Abstract
The effects of the interaction of the phase space trajectory of a modified van der Pol oscillator with a hyperbolic saddle are discussed. It is shown that the refractory period is obtained and that the saddle affects the way the model reacts both to parameter change and to external perturbation. We obtain results comparable to effects observed in recordings of heart rate variability.
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References
J. Keener, J. Sneyd, Mathematical Physiology, Interdisciplinary Applied Mathematics 8 (Springer, New York, 1998).
K. Grudzinski, J.J. Żebrowski, R. Baranowski, “A model of the sino-atrial and the atrio- ventricular nodes of the conduction system of the human heart”, Biomed. Eng. 51,210–214 (2006).
J.J. Żebrowski, K. Grudziński,. Buchner, P. Kuklik, J. Gac, G. Gielerak, P. Sanders, R. Baranowski, “A nonlinear oscillator model reproducing various phenomena in the dynamics of the conduction system of the heart”, to appear in Chaos 17, Focus Issue “Cardiovascular Physics”, (2007).
K. Grudziński, J.J. Żebrowski, “Modeling cardiac pacemakers with relaxation oscillators”, Physica A 336, 153–152 (2004).
G. Bub, L. Glass, “Bifurcations in a discontinuous circle map: a theory for a chaotic cardiac arrhythmia”, Int. J. Bifurcations and Chaos 5, 359–371 (1995).
D. di Bernardo, M.G. Signorini, S. Cerutti, “A model of two nonlinear coupled oscillators for the study of heartbeat dynamics”, Int. J. Bifurcations and Chaos 8, 1975–1985 (1998).
J.M.T. Thompson, H.B. Steward, Nonlinear Dynamics and Chaos(Wiley, New York, 2002).
B. van der Pol, J. van der Mark, “The heartbeat considered as a relaxation oscillation and an electrical model of the heart”, Phil. Mag. 6, 763–775 (1928).
R. FitzHugh, “Impulses and physiological states in theoretical models of nerve membrane”, Biophys. J. 1, 445–466 (1961).
C.R. Katholi, F. Urthaler, J. Macy Jr., T.N. James, “A mathematical model of automaticity in the sinus node and the AV junction based on weakly coupled relaxation oscillators”, Comp. Biomed. Res. 10, 529–543 (1977).
J. Honerkamp, “The heart as a system of coupled nonlinear oscillators”, J. Math. Biol. 18, 69–88 (1983).
B.J. West, A.L. Goldberger, G. Rovner, V. Bhargava, “Nonlinear dynamics of the heartbeat. The AV junction: Passive conduict or active oscillator?”, Physica D, 17, 198–206 (1985).
H. Zhang, A.V. Holden, I. Kodama, H. Honjo, M. Lei, T. Varghese, M.R. Boyett, “Mathematical models of action potentials in the periphery and center of the rabbit sinoatrial node”, Am. J. Physiol. Heart Circ. Physiol. 279, H397–H421 (2000).
D. Postnov, Kee H. Seung, K. Hyungtae, “Synchronization of diffusively coupled oscillators near the homoclinic bifurcation”, Phys. Rev. E 60, 2799–2807 (1999).
B.F. Hoffman, P.F. Cranefield, Electrophysiology of the Heart (Mc Graw Hill, New York, 1960).
J. Jalife, V.A. Slenter, J.J. Salata, D.C. Michaels, Circ. Res. 52, 642–656 (1983).
S.S. Demir, J.W. Clark, W.R. Giles, Am. J. Physiol. Heart. Circ. Physiol. 276, H2221–H2244 (1999).
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Grudziński, K., Żebrowski, J.J., Baranowski, R. (2009). Nonlinear Oscillations in the Conduction System of the Heart – A Model. In: Dana, S.K., Roy, P.K., Kurths, J. (eds) Complex Dynamics in Physiological Systems: From Heart to Brain. Understanding Complex Systems. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9143-8_8
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DOI: https://doi.org/10.1007/978-1-4020-9143-8_8
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