelet Transform have been applied to epileptic EEG records. All these approaches were performed to investigate the EEG synchronization in case of absence epilepsy for intracortical EEG series recorded from rats.11 In the present study, the linear synchronization quantity so called the Coherence Function (CF) is used to analyze these data and additional rat records where the cross spectra are computed by using several PSD predictors (Burg Method (BM), Yule Walker Method (YWM), Eigen Method (EM), and MUltiple Signal Classification (Music) Method (MM)). These spectral predictors are compared to each other owing to visual evaluation of CF estimations. Comparison of these spectral predictors by visual inspection has been studied for long time human EEG series in Subasi et al.12 This method reports that the MM detected the low frequency peaks whereas the BM could not in seizure where the methodologically selected AR model order is low (p = 8). However, the present study shows that the BM is very useful PSD predictor when the AR order is high (p ‡ 20) as indicated in past EEG analyses consist of AR modeling.2,6 In neurophysiology and cognitive neuroscience, phase synchronization has been practiced to insight the interactions between two different EEG bands.5,9 A linear synchronization measure so called the CF is used instead of the phase synchronization to compare the AR model based and subspace based PSD predictors by visual inspection. This paper is organized as follows: In section ‘‘Materials and Method’’, recording details of intracortical rat EEG series and basic definitions about the methods are given. CF estimations are shown in ‘‘Results’’ section.
Abstract—The present study compares two Auto-Regressive (AR) model based (Burg Method (BM) and Yule Walker Method) and two subspace based (Eigen Method and Multiple Signal Classification Method) power spectral density predictors in computing the Coherence Function (CF) to observe EEG synchronization between right and left hemispheres. For this purpose, two channels intracortical EEG series recorded from WAG/Rij rats (a genetic model for human absence epilepsy) are analyzed. In tests, AR modelbased predictors result the close performance such that the CF estimations are sensitive to the AR model order. Dealing with the subspace-based predictors; certain peaks in CF estimations can also be detected in case of low noise subspace dimension. Besides, they are more computational complexity. In conclusion, high order BM is proposed in EEG synchronization. The results support that each EEG sequence probably meets a high order AR model where the dimension of the related noise subspace is relatively low in comparison to the model order. Keywords—Coherence function, EEG synchronization, AR model, PSD predictors.
INTRODUCTION Synchronization is an active adjustment of rhythms of two oscillating subsystems due to some kind of interactions.1 This phenomenon has been an attractive field of research in both medicine and physics for a long history. However, there is no still unifying synchronization framework. In the present study, generalized EEG synchronization is studied instead of phase synchronization to observe some epileptic functional communications between brain hemispheres.
Address correspondence to Serap Aydın, Faculty of Engineering, Electrical and Electronics Engineering Department, Ondokuz Mayıs University, Kurupelit Campus, 55139 Kurupelit, Samsun, Turkey. Electronic mails:
[email protected],
[email protected]. URL: http://www2.omu.edu.tr/docs/english/1038.htm
192 0090-6964/09/0100-0192/0
2008 Biomedical Engineering Society
Comparison of Power Spectrum Predictors in Computing Coherence Functions
MATERIALS AND METHOD
Left electrodes
193 Right electrodes
Set A
Dataset The data described in website,13 which is publicly available, was used. In this section, only a short description is presented and refer to references7,11 for further details. The complete data consists of five sets each containing two channels intracortical EEG series.
