Abstract
Usually the models for the excitation and propagation of the nervous impulse are studied either in the space-clamp situation or on a model axon extended on both sides to infinity. Following Fitzhugh in the present paper the release of an impulse train at the axon hillock is studied within the scope of Fitzhugh's BVP model. The existence and stability of periodic oscillations are studied by direct methods, also the relation to Liénard's equation. The exact correspondence between the BVP model and the socalled Nagumo-equation is established. For typical examples the solutions are computed by numerical methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Evans, J.W.: Nerve axon equation I–IV, Indiana Univ. Math. J. 21, 877–885, 22, 75–90, 22, 577–593 (1972) & to appear
Fitzhugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)
Friedrichs, K.O.: Advanced ordinary differential equations. New York: Gordon & Breach 1965
Hastings, S.P.: The existence of periodic solutions to Nagumo's equation. (to appear)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.) 117, 500 (1952)
Hopf, E.: Abzweigung einer periodischen Lösung eines Differential-system. Ber. Math.-Phys. Klasse der Sächs. Akad. (Leipzig) XCIV, 1–22 (1942)
Hsü, I., Kazarinoff, N.D.: An applicable Hopf bifurcation formula and instability of small periodic solutions of the Field-Noyes model. J. Math. Anal. Appl. (to appear)
Katz, B.: Nerve, muscle and synapse. New York: McGraw-Hill 1966
Keller, J.B., Rinzel, J.: Traveling wave solutions of a nerve conduction equation. Biophys. J. 13, 1313–1336 (1973)
Kright, B.W.: Some questions concerning the encoding dynamics of neuron populations. IV. International Biophysics Congress, Academy of Sciences of the USSR, Pushchino 1973
Knobloch, H.W., Kappel, F.: Gewöhnliche Differentialgleichungen. Stuttgart: Teubner, 1974
McKean, H.P.: Nagumo's equation, Advan. Math. 4, 209–223 (1970)
Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE, 2061–2071, October 1962
Poore, A.B.: On the theory and applications of the Hopf-Friedrichs bifurcation theory. Arch. Rat. Mech. Analys. (to appear)
Troy, W.C.: Oscillation phenomena in the Hodgkin-Huxley equations, University of Pittsburgh, preprint 1975
Knobloch, H.W.: Hopf bifurcation via integral manifolds, preprint No 9, Mathem. Institute Univ. Würzburg 1975
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hadeler, K.P., an der Heiden, U. & Schumacher, K. Generation of the nervous impulse and periodic oscillations. Biol. Cybernetics 23, 211–218 (1976). https://doi.org/10.1007/BF00340337
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00340337