Phase transition in a healthy human heart rate.
ABSTRACT A healthy human heart rate displays complex fluctuations which share characteristics of physical systems in a critical state. We demonstrate that the human heart rate in healthy individuals undergoes a dramatic breakdown of criticality characteristics, reminiscent of continuous second order phase transitions. By studying the germane determinants, we show that the hallmark of criticality--highly correlated fluctuations--is observed only during usual daily activity, and a breakdown of these characteristics occurs in prolonged, strenuous exercise and sleep. This finding is the first reported discovery of the dynamical phase transition phenomenon in a biological control system and will be a key to understanding the heart rate control system in health and disease.
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Phase Transition in a Healthy Human Heart Rate
Ken Kiyono,1Zbigniew R. Struzik,1Naoko Aoyagi,1Fumiharu Togo,1and Yoshiharu Yamamoto1,2,*
1Educational Physiology Laboratory, Graduate School of Education, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
2PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan
(Received 24 February 2005; published 28 July 2005)
A healthy human heart rate displays complex fluctuations which share characteristics of physical
systems in a critical state. We demonstrate that the human heart rate in healthy individuals undergoes a
dramatic breakdown of criticality characteristics, reminiscent of continuous second order phase transi-
tions. By studying the germane determinants, we show that the hallmark of criticality—highly correlated
fluctuations—is observed only during usual daily activity, and a breakdown of these characteristics occurs
in prolonged, strenuous exercise and sleep. This finding is the first reported discovery of the dynamical
phase transition phenomenon in a biological control system and will be a key to understanding the heart
rate control system in health and disease.
DOI: 10.1103/PhysRevLett.95.058101PACS numbers: 87.19.Hh, 05.40.2a, 87.80.Vt, 89.75.Da
It has been suggested that the underlying mechanism
behind human heart rate regulation shares the general
principles of other complex systems [1,2]. Indeed, fractal
concepts , chaotic dynamics, and the statistical theory of
turbulence , have been shown to provide useful para-
digms for characterizing heart rate fluctuations, applicable
in the prognosis and diagnosis of cardiovascular diseases
[4,5]. Nevertheless, the nature of heart rate complexity has
eluded satisfactory explanation. Recently, critical phe-
nomena have been proposed as the likely paradigm to
explain the origins of heart rate fluctuations [6,7], suggest-
ing that the theory of phase transitions and critical phe-
nomena in nonequilibrium systems [8–10] may be useful
in elucidating the mechanism of complex heart rate dy-
namics. Characteristic features at a critical point of a
second order phase transition are the divergence of the
relaxation time with strongly correlated fluctuations and
the scale invariance in the statistical properties. Indeed, a
healthy human heart rate has been confirmed robustly to
show these types of behavior [1,5,7,11,12].
Here we seek a phase transition phenomenon in heart
rate in order to support the criticality hypothesis more
strongly. We provide evidence that a clear transition of
heart rate dynamics between distinct, stable ‘‘phases’’ can
be revealed in heart rate fluctuations in healthy individuals.
Within these phases—observed in experimentally con-
trolled behavioral states of sleep and prolonged strenuous
exercise—healthy human heart rate fluctuations undergo a
dramatic breakdown of critical characteristics, in particu-
lar, long-range correlations.
Using a modified random walk method with a detrend-
ing procedure, we study: (1) the correlation properties of
the interbeat intervals, (2) the scale dependence of the non-
Gaussian probability density function (PDF), and (3) the
magnitude correlation function which characterizes the
correlation properties of the local energy fluctuations
across different scales. All of these quantities confirm
that strongly correlated behavior is observed only in the
state of usual daily activity, and a breakdown of these
characteristics occurs in other states. In particular, an
essential difference among distinct states, which has not
been observed previously [1,5], is only observed by using
(3), i.e., the magnitude correlation function, suggesting the
need to account for the scalewise local energy correlations
in studying complex heart rate dynamics, and possibly
other complex real-world signals.
We analyze seven records of healthy subjects (mean age:
25.3 yr) in three behavioral states: (1) usual daily activity,
(2) experimental exercise, and (3) sleep. The data set
consists of the interbeat intervals between consecutive
heartbeats measured over 24 h [Fig. 1], in which the
subjects were initially asked to ride on a bicycle ergometer
for 2.5 h, as the exercise state, and maintain their heartbeat
intervals at 500–600 ms. After the exercise, the data were
continuously measured during usual daily activity in the
daytime and sleep at night, with regular sleep schedules.
As shown in Fig. 1, the data set is classified in four states:
a healthy subject measured over 24 h (13:00–). For analyses, we
selected four subintervals during different states from the records
of seven healthy subjects .
A representative record of heart interbeat intervals for
PRL 95, 058101 (2005)
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2005 The American Physical Society
(A) constant exercise, (B) usual daily activity after the
exercise, (C) sleep, and (D) usual daily activity the next
morning. The data for each state contains over 15000
In order to characterize the heart rate fluctuations, we
use a detrended random walk method. We consider a time
series of sequential heart interbeat intervals, fb?i?g (i ?
