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Nonlinear dynamics has been introduced to the analysis of biological data and increasingly recognized to be functionally relevant. The aim of this study is to quantify and compare the contribution of nonlinear and chaotic dynamics of human heart rate variability during two forms of meditation: (i) Chinese Chi (or Qigong) meditation and (ii) Kundalini Yoga meditation. For this purpose, Poincare plots, Lyapunov exponents and Hurst exponents of heart rate variability signals were analyzed. In this study, we examined the different behavior of heart rate signals during two specific meditation techniques. The results show that heart rate signals became more periodic and their chaotic behavior was decreased in both techniques of meditation. Therefore, nonlinear chaotic indices may serve as a quantitative measure for psychophysiological states such as meditation. In addition, different forms of meditation appear to differentially alter specific components of heart rate signals.
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I.J. Image, Graphics and Signal Processing, 2012, 2, 23-29
Published Online March 2012 in MECS (http://www.mecs-press.org/)
DOI: 10.5815/ijigsp.2012.02.04
Chaotic Behavior of Heart Rate Signals during
Chi and Kundalini Meditation
Atefeh Goshvarpour
Department of Biomedical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
E-mail: atefeh.goshvarpour@gmail.com
Ateke Goshvarpour
Department of Biomedical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
E-mail: ateke.goshvarpour@gmail.com
Abstract Nonlinear dynamics has been introduced to the
analysis of biological data and increasingly recognized to
be functionally relevant. The aim of this study is to
quantify and compare the contribution of nonlinear and
chaotic dynamics of human heart rate variability during
two forms of meditation: (i) Chinese Chi (or Qigong)
meditation and (ii) Kundalini Yoga meditation. For this
purpose, Poincare plots, Lyapunov exponents and Hurst
exponents of heart rate variability signals were analyzed.
In this study, we examined the different behavior of heart
rate signals during two specific meditation techniques. The
results show that heart rate signals became more periodic
and their chaotic behavior was decreased in both
techniques of meditation. Therefore, nonlinear chaotic
indices may serve as a quantitative measure for
psychophysiological states such as meditation. In addition,
different forms of meditation appear to differentially alter
specific components of heart rate signals.
Index Terms— Heart Rate Variability, Hurst Exponents,
Lyapunov Exponents, Meditation, Nonlinear Dynamics;
Poincare Plots.
I. INTRODUCTION
Meditation refers to a family of mental training
practices that is designed to familiarize the practitioner
with specific types of mental processes [1]. In addition,
meditation is essentially a physiological state of
demonstrated reduced metabolic activity – different
from sleep – that elicits physical and mental relaxation
and is reported to enhance psychological balance and
emotional stability [2].
There are dozens of specific techniques of
meditation practice; the word meditation may carry
different meanings in different contexts. Meditation has
been practiced since antiquity as a component of
numerous religious traditions. Two types of these
techniques are Chi and Kundalini meditation.
Qigong or Chi meditation is a practice of aligning
breath, movement, and awareness for exercise, healing,
and meditation [3]. With roots in Chinese medicine,
martial arts, and philosophy, qigong is traditionally
viewed as a practice to balance qi (Chi) or what has
been translated as "intrinsic life energy." [4] Typically a
qigong practice involves rhythmic breathing,
coordinated with slow stylized repetition of fluid
movement, and a calm mindful state [5].
Kundalini meditation is a physical, mental and
spiritual discipline for developing strength, awareness,
character, and consciousness. Its purpose is to cultivate
the creative spiritual potential of a human to uphold
values, speak truth, and focus on the compassion and
consciousness needed to serve and heal others [6-7].
Over the past years, a number of studies have
explored the physiological correlates of different types
of meditation. It is important to note here that
meditation refers to a broad variety of practices, ranging
from techniques designed to promote relaxation to
exercises performed with a more far reaching goal, such
as a heightened sense of well-being. This variation, in
itself, makes the study of such practices difficult.
