Detecting synchronization between the
signals from multivariate and univariate
biological data
Mikhail Prokhorov
Saratov Department of the
Institute of Radio Engineering & Electronics of
Russian Academy of Sciences
co-authors:
Vladimir Ponomarenko Saratov State University, Russia
Vladimir Gridnev
Institute of Cardiology, Saratov, Russia
Methods aimed at comparing the degree of complexity of different
time series. Usually the methods quantify the degree of regularity of
a time series by evaluating the appearance of repetitive patterns.
However, there is no straightforward correspondence between
regularity, which can be measured by entropy-based algorithms, and
complexity.
Complexity is associated with “meaningful structural richness” [1],
which, in contrast to the outputs of random phenomena, exhibits
relatively higher regularity.
[1] Information Dynamics, edited by H. Atmanspacher and H.
Scheingraber, Plenum, New York, 1991.
New quantitative measurements of complexity and regularity.
Method of multiscale entropy analysis [2].
[2] M. Costa, A. L. Goldberger, C.-K. Peng, Phys. Rev. E, 2005, V.71,
021906.
It has been applied to coding and noncoding DNA sequences. It has
been shown in [2] that the noncoding sequences are more complex
than the coding sequences. This result supports studies suggesting,
contrary to the “junk DNA” theory, that noncoding sequences
contain important biological information.
[2] M. Costa,
A. L. Goldberger,
C.-K. Peng,
Phys. Rev. E,
2005, V.71, 021906.
Concept of synchronization
The concept of synchronization is used to reveal interaction
between two or more systems from experimental data.
The case of phase synchronization of the systems or processes,
where only the phase locking is important, while no restriction
on the amplitudes is imposed.
Thus, the phase synchronization of coupled systems is defined
as the appearance of certain relation between their phases,
while the amplitudes can remain non-correlated.
1) One of the ways for introducing phases is to store all
time moments tk when the signal x(t) crosses some
threshold level B in one direction (for example, from
above to below) and then attribute to each such
crossing a phase increase of 2.
x(t)
B
t
Within the interval between these time moments the phase f of
the signal x(t) is linearly increasing as follows:
f  t   2
t  tk
 2 k , tk  t  tk 1
tk 1  tk
2) Another method is to reconstruct the phase portrait from
a signal, project it onto the phase plane
and introduce phase as a phase angle on this plane:
x( t   )
f  arctan
 2 k
x( t )
3) The third way to define the phase is to construct the
analytic signal  (t ) [3, 4], which is a complex function of
time defined as
 (t )  x(t )  ix(t )  A(t )eif ( t )
where A(t) and f(t) are respectively the instantaneous
amplitude and the instantaneous phase of the signal x(t) and
the function x(t ) is the Hilbert transform of x(t),

x( )
x(t )   
d
t 

1
[3] D. Gabor, J. IEE London, 1946, V.93, P.429–457
[4] A. Pikovsky, M. Rosenblum, J. Kurths
Synchronization: A Universal Concept in Nonlinear Science,
Cambridge University Press, Cambridge, 2001.
To detect synchronization between two signals we calculate the
phase difference
 n12,m  nf1  mf2 ,
where f1 and f2 are the phases of the first and the second signals,
12
n and m are integers, and  n,m is the generalized phase
difference, or relative phase.
The presence of n:m phase synchronization is defined by the
condition
 n12,m  C  const ,
where C is a constant. In this case the relative phase
12
difference  n,m fluctuates around a constant value.
(a)
(b)
Fig. 3. (a) Phase synchronization 1:1 between x and y.
(b) Absence of phase synchronization.
Phase synchronization in noisy systems can be understood in a
statistical sense as the appearance of a peak in the distribution
of the cyclic relative phase

12
n ,m

12
n ,m
mod 2
(a)
Fig. 4. (a) Phase synchronization in a statistical sense .
(b) Absence of phase synchronization.
(b)
Another technique widely used for the detection of
synchronization between two signals is based on the analysis of
the ratio of instantaneous frequencies f1 f 2 of these signals.
The instantaneous frequency can be calculated as the rate of
the instantaneous phase change.
In the region of frequency synchronization the ratio of
frequencies of noisy signals remains approximately constant.
The presence of synchronization between two signals can be
demonstrated by plotting a synchrogram.
To construct a synchrogram we determine the phase f2 of the
slow signal at times tj when the cyclic phase of the fast signal
attains a certain fixed value , f1 (t j ) mod 2   , and plot  m12 (t j )
versus tj, where
1
12
 m t j  
f2  t j  mod 2 m
2
and m is a number of adjacent
cycles of the slow signal.


