Bifurcation structure of a periodically driven nerve pulse equation modelling cardiac conduction. (English) Zbl 0967.92007

Summary: A novel quiescent nerve pulse equation has been used to model cardiac transmembrane action potential propagation. The bifurcation structure of this equation driven by a periodic train of Dirac delta spikes, modelling experimental action potential measurements, displays a complicated transition region which connects a conventional region of fully developed period doubling cascades to a conventional region of Arnold tongues. Within the transition region multistability is frequently encountered. Lyapunov exponents, winding numbers and firing rate maps are presented in dependence on amplitude-frequency parameters of driving. The rich variety of calculated arrhythmias and conduction blocks agrees well with measured behaviour of animal Purkinje fibres.


92C30 Physiology (general)
92C20 Neural biology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Full Text: DOI


[1] Mobitz, W., Über die unvollständige Störung der Erregungsüberleitung zwischen Vorhof und Kammer des menschlichen Herzens. Z. Gesamte Exp. Med, 1924,{\bf41,} 180-237.
[2] Van der Pol, B., van der Mark, J., The heartbeat considered as a relaxation oscillation and an electrical model of the heart. Phil. Mag. Suppl, 1928,{\bf6,} 763-775.
[3] Wiener, N., Rosenblueth, A., The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch. Inst. Cardiol. Mex, 1946,{\bf16,} 205-265. · Zbl 0063.08249
[4] Hodgkin, A. L., Huxley, A. F., A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol, 1952,{\bf117,} 500-544.
[5] Guevara, M. R., Glass, L., Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: A theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias. J. Math. Biology, 1982,{\bf14,} 1-23. · Zbl 0489.92007
[6] Glass, L., Sun, J., Periodic forcing of a limit-cycle oscillator: Fixed points, Arnold tongues, and the global organization of bifurcations. Phys. Rev. E, 1994,{\bf50,} 5077-5084.
[7] Glass, L., Goldberger, A. L., Courtemanche, M., Shrier, A., Nonlinear dynamics, chaos and complex cardiac arrhythmias. Proc. R. Soc. Lond. A, 1987,{\bf413,} 9-26. · Zbl 0626.92001
[8] Aihara, K., Matsumoto, G., Ikegaya, Y., Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator. J. Theor. Biol, 1984,{\bf109,} 249-269.
[9] Aihara, K., Matsumoto, G., Ichikawa, M., An alternating periodic-chaotic sequence observed in neural oscillators. Phys. Lett, 1985,{\bf111A,} 251-255.
[10] Aihara, K., Numajiri, T., Matsumoto, G., Kotani, M., Structures of attractors in periodically forced neural oscillators. Phys. Lett. A, 1986,{\bf116,} 313-317.
[11] Mettin, R., Parlitz, U., Lauterborn, W., Bifurcation structure of the driven van der Pol oscillator. Int. J. Bifurcation and Chaos, 1993,{\bf3,} 1529-1555. · Zbl 0887.34032
[12] West, B. J., Goldberger, A. L., Rovner, G., Bhargava, V., Nonlinear dynamics of the heartbeat. Physica D, 1985,{\bf17,} 198-206. · Zbl 0583.92005
[13] Nomura, T., Sato, S., Doi, S., Segundo, J. P., Stiber, M. D., A Bonhoeffer-van der Pol oscillator model of locked and non-locked behaviors of living pacemaker neurons. Biol. Cybern, 1993,{\bf69,} 429-437. · Zbl 0800.92066
[14] Braaksma, B., Grasman, J., Critical dynamics of the Bonhoeffer-van der Pol equation and its chaotic response to periodic stimulation. Physica D, 1993,{\bf68,} 265-280. · Zbl 0779.34032
[15] Wang, H.-X., de Paola, R., Norwood, W. I., Dynamics underlying the patterning of cardiac dysrhythmias. Phys. Rev. Lett, 1993,{\bf70,} 3671-3674.
[16] Glass, L., Cardiac arrhythmias and circle maps-A classical problem. Chaos, 1991,{\bf1,}13-19. · Zbl 0900.92093
[17] Matsumoto, G., Aihara, K., Hanyu, Y., Takahashi, N., Yoshizawa, S., Nagumo, J., Chaos and phase locking in normal squid axons. Phys. Lett. A, 1987,{\bf123,} 162-166.
[18] Takahashi, N., Hanyu, Y., Musha, T., Kubo, R., Matsumoto, G., Global bifurcation structure in periodically stimulated giant axons of squid. Physica D, 1990,{\bf43,} 318-334. · Zbl 0704.92007
[19] Guevara, M. R., Ward, G., Shrier, A. and Glass, L., Electrical alternans and period doubling bifurcations. In Computers in Cardiology. IEEE Computer Society, Silver Spring, 1984, pp. 167-170.
[20] Chialvo, D. R., Jalife, J., Non-linear dynamics of cardiac excitation and impulse propagation. Nature, 1987,{\bf330,} 749-752.
[21] Chialvo, D. R., Gilmour, R. F. Jr., Jalife, J., Low dimensional chaos in cardiac tissue. Nature,1990,{\bf343,} 653-657.
[22] Sato, S., Doi, S., Response characteristics of the BVP neuron model to periodic pulse inputs. Math. Biosci, 1992,{\bf112,} 243-259. · Zbl 0825.92046
[23] Doi, S., Sato, S., The global bifurcation structure of the BVP neuronal model driven by periodic pulse trains. Math. Biosci, 1995,{\bf125,} 229-250. · Zbl 0818.92003