However, we have tried to find similarities among these
practices, and feel that enough prior studies have
demonstrated changes associated with these practices
that seem worthwhile to continue to explore them. For
this purpose, heart rate variability of two specific
meditation techniques was studied.
Heart rate (HR) is a nonstationary signal; its
variation may contain indicators of psychophysiological
states, or warnings about impending cardiac diseases.
Heart rate variability is a measure of variations in the
HR. It is usually calculated by analyzing the time series
of beat to beat intervals from electrocardiogram or
arterial pressure tracings [8].
Heart rate variability (HRV) is a reliable reflection
of the many physiological factors modulating the normal
rhythm of the heart. In fact, it provides a powerful
means of observing the interplay between the
sympathetic and parasympathetic nervous systems. It
shows that the structure generating the signal is not only
simply linear, but also involves nonlinear contributions.
Various measures of HRV have been proposed,
which can roughly be subdivided into time domain,
frequency domain and nonlinear domain measures. Peng
et al [9] applied both spectral analysis and a technique
based on the Hilbert transform to quantify heart rate
dynamics during meditation. Some researchers have
shown that low and high frequency components have
Copyright © 2012 MECS I.J. Image, Graphics and Signal Processing, 2012, 2, 23-29
24 Chaotic Behavior of Heart Rate Signals during Chi and Kundalini Meditation
Copyright © 2012 MECS I.J. Image, Graphics and Signal Processing, 2012, 2, 23-29
increased during meditation [10-11]. Others revealed
that the high frequency power was increased during
meditation [12].
Recently, new dynamic methods of HRV
quantification have been used to uncover nonlinear
fluctuations in HR, which are not otherwise apparent.
Several methods have been proposed: Lyapunov
exponents [13], Poincare plots [14], 1/f slope [15],
approximate entropy (ApEn) [16] and detrended
fluctuation analysis (DFA) [17].
It is now generally recognized that these nonlinear
techniques are able to describe the processes generated
by biological systems in a more effective way, but a few
studies have investigated the nonlinear dynamics of
heart rate signals during meditation [14,18-19].
This study aims to evaluate the dynamic behaviors
of heart rate signals during, focused on the effects of
two different meditation techniques on the nonlinear
dynamics of heart rate variability. For this purpose, we
performed Lyapunov exponents, Hurst exponents and
Poincare plot analysis of heart rate variability signals.
The outline of this study is as follows. In the next
section, we briefly describe the heart rate signals applied
in our study. Then, the computations of the features
(Poincare plots, Lyapunov exponents and Hurst
exponents) are explained. Finally, the results of
application of the nonlinear features to the heart rate
signals on both datasets are presented and the study is
concluded.
II. BACKGROUND
A. Data selection
Two specific meditation techniques were studied: (i)
Chinese Chi (or Qigong) meditation (as taught by Xin
Yan) [9] and (ii) Kundalini Yoga meditation (as taught
by Yogi Bhajan) [9].
1) Chi meditation
The Chi meditators [9] were all graduate and post-
doctoral students. They were also relative novices in
their practice of Chi meditation, most of them having
begun their meditation practice about 1–3 months before
this study. The subjects were in good general health and
did not follow any specific exercise routines.
The eight Chi meditators, 5 women and 3 men (age
range 26–35, mean 29 years), wore a Holter recorder for
about 10 hours during which time they went about their
ordinary daily activities. At approximately 5 hours into
the recording each of them practiced one hour of
meditation. Meditation beginning and ending times were
delineated with event marks.
During these sessions, the Chi meditators sat quietly,
listening to the taped guidance of the Master. The
meditators were instructed to breathe spontaneously
while visualizing the opening and closing of a perfect
lotus in the stomach. The meditation session lasted
about one hour. The sampling rate was 360 Hz. Analysis
was performed offline and meditation beginning and
ending times were delineated with event marks.
