Chaos analysis of the electrical signal time series evoked by acupuncture

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1 Chaos, Solitons and Fractals 33 (2007) Chaos analysis of the electrical signal time series evoked by acupuncture Jiang Wang a, *, Li Sun a, Xiangyang Fei a, Bing Zhu b a School of Electrical Engineering, Tianjin University, Tianjin , PR China b Institute of Acupuncture and Moxibustion, China Academy of Traditional Chinese Medicine, Beijing , PR China Accepted 5 January 2006 Abstract This paper employs chaos theory to analyze the time series of electrical signal which are evoked by different acupuncture methods applied to the Zusanli point. The phase space is reconstructed and the embedding parameters are obtained by the mutual information and Cao s methods. Subsequently, the largest Lyapunov exponent is calculated. From the analyses we can conclude that the time series are chaotic. In addition, differences between various acupuncture methods are discussed. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction Four decades ago, the paradigm of classical chaos was introduced [1 4]. The interest in chaotic research has risen rapidly and several attempts have been made in many related fields such as power system and biomedical engineering as well as applications to the human brain and heart [5 7]. In the 1980s, Japanese scholars studied the repetitive firing of the action potential in squid giant axons stimulated by sinusoidal current and found various motions including periodic, quasi-periodic and chaotic through theoretical computations [8,9]. Chaos analysis, such as the embedding theorem, the mutual information method, Cao s method, the largest Lyapunov exponent and so on, is a powerful non-linear method [10] which enables the extraction of characteristic quantities. It offers tools that bridge the gap between experimentally observed irregular behavior and deterministic chaos theory [11 14]. Acupuncture is an important part of Chinese medicine theory and it is approved to be highly effective in treatment of more than 300 diseases [15]. Since the middle period of the 20th century, the applications of acupuncture have advanced in abirritation [16], quitting drug [17] and other problems. Acupuncture at the Zusanli point is not only utilized to treat common digestive system diseases such as duodenum ulcer, acute gastritis and gastroptosis, but also has auxiliary efficiency on enteritis, dysentery, constipation, hepatitis, gallstone, kidney stone, diabetes and hypertension [18]. When acupuncture is applied to the Zusanli point, electrical activity can be recorded from the spinal dorsal horn. Different kinds of acupuncture methods can evoke various electrical time series. * Corresponding author. Tel.: ; fax: address: jiangwang@tju.edu.cn (J. Wang) /$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi: /j.chaos

2 902 J. Wang et al. / Chaos, Solitons and Fractals 33 (2007) Fig. 1. The transmission path of the acupuncture signals at the Zusanli point. Fig. 2. The time series evoked by three acupuncture methods.

3 J. Wang et al. / Chaos, Solitons and Fractals 33 (2007) In this paper, the phase space of the time series at the spinal dorsal horn is reconstructed to obtain the embedding parameters by using the embedding theorem [19]. The largest Lyapunov exponents are calculated and turn out to be positive. Based on the two methods above, we conclude that the signals are chaotic. 2. Transmission path of the acupuncture signals According to previous studies, acupuncture signals follow a certain route from the acupuncture point to the spinal dorsal horn. The corresponding transmission path for acupuncture signals at the Zusanli point is shown in Fig. 1. Next, the electrical signal time series at the spinal dorsal horn is recorded and then non-linear dynamics is used to analyze the time series. 3. The time series evoked by acupuncture There are twelve alternative methods used in acupuncturing at the Zusanli point. This paper selects three of them, the twist method, the drag-plug method and the gradual method [20]. The time series at the spinal dorsal horn evoked by these three methods are shown in Fig. 2. Fig. 3. The embedding parameters of the twist method. Fig. 4. The embedding parameters of the drag-plug method.

4 904 J. Wang et al. / Chaos, Solitons and Fractals 33 (2007) Fig. 5. The embedding parameters of the gradual method. Fig. 6. The reconstructed attractors of the three methods.

