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Automated Quantification of the Synchrogram by Recurrence Plot Analysis Chinh Duc Nguyen*, Member, IEEE, Stephen James Wilson, Member, IEEE, and Stuart Crozier, Member, IEEE
Abstract—Recently, the concept of phase synchronization of two weakly coupled oscillators has raised a great research interest and has been applied to characterize synchronization phenomenon in physiological data. Phase synchronization of cardiorespiratory coupling is often studied by a synchrogram analysis, a graphical tool investigating the relationship between instantaneous phases of two signals. Although several techniques have been proposed to automatically quantify the synchrogram, most of them require a preselection of a phase-locking ratio by trial and error. One technique does not require this information; however, it is based on the power spectrum of phase’s distribution in the synchrogram, which is vulnerable to noise. This study aims to introduce a new technique to automatically quantify the synchrogram by studying its dynamic structure. Our technique exploits recurrence plot analysis, which is a well-established tool for characterizing recurring patterns and nonstationarities in experiments. We applied our technique to detect synchronization in simulated and measured infants’ cardiorespiratory data. Our results suggest that the proposed technique is able to systematically detect synchronization in noisy and chaotic data without preselecting the phase-locking ratio. By embedding phase information of the synchrogram into phase space, the phase-locking ratio is automatically unveiled as the number of attractors. Index Terms—Cardiorespiratory coupling, nonlinear analysis, phase synchronization, recurrence quantification analysis.
I. INTRODUCTION YNCHRONIZATION is an important phenomenon in nature and science, first discovered by Huygens in the 17th century. In general, synchronization implies the adjustment of frequencies of periodic self-sustained oscillators due to weak interactions. Synchronization in complex system has been extensively studied [1]–[5]. This phenomenon has been found in many natural [1], [6], [7] and engineering systems [2]. In physiology, the characteristics of synchronization have been shown to reflect different behaviors of coupled system [8]–[11]. The understanding of synchronization phenomenon, therefore, might provide important information on coupled systems. Unfortu-
S
Manuscript received June 24, 2011; revised October 29, 2011; accepted November 30, 2011. Date of publication December 15, 2011; date of current version March 21, 2012. Asterisk indicates corresponding author. *C. D. Nguyen is with the School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, Qld. 4072, Australia (e-mail:
[email protected]). S. J. Wilson and S. Crozier are with the School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, Qld. 4072, Australia (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBME.2011.2179937
nately, due to the presence of noise, it is difficult to detect and quantify synchronization in experimental data. Recently, the concept of phase synchronization has been developed in which only certain relationships of phases between weakly coupling oscillators are considered while the amplitudes remain noncorrelated. An advantage of this approach is that it is more robust to noise and allows detection of synchronization in weaker coupling; as the irregularity of amplitude can hide phaselocking information, leading to inaccurate results [3]. In physiology, the phase synchronization approach has been applied to study cardiorespiratory coupling in young healthy athletes [6], healthy adults [4], heart transplant patients [10], infants during sleep [8], and anesthetized rats [12]. It has been able to detect synchronization of weak coupling in noisy, nonstationary, and short data. Although phase synchronization is a promising approach to detect cardiorespiratory synchronization, it still has several limitations. First, phase synchronization of cardiorespiratory coupling was often studied by synchrogram analysis [3], [4], [6]–[8], which is a graphical tool representing the phase relationship of respiratory signal at the times of each heartbeat. If the system is synchronized, horizontal lines will appear on the synchrogram; several techniques have been proposed to automatically quantify synchrogram’s structure to characterize the synchrony [8], [10], [13]–[16]. Most techniques focused on measuring the variance of horizontal lines [10], [14], [16] or quantifying a statistical distribution of the cyclic relative phases in the synchrogram [8], [13]. However, all of them require a correct preselection of phase-locking ratio n:m by trial and error. The information of phase-locking ratio can be determined by the ratio between the instantaneous frequencies of two signals. Unfortunately, it has been known that the estimation of instantaneous frequency of a signal, in practice, is cumbersome [17]. The direct approach (differentiation of phase of the signal) often results large fluctuation due to noise and complicated form of the signal [3]. Moreover, the ratio between instantaneous frequencies does not provide the integer number of n:m ratio which is needed by synchrogram analysis. In cardiorespiratory coupling, the phase-locking ratio varies with physiological states and among individuals [8], [9]. This limitation, thus, can lead to inaccurate quantification and inadequate comparison between subjects. The only technique that does not require preselection of phase-locking ratio was proposed by Seidel [18]. This technique calculates the Fourier power spectrum of the cyclic relative phases in the synchrogram and measures the synchronization strength as the difference between maximum and minimum of power spectrum. However, the maximum peaks of power
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NGUYEN et al.: AUTOMATED QUANTIFICATION OF THE SYNCHROGRAM BY RECURRENCE PLOT ANALYSIS
spectrum are difficult to quantify, as it is often blurred in case of high-order phase-locking ratio or by the effect of noise. Furthermore, previous work has demonstrated that in the synchrogram not only horizontal lines but also its recurrence patterns might contain and provide important information about the coupled system [19]. Therefore, it is important to efficiently quantify recurrence patterns of the synchrogram. Recurrence plot (RP) analysis is a well-established tool to unveil the recurring patterns and nonstationarities in experimental data [20], [21]. The advantage of this technique is that it imposes no rigid requirement on data length, stationarity, or statistical distribution. This technique has been widely applied to analyze physiological data including heart rate [22], [23] and respiratory signals [21], [24]. However, the result of RP analysis strongly depends on the choice of many parameters making it difficult for comparison between studies. In this paper, we aim to introduce a new technique to automatically detect the phase synchronization with any n:m ratio by a combination of synchrogram and optimized RP analyses. Our technique is able to systematically quantify a synchrogram to unveil the synchronization ratio and detect phase synchronization. The remainder of this paper is organized as follows. Section II gives some background on quantification of the synchrogram and demonstrates its limitation by synthetic data. Section III describes our technique and optimization method to overcome the limitation. Finally, in Sections IV and V, our proposed technique is applied to detect synchronization in simulated and measured infants’ cardiorespiratory experiments. II. BACKGROUND
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Fig. 1. Synchrogram analysis and its limitation. Synchrograms generated by trial (a) m = 1 and (d) m = 2. (b), (e) Horizontal lines structure and (c), (f) synchronization strength quantified by automated synchrogram analysis with trial phase-locking ratio 3:1, 5:2, respectively. ψ denotes cyclic relative phase; λ is synchronization index. ψ, λ: normalized unit.
chosen by trial in order to determine the horizontal lines of the synchrogram.
