從先前的研究，我們認識到心律變異頻域分析中的「冪次關係」在臨床醫療上有著不小的使用價值，然而其背後的詳細成因仍然未定。為了更深入探討這個現象，我們使用同調節拍器來作為心律變異的等效模型。
同調節拍器除了在頻域空間中，我們所關注的頻率範圍內有著和心律變異有著相似的冪次關係外，其在時域空間亦展現與心律變異相似的一些關鍵表現。而後我們提出一個數學模型，可以同時描述心律變異與同調節拍器的時域空間行為，並在傅立葉轉換後，能夠在頻域空間中顯現出冪次關係。
另外，我們提出一個方案，可以有效縮短心律變異冪次分析的量測時間，從24小時縮至12小時，同時希冀這樣的方法，能夠使此種分析在醫療上的使用難度降低，並使這樣的分析更加有效且有力。

We learned from previous researches that power law in Heart Rate Variability (HRV) frequency domain analysis has been proven useful in clinical use. However, the reason behind this phenomenon is still unclear. To gain further insights into the source of the power law in HRV, we turn to synchronized metronomes as a toy model to study how fluctuations occur and affect the spectrum of a synchronized system.
The reason why we made an analogy between the complicated physiological phenomenon of heartbeats and the simple system like synchronized metronomes was motivated by a surprising finding that them both share similar time-domain properties and power-law behavior in their spectra. We proposed a mathematic model on time domain which is capable of describing the behavior of both HRV and synchronized metronomes and gives a power law on frequency domain when Fourier transformed.
We have another proposal to shorten the time required to perform HRV analysis from 24 to 12 hours, in hope to make the analysis more approachable and powerful in clinical use.

Abstract 2
摘要 3
Acknowledgement 4
Chapter 1. Introduction and What We Learned 7
1.1 What is Power Law 7
1.2 What is HRV 11
1.3 HRV and Power Law 13
1.4 Synchronization Phenomenon. 17
1.5 Metronomes 20
1.6 Summary of What We Learned from Previous Researches 22
1.6.1 The Power in HRV 22
1.6.2 Possible Sources of the Exponent 23
Chapter 2. Experiments and Results 24
2.1 The Transformation Between Auto-Correlation and Power Spectrum 24
2.2 Synchronized Metronomes 28
2.3 Power Law in HRV and Synchronized Metronomes 33
2.4 Tests on HRV Analysis 39
Chapter 3. Conclusion and Discussion 44
3.1 Proposals 44
3.1.1 A Temporal Model that Generates the Power Law in HRV and Synchronized Metronomes 44
3.1.2 A New Method to Shorten the Time of Measurement 47
3.2 Discussion 49
Reference 51
Appendix 55

[1] Zipf, George Kingsley (1932). Selected Studies of the Principle of Relative Frequency in Language. Cambridge, MA: Harvard University Press.
[2] Klaus, A., Yu, S., & Plenz, D. (2011). Statistical Analyses Support Power Law Distributions Found in Neuronal Avalanches. PLoS ONE, 6(5).
[3] Ribeiro, A. L. et al. (2002). Power-law behavior of heart rate variability in Chagas’ disease. The American journal of cardiology 89, 414-8.
[4] Kucera, J. P., Heuschkel, M. O., Renaud, P., & Rohr, S. (2000). Power-Law Behavior of Beat-Rate Variability in Monolayer Cultures of Neonatal Rat Ventricular Myocytes. Circulation Research 86(11), 1140-1145.
[5] Hausdorff, J. M., & Peng, C. K. (1996). Multiscaled randomness: A possible source of 1/f noise in biology. Physical review E, 54(2), 2154.
[6] Callen, H. B., & Callen, E. (1966). The present status of the temperature dependence of magnetocrystalline anisotropy, and the l (l+ 1) 2 power law. Journal of Physics and Chemistry of Solids, 27(8), 1271-1285.
