ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Pavlov A. N., Janson N. B. Application of method for mathematical model reconstruction to electrocardiogram. Izvestiya VUZ. Applied Nonlinear Dynamics, 1997, vol. 5, iss. 1, pp. 93-108. DOI: 10.18500/0869-6632-1997-5-1-93-108

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Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-1997-5-1-93-108

Application of method for mathematical model reconstruction to electrocardiogram

Autors: 
Pavlov Aleksej Nikolaevich, Saratov State University
Janson Natalia B., Lancaster University
Abstract: 

The algorithm for reconstruction of a mathematical model on one-dimensional realization of system under study is described. The results of its application to ECG are gathered and analysed. The possibility of modeling using the traditional methods for phase portraits reconstruction and the shortcomings of these methods in application to ECG are discussed.The results are presented obtained as the method for state vector coordinates restoration suggested by the authors was applied. The possibility for obtaining a robust mathematical model with the help of statistical approach is considered.The problems connected with mathematical description adequacy are discussed, the serious shortcomings of the global reconstruction algorithm and the prospects of such method for the aims of modeling are mentioned.

Key words: 
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Acknowledgments: 
Yanson N.B. thanks the international program “Sorov students and graduate students” for support (grant a 96-2593).
Reference: 
  1. Packard NH, Crutchfield JP,Farmer JD, Shaw RS. Geometry from а time series. Phys. Rev. Lett. 1980;45(9):712-716. DOI: 10.1103/PhysRevLett.45.712.
  2. Takens F. Detecting strange attractors in turbulence. In: Rand D, Young LS, editors. Dynamical Systems and Turbulence, Warwick 1980. Lecture Notes in Mathematics. Vol 898. Berlin: Springer, 1981. P. 366-381. DOI: 10.1007/BFb0091924.
  3. Farmer JD, Sidorowich JJ. Predicting chaotic time series. Phys. Rev. Lett. 1987;59(8):845-848. DOI: 10.1103/PhysRevLett.59.845.
  4. Casdagli M. Nonlinear prediction оf chaotic time series. Physica D. 1989;35(3):335-356. DOI: 10.1016/0167-2789(89)90074-2.
  5. Cremers J, Hubler А. Construction of differential equations from experimental data. Z. Naturforsh. А. 1987;42(8):797-802. DOI: 10.1515/zna-1987-0805.
  6. Glass L, Makey MС. From Clocks to Chaos: The Rhythms of Life. Princeton: Princeton University Press; 1988. 248 p.
  7. Babloyantz А, Destexhe А. Is а normal heart а periodic oscillator? Biol. Cybern. 1988;58(3):203-211. DOI: 10.1007/BF00364139.
  8. Zebrowski JJ, Poplawska W, Baranowski R. Pattern entropy а tool for non-linear dynamical analysis оf а biological nonstationary system: the human heart. Acta Physica Polonika В. 1995;26(6):1055-1064.
  9. Baranowski R, Zebrowski JJ, Poplawska W. et al. 3—dimensional Poincare plots of thе QT intervals — an approach to nonlinear QT analysis. In: Computers in Cardiology. 1995. Vienna, Austria. P. 789-792. DOI: 10.1109/CIC.1995.482783.
  10. Peng CK, Havlin S, Stanley HE, Goldberger AL. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos. 1995;5(1):82-87. DOI: 10.1063/1.166141.
  11. Kurths J, Voss А, Saparin Р, Witt А, Kleiner HJ, Wessel N. Quantitative analysis of heart rate variability. Chaos. 1995;5(1):88-94. DOI: 10.1063/1.166090.
  12. Voss А, Kurths J, Kleiner HJ, Witt А, Saparin P, Dietz R, Fiehring H, Wessel N. Neue Methoden fur die Erkennung von Hochrisikopatienten in der Kardiologie. Biomed. Tech. 1994;39(6):134-143. DOI: 10.1515/bmte.1994.39.6.134.
  13. Anishchenko VS, Safonova MA, Saparin PI. Measuring and computing complex for diagnostics of complex modes of self-oscillations. Sov. J. Commun. Tech. Electron. 1992;37(3):467-479. (in Russian).
  14. Saparin Р, Zaks MA, Kurths J, Voss А, Anishchenko VS. Reconstruction and structure of electrocardiogram phase portraits. Phys. Rev. E. 1996;54(1):737-742. DOI: 10.1103/physreve.54.737.
  15. Rosenblum M, Kurths J. A model of neural control оf the heart rate. Physica А. 1995;215:439-450. DOI: 10.1016/0378-4371(95)00041-5.
  16. Seidel H, Herzel H. Modelling Heart Rate Variability Due to Respiration and Baroreflex. In: Mosekilde E, Mouritsen OG, editors. Modelling the Dynamics of Biological Systems. Springer Series in Synergetics. Vol. 65. Berlin: Springer; 1995. P. 205-229. DOI: 10.1007/978-3-642-79290-8_11.
  17. Anishchenko VS, Yanson NB, Pavlov АN. A method for reconstructing nonuniform attractors. Tech. Phys. Lett. 1996;22(4):265-267.