In the case of n:m
12
synchronization,  m (t j ) attains
only n different values within m
adjacent cycles of the slow
signal, and the synchrogram
consists of n horizontal lines.
To characterize the degree of synchronization between signals
various synchronization measures have been proposed.
12

Analyzing the relative phases n ,m we calculate the
phase synchronization index
 n12,m   exp  i n12,m  t   t   cos n12,m (t ) t2  sin  n12,m (t ) t2
where brackets denote average over time.
By construction,  n12,m  0 if the phases are not synchronized at all
and  n12,m  1 when the phase difference is constant (perfect
synchronization).
The concept of synchronization is widely used for the analysis
of a variety of biological data.
It has been successfully applied to human posture control
data of healthy subjects and neurological patients, to
multichannel magnetoencephalography data and records of
muscle activity of a Parkinsonian patient, to human
cardiorespiratory data and many other physiological signals.
The presence or absence of synchronization can reflect
healthy dynamics.
Main rhythmic processes in the human
cardiovascular system
1) Main heart rhythm with a frequency of about 1 Hz
2) Respiration whose frequency is usually around 0.25 Hz
3) Process of blood pressure and heart rate regulation having in
humans the fundamental frequency close to 0.1 Hz (Mayer wave)
At first we examine synchronization between the main
rhythms using for the analysis multivariate data, i.e.,
the simultaneously measured signals of
electrocardiogram, respiration and blood pressure.
Measurements and data processing
Subjects: healthy young volunteers
Recorded signals: ECG, respiration and blood pressure
(with the sampling frequency 250 Hz and 16-bit resolution)
Regimes of breathing:
1) spontaneous respiration (10 minutes)
2) fixed-frequency breathing at 0.25 Hz (10 minutes)
3) fixed-frequency breathing at 0.2 Hz (10 minutes)
4) respiration with linearly increasing frequency from 0.05 Hz to
0.3 Hz (30 minutes)
  fh  4fr
hr
1,4
Fig. 11. Generalized phase difference (a) and the instantaneous
frequency ratio (b) of the signals of ECG and spontaneous
respiration, demonstrating 1:4 synchronization when 4
heartbeats occur within one respiratory cycle.
Phase synchronization between the main heart rhythm and
respiration has been demonstrated by several groups of
investigators [5–7].
[5] C. Schäfer, M.G. Rosenblum, J. Kurths, H.-H. Abel, Nature, 1998, V.392,
P.239–240.
[6] M. Bračič-Lotrič, A. Stefanovska, Physica A, 2000, V.283, P.451–461.
[7] S. Rzeczinski, N. B. Janson, A. G. Balanov, P. V. E. McClintock, Phys. Rev.
E, 2002, V.66, 051909.
We also observed phase synchronization between the main
heart rhythm and respiration lasting 30 s or longer for each of
the subject studied.
The duration of the longest epoch of synchronization within a
10-minute record has been about 2 minutes. Almost all subjects
demonstrated the presence of several different n:m epochs of
synchronization within one record.
vr
 2,1
 2fv  fr
2:1 synchronization, when two
adjacent respiratory cycles contain
one cycle of blood pressure slow
regulation,
vr
5,2
 5fv  2fr
5:2 synchronization
[8] M.D. Prokhorov,
V.I. Ponomarenko, V.I. Gridnev,
M.B. Bodrov, A.B. Bespyatov,
Phys. Rev. E, 2003, V.68, 041913.
Fig. 12. Generalized phase differences (a) and (b) and the
instantaneous frequency ratio (c) of the signal with basic frequency
fv  0.1 Hz and the signal of spontaneous respiration with average
frequency fr  0.25 Hz for one of the subjects.
Case of paced respiration with a fixed frequency
For the cases of breathing with the fixed frequency of 0.25 Hz
and 0.2 Hz we obtain the results coinciding qualitatively with
those obtained for the case of spontaneous respiration.
In comparison with the case of spontaneous respiration the case
of fixed-frequency breathing is characterized by longer epochs
of phase locking and higher index of phase synchronization.
Probably, it is explained by the fact that the variability of fixedfrequency respiration is several times smaller than the
variability of spontaneous respiration.
Case of paced respiration
with linearly increasing frequency
The signals were recorded continuously during 30 minutes under
respiratory frequency increasing linearly from 0.05 Hz to 0.3 Hz.
Fig. 13. Generalized phase difference of the signals of ECG and
respiration under paced respiration with linearly increasing
frequency.
Synchronization between the process whose basic frequency is
0.1 Hz and respiration
Fig. 14. Dependence of the frequency of the process of blood
pressure slow regulation fv on the frequency of respiratory fr for one
of the subjects.
Fig. 15. Generalized phase
difference (a), phase
synchronization index (b),
and the instantaneous
frequency ratio (c) of the
process of blood pressure
regulation and respiration
for one of the subjects
under linearly increasing
frequency of respiration fr.
(d) Synchrogram,
demonstrating one-band
structure (1:1
synchronization) and twoband structure (2:1
synchronization).
Synchronization is not a coincidence of phases or frequencies of
the rhythmic processes. Two uncoupled periodic processes
always have a constant frequency ratio, but they are not
synchronized.
Synchronization is an adjustment of rhythms due to interaction.
