Detecting synchronization from data

1
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
2
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.
3
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.
4
2 M. Costa, A. L. Goldberger, C.-K. Peng,
Phys. Rev. E, 2005, V.71, 021906.
5
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.
6
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
7
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
8
3) The third way to define the phase is to
construct the analytic signal 3, 4,
which is a complex function of time defined as
where A(t) and ?(t) are respectively the
instantaneous amplitude and the instantaneous
phase of the signal x(t) and the function
is the Hilbert transform of x(t),
3 D. Gabor, J. IEE London, 1946, V.93,
P.429457 4 A. Pikovsky, M. Rosenblum, J.
Kurths Synchronization A Universal Concept in
Nonlinear Science, Cambridge University Press,
Cambridge, 2001.
9
To detect synchronization between two signals we
calculate the phase difference
where ?1 and ?2 are the phases of the first and
the second signals, n and m are integers, and
is the generalized phase difference, or
relative phase. The presence of nm phase
synchronization is defined by the condition
where C is a constant. In this case the relative
phase difference fluctuates around
a constant value.
10
(a) (b)
Fig. 3. (a) Phase synchronization 11 between x
and y. (b) Absence of phase synchronization.
11
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
(b)
(a)
Fig. 4. (a) Phase synchronization in a
statistical sense . (b) Absence of phase
synchronization.
12
Another technique widely used for the detection
of synchronization between two signals is based
on the analysis of the ratio of instantaneous
frequencies 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.
13
The presence of synchronization between two
signals can be demonstrated by plotting a
synchrogram.
To construct a synchrogram we determine the phase
?2 of the slow signal at times tj when the cyclic
phase of the fast signal attains a certain fixed
value ?, , and plot
versus tj, where
and m is a number of adjacent cycles of the slow
signal.
In the case of nm synchronization,
attains only n different values within m adjacent
cycles of the slow signal, and the synchrogram
consists of n horizontal lines.
14
To characterize the degree of synchronization
between signals various synchronization measures
have been proposed. Analyzing the relative
phases we calculate the phase
synchronization index
where brackets denote average over time. By
construction, if the phases are
not synchronized at all and when
the phase difference is constant (perfect
synchronization).
15
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.
16
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)

17
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.
18
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
• spontaneous respiration (10 minutes)
• fixed-frequency breathing at 0.25 Hz (10 minutes)
• fixed-frequency breathing at 0.2 Hz (10 minutes)
• respiration with linearly increasing frequency
  from 0.05 Hz to 0.3 Hz (30 minutes)

19
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20
Fig. 11. Generalized phase difference (a) and
the instantaneous frequency ratio (b) of the
signals of ECG and spontaneous respiration,
demonstrating 14 synchronization when 4
heartbeats occur within one respiratory cycle.
21
Phase synchronization between the main heart
rhythm and respiration has been demonstrated by
several groups of investigators 57. 5 C.
Schäfer, M.G. Rosenblum, J. Kurths, H.-H. Abel,
Nature, 1998, V.392, P.239240. 6 M.
Bracic-Lotric, A. Stefanovska, Physica A, 2000,
V.283, P.451461. 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 nm epochs of synchronization within
one record.
22
21 synchronization, when two adjacent
respiratory cycles contain one cycle of blood
pressure slow regulation,
52 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.
23
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 fixed-frequency
respiration is several times smaller than the
variability of spontaneous respiration.
24
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.
25
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.
26
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 (11 synchronization) and
two-band structure (21 synchronization).
27
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 nm 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.
28
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
29
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.
30
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 2p 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
31
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.
32
Bandpass filtration and the Hilbert transform
To extract the respiratory component of HRV we
filter the sequence of R-R intervals with the
bandpass 0.150.4 Hz. Then, for the filtered
signal we apply the Hilbert transform and obtain
the phase . This phase we compare with the
phase computed in the similar way directly
from the respiratory signal filtered with the
same bandpass.
33
Fig. 18. Generalized phase difference
(a) and the instantaneous frequency
ratio (b) of the respiratory component extracted
from the HRV using filtration and the measured
respiratory signal.
34
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 (gt0.15 Hz) associated
predominantly with respiration, and very
low-frequency oscillations (lt0.05 Hz). After
this bandpass filtration we calculate the phase
of the signal using the Hilbert transform and
compare it with the phase computed using
the Hilbert transform of the blood pressure
signal filtered with the same bandpass.
35
Fig. 19. Generalized phase difference
(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.
36
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).
37
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 (iii) Repeat steps
(i) and (ii) for 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. (iv) Take the difference
and repeat steps (i)(iii) for it
to obtain the second IMF h2(t).
The procedure continues until the IMF hi(t)
contains fewer than two local extrema.
38
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 high-frequency
(respiratory) and low-frequency (about 0.1 Hz)
components of HRV.
Next, we calculate the phases ( and
) and frequencies ( and ) 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.
39
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 low-frequency component of the blood pressure
signal and corresponding IMF of the HRV.
40
Wavelet transform
We use the continuous wavelet transform of the
signal x(t)
As a complex basis function we choose the Morlet
wavelet
The wavelet spectrum
of the scalar signal
x(t) can be represented as two surfaces of the
amplitude 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.
41
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 and
of the HRV components are determined as the
frequencies corresponding to the maximum
amplitude of coefficients within
the intervals 0.150.35 Hz and 0.060.13 Hz,
respectively.
42
We determine the instantaneous phases and
as the phases ?(f,b) of the wavelet transform
coefficients computed for the same values of f
and b as the instantaneous frequencies and
, respectively. Comparing these phases with
the phases ?r and ?v calculated using the wavelet
transform of the respiratory and blood pressure
signals we obtained the results qualitatively
similar to those above presented.
43
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 11 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.
44
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 is computed using the
Hilbert transform of the HRV data filtered with
the bandpass 0.150.4 Hz.
45
Fig. 23. Generalized phase difference (a) and the
instantaneous frequency ratio (b) of the
heartbeat and respiration for a subject under
fixed-frequency breathing at 0.2 Hz.
The phase is determined using the method of
EMD and the Hilbert transform.
46
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 and are computed using
the wavelet transform of the sequence of R-R
intervals.
47
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.
48
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.239240. 6 M.
Bracic-Lotric, A. Stefanovska, Physica A, 2000,
V.283, P.451461. 7 S. Rzeczinski, N. B.
Janson, A. G. Balanov, P. V. E. McClintock, Phys.
Rev. E, 2002, V.66, 051909.
49
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).
50
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.
51
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.

52
References
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Bodrov M.B., Bespyatov A.B., Phys. Rev. E, 2003,
V.68, 041913. Bespyatov A.B., Bodrov M.B.,
Gridnev V.I., Ponomarenko V.I., Prokhorov M.D.,
Nonlin. Phen. in Compl. Syst., 2003, V.6, N.4,
P.885893. Ponomarenko V.I., Prokhorov M.D.,
Bespyatov A.B., Bodrov M.B., Gridnev V.I., Chaos,
Solitons Fractals, 2005, V.23, N.4,
P.1429-1438.