[24] Harmon, L. D., Studies with artificial neurons, I: Properties and functions of an artificial neuron. Kybernetik, 1961,{\bf1,} 89-101.
[25] Nagumo, J., Sato, S., On a response characteristic of a mathematical neuron model. Kybernetik,1972,{\bf10,} 155-164. · Zbl 0235.92001
[26] Holden, A. V. and El Naschie, M. S., (editors), Nonlinear phenomena in excitable physiological systems (special issue), Chaos, Solitons and Fractals, {\bf5}(3/4), March/April 1995.
[27] Mackey, M. C., Glass, L., Oscillation and chaos in physiological control systems. Science, 1977,{\bf197,} 287-289. · Zbl 1383.92036
[28] Engelbrecht, J., The evolution of nonlinear waves in active media. Wave Motion, 1986,{\bf8,} 93-100. · Zbl 0572.73038
[29] Engelbrecht, J., Tobias, T., On a model stationary nonlinear wave in an active medium. Proc.R.Soc. Lond, 1987,{\bfA411,} 139-154. · Zbl 0634.35040
[30] FitzHugh, R., Impulses and physiological states in theoretical models of nerve membrane. Biophys. J, 1961,{\bf1,} 445-466.
[31] Nagumo, J., Arimoto, S., Yoshizawa, S., An active pulse transmission line simulating nerve axon. Proc. IRE, 1962,{\bf50,} 2061-2070.
[32] Lieberstein, H. M., On the Hodgkin-Huxley partial differential equation. Math. Biosci, 1967,{\bf1,} 45-69.
[33] Katholi, C. R., Urthaler, F., Macy, J. Jr., James, T. N., A mathematical model of automaticity in the sinus node and AV junction based on weakly coupled relaxation oscillators. Comp. Biomed. Res, 1977,{\bf10,} 529-543.
[34] Goldberger, A. L., Bhargava, V., West, B. J., Mandell, A. J., Nonlinear dynamics of the heartbeat. Physica D, 1985,{\bf17,} 207-214. · Zbl 0583.92006
[35] Engelbrecht, J., von Hertzen, R. and Kongas, O., Driven nonlinear oscillators for modeling cardiac phenomena. Paper presented at Proc. IUTAM Symp. Chaos 97, Applications of Nonlinear and Chaotic Dynamics in Mechanics. Cornell University, USA, to be published. · Zbl 0963.92018
[36] Parlitz, U., Lauterborn, W., Period-doubling cascades and devils staircases of the driven van der Pol oscillator. Phys. Rev. A, 1987,{\bf36,} 1428-1434.
[37] Ketoja, J. A., Universal criterion for the breakup of invariant tori in dissipative systems. Phys.Rev. Lett, 1992,{\bf69,} 2180-2183. · Zbl 0968.37504
[38] Thompson, J. M. T. and Stewart, H. B., Nonlinear Dynamics and Chaos. John Wiley and Sons, New York, 1988. · Zbl 1174.37300
[39] Glass, L., Perez, R., Fine structure of phase locking. Phys. Rev. Lett, 1982,{\bf48,} 1772-1775.
[40] Hassard, B. D., Kazarinoff, N. D. and Wan, Y.-H., Theory and Applications of Hopf Bifurcation. London Math. Soc. Lecture Note Series No. 41. Cambridge Univ. Press, Cambridge, 1981. · Zbl 0474.34002
[41] Beiersdorfer, P., Universality of the topology of period doubling dynamical systems. Phys. Lett. A, 1984,{\bf100,} 379-382.
[42] Lewis, T. J., Guevara, M. R., Chaotic dynamics in an ionic model of the propagated cardiac action potential. J. Theor. Biol, 1990,{\bf146,} 407-432.
[43] Brown, D., Foweraker, J. P. A., Marrs, R. W., Dynamic equilibria and oscillations of a periodically stimulated excitable system. Chaos, Solitons and Fractals, 1995,{\bf5,} 359-369. · Zbl 0925.92040
[44] Landau, M., Lorente, P., Periodic behaviour in automatic and non-automatic cardiac cells. Chaos, Solitons and Fractals, 1995,{\bf5,} 347-357. · Zbl 0925.92045
[45] Guevara, M. R., Shrier, A. and Glass, L., Chaotic and complex cardiac rhythms. In CardiacElectrophysiology: From Cell to Bedside, ed. D. P. Zipes and J. Jalife. Saunders, Philadelphia, 1990.
[46] Guevara, M. R., Alonso, F., Jeandupeux, D. and van Ginneken, A. C. G., Alternans in isolated ventricular myocytes. In Cell to Cell Signalling: From Experiments to Theoretical Models, ed. A. Goldbeter. Academic Press, London, 1989, pp. 551-563.
[47] Hescheler, J., Speicher, R., Regular and chaotic behaviour of cardiac cells stimulated at frequencies between 2 and 20 Hz. Eur. Biophys. J, 1989,{\bf17,} 273-280.
[48] Matsumoto, G., Takahashi, N. and Hanyu, Y., Chaos, phase locking and bifurcation in normal squid axons. In Chaos in Biological Systems, ed. H. Degn, A. V. Holden and L. F. Olsen. Plenum, New York, 1987, pp. 143-156.
[49] Vinet, A., Chialvo, D. R., Jalife, J., Irregular dynamics of excitation in biologic and mathematical models of cardiac cells. Ann. N.Y. Acad. Sci, 1990,{\bf601,} 281-298.
[50] Glass, L., Hunter, P. J. and McCulloch, A., (editors), Theory of Heart: Biomechanics, Biophysics, and Nonlinear Dynamics of Cardiac Function. Springer-Verlag, New York, 1991.
[51] Jensen, M. H., Bak, P., Bohr, T., Transition to chaos by interaction of resonances in dissipative systems. I. Circle maps. Phys. Rev. A, 1984,{\bf30,} 1960-1969.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.