A. Phase Synchronization and Synchrogram Analysis
B. Limitation of Automated Synchrogram Analysis
In general, the phase synchronization of two weakly coupled oscillators is described as phase locking:
To demonstrate the limitation of synchrogram technique, we create the following model to generate synthetic data of synchrogram with n:m ratio: tk × mod(n) m mod (m ) + ξ ψ(tk ) = ψ0 + (3) n
|ϕn ,m | = |nφ1 − mφ2 | < constant
(1)
where φ1 and φ2 are the phases of oscillators, n and m describe the phase-locking ratio n:m, and ϕn ,m is the generalized phase difference between two oscillators. Two parameters n and m needs to be chosen by trial. With the synchrogram technique, the phase of the slower oscillator is observed not periodically in time, but at times tk when the phase of the faster oscillator attains a certain fixed value θ = φ1 (tk ) mod (2π). The synchrogram is constructed by plotting phase of the slower oscillator ψ(tk ) over tk , where 1 [φ2 (tk ) mod (2πm)]. (2) 2π If the system is synchronized by n:m ratio, n horizontal lines will be visually detected on the synchrogram. Hence, using the synchrogram, only one parameter m has to be chosen by trial. As the synchrogram is a graphical tool, it is difficult to visually examine the synchrogram of a long dataset to locate horizontal lines. Several techniques, therefore, have been proposed to automatically quantify its structure [8], [10], [13]–[16]. However, most of them require both n and m parameters to be ψm (tk ) =
where n and m are the integers describing the phase-locking ratio n:m; m is a value of m obtained by trial and error; N is the length of the synchrogram; tk = 1, 2, . . . , N ; ψ 0 is a constant; and ξ is the Gaussian white noise with standard deviation SD. It is known that due to modulation between two signals, the horizontal lines in the synchrogram are not equally spaced [3]. This phenomenon will reduce the ability to unveil horizontal lines of synchrogram analysis with certain m . To simulate this phenomenon, the horizontal lines can be rearranged with nonequal space after generating the synchrogram. Using the synchrogram model (3), we generated a synchrogram containing three episodes of strong noise and two episodes of synchronization with phase-locking n:m ratios 3:1 and 5:2 [see Fig. 1(a)]. In the episode of synchronization with 5:2 ratio, the effect of modulation is added by rearranging the horizontal lines with nonequal space. Thus, the synchrogram can reveal five horizontal lines with m = 2 [see Fig. 1(d)] but not with m = 1 [see Fig. 1(a)].
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An automated technique often used to quantify cardiorespiratory synchronization in experimental data [4], [8], [13] was then applied to measure the synchronization index λ: M 2 2 M 1 λ= sin(ψ(tk ) · 2π) + cos(ψ(tk ) · 2π) M t =1 t =1 k
k
(4) where M is the length of analysis window. The synchronization index λ has value from 0 to 1 in case of unsynchronized or synchronized system, respectively. In this experiment, λ was calculated by running windows, with window length M = 50 and step size = 10. Our results demonstrate that the automated synchrogram analysis can only detect synchronization if the phase-locking ratio n:m is correctly chosen by trial [see Fig. 1(b), (c), (e), and (f)]. With a phase-locking ratio 3:1, only phase synchronization in 3:1 ratio episode was detected [see Fig. 1(b) and (c)]. Similarly, only phase synchronization in 5:2 ratio episodes was detected with trial phase locking 5:2 [see Fig. 1(e) and (f)]. In order to have a correct n:m ratio, the analysis has to extensively scan all the possible combinations of n and m, then quantify many corresponding synchrograms. This limitation makes the interpretation of results difficult where all corresponding synchrogram with different n:m ratios are considered. III. SYNCHROGRAM-RP ANALYSIS A. Combination of Synchrogram and RP The limitation of synchrogram analysis demonstrated in the previous section might be due to the limited capability to unveil dynamic patterns within the 2-D synchrogram. We hypothesize that embedding the phase information of the synchrogram into phase space and quantifying its recurring patterns by RP analysis can eliminate this problem. The first step of our technique is to generate a synchrogram as (2) with m = 1. Hilbert transform, a well-known method to find instantaneous phase of signals, was used to calculate instantaneous phases of two signals to construct a synchrogram [25] (see Appendix A). From the synchrogram, the time series of cyclic relative phase Ψ = {ψ(1), ψ(1), . . . , ψ(M )} is obtained and then the RP can be constructed as R(i, j) = Θ(ε − ||X(i) − X(j)||)
(5)
where − is the symbol of norm. In our study, Euclidean norm is applied. Θ is the symbol of Heaviside function and ε is Euclidean threshold. X(i) and X(j) are phase space vectors obtained by embedding the time series Ψ into phase space using Taken’s theorem X(i) = {ψ(i), ψ(i + τ ), . . . , ψ(i + (D − 1) × τ )}
(6)
X(j) = {ψ(j), ψ(j + τ ), . . . , ψ(j + (D − 1) × τ )} (7) where i, j = 1, 2, 3, . . . , N with N = M − (D − 1) × τ . D is embedding dimension and τ is the time delay. If the distnce between X(i) and X(j) is less than ε, R(i, j) has a value of 1, else R(i, j) is 0. The RP is a binary array N × N marked as white with R(i,j) = 0 and black with R(i, j) = 1.