  18. Anishchenko VS. Complex Oscillations in Simple Systems. М.: Nauka; 1990. 312 p. (in Russian).
  19. Malinetskii GG, Potapov AB, Rakhmanov АI. Limitations of attractor reconstruction capabilities for chaotic dynamic systems. KIAM Academ. Sci. USSR Препринт No 10; 1993. 26 p.
  20. Маnе R. On the dimension of the compact invariant sets of certain nonlinear maps. In: Rand D, Young LS, editors. Dynamical Systems and Turbulence, Warwick 1980. Lecture Notes in Mathematics. Vol. 898. Berlin: Springer; 1981. P. 230-242.
  21. Sauer T, Yorke JA, Casdagli M. Embedology J. Stat. Phys. 1991;65:579-616. DOI: 10.1007/BF01053745.
  22. Berge Р, Pomeau Y, Vidal CH. Order Within Chaos. N.Y.: Wiley; 1986. 329 p.
  23. Landa PS, Rozenblum MG. On one method of estimating the dimension of the attractor attachment based on the results of the experiment. Sov. Phys. Tech. Phys. 1989;59(1):13-20. (in Russian).
  24. Fraser AM, Swinney HL. Independent coordinates from mutual information. Phys. Rev. А. 1986;33(2):1134-1140. DOI: 10.1103/physreva.33.1134.
  25. Liebert W, Schuster HG. Proper choice of the time delay for the analysis of chaotic time series. Phys. Lett. А. 1989;142(2-3):107-111. DOI: 10.1016/0375-9601(89)90169-2.
  26. Breeden JL, Packard NH. A learning algorithm for optimal representation of experimental data. Int. J. Bif. Chaos. 1994;4(2):311–326. DOI: 10.1142/S0218127494000228.
  27. Gouesbet С, Letellier С. Global vector—field reconstruction by using а multivariate polynomial L2 approximation оn nets. Phys. Rev. Е. 1994;49(6):4955-4972. DOI: 10.1103/physreve.49.4955.
  28. Kremliovsky M, Kadtke J, Inchiosa M, Moore P. Characterization оf dolphin acoustic echo—location data using a dynamical classification method. Int. J. Bifurc. Chaos. 1998;8(4):813-823. DOI: 10.1142/S0218127498000619.
  29. Anosov OL, Butkovskii OYa, Kravtsov YuA, Surovyatkina ED. Predictable nonlinear dynamics: advantages and limitations. AIP Conf. Proc. 1996;375:71-91. DOI: 10.1063/1.51009.
  30. Gribkov DА, Gribkova VV, Kravtsov YuА, Kuznetsov YuI, Rzhanov АG. Restoring the structure of the dynamic system by time series. J. Commun. Tech. Electron. 1994;39(2):269-277. (in Russian).
  31. Yanson NB, Anishchenko VS. Modeling of dynamic systems based on experimental data. Izvestiya VUZ. Applied Nonlinear Dynamics. 1995;3(3):112-121. (in Russian).
  32. Yanson NB, Pavlov АN, Balanov АG, Anishchenko VS. Reconstructing a mathematical model as applied to an electrocardiogram. Tech. Phys. Lett. 1996;22(8):669-671.
  33. Breeden JL, Packard NH. Nonlinear analysis оf data sampled nonuniformly in time. Physica D. 1992;58(1-4):273-283.
  34. Kadtke J, Kremliovsky M. Signal classification using global dynamical models. Part I: Theory. AIP Conf. Рrос. 1996;375:189-202. DOI: 10.1063/1.51029.
  35. Anishchenko VS, Smirnova NB. Analysis and synthesis оf dynamical systems from experimental data. In: SPIE. 1993. Vol. 2098. P. 137-142. DOI: 10.1117/12.175138.
  36. Janson NB, Anishchenko VS. Modeling the dynamical systems оn experimental data. AIP Conf. Proc. 1996;375:688-708. DOI: 10.1063/1.51006.
  37. Gribkov DA, Gribkova VV, Kuznetsov YuI, Rzhanov AG. Global dynamical modeling of time series and application to restoration of broadband signal characteristics. AIP Conf. Proc. 1996;375:181-188. DOI: 10.1063/1.51028.
  38. Gribkov DА, Gribkova VV, Kuznetsov YuI. Reconstruction of external signal from the time distribution of а variable of the autostochastic system. Moscow University Physics Bulletin. 1995;50(1):71-73.
  39. Brush JS. Classifying transient signals with nonlinear dynamic filter banks. AIP Conf. Proc. 1996;375(1):145-166. DOI: 10.1063/1.51024.
  40. Orlov VN. Guide to Electrocardiography. М.: Meditsina; 1983. 560 p.
  41. Vinogradova TS, editor. Instrumental Methods of Cardiovascular System Research. М.: Meditsina; 1986. 416 p. (in Russian).
  42. Anishchenko VS, Yanson NB, Pavlov АN. Saddle focus in a model of the electrical activity of the human heart. Tech. Phys. Lett. 1996;22(2):168.
  43. Anishchenko VS, Yanson NB, Pavlov АN. Work of the human heart: A regular process? J. Commun. Tech. Electron. 1997;42(8):936-940.
  44. Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical Recipes in C: the art оf scientific computing. Cambridge: Cambridge University Press. 1988. 735 p.
  45. Pavlov АN, Yanson NB, Anishchenko VS. Application of statistical methods to solve global reconstruction problems. Tech. Phys. Lett. 1997;23(4):297-299. DOI: 10.1134/1.1261854.
Received: 
07.02.1997
Accepted: 
28.03.1997
Published: 
18.05.1997
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