Set B
Set C
Subjects Set D
In experiments, subjects were adult male WAG/Rij rats weighing 265–325 g. These animals were born and received detailed care in the Comparative and Psychology Department Laboratory of Nijmegen University in Holland. At the time of surgery, they were over 6 months old.7 Data Acquisition Ibotenic acid (RBI, I-116) dissolved in phosphate buffer (pH = 7.3) was injected to two deep regions of the brain (6.5 mm and 7.5 mm from the scalp surface) for generation of lesions in the rostral pole of the reticular thalamic nucleus (RTN) for each rat. The needles (outer diameter is 0.4 mm) which were used for this injection were connected via polyethylene tube to a 2 u1 Hamilton micro syringe. The concentration of ibotenic acid was 5 lg/lL (or 6.5 lg/lL). Post-lesion EEG series were collected 3 days after this protocol to observe the base-line. Rats were anesthetized by 240–320 cc/min to implant the steel recording electrodes (Plastic One Inc. MS303/2) placed on the right and left frontal cortex. Later experiment, ibotenic acid was injected via a guide cannula (inner diameter is 0.5 mm, outer diameter is 1 mm and length is 3.5 mm) placed at the rostral pole of the RTN of the right hemisphere (AP = -1.6, L = -1.8). Electrode placement was made on clinical grounds. Coordinates of steel electrodes were AP = 2, L ¼ 3:5: The signals were referenced to an electrode placed at the cerebellum. During the acquisition the EEG series were bandpass filtered at 1–100 Hz and digitized at a rate of 200 Hz. The length of each data segment was 5 s. These records are shown in Fig. 1.
Set E
FIGURE 1. Intracortical two channels EEG series (The series recorded from right hemispheres are shown at the right side and the series recorded from left hemispheres are shown in the left side).
The cross-correlation is the most commonly used measure of linear synchronization between two possibly coupled dynamical systems, x and y. In frequency domain, interactions between these subsystems can be quantified by using the cross-spectrum in form, Cxy ðwÞ ¼ Px ðwÞPy ðwÞ
ð1Þ
Here, Px(w) and Py(w) are frequency spectra where w is discrete frequency index and the asterisk denotes complex conjugation.10 From the correlation theorem, the CF definition can be given as follow to observe the synchronization related to certain frequency ranges, Cxy ðwÞ Cxy ðwÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ Cxx ðwÞCyy ðwÞ Here, Cxx(w) and Cyy(w) are cross-spectrums created by x and y, respectively. This magnitude squared coherence values range from 1 to 0, indicating maximum and no synchrony, respectively. In the present study, parametric PSD predictors (BM and YWM) and subspace based PSD predictors (EM and MM) are comparatively used to obtain frequency spectra of two channels EEG series. Parametric PSD Estimations
Methods A linear synchronization measure so called the CF is used to observe EEG synchronization instead of the nonlinear measures. Reasons of this selection may be listed as follows: (1) To compare the AR model based and subspace based PSD estimations by visual inspection, (2) The CF is very useful measure when synchronization is limited to some particular frequency band, as it is usually the case in EEG signals.
From a statistical point of view, stochastic signals are generally characterized by their PSD estimations. In this application, parametric methods have been applied to intracortical short EEG series to obtain higher frequency resolution in comparison to the nonparametric estimation approaches such as Periodogram. Theoretically, it is stated that nonparametric PSD estimation methods suffer from spectral leakage effects due to inherent windowing where the
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autocorrelation function is assumed to be zero outside the window.8,10 Parametric PSD estimation methods are modelbased approaches. Therefore, the model parameters are estimated firstly and then PSD of the given data sequence is computed. Among three linear models (AR, AR-Moving Average (MA), MA), AR model is the mostly used model due to the fact that AR model having less coefficients is the most useful for representation of a narrow spectrum.8,10 Moreover, the AR model, which is a causal all-pole model driven by a white noise, has been already practiced in EEG analysis. This parametric modeling provides us to track changes in the source of EEG.10 In AR model based parametric PSD estimation methods, the observed data denoted by x(n) is assumed to be output of a linear system characterized by a transfer function denoted by H(z). These approaches so called the BM and YWM will be shortly introduced in the following sections. Burg Method The BM results a stable AR model with high frequency resolution. In addition, it has no spectral leakage problem.10 Assuming the signal to be analyzed meets an AR model with order p in form, xðnÞ ¼