The next step is to quantify the RP constructed from the synchrogram based on its diagonal lines. The diagonal lines in RP represent the dynamics recurring in phase space [21]. Processes with stochastic or chaotic behaviors result in no or very short diagonal lines, whereas deterministic processes have longer diagonal lines and less single isolated recurrence points [20]. A diagonal line of RP occurs when a segment of the trajectory runs almost in parallel to another segment for l time unit: X(i) ≈ X(j), X(i + 1) ≈ X(j + i), . . . , X(i + l − 1) ≈ X(j + l − 1).
(8)
The length of the diagonal line is determined by the duration of such similar local evolution of the trajectory segments [20]. In case of phase synchronization, horizontal lines will appear on the synchrogram and condition (8) is satisfied. Thus, the duration of phase synchronization, determined by the length of horizontal lines of the synchrogram, can be quantified by the length of diagonal line of the RP. Therefore, we have considered four typical output variables used to quantify RP based on its diagonal lines. Determinism (DET) measures the percentage of recurrence points that form diagonal lines:
N lP (l) (9) DET = Nl=l m i n i,j =1 R(i, j) where P (l) is the number of diagonal lines with the length l in RP; lm in is the minimum length of diagonal line in RP. Mean diagonal line length (L) measures the average of diagonal line length that is related to the average duration when all horizontal lines in the synchrogram remain horizontal. In case of synchronization, the RP will display long diagonal lines and if it is not synchronized, there will be more short diagonal lines. Thus, L will be high/low with/without synchronization:
N lP (l) m in . (10) L = l=l N l=l m i n P (l) This index can be normalized according to the analysis window size (W ) (see Appendix B) as Lnorm =
2L W −2
(11)
Lnorm has value from 0 to 1 in case of unsynchronized or synchronized system, respectively. Max diagonal line length (Lm ax measures the maximum length of diagonal lines in RP, which is related to maximum duration when a single horizontal line in synchrogram remains horizontal. However, as synchronization requires all lines in synchrogram to be horizontal, Lm ax can be sensitive to noise: l Lm ax = max({li }N i=1 )
(12)
where Nl is the total number of diagonal lines. Entropy (ENTR) measures the Shannon entropy of the probability distribution of the diagonal line lengths P (l). ENTR
NGUYEN et al.: AUTOMATED QUANTIFICATION OF THE SYNCHROGRAM BY RECURRENCE PLOT ANALYSIS
Fig. 2. Automated quantification of synchrogram by RP analysis. (a) Synchrogram is generated by m = 1. (b) Zoom-in windows of the synchrogram. (c) Synchrogram windows are embedded into phase space. (d) RPs are constructed. (e) Output variable of RP analysis. ψ denotes cyclic relative phase; D is embedding dimension, τ is delay time, and L is mean of diagonal line length in RP. ψ: normalized unit, L: sample number.
reflects the complexity of RP with respect to diagonal lines: EN T R = −
N
P (l) ln[P (l)].
(13)
l=l m i n
Our technique is, then, applied to quantify the synchrogram model in previous demonstration using running windows with window length = 50 and step size = 10. In this example, the embedding parameters used are D = 3, τ = 1, and ε = 0.2. Our results show that the recurring patterns in the synchrogram were systematically unveiled by the phase space embedding technique. Three, five, and zero attractors were automatically detected in phase space for 3:1, 5:2 phase-locking, and noisy segments, respectively [see Fig. 2(b) and (c)]. The constructed RPs demonstrated long and continuous diagonal lines in segments of 3:1, 5:2 phase-locking ratios, and few recurrence points in noisy segments [see Fig. 2(d)]. The recurring patterns of the RPs quantified by one of the RP quantification variables, correctly detected phase synchronization episodes of both 3:1 and 5:2 ratio [see Fig. 2(e)] with just a signal synchrogram [see Fig. 2(a)].
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value, the time delay τ = 1 might be an appropriate choice. With cardiorespiratory coupling, low-order phase-locking n:m ratios (m ≤ 3) are expected. Therefore, low embedding dimension D = 3 should be enough to unveil patterns of the synchrogram. Fig. 5 shows D = 3 and τ = 1 can reveal many high-order phase-locking ratios. The threshold ε is a crucial parameter of RP. If ε is too small, there will be very few recurrence points on RP and no recurring patterns will be quantified. On the other hand, if ε is too large, almost every point is included in the neighborhood of others and the quantification has many artifacts. Furthermore, noise smears the horizontal lines of a synchrogram and that strongly affects to the precision of all synchrogram quantification techniques. In this paper, ε must be appropriate to quantify the synchrogram in certain noise levels with any n:m ratios. To avoid false detection, it is necessary to define how “straight” the horizontal lines to be considered as synchronization, depending on noise level and application. In cardiorespiratory synchronization, previous work often defined strict thresholds to quantify horizontalness of synchrogram lines [8], [16], [26]. Hamann et al. considered horizontal lines with SD < ((m/5) × n) in analysis window of 30 s as synchronization [16]. Toledo et al. defined a threshold = 0.03 to characterize the variation of horizontal lines with their technique in 31-s window [26]. Using probability distribution index of synchrogram phase λ, Mrowka et al. consider index λ ≥ 0.95 as synchronization [8]. In this paper, our model (3) was used to evaluate the behavior of Lnorm index with different thresholds and noise level of synchrogram. We created the synchrogram with different synchronization ratios (2:1, 3:1, 4:1, 5:1, 6:1, 3:2, 5:2, 7:2, 9:2, 11:2, 5:3, 7:3, 10:3, 11:3, and 13:3). Gaussian white noise ζ is added into the synchrogram with different noise level SD = {0.01, 0.02, 0.03}. We only consider maximum noise level SD = 0.03 since it means that we consider a line with SD = 0.03 as a perfect straight line. For each threshold value and synchronization ratio, 100 synchrograms were generated. Each synchrogram is created to have 50 points and embedded into phase space with D = 3, τ = 1. Fig. 3(a) shows that with SD = 0.01, the suitable threshold should be chosen from 0.2 to 0.35. Phase synchronization can be detected with Lnorm ≥ 0.85. Fig. 3(b) shows that with SD = 0.02, Lnorm saturates with threshold ≥0.4 for low-order ratios and starts decreasing with threshold ≥0.31 for some higher order ratio (9:2, 11:2, 10:3, 11:3, and 13:3). Therefore, for SD = 0.02, we should keep threshold ≈0.3 and synchronization can be detected with Lnorm ≥ 0.5. With SD = 0.03, Lnorm attains low values from 0.2 to 0.4 with threshold ranging from 0.2 to 0.3, which indicates the quantification might not be accurate in this case [see Fig. 3(c)]. Our results suggest that the threshold ε can be chosen from 0.2 to 0.3 depending on noise level and the technique gives reliable results with noise level SD ≤ 0.02.
