Journal of Medical and Biological Engineering, 25(1): 7-13

7

Quantification of Pseudo-Periodicity of Alpha

Rhythm in Meditation EEG

Pei-Chen Lo* Jenq-Shiun Leu

Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu, Taiwan, 300, ROC

Received 20 Dec 2004; Accepted 24 Mar 2005

Abstract

As meditation demonstrates its significant healing power, researchers have been trying to find a more scientific way to study the mechanism. In our study of Zen meditation EEG (electroencephalogram), variation of alpha rhythmic frequency is considered as an important index to be monitored in the beginning of meditation session. It might be related to different meditation stages corresponding to various degrees of mindfulness states. Reliable estimate of the alpha rhythmic variation becomes crucial to the exploration of meditation scenario. Conventional methods for spectral analysis hardly discriminate subtle variation of alpha frequency. This paper describes a method based on the periodicity transform (PT) to track the time-varying, instantaneous frequency of alpha rhythm by analyzing the running principal period (RPP). The method is thus abbreviated as PBFT (period-based frequency tracker). Compared with the short-time Fourier transform (STFT) widely applied to the EEG, the proposed method contributes clearer and more proper results about the frequency variation of alpha rhythms and is useful to the study on meditation EEG.

Keywords: Meditation EEG, Periodicity transform, Period-based frequency tracker (PBFT), Variation of alpha rhythmic

frequency, Running principal period (RPP)

Introduction

Meditation is normally recommended as a stress management technique used to bring about a calm and relaxed state of mind. It is thus of great significance to investigate the physiological and mental characteristics of the human life system during the meditation state. The focus of our research is on the meditation EEG (electroencephalograph) analysis. In our study of the Zen-Buddhist meditation EEG, the EEG rhythms provide relevant features in grading the meditation stage [1]. Among various EEG rhythms, alpha (α) rhythm has the frequency ranging from 8Hz to ~13Hz. It was noticed that variation of α frequency provided an index of meditation-state transition from normal-consciousness relaxation to mindfulness meditation [2]. In the beginning of meditation, for instance, an increase in amplitude and a decrease in frequency are often observed when a meditator proceeds to attentively (mindfully) focus on a particular Chakra. Accordingly, tracking time-varying alpha frequency becomes significant in interpreting the meditation EEG record.

Our main objective is to quantify the subtle frequency change in alpha rhythmic band, with respect to the transition from normal consciousness to mindfulness attention in the beginning of Zen-Buddhist meditation. Conventional methods

* Corresponding author: Pei-Chen Lo

Tel: +886-3-573-1667; Fax: +886-3-571-5998 E-mail: pclo@faculty.nctu.edu.tw

based on Fourier analysis have the limitation in discriminating small frequency drift [3, 4]. It allows us to detect the existence of alpha rhythm in an EEG segment of reasonable duration. Nevertheless, such a segment may already involve frequency variation in alpha rhythm. Reducing the window size actually causes much trouble with poor frequency resolution, compared with the merit of feature localization and computational efficiency. Due to the issue mentioned, the short-time Fourier transform (STFT) [5-7], a widely used method in EEG spectral analysis, often fails in discriminating close spectral components. During the past two decades, time-frequency distribution [8-12] and multi-resolution signal analysis [13-19] have being intensively studied. Although they provide solutions to the problem, their demanding computation downgrades the feasibility in practical applications and on-line implementations. Parametric modeling like the autoregressive (AR) modeling approach [20-23] has the drawback of being highly dependent on the implementing parameters like the number of AR coefficients and the number of samples. Bigan and Woolfson introduced two novel methods for tracking variation in frequency [24], the polynomial modeling and phase compensation methods. They concluded with suggestions for combining the two methods to obtain a better performance. Their methods are useful in determining the frequency components of data segments. However, they have problems such as the choice of model order and the ability to converge.