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ð3Þ
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The corresponding transfer function H(z) is expressed in terms of AR model parameters (ap(.)) satisfying the Levinson Durbin recursion algorithm10 as follows, 1 1 Pp ¼ HðzÞ ¼ AðzÞ 1 þ i¼1 ap ðiÞzi
ð4Þ
Then, the least square error denoted by e^2t (sum of the forward and backward errors) is minimized in the BM.10 From the estimates of the AR model parameters, PSD is computed as e^2t
P^Burg ðfÞ ¼ 1 þ Pp ap ðiÞej2pif 2 i¼1
ð5Þ
Generally speaking, this model based parametric predictor has several advantages in estimating PSD: (1) Owing to no windowing, it is independent of the unrealistic assumption that the autocorrelation sequence is zero outside the window, (2) Its frequency resolution is high, (3) It yields a stable AR model, (4) There is no computational complexity.10 Besides those advantages, the model order selection highly affects the performance of the BM.8,10 Therefore, the real valued suitable model order must be correctly addressed to obtain true spectra.10 In past years, many techniques were proposed on selection of
p.10 However, forward prediction error criterion underestimated the AR model order in some experiments. In addition, both criterions of Akaike information criterion and minimum description length were found to be statistically inconsistent. As a result of the other experiments, it is proposed that AR model order should range from N/3 to N/2 for small data.10 Yule Walker Method The YWM estimates the AR model parameters from the Levinson Durbin Recursion algorithm. The resulting PSD estimation is defined by r2wp P^YuleAr ðfÞ ¼ 1 þ Pp ap ðiÞej2pif 2 i¼1
ð6Þ
where r2wp is the estimated minimum mean square value for the pth order predictor. r2wp ¼ rxx ð0Þ
p Y
2 1 ap ðiÞ :
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The BM was found to be superior to the YWM in reported applications for short data analysis under signal-to-noise ratio of 20 dB.10 Subspace-based PSD Estimations Eigen Method Assuming the signal to be analyzed consists of p complex sinusoids corrupted by additive a white noise sequence of the form, xðnÞ ¼
p X
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ð8Þ
i¼1
Here, the unknown variables, Ai, fi and /i are used to refer amplitude, frequency, and phase, respectively. In case of statistically independently random phase values uniformly distributed over (0, 2p), x(n) satisfies a widesense stationary process principles. So, its autocorrelation function is computed by, rxx ðmÞ ¼
p X
ðAi Þ2 expð2pjmfi Þ þ r2w dðmÞ
ð9Þ
i¼1
spectral o density of the white noise sequence where r2w is n 2 w(n), i.e., E jwðnÞj ¼ r2w : When x(n) has N samples, discrete index symbolized by m is ranged from -N to +N. Then, the related autocorrelation matrix of rank p is presented as follows, Rx ¼
p X
2 ðAi Þ2 si sH i þ rw I
ð10Þ
i¼1
Here, H denotes Hermitian (conjugation transpose) and si ¼ ½ 1 2pfi . . . 2pðN 1Þfi : Then, eigendecomposition of Rx gives the principal eigenvectors to span the signal subspace.10
Comparison of Power Spectrum Predictors in Computing Coherence Functions
Music Method The MM projects the given data onto the entire noise subspace by using the eigenstructure of the data. Supposing first that, the data sequence consists of complex sinusoids in presence of white noise, its autocorrelation function is defined by Rxx ¼ E xxH : Then, Rxx can be decomposed into matrix pairs as follows, X VH ; Rxx ¼U X where ¼ diag k1 ; k2 ; . . . ; kr ; r2w ; r2w ; . . . ; r2w Here, r indicates the dimension of the signal subspace which is orthogonal to the noise subspace. Then the MM invokes an eigen-filtering in form, 1 P^Music ðfÞ ¼ PK1 2 i¼0 jAi ðfÞj
ð11Þ
to estimate the PSD of x(n). Here, a desired polynomial, Ai ðfÞ; covers the eigenvectors where the noise subspace dimension is K = N - r.10
RESULTS The CF is considered as a linear synchronization measure ranges from 1 to 0, indicating maximum and no synchrony, respectively. The AR model-based PSD predictors (BM and YWM) and subspace-based spectral predictors (EM and MM) are provided to analyze the intracortical EEG series explained in section ‘‘Dataset’’. For the quantitative analysis, for each rat and condition, 10 data segments pre- and 10 segments post-lesion were analyzed, five of these segments corresponding to normal EEG’s and the other five containing spike discharges. In the previous study which cover the analysis of three examples (Set A, Set B, Set C), nonlinear synchronization measures (mutual information and phase synchronization) did not found to be superior to linear synchronization measures (cross-correlation and coherence).11 In this section, the most popular linear synchronization method so called the CF is used to observe synchronic oscillations between different locations of the brain. Four spectral predictors are compared to each other by visual inspection in detecting of peak frequencies via CF estimates. It is possible to choose a more restrictive analysis method to insight a certain frequency band. However, it will be studied in a future work. The best results are obtained by using the AR model-based predictors when the number of Fast Fourier Transform (FFT) is 128. The order of the considered AR model is empirically selected as high as possible due to the low signal-to-noise-ratio (SNR) and short data length as proposed in literature.10