B. Optimization of Embedding Parameters and Noise Effect An RP analysis has three parameters to be optimized: the embedding dimension D, the delay time τ , and the threshold ε. As the synchrogram is a series of discrete phases of an oscillator at the times when the phases of another oscillator attain a certain
IV. NUMERICAL EXPERIMENTS In this section, our proposed technique has been applied to several mathematical models that simulate phase synchronization of bivariate data. For validation, phase-randomized
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Fig. 3. Optimization of threshold ε and influence of noise on L n o rm index. Noise level of the synchrogram: (a) SD = 0.01, (b) SD = 0.02, and (c) SD = 0.03. Error bar: 95% CI.
surrogate data were created to check whether the detected synchronization is true or just artifact. We generated three surrogate datasets for each experiment by the Fourier-based surrogate method [27]. Each surrogate dataset has 100 realizations. Surrogate dataset 1: Phase of the first signal was randomized while the second signal was preserved as the original. Surrogate dataset 2: Phase of the second signal was randomized while the first signal was preserved as the original. Surrogate dataset 3: Phases of both two signals were randomized. A. Synchronization of Van der Pol Oscillator Driven by an External Force We consider a simple example of a periodic Van der Pol oscillator driven by an external force in noise free case: x ¨ − μ(1 − x2 )x˙ + ω02 x = F
(14)
where the external force F = A sin (vt), μ = 1, and natural frequency ω0 = 1 (rad/s). By varying the frequency v and amplitude A of the external force, phase synchronization can be observed [3]. Equation (14) was solved by Runge–Kutta method using MATLAB function ode45 (The Mathworks, Inc., Natick, MA). The bivariate dataset was created with 1000 data points for each signal. In this experiment, we intended to generate synchronization with phase-locking ratio 3:1 by setting the frequency v = 0.287 (rad/s) and varying the amplitude A of the external force. Phase synchronization is expected to appear with certain value of A. The synchrogram was then generated and has 150 data points. Finally, our technique was applied to quantify the synchrogram with embedding parameters (D = 3, τ = 1, and ε = 0.2) in both original and surrogate datasets. For comparison, we also computed the synchronization index λ (4) and frequency difference between two signals ΔΩ = 3v − Ωv dp , where Ωv dp is the mean rotation frequency of Vander Pol oscillator. Our result showed that the phase synchronization appeared when the amplitude A of the external force is greater than 0.77 (see Fig. 4). All of the synchronization indexes, except DET index, were able to detect phase synchronization in the original dataset. In the synchronization region, we observed index λ
Fig. 4. Synchronization of Van der Pol oscillator driven by an external forced with phase-locking ratio 3:1. Phase synchronization is detected by (a) frequency difference ΔΩ, (b) synchronization index λ, (c) normalized mean diagonal line length L n o rm , (d) maximum diagonal line length L m a x , (e) entropy ENTR, and (f) determinism DET. Black, red, green, and blue lines represent original, surrogate 1, surrogate 2, and surrogate 3 datasets, respectively. Error bars represent standard deviation.
attains value 0.99 in original dataset; mean value: 0.85 (SD = 0.22) in surrogate dataset 1; 0.74 (SD = 0.11) in surrogate dataset 2 and 0.65 (SD = 0.16) in surrogate dataset 3. Therefore, if consider λ ≥ 0.99 as synchronization, the detection by this index might not be accurate with surrogate dataset 1. On the other hand, index Lnorm is observed to attain value 0.99 in original dataset; mean value: 0.3 (SD = 0.16) in surrogate dataset 1; 0.09 (SD = 0.01) in surrogate dataset 2; 0.07 (SD = 0.01) in surrogate dataset 3. If Lnorm ≥ 0.5 is considered as synchronization, phase synchronization can be accurately detected by Lnorm . Interestingly, index Lnorm was also able to detect phase synchronization when A = [0.27, 0.29], 0.44, and 0.53 [see
NGUYEN et al.: AUTOMATED QUANTIFICATION OF THE SYNCHROGRAM BY RECURRENCE PLOT ANALYSIS
Fig. 5. High order of phase-locking ratio n:m quantified by synchrogramRP analysis technique. (a) Synchrograms with phase-locking ratio 13:4, 16:5, and 20:7 are embedded into (b) phase space and (c) corresponding RPs are constructed. ψ is the cyclic relative phase; D is the embedding dimension, and τ is the delay time.
