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The heart as a system of coupled nonlinear oscillators

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Abstract

In various approximations the heart is considered as a system of coupled nonlinear oscillators which are characterized by their phase transition curve (PTC) only. The system exhibits many phenomena which are known from ECG recordings such as multi-level-Wenckebach periodicity, dissociation, induction or removal of a tachycardia by an extrasystole, pseudo-block and the gap phenomenon. All these can be discussed within the model on a quantitative level.

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References

  • Csapo, G.: Erregungsbildungsund Erregungsleitungsstörungen. In: Roskamm, H., Reindell, H., (eds.). Herzkrankheiten. Berlin-Heidelberg-New York: Springer 1982a

    Google Scholar 

  • Csapo, G.: Konventionelle und intrakardiale Elektrokardiographie. In: Roskamm, H., Reindell, H., (eds.). Herzkrankheiten. Berlin-Heidelberg-New York: Springer 1982b

    Google Scholar 

  • Guevara, M. R., Glass, L.: Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oxcillator: A theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias. J. Math. Biology 14, 1–23 (1982)

    Google Scholar 

  • Guevara, M. R., Glass, L., Shrier, A.: Phase locking, period doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells. Science 214, 1350–1353 (1981)

    Google Scholar 

  • Grossmann, S., Thomae, S.: Invariant distributions and stationary correlation functions of onedimensional discrete processes. Z. Naturforschung 32a, 1353–1363 (1977)

    Google Scholar 

  • Heinecker, R.: EKG in Praxis und Klinik. Stuttgart-New York: Thieme-Verlag 1980

    Google Scholar 

  • Ikeda, N., Tsuruta, H., Sato, T.: Difference equation model of the entrainment of myocardial pacemaker cells based on the phase response curve. Biol. Cybern. 42, 117–128 (1981)

    Google Scholar 

  • Ikeda, N.: Model of bidirectional interaction between myocardial pacemakers based on the phase response curve. Biol. Cybern. 43, 157–167 (1982)

    Google Scholar 

  • Kai, T., Tomita, K.: Stroboscopic phase portrait and strange attractors. Phys. Lett. 66A, 91–93 (1978)

    Google Scholar 

  • Kai, T., Tomita, K.: Stroboscopic phase portrait of a forced nonlinear oscillator. Progr. Theor. Phys. 61, 54–73 (1979)

    Google Scholar 

  • Keener, J. P.: Chaotic cardiac dynamics. Lect. Appl. Math. 19, 299–325 (1981a)

    Google Scholar 

  • Keener, J. P.: On cardiac arrythmias: AV conduction block. J. Math. Biology 12, 215–225 (1981b)

    Google Scholar 

  • Lüderitz, B.: Elektrische Stimulation des Herzens. Berlin-Heidelberg-New York: Springer 1980

    Google Scholar 

  • Mandel, W. J.: Cardiac arrhythmias. Philadelphia-Toronto: J. B. Lippincott Comp. 1980

    Google Scholar 

  • May, R. M.: Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976)

    Google Scholar 

  • McAllister, R. E., Noble, D., Tsien, R. W.: Reconstruction of the electrical activity of cardiac Purkinje fibres. J. Physiol. (London) 251, 1–59 (1975)

    Google Scholar 

  • Moe, G. K., Rheinboldt, W. C., Abildskov, M. D.: A computer model of atrial fibrillation. Am. Heart J. 67, 200–220 (1964)

    Google Scholar 

  • Narula, O. S., Shantha, N.: Atrioventricular block: Clinical concepts and His bundle electrocardiography. In: Mandel, W. J., (ed.) Cardiac arrhythmias. Philadelphia-Toronto: J. B. Lippincott 1980

    Google Scholar 

  • Noble, D.: A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pacemaker potential. J. Phys. 160, 317–352 (1962)

    Google Scholar 

  • Noble, D.: The initiation of the heartbeat. Oxford: Clarendon Press 1979

    Google Scholar 

  • Ott, E.: Strange attractors and chaotic motions of dynamical systems. Rev. Mod. Phys. 53, 655–671 (1981)

    Google Scholar 

  • Pavlidis, Th.: Biological oscillators: Their mathematical analysis. New York-London: Academic Press 1973

    Google Scholar 

  • Roskamm, H., Reindell, H.: Herzkrankheiten. Berlin-Heidelberg-New York: Springer 1982

    Google Scholar 

  • Schamroth, L.: The disorders of cardiac rhythm, Vol. I, II. Oxford-London-Edinburgh-Melbourne: Blackwell Scient. Publ. 1980

    Google Scholar 

  • Seipel, L.: Atrioventrikuläre und intraventrikuläre Leitungsstörungen. In: Lüderitz, B., (ed.). Elektrische Stimulation des Herzens. Berlin-Heidelberg-New York: Springer 1980

    Google Scholar 

  • Van der Pol, D., Sc., Van der Mark, J.: The heartbeat considered as a relaxation oscillation, and an electrical model of the heart. Phil. Magazine 6, 763–775 (1928)

    Google Scholar 

  • Watanabe, Y., Dreifus, L. S.: Atrio-ventricular block: basic concepts. In: Mandel, W. J. (ed.). Cardiac arrythmias. Philadelphia-Toronto: J. B. Lippincott 1980

    Google Scholar 

  • Winfree, A. T.: The geometry of biological time. New York-Heidelberg-Berlin: Springer 1980

    Google Scholar 

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After completion of this work the author's attention was drawn to the work of Ikeda et al. (1981); Ikeda (1982), where a similar analysis of some cardiac arrhythmias has been done.

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Honerkamp, J. The heart as a system of coupled nonlinear oscillators. J. Math. Biology 18, 69–88 (1983). https://doi.org/10.1007/BF00275911

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  • DOI: https://doi.org/10.1007/BF00275911

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