2) Yoga meditation
The Kundalini Yoga [9] subjects were considered to
be at an advanced level of meditation training. The four
Kundalini Yoga meditators, 2 women and 2 men (age
range 20–52, mean 33 years), wore a Holter monitor for
approximately one and a half hours. 15 minutes of
baseline quiet breathing was recorded before the 1 hour
of meditation.
The meditation protocol consisted of a sequence of
breathing and chanting exercises, performed while
seated in a cross-legged posture. The beginning and
ending of the various meditation sub-phases were
delineated with event marks. The sampling rate was 360
Hz.
B. Poincare plot
The Poincare plot is a technique taken from
nonlinear dynamics and portrays the nature of R wave to
R wave interval (RR) interval fluctuations. Poincare plot
is a geometrical representation of a time series in a
Cartesian plane. A two dimensional plot constructed by
plotting consecutive points is a representation of the RR
time series on phase space or Cartesian plane [20].
A standard Poincare plot of RR interval is shown in
Figure 1. Poincare plot analysis is an emerging
quantitative-visual technique, whereby the shape of the
plot is categorized into functional classes, which
indicate the degree of the heart dynamics. The plot
provides summary information as well as detailed beat
to beat information on the behavior of the heart [21].
Figure 1. Standard Poincare plot. A standard Poincare plot (lag-1) of
RR intervals [14].
When the signal is steady and unchanging, the
phase space plot reduces to a point, but otherwise, the
trajectory spreads out to give some patterns on the
screen. The pattern that emerges can be interpreted for
finer details such as whether the signal is periodic,
chaotic, or random, etc.
Chaotic Behavior of Heart Rate Signals during Chi and Kundalini Meditation 25
Copyright © 2012 MECS I.J. Image, Graphics and Signal Processing, 2012, 2, 23-29
C.Lyapunov exponents
Consider two (usually the nearest) neighboring
points in phase space at time 0 and at a time t, distances
of the points in the ith direction being
()
0x i
δand
()
tx i
δ, respectively. The Lyapunov
exponent is then defined by the average growth rate λi of
the initial distance
()
() ()
()
()
0x
tx
log
t
1
lim
t2
0x
tx
i
i
2
t
i
tλ
i
ii
δ
δ
=λ
=
δ
δ
(1)
An exponential divergence of initially nearby
trajectories in phase space coupled with folding of
trajectories, ensures that the solutions will remain finite,
and is the general mechanism for generating
deterministic randomness and unpredictability.
Therefore, the existence of a positive λ for almost all
initial conditions in a bounded dynamical system is
widely used.
To discriminate between chaotic dynamics and
periodic signals Lyapunov exponent (λ) is often used. It
is a measure of the rate at which the trajectories separate
one from other. The trajectories of chaotic signals in
phase space follow typical patterns. Closely spaced
trajectories converge and diverge exponentially, relative
to each other. For dynamical systems, sensitivity to
initial conditions is quantified by the Lyapunov
exponent (λ). They characterize the average rate of
divergence of these neighboring trajectories.
A negative exponent implies that the orbits
approach a common fixed point. These systems are non
conservative (dissipative). The absolute value of the
exponent indicates the degree of stability. A zero
exponent means the orbits maintain their relative
positions on a stable attractor. Such systems are
conservative and in a steady state mode. Finally, a
positive exponent implies the orbits are on a chaotic
attractor [22-23]. The magnitude of the Lyapunov
exponent is a measure of the sensitivity to initial
conditions, the primary characteristic of a chaotic
system.
D. Hurst exponents
The Hurst exponent is used as a measure of the
long term memory of time series. It relates to the
autocorrelations of the time series and the rate at which
these decrease as the lag between pairs of values
increases. Studies involving the Hurst exponent were
originally developed in hydrology for the practical
matter of determining optimum dam sizing for the Nile
River's volatile rain and drought conditions that had
been observed over a long period of time [24]. The
name "Hurst exponent" or Hurst coefficient derives
from Harold Edwin Hurst (1880–1978), who was the
lead researcher in these studies, and the use of the
standard notation H for the coefficient relates to this
name also.