The presence of epochs where the instantaneous frequency ratio
of nonstationary signals remains stable while the frequencies
themselves vary, and the existence of several different n:m
epochs within one record count in favor of the conclusion that
the observed phenomena are associated with the process of
adjustment of rhythms of interacting systems.
Detecting synchronization from univariate data
Owing to interaction, the main rhythms of cardiovascular system
appear in various signals: ECG, blood pressure, blood flow and heart
rate variability (HRV).
Studying synchronization in the cardiovascular system from
univariate data it is favorable to choose for the latter the sequence
of R-R intervals containing information about different oscillating
processes governing the cardiovascular dynamics.
The sequence of R-R intervals
is the series of the time
intervals Ti between the two
successive R peaks
Fig. 17. Typical R-R intervals (a) and
their Fourier power spectra (b).
(c) Fourier power spectra of ECG.
fh is the frequency of the main heart
rhythm, fr is the respiratory
frequency, and fv is the frequency of
the process of slow regulation of
blood pressure and heart rate.
To calculate the phase of the main heart rhythm from the
sequence of R-R intervals we assume that at the time moments
tk corresponding to the appearance of R peak the heartbeat
phase fh is increased by 2 and within the interval between
these time moments the phase fh is linearly increasing. As the
result, the instantaneous phase of the main heart rhythm is
determined as
t  tk
fh  t   2
 2 k , tk  t  tk 1
tk 1  tk
To extract the instantaneous phases and frequencies of
respiration and the process of slow regulation of blood pressure
from the sequence of R-R intervals transformed to uniformly
time spaced data we apply the three different methods using
1) bandpass filtration and the Hilbert transform
2) empirical mode decomposition and the Hilbert transform
3) wavelet transform
The obtained values were compared with the values of
instantaneous phases and frequencies calculated directly from the
signals of respiration and blood pressure.
Bandpass filtration and the Hilbert transform
To extract the respiratory component of HRV we filter the
sequence of R-R intervals with the bandpass 0.15–0.4 Hz.
Then, for the filtered signal we apply the Hilbert transform and
obtain the phase fr1 . This phase we compare with the phase f r
computed in the similar way directly from the respiratory signal
filtered with the same bandpass.
Fig. 18. Generalized phase difference  1  fr1  fr (a)
and the instantaneous frequency ratio (b) of the
respiratory component extracted from the HRV using
filtration and the measured respiratory signal.
r r
To extract the low-frequency component of HRV with the basic
frequency close to 0.1 Hz we filter the sequence of R-R intervals
removing the high-frequency fluctuations (>0.15 Hz) associated
predominantly with respiration, and very low-frequency
oscillations (<0.05 Hz).
After this bandpass filtration we calculate the phase fv 1 of the
signal using the Hilbert transform and compare it with the
phase fv computed using the Hilbert transform of the blood
pressure signal filtered with the same bandpass.
Fig. 19. Generalized phase difference  1  fv 1  fv (a)
and the instantaneous frequency ratio (b) of the rhythm of
low-frequency regulation of blood pressure extracted from the
HRV using bandpass filtration and the blood pressure signal
filtered with the same bandpass.
v v
Empirical mode decomposition and the Hilbert transform
Empirical mode decomposition (EMD) is a signal processing
technique, which performs decomposition of a complicated
signal into the so-called intrinsic mode functions (IMFs) [9, 10],
i.e., the components with well-defined frequency.
[9] Huang N.E. et al., Proc. R. Soc. Lond. A (1998).
[10] Huang N.E. et al., Proc. R. Soc. Lond. A (2003).
To decompose the signal x(t) into IMFs we use the following
algorithm:
(i) Construct the upper xmax(t) and lower xmin(t) envelopes
connecting via cubic spline interpolation all the maxima and
minima of x(t), respectively.
(ii) Compute
x (t )  x (t )   xmax (t )  xmin (t )  2.
(iii) Repeat steps (i) and (ii) for x(t ) until the resulting signal
will possess the properties that the number of extrema is equal
(or differ at most by one) to the number of zero crossings, and
the mean value between the upper and lower envelope is equal
to zero at any point. Denote the resulting signal by h1(t), which
is the first IMF.
x1 (t )  x(t )  h1 (t )
(iv) Take the difference
(i)–(iii) for it to obtain the second IMF h2(t).
and repeat steps
The procedure continues until the IMF hi(t) contains fewer than
two local extrema.
Using the EMD technique for the cases of both spontaneous
and fixed-frequency breathing we decompose the heartbeat
time series and obtain IMFs corresponding to the highfrequency (respiratory) and low-frequency (about 0.1 Hz)
components of HRV.
Next, we calculate the phases ( fr 2 and fv 2 ) and frequencies
( f r 2 and f v 2 ) of these IMFs and compare them with the phases
and frequencies of the corresponding IMFs extracted directly
from the signals of respiration and blood pressure.
The instantaneous phases for all the signals were computed
using the Hilbert transform.