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In performing of the subspace-based predictors, the most smooth and sensitive results are obtained when the number of FFT is 256. Therefore, AR model-based predictors have less computational complexities. In applications, the first PSD predictor is so called the BM. The CF estimates obtained by using the BM for several AR model orders (p = 10, 20, 30, 40, 50) are shown in Fig. 2. For all sets, any expected synchronization cannot be observed in case of small p (i.e., p = 10). However, when p is increased, the peak frequencies of some synchronic oscillations can be determined where the magnitude of the CF is more close to 1. As p increase the distributions became more sharp. Set A is a normal EEG and the others have spike discharges. In particular, Set B and Set C show a similar global synchronization behavior as stated in Quiroga et al.11 Besides, both Set D and Set E show the highest synchronization as indicated in Luijtelear et al.7 The CF estimations are highly affected by changes in the AR model order for both Set D and Set E: Such that, high model degrees has led to high frequency resolution in estimations. It is well known that EEG phenomena range in frequency between 2 Hz up to 50 Hz in features observed in sleep and epilepsy.4 The recent analysis is investigated in accordance with this main EEG characteristic as follows: 1. For Set A: High degree of synchronization between 0–4 Hz and less synchronization between 5–7 Hz are observed. The CF estimations of Set A show a significant interaction concentrated between 1–10 Hz. 2. For Set B: High degree of synchronization between 6–8 Hz is observed and much less synchronal spikes oscillate at around 15 Hz. 3. For Set C: A certain peak placed between 6–10 Hz is observed. As reported by Quiroga et al.,11 in Set C the spikes have slightly different time lags between the right and left channels in comparison to Set B. Therefore, the CF estimations of Set C are slightly different from the CF estimations of the Set B. As reported in Quiroga et al.,11 in both Set B and Set C, it is commonly observed that there is a certain high peak between 7–10 Hz and a harmonic at about 15 Hz. 4. For Set D: High magnitude specified peaks placed between 7–10 Hz and 16–19 Hz are observed. Smaller correlation around 24 Hz and less synchronal oscillations at about 32 Hz are obtained in case of p ‡ 30. 5. For Set E: High synchronization for two specified frequency intervals (8–10 Hz and 18–20 Hz) are clearly provided. Smaller correlation around 28 Hz is also observed in case of p ‡ 20. The mentioned certain peaks in CF estimations driven by the BM can also be resolved by the YWM as
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shown in Fig. 3. The BM is not found to be superior to the YWM in this case study. Both BM and YWM exhibit sensitive to model order for short time EEG series where the frequency resolution of CF estimations improves as the model order increases (i.e., p ‡ 20). In applications, the third PSD predictor is so called the EM. The CF estimates obtained by using the EM for several subspace dimensions (K = 10, 20, 30, 40, 50) are shown in Fig. 4. For all sets, high degree of synchronizations, which are summarized in items above, can be provided for small dimensions, i.e., K £ 30. As K increases, some fluctuations are observed in CF estimations for Set C, Set D, and Set E. In addition, degree of some certain synchronizations (high synchronization
between 7–10 Hz in Set D and high synchronization between 18–20 Hz in Set E) decreases for K > 30. In summary, it can be concluded that the EM exhibits more sensitive to selection of K when the EEG data set consist of more synchronic oscillations. In other words, the EM does not exhibit sensitive to K if there is no abnormal synchronization for given EEG series. Figure 5 shows that the MM provides the results similar to the results created by the EM. However, the MM exhibits more sensitive to K for all data sets. For high dimensions, i.e., K > 30, the MM generates the more spurious peaks in all data sets. It can be concluded that the EM is more useful than the MM in estimating the CF for short time EEG series under low SNR.
