Detecting synchronization between the signals from multivariate and univariate biological data Vladimir Ponomarenko Saratov State University, Russia Vladimir Gridnev Institute of Cardiology, Saratov, Russia Mikhail Prokhorov co-authors: Saratov Department of the Institute of Radio Engineering & Electronics of Russian Academy of Sciences

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, 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,

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 t k 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 . Within the interval between these time moments the phase  of the signal x(t) is linearly increasing as follows: x(t)x(t) B t

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:

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 [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, 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),

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 n:m phase synchronization is defined by the condition where C is a constant. In this case the relative phase difference fluctuates around a constant value.

Fig. 3. (a) Phase synchronization 1:1 between x and y. (b) Absence of phase synchronization. (a) (b)

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 Fig. 4. (a) Phase synchronization in a statistical sense. (b) Absence of phase synchronization. (a) (b)

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.

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 t j when the cyclic phase of the fast signal attains a certain fixed value ,, and plot versus t j, where and m is a number of adjacent cycles of the slow signal. In the case of n:m synchronization, 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. 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).

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)

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, 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.

Fig. 12. Generalized phase differences (a) and (b) and the instantaneous frequency ratio (c) of the signal with basic frequency f v  0.1 Hz and the signal of spontaneous respiration with average frequency f r  0.25 Hz for one of the subjects. 2:1 synchronization, when two adjacent respiratory cycles contain one cycle of blood pressure slow regulation, 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,

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.

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 f v on the frequency of respiratory f r 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 f r. (d) Synchrogram, demonstrating one-band structure (1:1 synchronization) and two- band 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 T i 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. f h is the frequency of the main heart rhythm, f r is the respiratory frequency, and f v 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 t k corresponding to the appearance of R peak the heartbeat phase  h is increased by 2  and within the interval between these time moments the phase  h is linearly increasing. As the result, the instantaneous phase of the main heart rhythm is determined as

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. This phase we compare with the phase computed in the similar way directly from the respiratory signal filtered with the same bandpass.

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.

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 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.

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.

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).

(i) Construct the upper x max (t) and lower x min (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 h 1 (t), which is the first IMF. (iv) Take the difference and repeat steps (i)–(iii) for it to obtain the second IMF h 2 (t). The procedure continues until the IMF h i (t) contains fewer than two local extrema. To decompose the signal x(t) into IMFs we use the following algorithm:

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.

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.

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.

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 0.15–0.35 Hz and 0.06–0.13 Hz, respectively.

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.

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 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 fixed- frequency breathing at 0.2 Hz. The phase 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 and 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,

At Saratov cardiocenter we studied 32 patients (25 men and 11 women) aged 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. 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. healthy subjects patients after AMI first week after AMI three weeks after AMI

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 Prokhorov M.D., Ponomarenko V.I., Gridnev V.I., Bodrov M.B., Bespyatov A.B., Phys. Rev. E, 2003, V.68, 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.885–893. Ponomarenko V.I., Prokhorov M.D., Bespyatov A.B., Bodrov M.B., Gridnev V.I., Chaos, Solitons & Fractals, 2005, V.23, N.4, P