Fig. 4(e)]. It is when the system was synchronized with higher order n:m ratios: 13:4, 16:5, and 20:7, respectively. Fig. 5 showed the corresponding synchrograms, its phase spaces, and RPs in these regions. The result suggests that our proposed index Lnorm can automatically detect phase synchronization without the knowledge of the phase-locking ratio. Similar to Lnorm , Lm ax and ENTR can also detect synchronization; however, ENTR is not able to detect high-order synchronization. Moreover, both Lm ax and ENTR are not normalized that make them difficult for comparison. Fig. 6 illustrates the ability to unveil synchronization ratio in phase space of our technique. In the synchronization segment with n:m = 13:4 ratio, one can see the total number of attractors (n = 13), but m = 4 is not easily noticed. Since the phase difference of the synchrogram is embedded into three dimensions phase space with delay = 1, we have the corresponding position of these points in the phase space as shown in Fig. 6(b). We can observe that the groups {1,2,5,8,11}, {3,6,9,12}, and {4,7,10,13} form three “columns” in phase space. Each point in {3,6,9,12} or {4,7,10,13} or {2,5,8,11} belongs to a different cycle of a slower signal. Thus, the number m is equal to the minimum number of attractors in all “columns.” This explains the finding of 16:5, 20:7 ratios in Fig. 5 and 3:1, 5:2 ratios in Fig. 2. B. Synchronization of Two Coupled Rossler Oscillators We consider two coupled chaotic Rossler oscillators in a noise-free case [28], [29]: x˙ 1,2 = ω1,2 (−y1,2 − z1,2 )+ ∈ (x2,1 − x1,2 ) y˙ 1,2 = ω1,2 (x1,2 + 0.15y1,2 ) z˙1,2 = ω1,2 [0.2 + z1,2 (x1,2 − 10)]
(15)
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Fig. 6. Ability to unveil synchronization ratio 13:4 of a (a) synchrogram by (b) phase space embedding technique. 13 is the total number of attractors and 4 is the minimum number of attractors in all “columns.”
where ω 1 , ω 2 is the natural frequency of each Rossler oscillator and ∈ is the coupling strength. In this experiment, we intended to simulate synchronization when the system are close to state of 2:1 phase locking, so ω 1 = 0.98 (rad/s) and ω 2 = 2.02 (rad/s) were set. By increasing the coupling strength ε, phase synchronization could be observed and detected by our technique. System of (15) was solved by Euler’s technique with time step = 0.001. Initial data points were discarded so that the analysis only contained data of stationary state and each synchrogram will contain ∼200 data points. Similar to previous experiment, our technique was then applied to quantify the synchrogram with embedding parameters (D = 3, τ = 1, and ε = 0.2) in both original and three surrogate datasets. For comparison, we also computed the synchronization index λ (4) and frequency difference between two signals ΔΩ = 2Ω1 − Ω2 , where Ω1 and Ω2 are mean rotation frequencies of the Rossler oscillators. Our result showed that the phase synchronization appeared when the coupling strength ε is greater than 0.15 (see Fig. 7). All of the synchronization indexes were able to detect phase synchronization in the original dataset. In the synchronization region, λ attains value 0.99 in original dataset; mean value ranging from 0.58 (SD = 0.2) to 0.99 (SD = 0.01) in surrogate dataset 1; 0.58 (SD = 0.2) to 0.87 (SD = 0.1) in surrogate dataset 2; and 0.75 (SD = 0.17) to 0.99 (SD = 0.01) in surrogate dataset 3. Therefore, if consider λ ≥ 0.99 as synchronization, the detection by this index might not be accurate with both surrogate dataset 1 and 3. On the other hand, index Lnorm is observed to attain value 0.99 in original dataset; mean value ranging from 0.08 (SD = 0.05) to 0.29 (SD = 0.28) in surrogate dataset 1; 0.05 (SD = 0.03) to 0.16 (SD = 0.21) in surrogate dataset 2; and 0.03 (SD = 0.01) to 0.1 (SD = 0.06) in surrogate dataset
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Fig. 8. Histogram of difference between detected synchronized epochs of original dataset and the surrogate one. The difference tends to be positive (p < 0.001) indicating there are more synchronized epochs in original dataset compared to the surrogate one, and the technique can accurately detect synchronization.
Fig. 7. Synchronization of two coupled Rossler oscillators with phase-locking ratio 2:1. Phase synchronization is detected by (a) frequency difference ΔΩ, (b) synchronization index λ, (c) normalized mean diagonal line length L n o rm , (d) maximum diagonal line length L m a x , (e) entropy ENTR, and (f) determinism DET. Black, red, green, and blue lines represent original, surrogate 1, surrogate 2, and surrogate 3 datasets, respectively. Error bars represent standard deviation.