r2 r


 fr 2  fr mod 2

v2v


 fv 2  fv mod 2
Fig. 20. Distribution of the cyclic relative phase (a) and the
instantaneous frequency ratio (c) (blue line) of the associated
with respiration IMFs of the HRV and respiratory signal.
Distribution of the cyclic relative phase (b) and the
instantaneous frequency ratio (c) (red line) of the associated
with 0.1 Hz process IMFs of the HRV and blood pressure signal.
The more close correspondence is observed between the respiratory signal and the
HRV intrinsic mode function associated with respiration than between the lowfrequency component of the blood pressure signal and corresponding IMF of the
HRV.
Wavelet transform
We use the continuous wavelet transform of the signal x(t):
1
W ( a , b) 
a

t b
 x(t )  a  dt.

As a complex basis function we choose the Morlet wavelet
(t )   1/ 4 exp  i 2 f 0t  exp   t 2 2  .
The wavelet spectrum W (a, b)  W (a, b) exp  i( a, b)  of the scalar
signal x(t) can be represented as two surfaces of the amplitude W and
phase  of the wavelet transform coefficients in the three-dimensional
space. The projections of these surfaces into the (a,b) plane or the (f,b)
plane allow one to trace the variation of the amplitude and phase of the
wavelet transform coefficients at different scales and time moments.
Fig. 21. Distribution of the
amplitude of coefficients of the
HRV wavelet transform in the
time-frequency plane (a).
Instantaneous frequency ratio of
the HRV respiratory component
obtained using the wavelet
transform and the measured
respiratory signal (b).
Instantaneous frequency ratio of
the low-frequency components of
the HRV and blood pressure
signals obtained using the wavelet
transform (c).
The instantaneous frequencies f r 3
and f v 3 of the HRV components
are determined as the frequencies
corresponding to the maximum
amplitude of coefficients W ( f , b)
within the intervals 0.15–0.35 Hz
and 0.06–0.13 Hz, respectively.
We determine the instantaneous phases fr 3 and fv 3 as the
phases (f,b) of the wavelet transform coefficients computed for
the same values of f and b as the instantaneous frequencies f r 3
and f v 3 , respectively.
Comparing these phases with the phases fr and fv calculated using
the wavelet transform of the respiratory and blood pressure signals
we obtained the results qualitatively similar to those above
presented.
The instantaneous phase and frequency of the respiratory
component derived from the sequence of R-R intervals using each
of the three considered methods coincide closely with the
instantaneous phase and frequency of the respiratory signal itself.
We observed 1:1 phase and frequency synchronization between
the respiration and the HRV respiratory component for each
subject under both spontaneous and fixed-frequency breathing.
The phases and frequencies of the process with fundamental
frequency of about 0.1 Hz extracted from the HRV and blood
pressure signals of healthy humans are also sufficiently close but
demonstrate greater difference between themselves than the
respiratory oscillations.
Detecting synchronization between the rhythms of
cardiovascular system from R-R intervals
Fig. 22. Generalized phase
differences (a) and (b) and
the instantaneous
frequency ratio (c) of the
heartbeat and respiration
for one of the subjects
under spontaneous
breathing.
The respiratory phase fr1
is computed using the
Hilbert transform of the
HRV data filtered with the
bandpass 0.15–0.4 Hz.
Fig. 23. Generalized
phase difference (a)
and the instantaneous
frequency ratio (b) of
the heartbeat and
respiration for a
