The measured time history of the singing bowl sounds (top) and brain waves (middle) are presented in Fig. 2. The three bottom panels magnify the brain waves recorded at the characteristic temporal locations (beginning, halfway, and end) of the experiment, illustrating that the magnitudes of the brain waves increase with time and are significantly larger at the end of the experiment than those in the beginning. The increase was apparent in the low frequency components, as seen in the magnified figures. This type changes in EEG are known to be common in psychological relaxation or meditation[5].
3.1 Temporal and spectral characteristics of the singing bowl sound
Fig 3 shows a typical measured waveform of the singing bowl sound. It gradually diminishes in amplitudes for more than 40 seconds after hitting the percussion instrument and persists for approximately 50 seconds (Fig. 3a). A part of the waveform (marked by ‘A’) was expanded in the time axis to reveal its low frequency variation of the sound which is called as a beat (Fig. 3b). It was observed that the beat repeated at an interval of approximately 0.15 second.
Fig. 3c is the frequency spectrum of the singing bowl sound. The fundamental frequency (marked in ‘B’) that determines the pitch of the singing bowl sound was found to be 482.61 Hz. This frequency corresponds to B4 note in the musical scale. As seen in Fig. 3c, the singing bowl sound contains not only the fundamental frequency but also additional spectral components. The spectral components were observed at 773.15 Hz, 1102.56 Hz, 1464.81 Hz, and 1870.86 Hz, corresponding to the musical scales near to G5, C#6, F#6, and A#6, respectively. The number and magnitude of these spectral components determine the tonal property of the singing bowl sound. In addition, as seen in the box of ‘B’ in Fig. 3c, an additional frequency component (relatively small but significant) appears near the fundamental frequency (482.61 Hz). The minute frequency difference of 6.68 Hz between them causes beating phenomenon.
In order to calculate the frequency spectrum of the beat, we reconstructed its time domain signal using Hilbert transform, plotted in Fig. 3b as the envelope of the singing bowl sound. The envelope, Fig. 3d is the frequency spectrum of the beat rhythm plotted in the frequency range of 0 Hz~50 Hz, used in clinical brain waves. As shown in Fig. 3d, the strongest beat was observed to occur at 6.68 Hz, while a pair of minor beats appeared at the either side from about 1 Hz of 15 Hz. Note that the frequency of the strongest beat is located in the theta wave band (4 Hz~8 Hz), well observed in meditation. Fig. 3e is the time-frequency representation of the beat signal, showing the temporal variations of the multiple beat frequencies. The spectrogram was calculated using a short time FFT with a window length of 4 sec and at the time resolution of 0.5 sec. As expected, it is clearly seen that the strongest beat is shown at 6.68 Hz. Its loudness was maximum in the beginning of playing the singing bowl (t=0) and started to decrease rapidly from 10 sec until 30 sec. The minor multiple beats are seen at the frequency of near 10 Hz, 13.3 Hz, 16.2 Hz, and 36 Hz, disappearing within 10~20 sec.
3.2 Spectral magnitudes of brain waves
Fig. 4 shows the temporal changes in the spectral power of the measured brain waves, plotted at every 50 sec for which the singing bowl was repeatedly played. The seven spectral bands were considered in the study, including the five clinical frequency bands (delta: 0 Hz~4 Hz, theta: 4 Hz~8 Hz, alpha: 8 Hz~13 Hz, beta: 13 Hz~30 Hz, and gamma: 30 Hz~50 Hz), the entire frequency range (0 Hz~50 Hz) and the beat frequency (6.68 Hz). The mean and standard error of the spectral magnitude of the brain waves recorded for the 17 individuals are plotted at the temporal middle of each 50 se c singing bowl sound (t = 25, 75, 125, 175, 225, 275, 325, and 375 sec). The initial monitoring time ti = 25 sec represents the temporal middle of the 50 sec with no sound before the first singing bowl sound, and the final time tf = 375 sec is that after the last (6th) singing bowl sound. Note that the dotted and solid lines represent the measurement locations of the F3 and the F4, respectively. The ranges of the vertical axis were set in the 50 %~170 % of the overall mean values measured in the F4, so that they are all different. The spectral magnitude of the brain waves measured in the F4 were observed to be similar (Fig. 4e,f) or slightly larger (Fig. 4a~d,g) than those measured in the F3. However, there was no statistically significant difference observed between the measurement locations (F3 and F4) in all the frequency bands(Fig. 4). The ranges of the minimum to the maximum p values are presented in Fig. 4. and they are shown to be large enough for admitting that the location effects may not be significant. The data collected at each monitoring time are tested for their statistical normality using Shapiro-Wilk test.