The Hurst exponent is a measure that has been
widely used to evaluate the self-similarity and
correlation properties of fractional Brownian noise and
the time series produced by a fractional (fractal)
Gaussian process.
Hurst exponent is used to evaluate the presence or
absence of long-range dependence and its degree in a
time series. However, local trends (nonstationarities) are
often present in physiological data and may compromise
the ability of some methods to measure self-similarity.
Hurst exponent is the measure of the smoothness of a
fractal time series based on the asymptotic behavior of
the rescaled range of the process.
The autocorrelation function of a fractal noise {ξi}
is related to the Hurst exponent H via the (2):
()
2H2
2
i
rii rrc
+
ξ
ξξ
=
(2)
Or, equivalently, in the power spectrum
representation:
(3)
c
f
drercfxfS fri ===
β
π
1
)()()( 2
2
12H
=
+
β
(4)
1/f corresponds to chaotic behavior, and it is
characterized by Fractal properties not by Euclidean
properties. The harmonics of chaotic signals are
fractions of the main frequency of signal. The
corresponding spectral density of 1/f is continued and
described by power law. It suggests that the process is
self similar.
Hurst exponent of 0.5 represents the signal with the
characteristics of ordinary random walk or Brownian
motion. Values for H < 0.5, reflect the negative
correlation between the increments or anti persistent
time series, and H > 0.5, show the positive correlation
between the increments or persistent natural series. In
this study, in order to calculate the Hurst exponents,
Kaiser Window is used.
E. Statistical analysis
In this study, the t-test of the null hypothesis that
data in the vector x are a random sample from a normal
distribution with mean 0 and unknown variance, against
the alternative that the mean is not 0 is performed. The
result of the test is returned in p-value. P-value0
indicates a rejection of the null hypothesis at the 5%
significance level (p<0.05). P-value1 indicates a
failure to reject the null hypothesis at the 5%
significance level.
III. RESULTS
Heart rate signals (before and during meditation) in
two meditation practices and their Poincare plots are
shown in Figure 2. In these meditation techniques,
signals become more periodic and their chaotic behavior
was decreased. According to Figure 2, heart rate
26 Chaotic Behavior of Heart Rate Signals during Chi and Kundalini Meditation
Copyright © 2012 MECS I.J. Image, Graphics and Signal Processing, 2012, 2, 23-29
oscillations amplitude increased and the frequency of it
significantly decreased during meditation.
Different meditation techniques had different
effects on the mean of heart rate signals. In comparison
with before meditation, mean heart rate oscillations
were increased during Yoga meditation, but it was
significantly decreased during Chi meditation. It
suggests that different forms of meditation appear to
differentially alter specific components of heart rate
variability signals.
(a)
(b)
Figure 2. Effects of two meditation practices on heart rate signals.
The HRV signals are shown on the left planes (top: before meditation
and bottom: during meditation), and their respective Poincare plots are
shown on the right planes (top: before meditation and bottom: during
meditation).
(a) Chi meditation (b) Yoga meditation
Poincare plots (lag-1) of RR intervals (before and
during meditation) in two forms of meditation are
shown on the right planes of Figure 2. In comparison to
before meditation, the trajectories spread out in both
meditation techniques (Figure 2). These increments are
more significant in Yoga meditation than that of the
other meditation technique.
The Lyapunov exponents are computed in HRV
signals for each meditation practice (Chinese Chi
meditation and Kundalini Yoga meditation). The box
plots of estimated Lyapunov exponents are shown in
Figures 3. The average values of Lyapunov exponents
are summarized in the Table I.
(a)
(b)
Figure 3. Boxplots of Lyapunov exponents of HRV signals in before
meditation and meditation conditions.
(a) Chi meditation (b) Yoga meditation
According to results, there is a significant decrease
in the Lyapunov exponents during two forms of
meditation compared to before meditation states
(p<0.05), which shows that the HRV signals are less
chaotic during meditation than that of before meditation.
