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Nerve impulse propagation in a branching nerve system: a simple model. (English) Zbl 1194.37181
Summary: Local spatial changes of nerve axon geometry such as diameter increase and branching, may cause that action potential waves approaching a region of geometric change fail to propagate beyond it. In this paper, this effect will be examined for a special kind of nonuniformity, within the framework of a simple model: an initial value problem for a single nonlinear diffusion equation on an unbounded domain.

37N25 Dynamical systems in biology
92B20 Neural networks for/in biological studies, artificial life and related topics
Full Text: DOI
[1] Aronson, D.G.; Weinberger, H.F., Nonlinear diffusion in population genetics, combustion and nerve pulse propagation, (), 5-49 · Zbl 0325.35050
[2] Aronson, D.G.; Weinberger, H.F., Multidimensional nonlinear diffusion arising in population genetics, Advan. math., 30, 33-76, (1978) · Zbl 0407.92014
[3] Fife, P.C.; McLeod, J.B., The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. rat. mech. anal., 65, 333-361, (1977) · Zbl 0361.35035
[4] Fife, P.C.; Peletier, L.A., Nonlinear diffusion in population genetics, Arch. rat. mech. anal., 64, 93-109, (1977) · Zbl 0361.92020
[5] Fife, P.C.; Peletier, L.A., Clines induced by variable migration, (), New York · Zbl 0437.92019
[6] Goldstein, S.S.; Rall, W., Changes of action potential shape and velocity for changing core conductor geometry, Biophys. J., 14, 731-757, (1974)
[7] Hadeler, K.P.; Rothe, F., Travelling fronts in nonlinear diffusion equations, J. math. biol., 2, 251-263, (1975) · Zbl 0343.92009
[8] Hastings, S.P., The existence of homoclinic and periodic orbits for the Fitzhugh-Nagumo equations, Quarterly J. math., 27, 2, 123-134, (1976) · Zbl 0322.92008
[9] Kryzaňski, M., Certaines inégalités relatives aux solutions de l’equation parabolique lineaire normale, Bull. acad. polon. sci., ser. sci. math. astrom. phys., Vol. VII, No. 3, (1959) · Zbl 0085.08402
[10] Nagylaki, T., Clines with variable migration, Genetics, 83, 867-886, (1976)
[11] Pauwelussen, J.P., Nerve impulse propagation in a branching nerve system: a simple model, Mathematical centre report TW 203/80, (1980), Amsterdam · Zbl 0429.92012
[12] J.P. Pauwelussen, One way traffic of pulses in a neuron, J. Math. Biology, Submitted. · Zbl 0497.92007
[13] Protter, M.H.; Weinberger, H.F., Maximum principles in differential equations, (1967), Prentice Hall Englewood Cliffs, New Jersey · Zbl 0153.13602
[14] Reed, M.; Simon, B., Analysis of operators, (1978), Academic Press New York, San Francisco, London, part IV
[15] Rinzel, J., Repetitive nerve impulse propagation: numerical results and methods, () · Zbl 0363.35020
[16] Titchmarsh, E.C., Eigenfunction expansions part I, (1962), Oxford Univ. Press London · Zbl 0099.05201
[17] Titchmarsh, E.C., Eigenfunction expansions, part II, (1958), Oxford Univ. Press London · Zbl 0097.27601
[18] E.J.M. Veling, Traveling waves in an initial-boundary value problem, Proc. Roy. Soc. Edinburgh, to appear. · Zbl 0422.35046
[19] 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) · Zbl 0431.65069
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