Comparison of Power Spectrum Predictors in Computing Coherence Functions
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In a brief summary, among the AR model based and subspace based PSD predictors, the BM is found to be the most useful method to observe frequency domain EEG synchronization where the data length is short and the SNR is low.
CONCLUSIONS Overall results show that Set A consists of pre-lesion EEG series, whereas the others are related to epileptic lesions: There are more lesions in both Set D and Set E in comparison to Set B and Set C. All data sets show sleep-wave like oscillations.
Parametric PSD predictors (BM, YWM) are based on the AR model. The data is not windowed in performing these methods. Therefore, application of them eliminates the assumption that the autocorrelation sequence is zero outside the window. The BM and YWM are applied to short time EEG series at low SNR. It is stated that these predictors exhibit spectral line splitting in case of high SNR and sensitive to the initial phase in case of sinusoidal noisy signals.10 In the present application, the high order (p ‡ 20) parametric predictors provide us to useful CF estimations in the present EE synchronization where the number of FFT is 128. Subspace-based PSD predictors (EM, MM) are based on an eigen-decomposition of the correlation
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matrix of the noisy data. These methods assume that the data consists of some real sinusoids and then the frequencies of sinusoidal components are estimated. For the data at low SNR, it is difficult to determine the number of principal eigenvectors.10 In the present case study, subspace-based predictors can also provide useful estimations when the empirically selected subspace dimension is low (K = 10) such that the EM is less sensitive to this parameter. Therefore, the low dimensional EM can also be proposed in EEG synchronization for short time EEG series at low SNR. However, adequate estimations are obtained when the number of FFT is 256 at least.
In conclusion, it can be said that high order AR model-based predictors are more suitable than the subspace-based predictors. The reasons of this superiority can be summarized as: When the sampling frequency is low, short epochs are advised in EEG analysis as best guarantee for wide sense stationary. Therefore, a stable AR model can be yielded by the parametric estimations. High frequency resolution can be obtained by using both BM and YWM without consuming a large memory. Note that, the recent results support that the short time intracortical EEG series can be represented by a high order (p ‡ 20) AR model as indicated in past literature.2,6 In the former EEG study, p = 10 was
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provided for successful AR modeling with high accuracy. It was suggested in the latter study that p = 30 is the best order to verify normality in EEG series collected during anesthesia.2
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Manolakis, D. G., V. K. Ingle, and S. M. Kogon. Statistical and Adaptive Signal Processing. Artech House Press, Chaps. 1 and 9, 2005. 9 Nikulin, V. V., and T. Brismar. Phase synchronization between alpha and beta oscillations in the human electroencephalography. Neuroscience 137:647–657, 2006. doi: 10.1016/j.neuroscience.2005.10.031. 10 Proakis, J. G., and D. G. Manolakis. Digital Signal processing, Chap. 12, 3rd ed. Prentice Hall, pp. 925–956, 1996. 11 Quiroga, R. Q., A. Kraskov, T. Kreuz, and P. Grassberger. Performance of different synchronization measures in real
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