III. METHODS
The subjects of this study are divided into two groups: Zen-Buddhist practitioners (ex-perimental subjects) and non-practitioners (control subjects).
EEG background introduction:
In the study we aimed to quantitatively study the brain dynamics (Lo et al. 2000, 2001) in the course of meditation based on the brain electrical activities — EEG signals. The EEG sig-nals, discovered in 1924 by Hans Berger, represent the tracings of summated cortical electrical activity collected by applying multiple recording sensors (called the "EEG electrodes") on the scalp (non-invasive recording) or on the cortex (invasive recording). The cortical potentials ac-tually average the excitatory or inhibitory postsynaptic (EPSP or IPSP) potentials of hundreds of neural cells nearby the recording electrodes. After intensive research for several decades, the EEG has proved to be an important clinical tool for diagnosing and monitoring the central nervous system regarding normal or pathological conditions. For instance, sleep staging based on the EEG has been applied to the evaluation of sleep disorders.
Frequency is one of the important features of EEG waveforms. The EEG rhythmic bands were classified as follows: delta band (0.8 ~ 3.9Hz), theta band (3.9 ~ 7.8Hz), alpha band (7.8 ~ 13.3Hz), and beta band (13.3 ~ 25.0Hz). In our study, we first utilized the spectral contents to characterize and stage the meditation EEG. Method of spectral analysis is based on the short-time Fourier transform (STFT). In our investigation, we occasionally discovered the pos-sible correlation between perception of the inner light and the EEG alpha blocking. A series of experiments were thus performed. More than 60 experienced Zen-Buddhism meditators have been examined in the period from October of 1999 to August of 2003. The experiment contin-ues.
The search for physical and psychological correlates of meditation has centered essentially on three methods: Yoga in India, Transcendental Meditation (TM) in the United States, and Zen Buddhism in Japan. Two major techniques for a beginner to get into good-quality meditation are:
1) switching the breathing habit from chest to abdominal breathing so that the breathing be-comes smooth, deep, and quiet, and 2) guarding some important apertures like the Zen Chakra (inside the third ventricle), the Wisdom Chakra (corpora quadrigemina), and the Dharma-eye Chakra (hypophysis). Figure 1 illustrates the locations of these Chakras. Gradually, the human life system enters a unique status in harmony with the nature and the universe (called “the uni-fication of heaven, earth, and human”). In the past decade, the Zen-Buddhist meditation, as an unconventional therapy, has proved efficacious for many chronic diseases, infections, and even some malignant tumors. Consequently, more people began to practice the Zen-Buddhist medita-tion in Taiwan. It aroused our attenmedita-tion to the EEG investigamedita-tion on the Zen-Buddhist disciples.
New findings have been continuously observed and reported (Lo et al., 2003).
Alpha waves usually occur during relaxed wakefulness with eyes closed. When opening the eyes, occipital alpha blocking normally follows. This phenomenon has been well observed and accepted as a convention in EEG research. Investigators accordingly deduced that the oc-cipital alpha rhythm blocking is associated with increased visual attention. However, the mechanism at the neuronal level is still unknown because there are still uncertainties about the
Biological mechanism of meditation model:
By seeking Zen, one is actually seeking the true energy of life. The only entity being prom-ulgated in the Zen-Buddhism Sect is the truth, the wisdom, and the power of Zen in nature.
Based on the essence of orthodox Zen Buddhism, we hypothesize that its pivotal technique of meditation can be comprehended via the resonance phenomenon.
In a series RLC electrical circuit, the input sinusoid is amplified the most when its fre-quency equals the resonance frefre-quency of the circuit. No resonance occurs in the physiological, mental, conscious, or subconscious states due to the existence of selfhood. To be in resonance with the inner light, disciples of the Zen-Buddhism Sect spend years preparing themselves for the moment of resonance. One of the preparations, for instance, involves transcending the physiological habituation. The first step is to switch the breathing habit from chest to abdominal breathing. Then by guarding some important apertures, the qi-energy starts penetrating, from the corpora quadrigemina (the Wisdom Chakra), through the pineal gland (Figure 1), bridging the energy passage between cerebellum and cerebrum. Gradually, the human life system enters a unique status in harmony with the nature and the universe (called “the unification of heaven, earth, and human”). The physical body will change its constitution and, thus, become totally free from diseases.
