Abstract
In this paper we investigate the effect of a change in geometry of a nerve axon on the propagation of potential waves along the axon. In particular we show that potential waves are stopped at a sudden large increase of cross-section area such as increase of diameter or branching. Some special examples are treated. The results do also apply to problems in population genetics and chemical reaction theory.
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Chueh, K., Conley, C., Smoller, J.: Positively invariant regions for systems of nonlinear parabolic equations. Ind. Univ. Math. J. 26, 273–392 (1977)
Fife, P. C., Tyson, J. J.: Target patterns in a realistic model of the Belousov-Zhabotinskii reaction. J. Chem. Phys. 73(5) (1980)
FitzHugh, R.: Mathematical models of excitation and propagation in nerve. In: Biological engineering, Schwan, H. P. (ed.). New York: McGraw-Hill 1969
Goldstein, S. S., Rall, W.: Changes of action potential shape and velocity for changing core conductor geometry. Biophys. J. 14, 731–757 (1974)
Mitchell, A. R.: Numerical studies of travelling waves in nonlinear diffusion equations. Bull. Inst. Math. Applies. 17, 14–20 (1981)
Miura, R. M.: Accurate computation of the stable solitary wave for the FitzHugh-Nagumo equations. J. Math. Biol. 13, 247–269 (1982)
Nagylaki, T.: Clines with variable migration. Genetics 83, 867–886 (1976)
Pauwelussen, J. P.: Nerve impulse propagation in a branching nerve system: A simple model. Physica 4D, 67–88 (1981)
Pauwelussen, J. P.: Existence and uniqueness for a nonlinear diffusion problem arising in neurophysiology, Mathematical Centre Report, Amsterdam, 1981
Protter, M. H., Weinberger, H. F.: Maximum principles in differential equations. Englewood Cliffs, NJ: Prentice-Hall 1967
Rauch, J., Smoller, J.: Qualitative theory of the FitzHugh-Nagumo equations. Adv. in Math. 27, 12–44 (1978)
Rinzel, J.: Repetitive nerve impulse propagation: Numerical results and methods. In: Fitzgibbon, W. E., Walker, H. F. (eds.). Nonlinear Diffusion. London: Pitman 1977
Rinzel, J.: Models in Neurobiology, Preprint
Verwer, J. G.: An implementation of a class of stabilized explicit methods for the time integration of parabolic equations. ACM Trans. Math. Software 6(2), 188–205 (1980)
Miller, R. N.: A simple model of delay, block and one way conduction in Purkinje Fibers. J. Math. Biol. 7, 385–398 (1979)
Rinzel, J., Terman, D. H.: Propagation phenomena in a bistable reaction diffusion system. MRC Technical Summary Report 2225 (1981)
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Pauwelussen, J. One way traffic of pulses in a neuron. J. Math. Biol. 15, 151–171 (1982). https://doi.org/10.1007/BF00275071
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DOI: https://doi.org/10.1007/BF00275071