J. Med. Biol. Eng., Vol. 25. No. 1 2005 8

Figure1. Flow chart of S-to-L algorithm. The periodicity transform (PT) [25] introduced by

Sethares and Staley quantifies the spectral characteristics of a signal according to its period instead of frequency. Unlike the Fourier transform, the PT is linear in period rather than in frequency. It can detect the periodicities, repetitions, and regularities in a data sequence and decompose the sequence into a set of periodic-basis elements. There is thus no need for further decomposition should the sequence itself contain only periodic elements. To identify the subtle variation in alpha rhythmic band, this paper applies the concept of periodicity transform to meditation EEG signals. Although EEG rhythms do not exhibit perfect periodicity, our study demonstrates that the quantitative results derived by the PT approach reliably reflect the frequency drift of the repetitive patterns or activities. Similar to the wavelet transform that decomposes a signal into components of various scales (resolutions), the PT describes a data sequence as a summation of various periodic elements. Substantially speaking, the PT will extract those elements possessing some form of repetitive, periodic, or regular activities. Due to the non-ideal periodicity of electrophysiological signals, each element extracted corresponds to the average of approximate periods in an EEG segment. As a consequence, period of the major rhythmic pattern can be effectively identified by the PT quantification. Exploring the virtues of PT, the period-based frequency tracker (PBFT) proposed in this paper contributes to the detection of subtle variation in alpha rhythm. From the results of tracking the alpha variation, we might be able to study the mechanism of mindfulness during Zen meditation. In this paper, the proposed method is demonstrated to be superior to the conventional spectral analysis method based on Fourier transform (such as the STFT).

Theory and Method

This section begins with a brief description of the PT [25] introduced by Sethares and Staley. The PBFT method for quantifying the rhythmic variation in a narrow frequency range, such as the alpha band, is then proposed.

An infinite sequence of real numbers x[n] is called p-periodic if there exists an integer p such that x[n+p]= x[n], ∀n∈I (integer). In practical situation, one may focus on evaluating the periodic behavior over a finite length of N points. Consider a data sequence x[n], 0≤n≤N−1 (N is even). The p-periodic element xp[n], 2≤p≤N/2, of x[n] can be derived by

( ) ,][][],[][1

0,∑

−

=⋅=∆=

p

iipppp nuixnxnx Ρ 10 −≤≤ Nn (1)

where P(x[n], ∆p) is the function to calculate the p-periodic element of x[n]. The p-periodic basis vectors, up,j[n], j= 0, 1, …, p−1, are defined as

∑∞

−∞=+⋅−=

ljp jlpnnu )]([][, δ , (2)

where δ[⋅] is the unit impulse sequence. The sequence 10],[ −≤≤ piix p represents the average periodic pattern

of all the periods in x[n]. The number of periods is computed by Np= round{N/p}, that is, the round-off integer of N/p. Then, the average period is

∑−

=+=

1

0][1][

pN

mpp mpix

Nix , (3)

To search for the best periodic characterization of x[n], the “degree of significance (DOS)” of p-periodicity in x[n] is evaluated for all the possible values of p. The DOS is estimated by computing the norm ratio of the p-periodic vector xp: (xp[−k], …, xp[0], …xp[k]) to that of the vector x: (x[−k], …, x[0], …x[k]). The norm is defined below:

( )∑−=∞→

⋅+

==k

knknxnx

k][][

121lim, xxx , (4)

where ⟨a, b⟩ is the inner product of vectors a and b. In x[n], DOS (denoted by ζp) of the p-periodic element xp[n] is thus evaluated by

x

x p=pς , (5)

A larger value of ζp indicates a higher degree of significance of p-periodicity in x[n].

The PT considers a data sequence to be the summation of a number of different periodic basis elements. It is linear in period, but does not in general provide a unique representation of decomposition. Sethares and Staley proposed some algorithms for periodic decomposition. Among them, the algorithm to extract rhythmic patterns from the shortest to the longest periods (S-to-L algorithm, Fig. 1) is adopted and further modified for analyzing the time-varying α rhythms in

Pseudo-Periodicity of Alpha Rhythm

9

Figure 2. (a) A 2-second EEG signal, (b)-(f) the first five periodic

components derived for p = 54, 55, 53, 56, and 107 samples, (g) residual component.

EEG. Acknowledgment of the presence of p-periodicity in x[n] is determined by the threshold Thd in the algorithm.

To track the variation of alpha rhythmic frequencies, the PBFT algorithm adopts a running window with a fixed window size and a constant moving step. The instantaneous period characterizing a particular framed epoch is the running principal period (RPP) at the time of the mid frame.

