Distinguishing Deterministic Chaos and Periodicity ...

2014 International Symposium on Nonlinear Theory and its Applications NOLTA2014, Luzern, Switzerland, September 14-18, 2014

Distinguishing Deterministic Chaos and Periodicity in Human Photoplethysmogram Nina Sviridova† and Kenshi Sakai† †Environmental and Agricultural Engineering Department, Tokyo University of Agriculture and Technology 3-5-8 Saiwai-cho, Fuchu-shi, Tokyo 183-8509, Japan Email: [email protected], [email protected] Abstract– The photoplethysmogram (PPG) is widely used in medical settings and sports equipment biological signal. PPG, which is measured noninvasively, can provide valuable information about cardiovascular system performance. The present study sought to investigate the underlying dynamics of healthy young human subjects’ PPG. In previous studies PPG was claimed to be driven by deterministic chaos; however, methods applied for chaos detection were noise sensitive and inconclusive. In this paper, methods of nonlinear time series analysis, including time delay embedding, nonlinear deterministic prediction and Wayland test were applied to PPG to find out whether the signal is driven by deterministic chaos, limit cycle process or nondeterministic process. Results demonstrated the presence of both deterministic chaos and limit cycle in the investigated PPG time series. 1. Introduction Physiological signals derived from the cardiovascular system show an extreme intricacy that arises from the interaction of many processes, structure units and feedback loops in humans. Attempts to improve our understanding of physiological complexity and develop new tools for promising applications for human mental and physical health monitoring have made physiological signals such as Electrocardiogram (ECG), electroencephalogram (EEG), blood pressure, heart rate variability (HRV) and Photoplethysmograph (PPG) the subject of recent studies [4-6]. Due to an increase in the successful use of nonlinear time series analysis (NTSA) methods in many scientific disciplines to quantify the complexity of signals [8], methods of nonlinear dynamics analysis have become a new and powerful tool for physiological signal investigation. NTSA allows one not only to quantify, but to qualify data. Although there are many studies that have investigated ECG and HRV signals obtained from healthy human subjects, as well as from patients with mental or heart illnesses, it is still quite controversial whether these signals are chaotic or not [3]. It is also not known whether the PPG signal, which is commonly measured by commercially available medical and sport equipment devices to obtain HRV data, oxygen saturation and blood pressure, is chaotic. Photoplethysmography is a simple and low-cost optical technique that can be used to detect blood volume changes

in the microvascular bed of tissue. The PPG wave form comprises a pulsatile physiological waveform attributed to cardiac synchronous changes in the blood volume with each heart beat and is superimposed on a slowly varying baseline with various lower frequency components attributed to respiration, sympathetic nervous system activity and thermoregulation [2]. Even though pulsation in a finger’s capillary vessels (i.e. PPG obtained from finger) in normal subjects was claimed to be chaotic [12], only classical tests such as power spectrum, correlation dimension (CD), the Lyapunov exponent (LE), and measurement of predictability by sample entropy have been applied to characterize PPG time series [7, 12]. However, these tests (CD and LE) are inconclusive since they may indicate chaos even in nonchaotic systems [11]. Therefore, a clear answer regarding the nature of the PPG signal cannot be obtained by applying these types of classical measurements, although they may provide useful results for medical applications. Moreover, in some cases not only the classification between stochastic and deterministic processes becomes an issue, but also distinguishing between deterministic chaos and periodic motion [9] is not straightforward. Periodic heart beating is one of the main mechanisms driving blood circulation that is why it is necessary to pay close attention to the differentiation between the possible periodicity of PPG masked by noise and deterministic chaos. Additionally in theoretical studies, bifurcation obtained for the SeideiHerzel model [10] demonstrated that changes in the system control parameters can cause a system dynamics shift between limit cycle and chaos. In this paper we have performed a time delay embedding method, calculated the power spectrum, nonlinear prediction’s Relative Route Mean Square Error (RRMSE) and Wayland test translation error to investigate whether the underlying dynamics of the PPG signal involve deterministic chaos, limit cycle or nondeterministic process. 2. Methods and Materials The PPG signal was recorded using a finger PPG recorder by detecting the near infrared light reflected by vascular tissue following illumination with a LED. Data were collected from nine healthy 19- to 27-year old volunteers among Tokyo University of Agriculture and