In both groups, the mean values of Lyapunov
exponents are about 0.7 before meditation (Figure 3 and
Table I). However, during meditation these values are
decreased to 0.42 and 0.23 in Chi and Yoga meditation
respectively. Furthermore, heart rate time series are
Chaotic Behavior of Heart Rate Signals during Chi and Kundalini Meditation 27
Copyright © 2012 MECS I.J. Image, Graphics and Signal Processing, 2012, 2, 23-29
TABLE II. AVERAGE HURST EXPONENTS OF HRV IN TWO FORMS OF
MEDITATION.
chaotic both before and during meditation, as suggested
by the positive Lyapunov exponents in either state. Hurst exponents
t-test (p-
value)
During
meditation
Before
meditation
2.7578e-005-0.2409 0.3302
Chi meditation
0.0186 -0.0121 -0.5973
Yoga
meditation
TABLE I. AVERAGE LYAPUNOV EXPONENTS OF HRV IN TWO FORMS OF
MEDITATION.
Lyapunov exponents
t-test (p-
value)
During
meditation
Before
meditation
0.0075 0.4222 0.6827
Chi meditation
0.0005 0.2309 0.7174
Yoga
meditation
IV. DISCUSSION
Heart rate variability analysis has become an
important tool in cardiology, because its measurements
are noninvasive and easy to perform, have relatively
good reproducibility and provide the great information;
information was found directly linked to health [9] and
prognostic information on patients with heart disease.
Different measures of heart rate signals are
designed to reflect different aspects of these signals and
thus do not always provide the same information about
the cardiovascular system [14, 25-27]. Thus, some
measures of heart rate signals may be more attuned to
the effects of meditation on the cardiovascular system
than others.
The framework of the theory of nonlinear dynamics
provides new concepts and powerful algorithms to
analyze such time series. The main goal of this study
was to investigate the nonlinear dynamics of heart rate
signals in the specific psychophysiological state. In this
study, the different behavior of heart rate signals before
and during two specific meditation techniques are
examined (Chi meditation and Kundalini Yoga
meditation).
The results show that signals became more periodic
and their chaotic behavior was decreased in both
techniques of meditation. Furthermore, heart rate
oscillation amplitude increased and the frequency of it
significantly decreased during meditation (Figure 2).
These results were similar to those in previous
meditation studies [9, 18-19, 28-29].
According to Figure 2, both forms of meditation
had different effects on the mean of heart rate signals. In
comparison with before meditation, mean heart rate
oscillations were increased during Yoga meditation
(p<0.05), but it was significantly decreased during Chi
meditation (p<0.05).
In a previous study [14], we demonstrated that the
comparative dynamic measures of the lagged Poincare
plots during meditation give more insight of the heart
rate signals in a specific psychophysiological state. The
Poincare plot is a quantitative visual tool which can be
applied to the analysis of RR interval data gathered over
relatively short time periods. In this study, Poincare
plots of HRV signals were constructed. The results show
that the trajectories spread out during meditation in both
meditation techniques (Figure 2). These increments are
more significant in Yoga meditation than that of the
other meditation technique.
The Lyapunov exponent is a quantitative measure
of separation of the divergence of the trajectories from
their initial close positions. The magnitude of this
exponent indicates the intensity of the chaotic system.
The box plot of Hurst exponents of heart rate
variability signals in two meditation techniques is shown
in Figures 4.
(a)
(b)
Figure 4. Box plot of Hurst exponents of HRV signals in before
meditation and meditation conditions.
(a) Chi meditation (b) Yoga meditation
According to Figures 4, different forms of
meditation have different Hurst exponents. The Hurst
exponents are increased during Yoga meditation, but
decreased during Chi meditation. The average values of
Hurst exponents of HRV signals in two forms of
meditation are summarized in the Table II.
28 Chaotic Behavior of Heart Rate Signals during Chi and Kundalini Meditation
Copyright © 2012 MECS I.J. Image, Graphics and Signal Processing, 2012, 2, 23-29
Figure 3 demonstrates that heart rate variability
time series are chaotic both before and during
meditation, as suggested by the positive Lyapunov
exponents in either state.