Experimental protocol and setup:
The subjects of this study are divided into two groups: Zen-Buddhist practitioners (experi-mental subjects) and non-practitioners (control subjects). The EEG signals were recorded by a 30-channel SynAmps amplifiers (manufactured by NeuroScan, Inc.), using an electrode array with a common linked-mastoid (MS1-MS2) reference according to the international 10-20 method. Figure 2 displays the recording montage. The signals were sampled at 200 Hz and fil-tered at 0.1-50Hz. Experimental subjects were asked to meditate for 30-35 minutes. Controls were asked to sit and rest quietly with their eyes closed for the same period without falling asleep. Each session of EEG recording lasted 45 minutes, including the first and the last 5-minute background EEG recording (in normal relaxed position) and the mid 30-minute medi-tation EEG recording. During the medimedi-tation session, the subject sat, with eyes closed, in the full-lotus or half-lotus position, putting each hand on the unilateral lap (Figure 3). Both hands formed a special mudra (called the Grand Harmony Mudra). The subject focused on the Zen Chakra and the Dharma Eye Chakra (also known as the “Third Eye Chakra”) in the beginning of meditation till he or she transcended the physical and mental realm. The Zen Chakra locates in-side the third ventricle, while the Dharma Eye Chakra locates at the hypophysis (Figure 1).
Methods and algorithms for EEG analysis dimensional analysis and complexity index:
Let X=
{ }
Xi iN=1 be the set of points on the EEG trajectory, where Xi is an n-dimensional point constructed from (1) the single-channel EEG, or (2) the n-channel EEG signals (Lo &Chung, 2000). In the single-channel case, the n-dimensional phase-space point Xi is constructed according to the Takens (1981) embedding theory: a smooth map from the time series (e.g., EEG) {x[i], i=1, …, N+(n−1)τ} to the phase-space trajectory
X=
{
Xi =(x[i],x[i+τ],K,x[i+(n−1)τ])}
iN=1 preserves some of topological invariants of the original system. Here τ represents the time delay in number of samples.For each point in the set X (e.g., Xi), a KNN hypersphere is determined and formed by the K’s nearest neighboring (NN) points
{ }
Vij Kj=1, Vij∈X and Vi1=Xi. The Xi is called the seed point of the ith hypersphere. Inside the ith hypersphere, the largest distance to the seed point Xi is:i iK K
di, NN = V −X (1)
where the operator ⋅ evaluates the Euclidean distance. It was reported (Lo & Chung, 2000) that
where E{dKNN} is the first order moment of dKNN, the Kth NN distance of any hypersphere in X.
Thus, we proposed quantifying the global waveform complexity of multi-channel EEG by esti-mating the complexity index δ as follows (Lo & Chung, 2000):
1
To obtain a reliable estimate of δ, we normally average the δ’s over a moderate range of K’s to obtain the final estimate. The average δ is denoted by δ .
Although Eq.(3) can be easily implemented, a large portion of computer time is spent on searching for the KNN and (K+1)NN distances. Undoubtedly, computer time required by the algorithm implemented in this manner is highly dependent on the values of K and N. A large K costs more effort in the competition process. A large N indicates a large number of distances to be computed.
The authors proposed an approach that does not require computing all the inter-point dis-tances and reduces the exhaustedly sorting process (Lo & Chung, 2001). The approach mainly adopted the eigenfunction and principal-axis analysis. Let [X] be an N×n matrix with its ith row vector representing the ith n-dimensional point Xi on the EEG trajectory. The eigenvector asso-ciated with the largest eigenvalue of the covariance matrix of [X] is denoted by [Φ], a 1×n row matrix. And Φ=1. Then the transformation [Y]=[X][Φ]T results in an N×1 column matrix containing N scalars yi, i=1, …, N on the principal axis (the largest eigenvector). As a result, the inter-point distances of Xi: dij, j=1, …, N are mapped to
Using ρij as reference, the sorting process for determining the di,KNN and di,(K+1)NN becomes less laborious. Details of implementation were given in (Lo & Chung, 2001).
Methods and algorithms for EEG analysis AR model and subband filtering:
The method proposed is focused on monitoring the time-varying characteristic frequency in meditation EEG. The AR model is applied to the subband component to quantify the character-istic frequency. Consider that the EEG signal, x[n], is generated by an autoregressive (AR(p)) process driven by the unit-variance white noise [28]. A pth order all-pole model is formulated by
∑
= obtained if the coefficients ap[k] are known. A number of techniques have been proposed to estimate parameters ap[k]. We apply the autocorrelation method in which the AR coefficients] [k
ap are determined by solving the autocorrelation normal equations
where ε is the modeling error and γx[k] is the estimated autocorrelation function defined be-low:
As addressed previously, a higher model order (for example, p ranges from 6 to 14) is normally required to better estimate the low-frequency component in EEG. According to (7), the coefficients of AR(6) model are determined by {γx[k]0≤k≤6}. As demonstrated in Table 1, AR(6)’s coefficients γx[0] ~ γx[6] for θ and ∆ rhythms are too close to distinguish between each other, whereas the coefficients for α and β exhibit significant deviation.