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Technology (TUAT) students. Experimental data collection was approved by TUAT authorities. Written informed consent was given to participants prior the experiment. At the time of the study all subjects were healthy non-smokers, physically active to similar levels, were not taking any medication, and none declared a history of heart disease. For each subject five measurement repeats were done. The measured period was 5 min with 5 msec sampling steps. For all data collection sessions, the sensor was located on the right forefinger. Every measurement was preceded by a blood pressure check and was done with the subject in a relaxed sitting position in a room with temperature, noise and vibration control. An example of a 30-second long portion of the obtained PPG signal is shown in Fig. 1. Although the origins of the components of the PPG signal are not fully understood, it is generally accepted that PPG can provide valuable information about the cardiovascular system [2]. For example in the PPG signal shown in Fig. 1, fluctuations of 1st order corresponding to one heart cycle, and 2nd order, which have periods multiple to the 1st order wave period, can be distinguished. 1st order fluctuations are caused by the heartbeat and 2nd order fluctuations by respiration.

FIG. 1. Example of PPG time series

FIG. 2. Examples of typical spectra 3.2. Time Delay Embedding By using the embedding technique, the presence of deterministic chaos has been clarified in many complex systems [12]. Although the existence of an attractor is not a sufficient indicator of chaos in dynamical systems, it is necessary; furthermore, it is considered to be an important characteristic of time series. The obtained geometric pattern of an attractor may provide valuable information about biological signal properties. For example, the features of the chaotic attractor can reflect the degree of physical or mental activity or the degree of maturity [12]. For time delay reconstruction, the time delay needs to be sufficiently large to ensure that the resulting individual coordinates are relatively independent; however, it should not be so large that it is completely independent statistically [1]. In this study we have defined the time lag for further calculations as a quarter of the period of the spectrum predominant component [9]. In this study we have performed a time-delay embedding technique with dimension 4 to obtain the experimental attractor. An example of our typical data for a time-delay reconstructed attractor is shown in Fig. 3. The presence of a reconstructed experimental attractor with clear structure can be seen.

3. Results In this paper to analyze data sets obtained in the experiments described in section 2, we have applied “classical” analysis tools such as power spectrum and time-delay embedding, along with nonlinear deterministic prediction and the Wayland test. 3.1. Spectral analysis Spectral analysis is one of the widely used tools for analysis of time series. By using Fourier analysis, frequency components of almost any arbitrary signal can be identified regardless of whether or not the signal was produced by a linear system. In this study we have applied spectral analysis to PPG data. Examples of typical plots of the spectrum in the studied time series are shown in Fig. 2. In Fig. 2, small fluctuations, which indicate noise, can be distinguished around the predominant component whose period is approximately equal to the heart cycle period. Lower frequency components correspond to respiration and other effects, such as thermoregulation and nervous system activity.

FIG. 3. Example of time-delay reconstructed attractor 3.3. Nonlinear Deterministic Prediction The impossibility of long-term prediction is one of defining features of chaos. If a chaotic system is deterministic, it shows predictability in the short term that should decay rapidly. Since different systems demonstrate different predictabilities, one can use this information to distinguish different behavior and unlike, for example, the correlation dimension, forecasting results are often readily interpretable and point directly to the determinism of the system under investigation [11].

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To examine whether PPG time series are predictable in the short-term and investigate how forecasting quality changes with increasing prediction time, we have conducted direct nonlinear deterministic prediction (NDP) in this study. Fig. 4 shows two typical data plots of RRMSE between real and predicted data. Fig. 5 shows an example of actual vs. 0.8s predicted PPG. 1.4 1-3 9-2

1.2

RRMSE

1 0.8 0.6

driving mechanism of the cardiovascular system, we may expect that the PPG signal might be driven by periodic or quasi-periodic motion, which is difficult to classify in circumstances of noisy data. In this study NDP became an important method for separating data with periodic motion. In several studies it was claimed that PPG time series of young healthy subjects are deterministic chaos; however, as seen in Figures 4, 5 and Table I, under some conditions (which are not the subject of the investigation in this paper), young healthy subjects’ PPG data may demonstrate longterm predictability, which is a sign of limit cycle.

0.4

3.4. Wayland test

0.2 0 0

50

100

150

200 N

250

300

350

400

Table I shows values of RRMSE for nonlinear deterministic prediction: short-term (ST) for 5 steps prediction and long-term (LT) for 80 (0.4 s, which is half of an average heart cycle) steps. Highlighted values in Table I indicate the presence of long-term prediction.