Furthermore, there is a significant decrease in
Lyapunov exponents during two forms of meditation
compared to before meditation (p<0.05) (Figures 3),
which states that the heart rate variability signals are less
chaotic during meditation than that of before meditation.
According to the results, in both groups the mean values
of Lyapunov exponents are about 0.7 before meditation
(Figure 3 and Table I). However, during meditation
these values are decreased to 0.42 and 0.23 in Chi
meditation and Yoga meditation respectively. These
results were similar to our previous meditation studies
[18-19].
In addition, the predictability of heart rate time
series is estimated by using the Hurst exponent. The
results of the present study demonstrate that the Hurst
exponents are decreased during Chinese Chi meditation
(p<0.05), but increased during Yoga meditation (p<0.05)
(Figures 4).
V. CONCLUSION
The results suggest that nonlinear chaotic indices
may serve as a quantitative measure for
psychophysiological states such as meditation. In
addition, different forms of meditation appear to
differentially alter specific components of heart rate
signals, and they may have their specific benefits to
improve health.
Other nonlinear indices of heart rate signals like
Fractal dimension can be studied in future. Furthermore,
other biological signals like electroencephalogram
signals and respiration can be analyzed during different
types of meditation.
REFERENCES
[1] Brefczynski-Lewis J.A., Lutz A., Schaefer H.S., Levinson
D.B., Davidson R.J. (2007). Neural correlates of
attentional expertise in long-term meditation practitioners,
PNAS, 104, 11483-11488.
[2] Jevning R., Wallace R.K., Beidebach M. (1992). The
physiology of meditation – a review – a wakeful
hypometabolic integrated response, Neurosci Biobehav
Rev, 16, 415–424.
[3] Cohen K.S. (1999). The Way of Qigong: The Art
and Science of Chinese Energy Healing. Random
House of Canada. ISBN 0345421094.
[4] Yang J.M. (1987). Chi Kung: health & martial arts.
Yang's Martial Arts Association. ISBN 0940871009.
[5] Frantzis B.K. (2008). The Chi Revolution:
Harnessing the Healing Power of Your Life Force.
Blue Snake Books. ISBN 1583941932.
[6] Maheshwarananda P.S. The hidden power in
humans, Ibera Verlag, 47-48. ISBN 3-85052-197-4.
[7] Radha S.S. (2004). Kundalini Yoga for the West,
timeless, 13-15. ISBN 1932018042.
[8] Acharya U.R., Joseph K.P., Kannathal N., Lim C.M., Suri
J.S. (2006). Heart rate variability: a review. Med Bio Eng
Comput, 44, 1031–1051.
[9] Peng C-K., Mietus J. E., Liu Y., Khalsa G., Douglas P.S.,
Benson H., Goldberger A.L. (1999). Exaggerated heart
rate oscillations during two meditation techniques, Int J
Cardiol, 70, 101-107.
[10] Hoshiyama M., Hoshiyama A. (2008). Heart rate
variability associated with experienced Zen meditation,
Computers in Cardiology, 35, 569–572.
[11] Phongsuphap S., Pongsupap Y., Chandanamattha P.
(2008). Changes in heart rate variability during
concentration meditation, Int J Cardiol, 130, 481–484.
[12] Takahashi T., Murata T., Hamada T., Omori M., Kosaka
H., Kikuchi M., Yoshida H., Wada Y. (2005). Changes in
EEG and autonomic nervous activity during meditation
and their association with personality traits, Int J
Psychophysiol, 55, 199-207.
[13] Rosenstien M., Colins J.J., De Luca C.J. (1993). A
practical method for calculating largest Lyapunov
exponents from small data sets, Physica D, 65, 117–134.
[14] Goshvarpour A., Goshvarpour A., Rahati S. (2011).
Analysis of lagged Poincare plots in heart rate signals
during meditation, Digital Signal Processing, 21, 208-
214.