1;...;Nmax), where i is the beat number and the time series
is normalized to have zero mean and unit variance.
Because the original time series of the interbeat intervals
is not a diffusive random walk, we first integrate the time
series fb?i?g, and analyze the ‘‘walk’’ of B?i?:
To eliminate the ?d ? 1?th order polynomial trends in-
cluded in the time series fb?i?g, in each subinterval ?1 ?
s?k ? 1?;s?k ? 1?? of length 2s, where k is the index of the
subinterval, we fit B?i? using a polynomial function of the
order d, which represents the trend in the corresponding
segment. The fluctuation statistics show no significant
difference with respect to the order of detrending polyno-
mials if the order is greater than two . Thus, in the
following analysis, we use the third order detrending ?d ?
3?. After this detrending procedure, we define heart rate
increments at a scale s as ?sB?i? ? B??i ? s? ? B??i?,
where 1 ? s?k ? 1? ? i ? sk and B??i? is a deviation
from the polynomial fit.
One of the widely used methods to characterize long-
range power-law correlations is a scaling analysis of the
mean square displacement , S2?s? ? hjB?i ? s? ?
B?i?j2i, where h?i denotes a statistical average; the S2?s?
is related to the so-called Hurst exponent H  as S2?s? ?
s2H.Consider a stationary stochastic processwith hbii ? 0,
ordinary Brownian motion, no correlations of b?i? exist
and H ? 1=2. If the autocorrelation function C??? of
b?i? scales as C??? ? ???(0 < ? < 1) for a large time
lag ?, the mean square displacement is evaluated as
S2?s? ? s2??. Thus, the H represents the long-range
H ? ?2 ? ??=2 > 1=2.
The scale dependence of the square root of S2?s? in each
of the behavioral states is shown in Fig. 2. For the usual
daily activity [Figs. 2(b) and 2(d)], we can see scaling
behavior with a slope 1, which implies 1=f scaling in the
power spectrum and the long-range correlated behavior in
a wide range of scales. Onthe other hand, constant exercise
[Fig. 2(a)] and sleep [Fig. 2(c)] states exhibit a crossover in
for both states, and the slopes for larger scales s approach
0.5, which signifies uncorrelated behavior. This result
demonstrates an almost complete breakdown of the long-
range correlated behavior in these states with higher (ex-
ercise) and lower (sleep) heart rates.
ii ? ?2. If the walk of B?i? is totally random, as in
of the signalas
We previously reported  the robust scale-invariance in
the non-Gaussian PDF of ?sB?i? for a healthy human heart
rate, which is preserved not only in a quiescent waking
state, but also during usual daily activity. In Refs. [15–18],
it has been demonstrated that a non-Gaussian PDF with fat
tails can be modeled by random multiplicative processes.
Let us represent the increment ?sB by the following multi-
plicative form :
?sB?i? ? ?s?i?e!s?i?;
where?sand !sare random variables, independent ofeach
other. If weassume ?sto be a fractional Gaussiannoise and
thevariance of !sis small enough, the correlation property
(or the Hurst exponent H) of ?sB is approximated by that
of ?s. This follows from the observation that for fractional
Brownian motion, a long-range correlation property is
generated from the sign of their increments rather than
from the amplitudes . Note that, in Eq. (2), the sign
of ?sB is determined only by the sign of the ?s, and the
amplitudes of ?sare modulated by exp?!s? (>0).
In addition, non-Gaussian PDF’s can be described by the
PDF form of Eq. (3). If we assume the PDF of !sto be
Gaussian, the PDF of ?sB?i? is expressed by
where Fsand Gsare both Gaussian with zero mean and
converges to a Gaussian when ? ! 0. On the contrary, as
? increases, fat tails and a peak around the mean value
For a quantitative comparison, we fit the data to the
above function [Eq. (3)] and estimate the variance ?2
s, respectively . In this case, Ps
order structure function of seven records from healthy subjects
during: (a) constant exercise, (b) usual daily activity after the
exercise, (c) sleep, and (d) usual daily activity the next morning.
Scale dependence of the square root of the second
PRL 95, 058101 (2005)
PHYSICAL REVIEW LETTERS
29 JULY 2005
!sin Eq. (2) . The deformation of the standardized
PDF’s of the increment ?sB across scales is shown in
Fig. 3(a) for the usual daily activity, showing the non-
Gaussian PDF [Fig. 3(a)], which is well described by
Eq. (3), as reported in our previous study . The scale
dependence of ?2, as shown in Fig. 3(b), indeed exhibits
distinctive features. The non-Gaussian PDF’s for the two
records of daily activity show a striking resemblance. The
scale invariance of the non-GaussianPDFin a range of 20–
1000 beats, however, disappears in sleep states, in which
non-Gaussian fluctuations at a characteristic scale of
?100 beats are dominant. The ?2for constant exercise is
much smaller than for the other states, implying near
It is important to note that, from the point of view of the
increment PDF and multifractal formalism, the non-
Gaussian noise with uncorrelated !sin Eq. (2) and multi-
fractal random walk-type process with long-range corre-
lated !s are indistinguishable because their one-point
statistics at a given scale can be identical; e.g., we can
see similar non-Gaussian PDF’s at certain scales for both
usual daily activity and sleep states [Fig. 3(b)]. Thus, to
find the origin and possible mechanisms of the non-
Gaussian fluctuations, we have to establish another aspect
such as correlation properties of !sin Eq. (2). To do this,
we further introduce an alternative method to study corre-
lation functions of the local energy fluctuations .