In order to explain the variability observed in complex time series and distinguish whether it is due to external stochastic noise, internal deterministic dynamics, or a combination of both, we have applied the Wayland test, which is a computationally simple variation of the Kaplan-Glass method and uses the phase space continuity observed in time series to measure determinism [13]. As mentioned above, results of LE are inconclusive due to the sensitivity of the Lyapunov exponent to noise. We therefore calculated Wayland test Translation Error (WTE), which performs well in high levels of uncorrelated noise and provides a robust measure of the determinism in the attractor. WTE is insensitive to an overall scaling of the original time series. If the time series is deterministic, then the WTE will be small [13]. For instance, WTE for time series produced by a system of Lorentz equations is 0.002 and for a white noise is 0.844. Table II shows the results of WTE calculations.

Table I. RRMSE for nonlinear deterministic prediction

TABLE II. Wayland test translation error

FIG. 4. Examples of two typical RRMSE curves 3500 X - actual signal (9-2) Y - predicted signal 3000

X, Y

2500 2000 1500 1000 0

500

1000

1500 N

2000

2500

3000

FIG. 5. Example actual vs. 0.8s predicted PPG

1 № 1 2 3 4 5 6 7 8 9

ST 5 0.02 0.08 0.05 0.04 0.06 0.02 0.06 0.06 0.03

2 LT 80 0.34 0.99 1.10 0.80 1.18 0.67 0.88 0.82 0.50

ST 5 0.03 0.05 0.05 0.06 0.08 0.05 0.02 0.09 0.02

3 LT 80 0.60 0.63 0.95 0.94 1.44 1.23 0.66 1.52 0.36

ST 5 0.03 0.06 0.04 0.04 0.08 0.01 0.02 0.05 0.03

4 LT 80 0.72 0.66 0.93 0.94 1.44 0.38 0.61 0.86 0.41

ST 5 0.03 0.04 0.03 0.03 0.08 0.03 0.02 0.05 0.03

5 LT 80 0.76 0.73 0.78 0.61 1.59 0.88 0.57 0.85 0.76

ST 5 0.04 0.04 0.04 0.03 0.08 0.03 0.02 0.03 0.03

LT 80 1.03 0.79 0.76 0.61 1.55 0.48 0.59 0.89 0.44

As seen in Table I, in all cases time series demonstrated the presence of short-term prediction, which is indicative of underlying determinism, and in most cases the absence of long-term forecasting typical for deterministic chaos. However, as seen from the data in Fig. 4, there is a different tendency in the rise of the RRMSE curve. Line “9-2” illustrates quite low RRMSE during two heart cycles, which might be a sign of periodic motion, while RRMSE for “1-3” has a tendency to increase more rapidly. As shown previously [9], sometimes it is important to distinguish not only stochastic and deterministic processes, but in some cases the differentiation between periodic and chaotic motion becomes a serious issue. As the heart beating is the main

No 1 2 3 4 5 6 7 8 9

1 0.007 0.041 0.023 0.016 0.092 0.015 0.012 0.031 0.018

2 0.006 0.041 0.012 0.024 0.024 0.010 0.017 0.030 0.019

3 0.005 0.045 0.013 0.020 0.059 0.019 0.016 0.015 0.027

4 0.010 0.023 0.015 0.034 0.027 0.019 0.026 0.039 0.045

5 0.024 0.048 0.050 0.085 0.017 0.037 0.035 0.048 0.166

According to Table II, most of the WTE values are small, which is an indication of determinism in the PPG data. Values exceeding 0.1 are highlighted. 4. Discussion The main goal of this study was to find out whether PPG signals from healthy human subjects are driven by nonlinear deterministic chaos, limit cycle or a nondeterministic process. Produced by sophisticated mechanisms of the cardiovascular system, the PPG signal is still not fully understood. Also, it is well known that physiological data can be contaminated by environmental noise and movement artifacts, which makes PPG signal analyses even more challenging. In an attempt to reach