[15] Kobayashi M., Musha T. (1982). 1/f fluctuation of heart
beat period, IEEE Trans Biomed Eng, 29, 456–457.
[16] Pincus S.M. (1991). Approximate entropy as a measure
of system complexity, Proc Natl Acad Sci USA, 88,
2297–2301.
[17] Peng C-K., Havlin S., Hausdorf J.M., Mietus J.E., Stanley
H.E., Goldberger A.L. (1996). Fractal mechanisms and
heart rate dynamics, J Electrocardiol, 28, 59–64.
[18] Goshvarpour A., Rahati S., Saadatian V. (2010).
Estimating depth of meditation using
electroencephalogram and heart rate signals, [MSc. Thesis]
Department of Biomedical Engineering, Islamic Azad
University, Mashhad Branch, Iran. [Persian]
[19] Goshvarpour A., Rahati S., Saadatian V. (2010). Analysis
of electroencephalogram and heart rate signals during
meditation using Hopfield neural network, [MSc. Thesis]
Department of Biomedical Engineering, Islamic Azad
University, Mashhad Branch, Iran. [Persian]
[20] Karmakar C., Khandoker A., Gubbi J., Palaniswami M.
(2009). Complex correlation measure: a novel descriptor
for Poincare plot, Biomed Eng, 8, 17.
[21] Kamen P.W., Krum H., Tonkin A.M., (1996). Poincare
plot of heart rate variability allows quantitative display of
parasympathetic nervous activity, Clin Sci, 91, 201–208.
[22] Haykin S., Li X.B. (1995). Detection of signals in chaos,
Proc IEEE, 83(1), 95–122.
[23] Abarbanel H.D.I., Brown R., Kennel M.B. (1991).
Lyapunov exponents in chaotic systems: their importance
and their evaluation using observed data, Int J Mod Phys
B, 5(9),1347–1375.
[24] Hurst H.E., Black R.P., Simaika Y.M. (1965). Long-term
storage: an experimental study Constable, London.
[25] Storella R.J., Shi Y., O'Connor D.M., Pharo G.H.,
Abrams J., Levitt T.J. (1999). Relief of chronic pain may
be accompanied by an increase in a measure of heart rate
variability, IARS, 89, 448-450.
[26] Vandeput S., Verheyden B., Aubert A.E., Huffel S.V.
(2008). Nonlinear heart rate variability in a healthy
population: influence of age, Computers in Cardiology,
35, 53-56.
[27] Xiao D., He W., Yang H., Yu C. (2006). Study on
correlative dimension of HRV signals and its clinical
Chaotic Behavior of Heart Rate Signals during Chi and Kundalini Meditation 29
Copyright © 2012 MECS I.J. Image, Graphics and Signal Processing, 2012, 2, 23-29
applications, IEEE- EMBC, 17, 4522-4525.
[28] Cysarz D. (2005). Cardiorespiratory synchronization
during Zen meditation, Eur J Appl Physiol, 95, 88-95.
[29] Lehrer P., Sasaki Y., Saito Y. (1991). Zazen and cardiac
variability, Psychosom Med, 61, 812-821.
Atefeh Goshvarpour: Obtained a Masters in
Biomedical Engineering from Islamic Azad University,
Mashhad Branch, Iran in 2010. Her thesis research
focused on analyzing biomedical signals during
meditation. In addition, her research interests include
biomedical signal processing, mathematical modeling,
nonlinear analysis and neural networks.
Ateke Goshvarpour: Obtained a Masters in Biomedical
Engineering from Islamic Azad University, Mashhad
Branch, Iran in 2010. Her thesis research focused on
analyzing biomedical signals during meditation. In
addition, her research interests include biomedical signal
processing, mathematical modeling, nonlinear analysis
and neural networks.