5 J. Wang et al. / Chaos, Solitons and Fractals 33 (2007) Time series analysis 4.1. Phase space reconstruction theory The phase space is reconstructed according to the delay coordinate method proposed by Takens [21] and Packard [22]. Here, define a discrete time array x(i), i = 1,2,..., n obtained by measurement or simulation, reconstruct the m-dimension state vector X n by the delay coordinate method; X n ¼ xðnþ; xðn þ T Þ; xðn þ 2T Þ;...; xðn þðm 1ÞTÞ ð1þ where T is called time delay and m is embedding dimension. T and m are two important parameters in the phase space reconstruction. Values of T, m are obtained by the mutual information method [23] and Cao s method [24], respectively Largest Lyapunov exponents Based on the reconstructed phase space, we analyze the spatio-temporal behavior of the time series. The Lyapunov exponent is an important parameter for describing the non-linear system behavior. It states the rate of exponential divergence from initial perturbed conditions. Consider a one-dimensional map x nþ1 ¼ f ðx n Þ ð2þ Assume the difference of the initial value x 0 is dx 0. The n-time iteration value is; dx n ¼jf n ðx 0 þ dx 0 Þ f n ðx 0 Þj ¼ df n ðx 0 Þ dx 0 ¼ e kn dx 0 ð3þ dx 0 Fig. 7. The LLEs of the three methods.

6 906 J. Wang et al. / Chaos, Solitons and Fractals 33 (2007) where k is Lyapunov exponent. The magnitude of the Lyapunov exponent is a measurement of the sensitivity to initial conditions. The system is chaotic and unstable when the Lyapunov exponent is positive. For an n-dimensional map, the largest Lyapunov exponent (LLE) is preferred to estimate whether the system is chaotic or not. This paper adopts the method introduced by Wolf [25] to calculate the LLE of the time series at the spinal dorsal horn. 5. Experimental data processing We select data points within 20 s and reconstruct the phase space for these experimental data [26 29]. As to the data of the twist method shown in Fig. 3, the time delay T = 3 and the embedded dimension m = 3, while T =3, m = 3 and T =4,m = 3 are chosen for the drag-plug method shown in Fig. 4 and the gradual method shown in Fig. 5, respectively. The attractors of the reconstructed phase space are shown in Fig. 6. They are all strange attractors for all the three methods even with different shapes according to the figures. So we confirm these signals are chaotic preliminarily. In addition, the LLEs according to the Wolf s algorithm are calculated to quantitatively describe the time series. The calculation results of the LLEs are shown in Fig. 7. The LLEs of the three methods are ± , ± , ± , respectively. Obviously, the different LLEs show the distinction of the time series for the different methods. Based on the attractors of the reconstructed phase space and the LLEs, we can conclude that these signals are chaotic. 6. Conclusion In this paper, methods of non-linear dynamics are employed to analyze the electrical signal time series evoked by acupuncture. The time delay and the embedding dimension are obtained by the mutual information method and Cao s method respectively. According to the attractors of the reconstructed phase space, it is found that the time series are chaotic. The largest Lyapunov exponents, which are positive, are calculated to quantitatively describe the chaotic behavior. Thus we conclude that the signals are chaotic. The largest Lyapunov exponents of the signals obviously vary in different acupuncture methods, which mean that different acupuncture methods will have different influences on the same disease. On the base of the chaotic analysis of the acupuncture electrical signals, we can understand the influence of various acupuncture methods on the same disease more comprehensively. Future research should be focused on analyzing the cause of chaos of various acupuncture methods. It will definitely lead the acupuncture theory to a more advanced level. Acknowledgement The authors gratefully acknowledge the support of the NSFC (Nos and ). References [1] Lorenz EN. Deterministic non-periodic flow. J Atmos Sci 1963;20: [2] Chen G, Dong X. From chaos to order: methodologies, perspectives and applications. Singapore: World Scientific; May. [3] Bolis CL, Licinio J, editors. Introduction to chaos theory, fractals, and complexity in clinical medicine: the autonomic nervous system. Geneva: World Health Organization; [4] Li Guo-Hui. Synchronization of chaotic systems with parameter driven by a chaotic signal. Chaos, Solitons & Fractals 2005;26(5): [5] Yang Xiao-Song, Yangi Fangyan. A rigorous verification of chaos in an inertial two-neuron system. Chaos, Solitons & Fractals 2004;20(3): [6] Jin Wu-Yin, Xu Jian-Xue, Wu Ying, Hong Ling. Rate of afferent stimulus dependent synchronization and coding in coupled neurons system. Chaos, Solitons & Fractals 2004;21(5): [7] Gabryś El_zbieta, Rybaczuk Marek, Keßdzia Alicja. Blood flow simulation through fractal models of circulatory system. Chaos, Solitons & Fractals 2006;27(1):1 7. [8] Matsumoto G, Aihara K, Utsunomiya T. A spatially ordered pacemaker observed in squid giant axons. J Phys Soc Jpn 1982;51(3):

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