The spectral magnitude of the delta and theta waves were shown to significantly increase with time (Fig 4a and Fig 4b). The increase was found to be the largest at the beat frequency (Fig. 4f). The beta and gamma waves did not show significant changes, while the gamma wave had variability larger than the beta wave (Fig. 4d,e). In contrast, the alpha wave initially increased a bit for 100 sec, followed by continuous decreases with time(Fig 4c). The magnitude of the overall frequency band (0 Hz~50 Hz) increased for the initial 100 sec, and it remained almost unchanged until the second noticeable increase at t = 275 sec, followed by decrease to the end of the experiment (Fig. 4g).
3.3 Synchronized activation of brain waves at the beat frequency
The spectral magnitude of each frequency band of brain waves shown in Fig. 4 are different one another in its initial value. This makes it difficult in comparing their temporal changes one another. To remove the effect of the initial value difference, the magnitude of each frequency band needs to be normalized to the initial value. In addition, the magnitude of the measured brain waves varies from subject to subject. The spectral power of a particular clinical frequency bands is often expressed as a ratio (in %) to the total power of the overall frequency range (0 Hz~50 Hz), to compensate for the differences by subjects.
A new parameter of the spectral magnitude of brain waves was introduced to effectively remove the effects of not only the initial value difference but also the subject dependence. Let M(fb,t) be the spectral magnitude of a frequency band of the brain wave at a time t. The new parameter A(fb,t) was introduced in the present study and defined in Eq.1, which is the magnitude of a frequency band of the brain wave normalized to its initial value and to the magnitude of the overall frequency range.
where fb represents the frequency band; t is the time variable; ti stands for the initial time that is 25 sec in the present study as illustrated in Fig. 4. The numerator of the right hand side of Eq.1 represents the temporal history of the magnitude of each frequency band relative to its initial value, while the denominator is the temporal magnitude of the overall frequency band relative to its initial value. The A(fb,t) stands for the rate of changes in the spectral magnitude of each frequency band that is normalized to that of the whole frequency range (0 Hz~50 Hz).
In order to compare more effectively the temporal changes in the magnitude of each frequency band of the brain wave, we averaged the values measured from the two locations of F3 and F4. This unification was justified by the statistical test that the spectral magnitudes of every frequency band are not different between the two locations for the entire experimental duration, as the maximum and the minimum p values are presented in Fig. 4.
Fig. 5 shows A(fb,t) in %, the rate of changes in the spectral magnitude of each frequency band ((a) delta: 0 Hz~4 Hz, (b) theta: 4 Hz~8 Hz, (c) alpha: 8 Hz~13 Hz, (d) beta: 13 Hz~30 Hz, (e) gamma: 30 Hz~50 Hz, (f) beat: 6.68 Hz), normalized to that of the whole frequency range (0 Hz~50 Hz) and averaged with the data measured at the two locations of F3 and F4 for the 17 subjects. The temporal changes were plotted at every 50 sec for the time from ti = 25 sec to tf = 375 sec, and the error bar represents the standard error. The data are provided in Table 1, together with the p values resulted from the statistical test on each temporal change to the initial value at t=ti. The p value (at t= tf) after the experiment is presented Fig. 5, and, if it is not the minimum value, the minimum is also provided at its time location.