... Previously, some nonlinear methods have been studied to reveal HRV dynamics during meditation. Nonlinear features applied in meditation studies are Hilbert transform (Peng et al., 1999), dynamical complexity (Li, Hu, Zhang, & Zhang, 2011), Poincare plots (Goshvarpour et al., 2011;Goshvarpour & Goshvarpour, 2015), higher order spectra , LE (Goshvarpour and Goshvarpour, 2012a;2012b), recurrence plots (Goshvarpour & Goshvarpour, 2012c), fractal scaling (Alvarez-Ramirez, Rodríguez, & Echeverría, 2017), multifractal analysis (Song, Bian, & Ma, 2013), wavelet entropy (Gao et al., 2016), and approximate entropy (ApEn) (Goshvarpour & Goshvarpour, 2012a). ...
... Previously, some nonlinear methods have been studied to reveal HRV dynamics during meditation. Nonlinear features applied in meditation studies are Hilbert transform (Peng et al., 1999), dynamical complexity (Li, Hu, Zhang, & Zhang, 2011), Poincare plots (Goshvarpour et al., 2011;Goshvarpour & Goshvarpour, 2015), higher order spectra , LE (Goshvarpour and Goshvarpour, 2012a;2012b), recurrence plots (Goshvarpour & Goshvarpour, 2012c), fractal scaling (Alvarez-Ramirez, Rodríguez, & Echeverría, 2017), multifractal analysis (Song, Bian, & Ma, 2013), wavelet entropy (Gao et al., 2016), and approximate entropy (ApEn) (Goshvarpour & Goshvarpour, 2012a). ...
... However, in this part of the literature, a limited number of features have been extracted. For example, lagged Poincare indices (lags of 1-6) were studied in Chi meditators in Goshvarpour and Goshvarpour (2015), LE and ShEn were used while subjects focused on breathing (Goshvarpour & Goshvarpour, 2012a). ...
Article
Purpose The heart is a complex system and many researchers have been recently studying cardiac behavior using the theory of nonlinear dynamical systems. One of the most appealing tools for analyzing heart function is the heart rate variability (HRV) signal. This study aimed to elucidate the HRV dynamics of six distinct states: spontaneous normal breathing (SNB) and metronomic breathing (MB), as non-meditator groups, before Chinese Chi meditation (CCM), during CCM, before Kundalini yoga meditation (KYM), and during KYM, as meditator groups. Methods The HRV data were obtained from the Physionet database. Lagged Poincare indices, Lyapunov exponent (LE), Lempel-Ziv complexity (LZ), and 4 types of entropy were calculated. Results The results showed the greatest discrepancies in the lagged Poincare indices for KYM and CCM. In contrast, the least variations were achieved for MB. Compared to SNB, an enhancement in the log energy entropy and a reduction in the LZ and other entropies were concluded during KYM and CCM practices. In contrast, a reverse pattern was observed for MB. Using support vector machine, HRV dynamics were classified with average accuracies of 99.14 and 98.2% and average sensitivities of 99.87 and 99.57% for pre-KYM and during KYM, respectively. Conclusion It was shown that the HRV dynamics were significantly different in meditators and non-meditators.
... Lyapunov exponents widely apply to the analysis of dynamic system qualitative behavior. They allow estimating trajectory behavior of various objects in physics [1], medicine [2][3][4], economy [5], astronomy [6]. LE determine on the basis of time series analysis most often. ...
Article
Lyapunov exponents (LE) identification problem of dynamic systems with periodic coefficients is considered under uncertainty. LE identification is based on the analysis of framework special class describing dynamics of their change. Upper bound for the smallest LE and mobility limit for the large LE are obtained and the indicator set of the system is determined. The graphics criteria based on the analysis of framework special class features are proposed for an adequacy estimation of obtained LE estimations. The histogram method is applied to check for obtained estimation set. We show that the dynamic system can have the LE set.
... Lyapunov exponents widely apply to the analysis of dynamic system qualitative behavior. They allow estimating trajectory behavior of various objects in physics [1], medicine [2][3][4], economy [5], astronomy [6]. LE determine on the basis of time series analysis most often. ...
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