Abstract
We present a model of excitable media with the feature that it has a vulnerable phase during which a premature current stimulus will result in the formation of a reentrant selfsustained wave of excitation. The model exploits anisotropic coupling of identical cells, and is therefore useful as a model for the myocardium. We give rigorous verification that there is a vulnerable phase, and demonstrate numerically that permanently rotating waves are formed. Finally, it is shown that the direction of fastest propagation in myocardium is not necessarily the direction of highest safety factor, contrary to commonly accepted opinion.
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Chi, H., Bell, J., Hassard, B.: Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory, J. Math. Biol. 4, 583–607 (1986)
Clerc, L.: Directional differences of impulse spread in trabecular muscle from mammalian heart. J. Physiol. 255, 335–346 (1976)
Cranefield, P. F.: The conduction of the cardiac impulse. Mt. Kisco, N.Y.: Futura 1975
Diaz, P. J., Rudy, Y., Plonsey, R.: Intercalated discs as a cause for discontinuous propagation in cardiac muscle: a theoretical simulation. Ann. Biomed. Eng. 11, 177–189 (1983)
Dillon, S., Ursell, P. C., Wit, A. L.: Pseudo-block caused by anisotropic conduction: a new mechanism for sustained reentry. Circ. Suppl. 72, III-279 (1985)
FitzHugh, R.: Impulse and physiological states in models of nerve membrane. Biophys. J. 1, 445–466 (1961)
Han, J., Garcia de Jalon, P. D., Moe, G. K.: Fibrillation threshold of premature ventricular responses. Circ. Res. 43, 18–25 (1966)
Joyner, R. W.: Effects of the discrete pattern of electrical coupling on propagation through an electrical synctium. Circ. Res. 50, 192–200 (1982)
Keener, J. P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47, 556–572 (1987)
Keener, J. P.: Propagation and its failure in the discrete Nagumo equation. In: Sleeman, B. D., Jarvis, R. J. (eds.) Ordinary and partial differential equations. Pitman Res. Notes Math., vol. 157, New York: Longman 1987.
Krinskii, V. I.: Spread of excitation in an inhomogeneous medium (state similar to cardiac fibrillation). Biofizika 11, 676–683 (1966)
Miller, R., Rinzel, J.: The dependence of impulse propagation speed on firing frequency, dispersion for the Hodgkin-Huxley models. Biophys. J. 34, 227–259 (1981)
Moe, G. K., Rheinholdt, W. C., Abildshov, J. A.: A computer model of atrial fibrillation. Am. Heart J. 67, 200–220 (1964)
Rinzel, J., Keller, J. B.: Traveling wave solutions of a nerve conduction equation. Biophys. J. 13, 1313–1337 (1973)
Smith, J. M., Cohen, R. J.: Simple finite-element model accounts for wide range of cardiac dysrhythmias. Proc. Natl. Acad. Sci. USA 81, 233–237 (1984)
Spach, M. S., Kootsey, J. M.: The nature of electrical propagation in cardiac muscle. Am. J. Physiol. 244, H3-H22, (1983)
Spach, M. S., Miller, W. T., Geselowitz, D. B., Barr, R. C., Kootsey, J. M., Johnson, E. A.: The discontinuous nature of propagation in normal canine cardiac muscle. Circ. Res. 48, 39–54 (1981)
Van Cappelle, F. J. L., Durrer, D.: Computer simulation of arrhythmias in a network of coupled excitable elements. Circ. Res. 47, 454–466 (1980)
Winfree, A. T.: Sudden cardiac death, a problem in topology. Sci. Am. 248, 114–161 (1983)
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Keener, J.P. On the formation of circulating patterns of excitation in anisotropic excitable media. J. Math. Biology 26, 41–56 (1988). https://doi.org/10.1007/BF00280171
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DOI: https://doi.org/10.1007/BF00280171