[7] Witten Jr, T. A., & Sander, L. M. (1981). Diffusion-limited aggregation, a kinetic critical phenomenon. Physical review letters, 47(19), 1400.
[8] Gabaix, X., Gopikrishnan, P., Plerou, V., & Stanley, H. E. (2003). A theory of power-law distributions in financial market fluctuations. Nature, 423(6937), 267.
[9] Yaneer Bar-Yam. Concepts: Power Law. New England Complex Systems Institute. Retrieved 18 August 2015.
[10] Bak, P., & Tang, C. (1989). Earthquakes as a self‐organized critical phenomenon. Journal of Geophysical Research: Solid Earth, 94(B11), 15635-15637.
[11] Tsai, S. T., Wang, L. M., Huang, P., Yang, Z., Chang, C. D., & Hong, T. M. (2016). Acoustic emission from breaking a bamboo chopstick. Physical review letters, 116(3), 035501.
[12] Bak, P., Tang, C., & Wiesenfeld, K. (1987). Self-organized criticality: An explanation of the 1/f noise. Physical review letters, 59(4), 381.
[13] Malik, M., Bigger, J. T., Camm, A. J., Kleiger, R. E., Malliani, A., Moss, A. J., & Schwartz, P. J. (1996). Heart rate variability: Standards of measurement, physiological interpretation, and clinical use. European heart journal, 17(3), 354-381.
[14] Bigger JT Jr; Fleiss JL; Steinman RC; Rolnitzky LM; Kleiger RE; Rottman JN. (1992). Frequency domain measures of heart period variability and mortality after myocardial infarction. Circulation. 85 (1): 164–171.
[15] Kleiger RE, Miller JP, Bigger JT Jr, Moss AJ (1987). Decreased heart rate variability and its association with increased mortality after acute myocardial infarction. Am J Cardiol. 59 (4): 256–262.
[16] Abildstrom SZ, Jensen BT, Agner E, et al. (2003). Heart rate versus heart rate variability in risk prediction after myocardial infarction. Journal of Cardiovascular Electrophysiology. 14 (2): 168–73.
[17] Rivera-Ruiz M, Cajavilca C, Varon J (29 September 1927). Einthoven's String Galvanometer: The First Electrocardiograph. Texas Heart Institute journal / from the Texas Heart Institute of St. Luke's Episcopal Hospital, Texas Children's Hospital. 35 (2): 174–78.
[18] Hurst JW (3 November 1998). Naming of the Waves in the ECG, With a Brief Account of Their Genesis. Circulation. 98 (18): 1937–42.
[19] Malmivuo, P., Malmivuo, J., & Plonsey, R. (1995). The Heart. Bioelectromagnetism: principles and applications of bioelectric and biomagnetic fields. Oxford University Press, USA.
[20] Bigger, J. J., Steinman, R. C., Rolnitzky, L. M., Fleiss, J. L., Albrecht, P., & Cohen, R. J. (1996). Power law behavior of RR-interval variability in healthy middle-aged persons, patients with recent acute myocardial infarction, and patients with heart transplants. Circulation, 93(12), 2142-2151.
[21] Kuo, T. B., Lin, T., Yang, C. C., Li, C. L., Chen, C. F., & Chou, P. (1999). Effect of aging on gender differences in neural control of heart rate. American Journal of Physiology-Heart and Circulatory Physiology, 277(6), H2233-H2239.
[22] P. Stoica and R. Moses (2005). Spectral Analysis of Signals. Prentice Hall, Upper Saddle River.
[23] Shannon, Claude E. (January 1949). Communication in the presence of noise. Proceedings of the Institute of Radio Engineers. 37 (1): 10–21.
[24] Goldberger, A. L., Amaral, L. A., Glass, L., Hausdorff, J. M., Ivanov, P. C., Mark, R. G., ... & Stanley, H. E. (2000). PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals. Circulation, 101(23), E215-20.