subject under fixedfrequency breathing
at 0.2 Hz.
The phase fr 2 is determined using the method of EMD and the
Hilbert transform.
Fig. 24. Generalized phase
differences (a) and (b) and
the instantaneous
frequency ratio (c) of the
process of blood pressure
regulation and respiration.
The phases fr 3 and fv 3 are computed using the wavelet
transform of the sequence of R-R intervals.
The results of our investigation of synchronization between the
rhythms of CVS obtained from the analysis of univariate data
in the form of R-R intervals of healthy subjects coincide
qualitatively with the results of synchronization investigation
from multivariate data.
The feasibility of detecting the presence of synchronization
between the rhythms in the cardiovascular system and
measuring the duration of this synchronization having at the
disposal only univariate data in the form of R-R intervals
opens up new possibilities for applying this measure in
practice. In this case it is not necessary to record
simultaneously the signals of ECG, respiration and blood
pressure. Instead of this one can analyze, for example, the data
of Holter monitoring widely used in cardiology.
The epochs of cardiorespiratory synchronization has been
found to be longer in athletes [5] than in subjects performing
recreative activity only [6, 7]. The more significant distinction of
duration and the presence itself of synchronization of the
cardiovascular rhythms is expected between the healthy
subjects and the subjects with the disfunctions of the
cardiovascular system, having usually the low HRV.
[5] C. Schäfer, M.G. Rosenblum, J. Kurths, H.-H. Abel, Nature, 1998, V.392,
P.239–240.
[6] M. Bračič-Lotrič, A. Stefanovska, Physica A, 2000, V.283, P.451–461.
[7] S. Rzeczinski, N. B. Janson, A. G. Balanov, P. V. E. McClintock, Phys. Rev.
E, 2002, V.66, 051909.
At Saratov cardiocenter we studied 32 patients (25 men and
11 women) aged 41-80 years after acute myocardial
infarction (AMI).
The ECG and blood pressure signals were simultaneously
recorded twice:
1) after 3 to 5 days after AMI
2) after 3 weeks after AMI
Control group: young healthy men without any cardiac pathology
(23 records).
The duration of synchronization regions in healthy subjects has
been found to be in 2.5 times longer on the average than in patients
after AMI.
healthy subjects
patients after AMI
first week after AMI
three weeks after AMI
The duration of synchronization regions in patients after 3 weeks
after AMI increases in 1.5 times on the average in comparison
with the same patients during the first week after AMI.
However, the duration of synchronization regions remains
significantly lower than in healthy subjects.
Conclusion
The concept of synchronization can be successfully applied to
the analysis of physiological and biological data.
Synchronization between different rhythmic processes can be
detected even from the analysis of univariate data.
 The phases and frequencies of the rhythmic components can
be extracted from the complicated signal using the methods
based on bandpass filtration and the Hilbert transform,
empirical mode decomposition and the Hilbert transform, and
wavelet transform.
The main rhythmic processes in the human cardiovascular
system can be synchronized with each other.
The presence and duration of synchronization can reflect
healthy dynamics and can be used for diagnostics.
References
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Bespyatov A.B., Phys. Rev. E, 2003, V.68, 041913.
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