In fact, increasing the model order up to p=12 even cannot discriminate ∆ rhythm from θ rhythm. The dominant pole pair for ∆ and θ modeled by AR(12) are 0.964∠±0.131 and 0.959∠±0.129, respectively, which results in a close estimate of the spectral frequencies (sym-bol ‘∠’ denotes the phase in radian). In addition, the computational time required by AR(12) becomes quadruple, in compared with that for AR(6).
Downsampling process makes the AR modeling better characterize the low frequency ac-tivities. This is revealed by the AR(6)’s coefficients estimated for ∆ and θ rhythms downsam-pled by 8:
γx[k]∆: {1.00, 0.48, −0.18, −0.48, −0.39, −0.03, 0.06}, (8a)
γx[k]θ: {1.00, −0.16, 0.26, 0.06, −0.42, 0.02, −0.23}. (8b)
Moreover, according to Gabor’s uncertainty principle, downsampling operation improves fre-quency resolution that is desired for the narrow-band EEG. We accordingly employ the sub-band-filtering scheme prior to the frequency analysis by AR modeling.
In summary, EEG signals are firstly decomposed into different subband components by downsampling and filtering. Then the characteristic frequency (root frequency) of each subband component is estimated by AR(2) model. The entire scheme is called the Subband-AR EEG Viewer. The Subband-AR-EEG Viewer can be illustrated by the tree-structural filter bank. In the
Subband-AR EEG Viewer, a linear-phase lowpass FIR filter H(z) with cutoff frequency 30Hz is used as an anti-aliasing filter before the downsampling operation. Then the AR(2) model is ap-plied to the decimated signal. The filtering-and-downsampling process is repeated until the equivalent cutoff frequency equals 1.875Hz.
Differing from the wavelet decomposition, the Subband-AR-EEG-Viewer only employs the lowpass linear-phase filter. A nonlinear-phase filter frequently employed in the wavelet analysis (Vaidyanathan 1993) will distort the temporal information of EEG and consequently affect the AR model coefficients estimated by the autocorrelation method.
The Subband-AR-EEG-Viewer structure can be rearranged as the one shown in Figure 4, which is a six-channel filter bank with two’s-power decimation ratios.
The cutoff frequencies of H1(z), …, H5(z) are, respectively, 30Hz, 15Hz, 7.5Hz, 3.75Hz, and 1.875Hz (sampling rate: 200Hz).
Note that the cutoff frequencies approximate the upper boundaries of the four well-known EEG rhythms β (13~30Hz), α (8~13Hz), θ (4~8Hz), and ∆ (below 4Hz). Therefore, changes of the characteristic frequency in meditation EEG can be traced by quantifying the root fre-quency (fr) of each subband filtered component. For example, when fr’s of output1, output2 and output3 are all within the range 8~12 Hz, the dominated pattern of this windowed segment is identified, to a great degree, as the α rhythm. When fr’s of output1 and output2 are greater than 15Hz and fr of output4 is between 4Hz and 7Hz, the particular segment most likely contains the θ intermixed with β rhythm.
After the subband decomposition, the AR(2) model coefficients are computed. An AR(2) model can be expressed as
]
The model coefficients are directly computed by
where ]γx[k is the autocorrelation function estimated by (3).
The characteristic frequency, also called the “root frequency,” of outputi can be estimated from the phase of the pole, or the root of the model equation (9) expressed in frequency domain.
After obtaining the model coefficients, the conjugated pole pair are
2
Therefore, the root frequency fr can be formulated as
]
Because the root frequency is much smaller than the sampling frequency, the result of sin−1x can be approximated by x. For example, the α rhythm having a higher frequency of 12Hz results in a normalized radian frequency of 0.12π (assume fs = 200Hz). The approximation only causes a 2.35% deviation from the true value. Note that output2~output6 are the results of downsampling (Figure 4), the root frequency fr,i should be further divided by 2i. According to equations (10) to (13), root frequency of each subband component depends on γx[0], ]γx[1 , and γx[2].
Tracking the root frequency of each subband component provides an efficient way to illus-trate the time evolution of characteristic frequency in meditation EEG. The following section presents two algorithms for the EEG feature extraction and the signal segmentation based on the ideas and methods introduced in this section.