[25] Bennett, M., Schatz, M., Rockwood, H., & Wiesenfeld, K. (2002). Huygens's Clocks. Proceedings: Mathematical, Physical and Engineering Sciences, 458(2019), 563-579.
Retrieved from http://www.jstor.org/stable/3067433
[26] Kuramoto, Yoshiki (1975). H. Araki, ed. Lecture Notes in Physics, International Symposium on Mathematical Problems in Theoretical Physics. 39. Springer-Verlag, New York. p. 420
[27] Strogatz S (2000). From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D. 143 (1–4): 1–20.
[28] Cumin, D.; Unsworth, C. P. (2007). Generalising the Kuromoto model for the study of neuronal synchronisation in the brain. Physica D. 226 (2): 181–196.
[29] Breakspear M, Heitmann S, Daffertshofer A (2010). Generative models of cortical oscillations: Neurobiological implications of the Kuramoto model. Front Hum Neurosci. 4.
[30] Sivashinsky, G.I. (1977). Diffusional-thermal theory of cellular flames. Combust. Sci. and Tech. 15 (3–4): 137–146.
[31] Osipov, G. V., Pikovsky, A. S., Rosenblum, M. G., & Kurths, J. (1997). Phase synchronization effects in a lattice of nonidentical Rössler oscillators. Physical Review E, 55(3), 2353.
[32] Wang, X. F. (2002). Complex networks: topology, dynamics and synchronization. International journal of bifurcation and chaos, 12(05), 885-916.
[33] Murray, James D. (2002). Mathematical Biology. I. An Introduction (3rd ed.). Springer. pp. 295–299.
[34] Z.Néda, E. Ravasz, Y. Brechet, T.Vicsek, and A. L. Barabási, (2000). Physics of the rhythmic applause, Phys. Rev. E, vol. 61, pp. 6987–6992.
[35] Glass, L. (2001). Synchronization and rhythmic processes in physiology. Nature, 410(6825), 277.
[36] Mirollo, R. E., & Strogatz, S. H. (1990). Synchronization of Pulse-Coupled Biological Oscillators. SIAM Journal on Applied Mathematics, 50(6), 1645-1662.
[37] Bottani, S. (1995). Pulse-Coupled Relaxation Oscillators: From Biological Synchronization to Self-Organized Criticality. Physical Review Letters, 74(21), 4189-4192.
[38] Peskin, C. S. (1975). Mathematical aspects of heart physiology. Courant Inst. Math, New York University.
[39] Imbs, J. (2004). Trade, finance, specialization, and synchronization. Review of Economics and Statistics, 86(3), 723-734.
[40] Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., & Zhou, C. (2008). Synchronization in complex networks. Physics reports, 469(3), 93-153
[41] Barthélémy, M., & Amaral, L. A. N. (1999). Small-world networks: Evidence for a crossover picture. Physical Review Letters, 82(15), 3180.
[42] Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’networks. nature, 393(6684), 440.
[43] Blasius, B., Huppert, A., & Stone, L. (1999). Complex dynamics and phase synchronization in spatially extended ecological systems. Nature, 399(6734), 354.
[44] Pantaleone, J. (2002). Synchronization of metronomes. American Journal of Physics, 70(10), 992-1000.
[45] Ashkenazy, Y., Ivanov, P. C., Havlin, S., Peng, C. K., Goldberger, A. L., & Stanley, H. E. (2001). Magnitude and sign correlations in heartbeat fluctuations. Physical Review Letters, 86(9), 1900.
[46] Fearnley CJ, Roderick HL, Bootman MD. (2011). Calcium Signaling in Cardiac Myocytes. Cold Spring Harbor Perspectives in Biology. 3(11).
[47] Robert, V., Gurlini, P., Tosello, V., Nagai, T., Miyawaki, A., Di Lisa, F., & Pozzan, T. (2001). Beat‐to‐beat oscillations of mitochondrial [Ca2+] in cardiac cells. The EMBO journal, 20(17), 4998-5007.