3. If Lnorm ≥ 0.6 is considered as synchronization, phase synchronization can be accurately detected by Lnorm . V. INFANTS CARDIORESPIRATORY SYNCHRONIZATION In this section, our proposed technique has been applied to infants’ cardiorespiratory signals during quiet sleep to test its ability in detecting synchronization of real and noisy data. To validate whether the detected synchronization is true or just artifact, a surrogate dataset is constructed by associating the respiration signal of every subject with the heart rate of all others [26]. If the detected synchronization is correct, the original dataset is expected to have more synchronized epochs than the surrogate one. Twenty-five healthy infants aged six months have been studied. This study was conducted from March 2006 to January 2009 and under the approval of the Mater Health Services Human Research Ethics Committee (Number 952C). For each infant, 9 min of artifact free data were extracted during quiet sleep. The electrocardiography (ECG) and respiratory (airflow) signals were extracted from polysomnography recordings. ECG and flow signals were sampled at 200 Hz. We extracted the R wave from the ECG by an automatic algorithm [30]. The synchrogram technique requires narrow banded input signals; therefore, a second-order Savitzky–Golay filter was applied to respiratory signal to remove high-frequency noise [3]. Our technique is applied using overlapping windows, epoch length = 20 s, with step size = 5 s. For each epoch, a synchrogram is quantified with embedding dimension D = 3, time delay τ = 1,
and threshold ε = 0.3. An epoch with Lnorm ≥ 0.5 is considered as synchronized. Surrogate dataset has 625 surrogates. We define the recording ij as the recording composed of the respiratory signal of subject j and the ECG signal of subject i. The recording ii then represents the original ones and ij represents the surrogate ones (i = j). Let nii and nij (i = j) be the number of synchronized epoch detecting in original and surrogate dataset, respectively. We formed the set A = {nii − nij | i, j = 1, 2, . . . , 25, i = j} and used the one-sided Wilkinson rank sum test to determine whether it tends to be positive [31]. Fig. 8 illustrates the probability density function of set A, indicates the tendency toward positive (p < 0.001). The average difference of synchronized epoch between original and surrogate dataset is 2.42. The result proves that some of the cardiorespiratory synchronization found is true and the proposed technique should be able to detect cardiorespiratory synchronization in real data. VI. DISCUSSION In the context of phase synchronization, automated quantification of synchrogram analysis is problematic due to the preselection of phase-locking ratio by trial and error. Previous work had to repeatedly construct and quantify many synchrograms corresponding to all combinations of phase-locking ratio (n:m) in order to measure the synchrony. This study provides a unique technique to automatically quantify the synchrogram without the trial selection of phase-locking ratio. Our work suggests that by embedding phase information of the synchrogram into phase space, the phase-locking ratio is automatically unveiled as the number of attractors (see Figs. 5 and 6). This feature enables the automated detection of any phaselocking ratio with just one single construction of synchrogram. Our results suggest that the proposed technique is able to detect any phase-locking ratio in numerical experiments (see Figs. 2 and 5). This time delay embedding process has two parameters to be optimized: embedding dimension D and time delay τ .
NGUYEN et al.: AUTOMATED QUANTIFICATION OF THE SYNCHROGRAM BY RECURRENCE PLOT ANALYSIS
When it is applied directly to raw signals, these parameters are estimated regarding nature of the investigated signals (intrinsic frequency, sampling frequency, noise level, etc.). Different approaches for the estimation of the embedding dimension (e.g., the false nearest-neighbors algorithm [25]) and time delay (e.g., the autocorrelation function and the mutual information function [25], [32]) have been proposed. However, the differences of laboratory environments, measuring techniques, and subjects’ variance strongly affect these embedding parameters. In this study, the time delay embedding process is not directly applied on raw signal, but on its “stroboscopic map”—the synchrogram, which represents the cyclic phases of slower at times when the phase of the faster oscillator attains a certain fixed value. Therefore, it is natural to set the time delay τ = 1 and low embedding dimension D is adequate to unfold the synchrogram’s structure. In our example, D = 3 is able to detect high-order phase-locking ratio 20:7 (see Fig. 5). Differing from previous techniques which often quantify the synchrogram by measuring the variance of horizontal lines [10], [14], [16] or quantified a statistical distribution of the cyclic relative phases in the synchrogram [8], [13], [18], this study systematically quantifies the synchrogram by characterizing its dynamic structure. Our technique exploits RP analysis, which is a well-known nonlinear tool to discover recurring patterns and nonstationarities of experimental data for this task. The advantage of this process is that RP analyses enable dynamic quantification of all synchrogram’s patterns (not only horizontal lines) and it imposes no rigid requirement on data length, stationarity, or statistical distribution [20]. The crucial parameter of RP analysis is the threshold ε. Our synchrogram model (3) has been proposed to optimize this threshold to detect phase synchronization with many phase-locking ratios. Our result suggests that the threshold ε ≤ 0.3 is appropriate to detect low-order synchronization with different noise levels (see Fig. 3). There are several indexes, which have been proposed to quantify the RP [20]. In our case, when the system is synchronized, horizontal structure will appear on the synchrogram leading to the appearance of diagonal lines on the corresponding RP. The duration of phase synchronization, determined by the length of horizontal lines of the synchrogram, can be quantified by the length of diagonal line. Thus, we have applied four indexes quantifying the RP based on its diagonal lines: determinism (DET), mean diagonal line length (L), maximum diagonal line length (Lm ax ) and entropy (ENTR). Our results with several mathematical models show that three of these indexes (L, Lm ax , and ENTR) are able to detect the phase synchronization in coupled system (see Figs. 2, 4, and 6). The surrogate test suggests that the mean diagonal line length L provides the most reliable results (see Figs. 4 and 6). It is an important feature of our technique as most synchronization indexes (such as λ) show inaccurate results with surrogate test (see Figs. 4 and 6). In physiology, synchrogram analysis is often applied to detect phase synchronization of cardiorespiratory coupling [7], [8], [16], [33]. Our result demonstrates that the proposed technique is able to detect cardiorespiratory synchronization in infant’ data (see Fig. 8). It is interesting to study the behavior of cardiorespiratory synchronization in the period preceding/during different