To track the RPP and instantaneous frequency of α rhythm, the PBFT algorithm follows two strategies to resolve some practical issues. First, p-periodic components with 2≤p≤pm (pm is the largest period that causes undesired interference) are removed from each windowed segment to reduce the interference from these short-period activities. Note that a rhythmic pattern of period-p may also be identified as a 2p-, 3p-, …, or lp-periodic component (l: integer), similar to the harmonic phenomenon in Fourier-series decomposition. Should an lp-periodic element be within the α band (l≠1), it behaves as a noise component interfering with the true α-periodic pattern in PBFT analysis. As a consequence, the first strategy functions as a lowpass filtering process. We select a pm of 33 for the sampling rate of 500Hz in our experiment.

The second strategy is to identify the first three periods p1, p2, and p3, that is, the periods of the three largest DOS’s (ζp1 > ζp2 > ζp3 > ζpi for all other periods pi’s). Unlike the S-to-L algorithm in Fig. 1, the pi-periodic component is not removed from x[n] even ζpi>Thd. Because unbiased estimate of DOS is required in this part of study, removal of any component will alter the quantitative result of RPP.

Finally, RPP (pr) of each windowed segment is estimated by the first three periods, p1, p2, and p3, and the corresponding DOS’s, ζp1, ζp2, and ζp3:

321

332211

ppp

pppr

pppp

ςςςςςς

++

⋅+⋅+⋅= , (6)

The instantaneous frequency is then derived from the reciprocal of RPP. In the following section, we present the analyzing results for both the simulated signal and the empirical EEG data.

Experiments and Results

A. Extraction of periodic components of in α-rhythmic band Frequency of the α rhythm ranges from 8Hz to about

13Hz. The modified PT method introduced here allows us to extract the M’s most significant components xp[n] by searching for the largest M ζp′s (DOS’s). Furthermore, weights of ζp′s within a particular EEG rhythmic band provide a quantitative evaluation of time-varying pseudo-periodicity. Apparently, resolution of this method is limited by the sampling rate. A higher sampling rate of 500Hz was used to better differentiate subtle frequency variation. Consider an N-sample EEG segment, the PT method is modified to extract the first M periodic components. The proposed algorithm involves the following three steps:

Time(Sec)

Step 1: Select a range of values of periods (P1≤p≤P2) including all the possible periods for the particular EEG rhythm.

Step 2: Employ the S-to-L algorithm to derive period-p component xp[n] and calculate the corresponding DOS (ζp) for all p’s.

Step 3: Arrange xp[n] in DOS order and keep the first M pairs of {xp[n], ζp}. The residue component rM[n] is

∑=

⋅−=MP

PpppM nxnxnr

1

][][][ α , (7)

which represents the residual signal after removing the first M periodic components from the original signal. The coefficient αp is the weighting factor for the periodic element xp[n] (αp =1 ∀p in this experiment).

Fig. 2(a) illustrates a 2-second EEG signal that apparently is dominated by low-frequency α oscillating at about 9Hz. Following the above algorithm, the first five periodic components are derived for p = 54 (ζ54 = 0.387), 55 (ζ55 = 0.382), 53 (ζ53 = 0.341), 56 (ζ56 = 0.199), and 107 samples (ζ107 = 0.132), as displayed in Figs. 2(b)~2(f) respectively. The first four p values correspond exactly to the α frequency ranging from 8.9Hz to 9.4Hz. One extraordinary case is the fifth component with a period of 107 samples (Fig. 2(f)), twice the main period of the original signal (Fig. 2(a)). This component evidently characterizes the rhythmic pattern of period 53~54 samples (9.4~9.3Hz) since a p-periodic sequence is also lp-periodic (l: positive integer). The residual component in Fig. 2(g) exhibits much less α-rhythmic than the original signal in Fig. 2(a).

The following experiments demonstrate the capacity of PT for quantifying pseudo-periodicity and tracking time-varying period in EEG. Consider the same EEG in Fig. 2(a), magnitude of its Fourier spectrum is displayed in Fig. 3(a). Fig. 3(b) illustrates the DOS-versus-p fluctuation when applying the S-to-L algorithm (Fig. 1) with Thd=0.01. In the situation, each p-periodic component is removed when ζp<0.01. The maximum peak occurs at p = 54 samples (ζp=0.387), specifying the dominant frequency of 9.26Hz in this particular EEG segment. Dependence of the DOS on frequency can be examined by DOS-versus-p−1 (Fig. 3(c)), which appears to highly consist with the Fourier magnitude spectrum in Fig. 3(a). Note that the unit of p is sample. However, PT is superior

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Figure 3. (a) Fourier magnitude spectrum of the EEG in Fig. 2(a), (b)

DOS-versus-p (Thd=0.01) curve, and (c) the dependence of DOS on frequency (DOS-versus-p−1 curve).