Table 1. Temporal variations (in %) in the spectral band brain wave magnitudes relative to their initial values(0 sce~50 sec), normalized to those of the overall frequency band, and averaged the data measured at the two positions (F3, F4) of the subjects (N=17) who heard the strongly beating singing bowl sounds repeated 6 times at every 50 sec for t=50 sec~350 sec. (†: maximum change)
Time (sec)
|
Delta wave
|
Theta wave
|
Beat
Frequency
|
Alpha wave
|
Beta wave
|
Gamma wave
|
changes in
EEG (%)
|
p
|
changes in
EEG (%)
|
p
|
changes in
EEG (%)
|
p
|
changes in
EEG (%)
|
p
|
changes in
EEG (%)
|
p
|
changes in
EEG (%)
|
p
|
Before
exp.
|
0~50
|
100
|
-
|
100
|
-
|
100
|
-
|
100
|
-
|
100
|
-
|
100
|
-
|
experiment
|
50~100
|
93.52
|
0.014
|
103.16
|
0.176
|
163.90
|
0.006
|
103.69
|
0.137
|
98.48
|
0.248
|
87.99
|
0.000
|
100~150
|
100.09
|
0.985
|
108.57
|
0.015
|
144.40
|
0.160
|
97.69
|
0.411
|
97.54
|
0.280
|
85.18
|
0.000
|
150~200
|
100.37
|
0.933
|
106.85
|
0.037
|
227.66
|
0.006
|
97.57
|
0.533
|
99.16
|
0.579
|
85.53
|
0.000
|
200~250
|
102.71
|
0.641
|
108.09
|
0.031
|
199.67
|
0.032
|
95.47
|
0.275
|
98.78
|
0.586
|
84.10
|
0.000
|
250~300
|
109.62
|
0.220
|
116.16
|
0.003
|
251.98†
|
0.021
|
87.40
|
0.003
|
95.19
|
0.067
|
85.65
|
0.006
|
300~350
|
117.95
|
0.029
|
114.34
|
0.004
|
182.19
|
0.001
|
89.07
|
0.010
|
95.09
|
0.062
|
81.86 †
|
0.000
|
After
exp.
|
350~400
|
135.18†
|
0.001
|
117.07†
|
0.002
|
157.06
|
0.049
|
85.28 †
|
0.005
|
93.75 †
|
0.012
|
90.41
|
0.047
|
As expected, the rate of changes increase the most at the beat frequency. with time (Fig. 5f). Among the clinical frequency bands, the increase rate was the largest in the delta wave (135.18 %, p=0.001), followed by the theta wave (117.07 %, p=0.002). In those two waves located in the low frequency ranges the rate of changes in the spectral magnitude increase with time, whereas they decrease with time in the high frequency ranges including alpha, beta and gamma waves. The tendency of the changes was kept to be extended during the silent time after the last singing bowl sound, except the gamma wave and the beat frequency. This trend implies that the largest changes are observed after the last singing bowl sound rather than when the subjects heard the last singing bowl sound. This is why the p value was the minimum at t = 375 sec rather than at t=325 sec (Figures 5a~d). At the beat frequency, however, the largest increase in the spectral magnitude was observed at the time when the subjects heard the 5th singing bowl sound just before the final one. This would be understood as an extension of the preceding repeated pattern of the (large and rapid) jump and (small and slow) fall, and it is expected to have the spectral magnitude larger than the previous maximum if the subjects may hear the additional (7th) singing bowl sound after the last one.
Fig. 6 compares the maximum rate of the relative changes in the spectral magnitude of each spectral band (A(fb,t) in %), together with the frequency spectrum of the beat of singing bowl sound. The rate of the increase is predominant at the beat frequency, which reaches 251.98 % (p=0.021) of its initial value at the time (t = 275 sec) approaching to the end of the experiment. This implies that brain waves are most effectively synchronized at its beat frequency and activated by the singing bowl sound. Among the five clinical EEG frequency bands, the delta wave rose up the most to 135.18 % (p = 0.001) from its initial state, followed by the theta wave with a rise to 117.07 % (p = 0.002). In contrast, the other three spectral bands decrease after the experiment. The gamma wave was down to 81.86 % (p = 0.000) from its initial state, the alpha wave down to 85.28 % (p = 0.005), and the beta wave down to 93.75 % (p = 0.012).