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physiological or pathological events such as sleep apnea, sighs, arousals, etc. or in subjects with different diseases. Because of the limitation of previous synchrogram techniques, it is difficult to study phase synchronization produced by all phase-locking ratios in the synchrogram, so that the ability to discover synchronization in these interesting events is limited. Our proposed technique enables automatic detection of phase-locking ratio and systematic quantification of the synchrogram; hence, it can be applied to study these events in future work. This study still has some limitations. The phase synchronization is understood in the statistical sense as the existence of phase locking, which is visually detected as horizontal lines in synchrogram. However, strong noise smears these lines leading to the inaccuracy of quantification techniques, especially when high-order synchronization is considered. Our work quantifies the recurring patterns of those horizontal lines so that it is also affected by noise. A Hilbert transform technique, used to calculate the instantaneous phase of the signal, requires narrow banded signal to operate. In this study, a Savitzky–Golay smoothing filter has been applied to the signal, but this process might not be the most optimal one. There are several avenues for future research. First, the embedding parameters and threshold in this study are optimized for detecting synchronization with low-order phase-locking ratio n:m (m ≤ 3) in cardiorespiratory coupling. Further study is needed to validate the ability to detect synchronization with higher locking ratios. Second, the proposed technique has been validated by numerical and healthy infants experiments. The performance of this technique on human cardiorespiratory data in different physiological/pathological condition will be tested in the near future. VII. CONCLUSION In this study, we have proposed a new technique to automatically detect the phase synchronization of coupled system with any phase-locking n:m ratio. The advantage of our technique is that it offers a fully automated quantification of the synchrogram, where preselection of phase-locking ratio is not required. This technique has overcome the limitation of traditional synchrogram analysis and has been able to detect phase synchronization in simulated and measured healthy infants’ cardiorespiratory data. Future research to validate this technique with human cardiorespiratory data in different physiological/pathological condition is necessary. APPENDIX A A popular method to define the instantaneous phase of a signal is known as analytic signal approach [25], based on Hilbert transform. By this method, the analytic signal ζ(t) is constructed by adding to signal s(t) its imaginary part i˜ s(t): ζ(t) = s(t) + i˜ s(t) = A(t)eiφ(t) where the function s˜(t) is the Hilbert transform of s(t): +∞ s(τ ) s˜(t) = π −1 P dτ −∞ t − τ
(A1)
(A2)
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where P is the Cauchy principal value, A(t) is the instantaneous amplitude, and φ(t) is the instantaneous phase of the signal s(t). An important advantage of this method is that the instantaneous phase can be easily measured from experimental data. APPENDIX B To normalize L: In general case, let x be the length of the longest diagonal line in the RP, r be the recurrence frequency of the diagonal lines, W be the length of analysis window, and n be the number of diagonal lines in the upper part of RP. We have x−α , (0 ≤ α < r). (B1) n= r If it is fully synchronized L=
(x − r) + (x − 2r) + · · · + (x − nr) n
(B2)
or n+1 r. (B3) 2 We consider r ≈ α for low-order synchronization so that from (B1) and (B3), in case of full synchronization L=x−
x W −2 = . 2 2 Therefore, we can normalize L as L≈
Lnorm =
2L . W −2
(B4)
(B5)
ACKNOWLEDGMENT The authors gratefully acknowledge the Department of Respiratory and Sleep Medicine, Mater Children’s Hospital, Brisbane, Qld., Australia, for providing them healthy infants data and P. I. Terrill for fruitful discussion. REFERENCES [1] L. Glass and M. C. Mackey, From Clocks to Chaos : The Rhythms of Life. Princeton: Princeton University Press, 1988. [2] A. Pikovsky, J. r. Kurths, and M. Rosenblum, Synchronization : a universal concept in nonlinear sciences. Cambridge, [u.a.]: Cambridge Univ. Press, 2003. [3] C. Schafer, M. G. Rosenblum, H. H. Abel, and J. Kurths, “Synchronization in the human cardiorespiratory system,” Physical Review E, vol. 60, no. 1, p. 857-870, Jul, 1999. [4] M. B. Lotric and A. Stefanovska, “Synchronization and modulation in the human cardiorespiratory system,” Physica a-Statistical Mechanics and Its Applications, vol. 283, no. 3-4, p. 451-461, Aug, 2000. [5] R. Quian Quiroga, A. Kraskov, T. Kreuz, and P. Grassberger, “Performance of different synchronization measures in real data: A case study on electroencephalographic signals,” Phys. Rev. E, vol. 65, no. 4, p. 041903, 2002. [6] C. Schafer, M. G. Rosenblum, J. Kurths, and H. H. Abel, “Heartbeat synchronized with ventilation,” Nature, vol. 392, no. 6673, p. 239-240, Mar 19, 1998. [7] T. Penzel, N. Wessel, M. Riedl, J. W. Kantelhardt, S. Rostig, M. Glos, A. Suhrbier, H. Malberg, and I. Fietze, “Cardiovascular and respiratory dynamics during normal and pathological sleep,” Chaos, vol. 17, no. 1, p. 015116, Mar, 2007. [8] R. Mrowka, A. Patzak, and M. Rosenblum, “Quantitative analysis of cardiorespiratory synchronization in infants,” Int. J. Bifurcation Chaos, vol. 10, no. 11, p. 2479-2488, Nov, 2000.
