The bifurcation of periodic solutions in the Hodgkin-Huxley equations

Troy, William C.

doi:10.1090/qam/472116

Author: William C. Troy
Journal: Quart. Appl. Math. 36 (1978), 73-83
MSC: Primary 92A05; Secondary 35K55
DOI: https://doi.org/10.1090/qam/472116
MathSciNet review: 472116
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Abstract: We consider the current clamped version of the Hodgkin-Huxley nerve conduction equations. Under appropriate assumptions on the functions and parameters we show that there are two critical values of $I$, the current parameter, at which a Hopf bifurcation of periodic orbits occurs.

K. S. Cole, H. A. Antosiewicz, and P. Rabinowitz, Automatic computation of nerve excitation, J. Soc. Indust. Appl. Math. 3 (1955), 153–172. MR 74139

Cooley, Dodge and Cohen, J Cellular Comp. Phys. 66, 99–110 (1965)

R. FitzHugh, Thresholds and plateaus in the Hodgkin-Huxley nerve equations, J. Gen. Physiology 43, 867–896 (1960)

R. Fitzhugh and H. A. Antosiewicz, Automatic computation of nerve excitation-detailed corrections and additions, J. Soc. Indust. Appl. Math. 7 (1959), 447–458. MR 130012
S. P. Hastings, On travelling wave solutions of the Hodgkin-Huxley equations, Arch. Rational Mech. Anal. 60 (1975/76), no. 3, 229–257. MR 402302, DOI https://doi.org/10.1007/BF01789258

O. Hauswirth, D. Noble, and R. W. Tsien, The mechanism of oscillatory activity at low membrane potentials in cardiac purkinje fibres, J. Physiol. 200, 255–265 (1969)

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J, Physiol. 117, 500–544 (1952)

Eberhard Hopf, Abzweigung einer periodischen Lösung von einer stationären eines Differentialsystems, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Nat. Kl. 95 (1943), no. 1, 3–22 (German). MR 39141
In Ding Hsü and N. D. Kazarinoff, An applicable Hopf bifurcation formula and instability of small periodic solutions of the Field-Noyes model, J. Math. Anal. Appl. 55 (1976), no. 1, 61–89. MR 466758, DOI https://doi.org/10.1016/0022-247X%2876%2990278-X

R. E. McAllister, D. Noble, and R. W. Tsien, Reconstruction of the electrical activity of cardiac purkinje fibres, J. Phys. 251, 1–59 (1975)

N. H. Sabah and R. A. Spangler, Repetitive response of the Hodgkin-Huxley model for the squid giant axon, J. Theor. Biol. 29, 155–171 (1970)

W. C. Troy, doctoral dissertation, SUNY Buffalo, 1974

William C. Troy, Oscillation phenomena in the Hodgkin-Huxley equations, Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 299–310 (1976). MR 442444, DOI https://doi.org/10.1017/s0308210500016735

J. Uspensky, Theory of equations, New York, 1948