(a)

(b)

Figure 4. (a) EEG with complex rhythmic pattern in alpha band and (b) its Fourier magnitude spectrum.

to Fourier spectral analysis in a complex rhythmic situation. Considering the example of irregular α rhythm, conventional Fourier spectral analysis often results in multiple peaks that obscure the identification of dominant EEG pattern. As illustrated in Fig. 4, Fourier magnitude spectrum of a 2-second EEG involves three major peaks locating at 4.0Hz, 9.0Hz, and 10.5Hz. It is difficult to determine the main rhythm from the magnitude spectrum. On the other hand, the DOS analysis results in an estimate of the principal period of p=54 (9.26Hz). As a consequence, the PT approach provides an ideal tool for long-term EEG rhythmic tracking and interpretation.

The other merit of DOS analysis is its ability of achieving better frequency resolution with fewer data. In consideration of the aspect of computational efficiency, DOS analysis is accordingly more robust than Fourier spectral analysis. Gabor’s uncertainty principle further demonstrates the conflict between frequency resolution and temporal window size when applying the Fourier analysis. For instance, the Fourier

(a)

(b)

(a) (c)

(b)

Figure 5. (a) Six one-second EEG segments, including 3 slow-α and3 fast-α epochs, and (b) the corresponding DOS-versus-p curves.

magnitude cannot resolve spectral component closer than 0.5Hz with the FFT size reduced to 2 seconds (1000 samples), that leads to an estimate of the major spectral peak at 9Hz. On the other hand, DOS analysis of the same 2-second EEG segment results in a more precise estimate of the principal period: p=54 samples (9.26Hz).

As addressed in the beginning, this study was aimed at tracking the α-band rhythmic drift in a long-term meditation EEG record. To demonstrate the feasibility of distinguishing between the low-frequency α and the high-frequency α, we analyze six one-second EEG segments, including three slow-α and three fast-α epochs (Fig. 5(a)). Fig. 5(b) illustrates the results of DOS analysis. Two distinct groups of DOS peaks locating in the p ranges: 54≤p≤58 and 45≤p≤47 (in number of samples) correspond respectively to the slow-α and the fast-α rhythms. The first 5 periodic components for the six α epochs in Fig. 5(a) are listed in Table 1, with the corresponding ζp’s specified within the parentheses. Based on the first three ζp’s, we note that the slow-α epochs consist of periods mainly in the range 53≤p≤59, while the p interval is 44≤p≤48 for the fast α’s. These two p-intervals quantitatively differentiate between the slow-α and fast-α rhythms, and consequently provide an adequate criterion for tracking the time-varying alpha frequency.

Pseudo-Periodicity of Alpha Rhythm

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Table 1. The first five periods and corresponding ζp for the six α epochs in Fig. 5(a).

order slow1 slow2 slow3 fast1 fast2 fast3

1 54(0.894) 56(0.867) 58(0.853) 47(0.837) 45(0.761) 46(0.814)

2 55(0.858) 57(0.820) 57(0.843) 48(0.810) 46(0.721) 45(0.728)

3 53(0.853) 55(0.818) 59(0.806) 46(0.765) 44(0.588) 47(0.716)

4 56(0.757) 54(0.723) 56(0.783) 49(0.696) 47(0.576) 44(0.560)

5 52(0.750) 58(0.714) 60(0.701) 43(0.545) 53(0.451) 48(0.552)

(a)

(b)

(c)

Figure 6. (a) The simulated signal and (b) its Fourier magnitude spectrum; (c) the frequency-versus-time curves derived by PBFT and STFT.