[9] R. Mrowka, L. Cimponeriu, A. Patzak, and M. G. Rosenblum, “Directionality of coupling of physiological subsystems: age-related changes of cardiorespiratory interaction during different sleep stages in babies,” Am J Phys. Regul Integr Comp Phys., vol. 285, no. 6, p. R1395-401, Dec, 2003. [10] E. Toledo, M. G. Rosenblum, C. Sch¨afer, J. Kurths, and S. Akselrod, “Quantification of cardiorespiratory synchronization in normal and heart transplant subjects,” in Proc. Int Symposium on Nonlinear Theory and Its Applications, vol. 1, Lausanne, 1998, pp. 171–174. [11] F. Wallois, A. Aarabi, G. Kongolo, A. Leke, and R. Grebe, “Inverse coupling between respiratory and cardiac oscillators in a life-threatening event in a neonate,” Auton Neurosci, vol. 143, no. 1-2, p. 79-82, Dec 5, 2008. [12] A. Stefanovska, H. Haken, P. V. E. McClintock, M. Hozic, F. Bajrovic, and S. Ribaric, “Reversible transitions between synchronization states of the cardiorespiratory system,” Phys. Rev. Lett., vol. 85, no. 22, p. 4831-4834, Nov 27, 2000. [13] P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, “Detection of n : m phase locking from noisy data: Application to magnetoencephalography,” Phys. Rev. Lett., vol. 81, no. 15, p. 3291-3294, Oct. 12 1998. [14] H. Bettermann, D. Cysarz, and P. Van Leeuwen, “Comparison of two different approaches in the detection of intermittent cardiorespiratory coordination during night sleep,” BMC Phys., vol. 2, no. 18, Dec 4, 2002. [15] H. Seidel and H. Herzel, “Analyzing entrainment of heartbeat and respiration with surrogates,” IEEE Eng. Med. Biol. Mag., vol. 17, no. 6, pp. 54–57, Nov-Dec. 1998. [16] C. Hamann, R. P. Bartsch, A. Y. Schumann, T. Penzel, S. Havlin, and J. W. Kantelhardt, “Automated synchrogram analysis applied to heartbeat and reconstructed respiration,” Chaos, vol. 19, no. 1, p. 015106, Mar, 2009. [17] A. Pikovsky, M. Rosenblum, J. Kurths, R. C. Hilborn, and Reviewer, “Synchronization: A universal concept in nonlinear science,” Amer. J. Phys., vol. 70, no. 6, p. 655, 2002. [18] H. Seidel, “Nonlinear dynamics of physiological rhythms: the baroreflex,” Berlin: Logos, 1998. [19] M. Vejmelka, M. Palus, and W. T. Lee, “Phase synchronization analysis by assessment of the phase difference gradient,” Chaos, vol. 19, no. 2, p. 023120, Jun, 2009. [20] N. Marwan, M. C. Romano, M. Thiel, and J. Kurths, “Recurrence plots for the analysis of complex systems,” Physics Reports-Rev. Section of Phys. Lett., vol. 438, no. 5-6, p. 237-329, Jan, 2007. [21] C. L. Webber and J. P. Zbilut, “Dynamical assessment of physiological systems and states using recurrence plot strategies,” J. Appl. Physiol., vol. 76, no. 2, p. 965-973, Feb, 1994. [22] N. Marwan, N. Wessel, U. Meyerfeldt, A. Schirdewan, and J. Kurths, “Recurrence-plot-based measures of complexity and their application to heart-rate-variability data,” Phys. Rev. E, vol. 66, no. 2, p. 026702, Aug, 2002. [23] H. Ding, S. Crozier, and S. Wilson, “A new heart rate variability analysis method by means of quantifying the variation of nonlinear dynamic patterns,” IEEE Trans. Bio-Med. Eng., vol. 54, no. 9, p. 1590-7, Sep, 2007. [24] P. I. Terrill, S. J. Wilson, S. Suresh, D. M. Cooper, and C. Dakin, “Attractor structure discriminates sleep states: Recurrence plot analysis applied to infant breathing patterns,” IEEE Trans. Biomed. Eng., vol. 57, no. 5, p. 1108-1116, May, 2010. [25] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis. Cambridge, [u.a.]: Cambridge Univ. Press, 2004. [26] E. Toledo, S. Akselrod, I. Pinhas, and D. Aravot, “Does synchronization reflect a true interaction in the cardiorespiratory system?,” Med. Eng. Phys., vol. 24, no. 1, p. 45-52, Jan, 2002. [27] T. Schreiber and A. Schmitz, “Surrogate time series,” Phys. D-Nonlinear Phenomena, vol. 142, no. 3-4, p. 346-382, Aug 15, 2000. [28] M. G. Rosenblum and A. S. Pikovsky, “Detecting direction of coupling in interacting oscillators,” Phys. Rev. E, vol. 6404, no. 4, p. 045202, Oct, 2001. [29] M. Palus and M. Vejmelka, “Directionality of coupling from bivariate time series: How to avoid false causalities and missed connections,” Phys. Rev. E, vol. 75, no. 5, p. 056211, May, 2007. [30] P. S. Hamilton and W. J. Tompkins, “Quantitative Investigation of QRS detection rules using the Mit/Bih arrhythmia database,” IEEE Trans. Biomed. Eng., vol. 33, no. 12, p. 1157-1165, Dec, 1986. [31] L. Ott, An introduction to statistical methods and data analysis, 4th ed. Belmont, Calif: Duxbury Press, 1993.
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[32] L. Y. Cao, “Practical method for determining the minimum embedding dimension of a scalar time series,” Physica D, vol. 110, no. 1-2, p. 43-50, Dec 1, 1997. [33] K. Kotani, K. Takamasu, Y. Ashkenazy, H. E. Stanley, and Y. Yamamoto, “Model for cardiorespiratory synchronization in humans,” Phys Rev E Stat Nonlin Soft Matter Phys, vol. 65, no. 5, p. 051923, May 2002.
Chinh Duc Nguyen (M’08) received the B.Eng. (Hons.) degree in electrical and electronics engineering from the University Technology PETRONAS, Seri Iskandar, Malaysia, in 2008. He is currently working toward the Ph.D. degree in biomedical engineering at the University of Queensland, Brisbane, Qld., Australia. His research interests include mathematical analysis of physiological signals, cardiorespiratory synchronization, and the signal-processing technique applied to sleep medicine.
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Stephen James Wilson (M’96) received the M.B.B.S. and Ph.D. degrees from the University of Queensland, Brisbane, Qld., Australia, and the M.Biomed.Eng. degree from the University of New South Wales, Sydney, N.S.W., Australia. He is currently an Associate Professor in the Biomedical Engineering Program, School of Information Technology and Electrical Engineering, University of Queensland, where he is also engaged in electronics teaching, cardiopulmonary and musculoskeletal instrumentation, and associated biosignal analysis.
Stuart Crozier (M’96) received the B.Eng. (Hons.), M.Sc. (medical physics), Ph.D., and D.Eng. degrees from the University of Queensland, Brisbane, Australia. He currently holds the position of a Professor and Director of biomedical engineering at the University of Queensland, Brisbane, Qld., Australia. He has published more than 170 journal papers in the biomedical field. He holds more than 30 patents. Dr. Crozier is currently an Associate Editor of the IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING and was elected as a Fellow of the Institute of Physics in 2004 and a Fellow of the Australian Academy of Technological Sciences and Engineering in 2007.