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reliable conclusions, we have applied various methods of NTSA, including classical and widely used ones – power spectrum, TDE, and have conducted NDP and WTE, the results of which are considered to be more reliable in the case of noise-contaminated data. The investigated data primarily demonstrated the presence of an experimental attractor with clear structure, which is necessary for a deterministic process; however, attractor structure might be affected by noise, and on the other hand the attractor might be produced by a nonchaotic system as well. All of the data had short-term predictability and in most cases did not have long-term predictability. In a few cases long-term predictability was observed, which might be a sign of periodicity in the data; therefore, these cases were inconclusive and we were unable to firmly distinguish chaos and limit cycle. Results of spectral analysis were in correspondence with the obtained experimental attractor and NDP. In a few cases the data demonstrated spectra with several spikes, indicating periodic components, and other results showed the presence of a dominant peak corresponding to periodic heart beating, and low frequency components that may arise from respiration and other effects comprised in the PPG signal waveform. To reach a conclusion about the PPG underlying process we applied the Wayland test. Part of the data showed a general trend to have short-term but not longterm prediction, and a small WTE. Based on these results part of the PPG time series obtained from healthy, young human subjects should be concluded to be deterministic chaos. Nevertheless several time series from the healthy group showed signs of strong periodicity; therefore, we have to admit that periodicity may arise in healthy, young subjects. The reason for the apparent periodicity in the data is currently under investigation. Under the conditions of our experiment, periodicity might be due to internal or external effects, such as emotional disturbance, fatigue, or uncomfortable position during measurements, etc. 5. Conclusion In this study we have discovered that significant changes in PPG signals may arise not only with aging or disease [7, 12] but as well with minor changes in the subject’s internal or external conditions. We found that in several cases healthy young subjects demonstrated underlying deterministic chaos, which is in agreement with other studies, as a few time series clearly showed periodicity. All of the measurements were done consecutively, and for each subject total data collection time did not exceed 40 min. In addition, all data collection was done under nearly the same environmental conditions. Therefore, changes of underlying properties of time series for the same subject within a short time under the same external conditions from deterministic chaos to limit cycle might be produced by internal natural processes. This finding brought us to the conclusion that healthy young human subjects can naturally demonstrate both nonlinear

determinism and strong periodicity depending on slight internal changes. This conclusion based on nonlinear time series analysis of experimental data is actually consistent with theoretical results shown previously [10], where Hopf bifurcation for the Seidel-Herzel model of baroreceptor-cardiac reflex demonstrated that with changes of system parameters, such as baroreceptor-vagal nerves coupling, system behavior can vary from limit cycle to chaos. Acknowledgments This work was supported by JSPS Grant-in-Aid No.25660204. References [1] H. D. I., Abarbanel, “The analysis of observed chaotic data in physical systems,” Rev. of modern physics, vol. 65, pp. 1331-1392, 1993. [2] J. Allen, “Photoplethysmography and its application in clinical physiological measurement,” Physiological measurement, vol. 28, pp. 1-39, 2007. [3] L. Glass, “Introduction to controversial topics in nonlinear science: is the normal heart rate chaotic?” Chaos, vol. 19, pp. 1-4, 2009. [4] P. Ch. Ivanov, L. A. Nunes Amaral, A.L. Goldberger and et. al., “Multifractality in human heartbeat dynamics,” Nature, vol. 399, pp 461-465, 1999. [5] S.A. Mascro, H.H. Asada, “Photoplethysmograph Fingernail Sensors for measuring Finger Forces Without Haptic Obstruction,” IEEE transaction on robotics and automation, vol. 17, pp. 698-708, 2001. [6] P. V. E. McClintock, A. Stefanovska, “Noise and determinism in cardiovascular dynamics,” Physica A, vol. 314, pp. 69-76, 2002. [7] T.D. Pham, T.C. Thang, M. Oyama-Higa, M. Sugiyama, “Mental-disorder detection using chaos and nonlinear dynamical analysis of photoplethysmographic signal,” Chaos, Solitons & Fractals, vol. 51, pp. 64-74, 2013. [8] C.-S. Poon, C.K. Merrill, “Decrease of cardiac chaos in congestive heart failure,” Nature, vol. 389, pp. 492 – 495, 1997. [9] K. Sakai, Nonlinear Dynamics and Chaos in Agricultural Systems, Elsevier, Netherlands, 2001. [10] H. Seidel, H. Herzel, “Bifurcations in a nonlinear model of the baroreceptor-cardiac reflex,” Physica D, vol. 115, pp. 145-160, 1998. [11] M. Shelhamer, Nonlinear Dynamics in Physiology. A state-Space Approach, World Scientific, Singapore, 2007. [12] I. Tsuda, “Chaotic pulsation in human capillary vessels and its dependence on mental and physical conditions,” International journal of Bifurcation and chaos, vol. 2, No 2, pp. 313-324, 1992. [13] R. Wayland, D. Bromley, D. Pickett, A. Passamante, “Recognizing Determinism in a Time Series,” Physical review letters, vol. 70, pp. 580-582, 1993.

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