B. Quantification of time-varying frequency in the simulated sequence

To track the long-term frequency drift in meditation EEG, we employ the running-window method with a fixed window size and a constant moving step. A window size of 0.5 − 2 seconds is normally used when applying STFT to the EEG. The PBFT method, entirely implemented in time domain, is expected to be more sensitive to local waveform variation. Accordingly, we first investigate the effect of window size on the estimate of the RPP. The simulated signal consists of two components. The major component is a chirp signal generated by Matlab (frequency varying linearly with time from 8Hz to

(a)

(b)

(c)

Figure 7. (a) Running frequency of 5-minute EEG (60 seconds per tracing), (b) a detailed illustration of a 30-second EEG and its running frequency (10 seconds per tracing), and (c) the largest DOS for the 30-second EEG in (b).

10Hz in 5 seconds). The minor component of one half the amplitude of the chirp has a frequency in the upper α band (an 11Hz sinusoid). The simulated signal and its Fourier magnitude spectrum are shown in Fig. 6(a) and 6(b). In the proposed PBFT method, results of the RPP analysis are displayed in the form of time-varying frequency (the reciprocal of RPP) curve as shown in Fig. 6(c). The result of PBFT in alignment with the theoretical result demonstrates the effectiveness of PBFT in tracking the linearly varying frequency. The STFT fails to track the time-varying frequency due to poor frequency resolution based on a window size of 500 points. Next, the PBFT method and strategies proposed

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are applied to meditation EEG to estimate the time-varying frequency within the α rhythmic band.

C. Quantification of fluctuation in α rhythmic frequency Conventional approach for tracking the time-varying,

instantaneous frequency in EEG normally involves the Fourier transformation computation. Although fast algorithm provides efficient computation, smaller window size imposes restrictions on the frequency resolution that causes difficulties in discriminating nearby spectral components, as shown in Fig. 6(b). The PBFT approach directly calculates the principal period in time domain and, accordingly, can identify changes in period down to the scale of number of points.

In this experiment, five-minute EEG consisting mostly of α rhythms is analyzed by PBFT. The window size is one second (500 samples). Using a step size of 0.2 second, subtle changes in α rhythms can be readily observed from the result of PBFT, as shown in Fig. 7(a). Apparently, this EEG segment is dominated by high-frequency α oscillating faster than 10Hz. Within the 300-second EEG record, the rhythmic frequency rises from around 10Hz to the range above 11Hz. Fig. 7(b) displays a detailed illustration of a 30-second EEG and its running frequency to demonstrate the efficacy of the method in tracing instantaneous frequency drift. Good sensitivity of PBFT can be observed. For example, a transition from high to low frequency (marked by the downward-arrow symbols, ↓) always lowers down the running curve. The time-varying ζp1, the largest DOS, provides an index of the degree of significance of the dominant period p1 (Fig. 7(c)). For a comparison, the ζp1 curve for the first 10 seconds is plotted in Fig. 7(b) (dashed, thin curve) to demonstrate its significance in reflecting complex rhythm. This value often drops at the transitions or epochs involving complex rhythmic patterns. Nevertheless, ζp1 mostly stays above 0.5 except for those epochs exhibiting remarkably irregular periodicity (e.g., 19s−20s and 28s−29s).

Discussion and Conclusion

In this paper, we demonstrated that the proposed PBFT method, based on PT theorem, can effectively discriminate frequency variation within a narrow α band that had been often observed in the meditation EEG study. This method also helps resolve the difficulty caused by irregular periodicity of EEG. Estimation of the running, instantaneous period in time domain further avoids the problem of poor frequency resolution encountered in the short-time Fourier analysis. In addition, DOS analysis allows us to gain access to rhythmic complexity of EEG that may induce a new view on those well-defined EEG rhythms like the ∆ (0−4Hz), θ (4−8Hz), α (8−13Hz), and β (>13Hz). More substantially, composition of various EEG rhythms may be redefined in an alternative way other than the convention. For instance, we noted that a slow α epoch was characterized as a pseudo-periodic pattern with period drifting between 53 samples (0.106s) and 59 samples (0.118s). Results of analyzing both the simulated signal and meditation EEG show the feasibility and efficacy of this method in tracking the time-varying frequencies. PBFT

facilitates the detection of alpha rhythmic variation that has been recognized as the cue at the early meditation stage.

Acknowledgments

The invaluable assistance of the Taiwan Zen-Buddhist Association is greatly appreciated.

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