Ata scale s,wedefine the local energyanditsmagnitude
? !s?i? ?1
respectively. Using the magnitude ? !swe evaluate the
fluctuation properties of !sin Eq. (2). The local energy
fluctuations computed from data from a single subject
(Fig. 1) are shown in Fig. 4(a) as the function of the beat
number and the scale. In Fig. 4(a), we see quite different
patterns in the local energy fluctuations depending on the
behavioral states. Further, by using the magnitude correla-
tion function as defined by
C??;s1;s2? ? h? ? !s1?i? ? h ? !s1i?? ? !s2?i ? ?? ? h ? !s2i?i;
we are able to study nontrivial correlation properties of
heart rate fluctuations. Higher values of C??;s1;s2? imply
more correlated behavior of !s. Thus, the C??;s1;s2?
quantifies clustering of the local energy fluctuations.
The ‘‘one-scale (s1? s2)’’ and ‘‘two-scale (s1< s2)’’
magnitude correlation functions for each state are shown in
Figs. 4(b) and 4(c), respectively. This analysis clearly
reveals the difference in the origin of the non-Gaussian
fluctuations between the usual daily activity and sleep
states. The magnitude fluctuations for the daily activity
state are strongly correlated, not only at a fixed scale but
also at different scales, which means the magnitude fluc-
tuations at a short time scale strongly influence the sub-
sequently generated magnitude fluctuations across a wide
range of time scales [Fig. 4(c)]. On the other hand, the
magnitude correlations for the sleep state are very weak
between different time scales, although correlated behavior
is observed at small scales <30 beats. This fact shows that,
for the sleep state, the non-Gaussian fluctuations at a
characteristic scale of ?100 beats have a different mecha-
nism from that for the waking state.
To date, no significant difference in the multifractal
scaling property  or the PDF of the local energy fluctua-
tions  between sleep and waking states has been ob-
served. In contrast, using the magnitude correlation
functions, we demonstrate that there is indeed an essential
difference between sleep and waking states. This result
indicates that one-point statistics are not sufficient fully
to characterize heart rate fluctuations. Similarly, no signifi-
cant change in the correlation properties of the magnitude
of increment has been reported for exercise and resting
conditions , whereas we now demonstrate a striking
difference between constant exercise and usual daily
In conclusion, we have demonstrated that a healthy
human heart rate exhibits phase transitionlike dynamics
between different behavioral states, with a dramatic depar-
ture from criticality. We have discussed relevant character-
istics in the significant behavioral states, showing that
strongly correlated fluctuations—a hallmark of critical-
ity—are observed only in the narrow region of usual daily
activity, while a breakdown follows the transition to other
FIG. 3 (color online).
(the standard deviation has been set to one) of the increment ?sB
across scales, where the ?sis the standard deviation of ?sB?i?.
The standardized PDF’s (in logarithmic scale) for different time
8;16;32;64;128;256;512;1024 beats. The PDF’s are estimated
from all the records for usual daily activity after the exercise.
The dashed line is a Gaussian PDF for comparison. In solid lines,
we superimpose the deformation of the PDF using Castaing’s
equation  with the log-normal self-similarity kernel, provid-
ing an excellent fit to the data. (b) Dependence of the fitting
parameter ?2on the scale s during constant exercise, usual daily
activity after the exercise (DA1), sleep, and usual daily activity
the next morning (DA2).
(a) Deformation of standardized PDF’s
PRL 95, 058101 (2005)
29 JULY 2005
states. This transition is reminiscent of a second order
phase transition, and supports the hypothesis  that a
healthy human heart rate is controlled to converge continu-
ally to a critical state during usual daily activity. We
believe this feature to be a key to understanding the heart
rate control system .
This work was in part supported by Grants from the
Japan Society for the Promotion of Science for Young
Scientists (to K.K.), and
Technology Agency (to Y.Y.).
the Japan Scienceand
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FIG. 4 (color).
(Fig. 1) at each scale (resolution) s; (b) one-scale magnitude correlation functions, C??;4;4?=C?0;4;4?; (c) two-scale magnitude
correlation functions for each state. The color scales and contour lines (at 0.05 resolution increment levels) represent values of
C??;4;s1?=?4?s2. For exercise and sleep states, contour lines of only C??;4;s2? ? 0 can be seen, as the correlation levels remain
within jC??;4;s2?j ? 0:05. The one-scale and two-scale magnitude correlation functions are estimated from all the records for each
(a) Local energy fluctuations ?2
s?i?i of the heart rate increments ?sBs?i? obtained from a representative record
PRL 95, 058101 (2005)
29 JULY 2005