Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Perturbation analysis of an approximation to the Hodgkin-Huxley theory

Authors: Richard G. Casten, Hirsh Cohen and Paco A. Lagerstrom
Journal: Quart. Appl. Math. 32 (1975), 365-402
MSC: Primary 35B99; Secondary 92A05
DOI: https://doi.org/10.1090/qam/445095
MathSciNet review: 445095
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • [1] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol (London) 117, 500 (1952)
  • [2] A. F. Huxley, Can a nerve propagate a sub-threshold disturbance? Proc. Phys. Soc. 10-11, July 1959, J. Physiol. 148: 80-81P
  • [3] J. Cooley and F. Dodge, Digital computer solutions for excitation and propagation of the nerve impulse, Biophysics J. 6, 583 (1966)
  • [4] J. Evans and N. Shenk, Solutions to axon equations, Biophysics J. 10, 1090-1101 (1970)
  • [5] R. Fitz Hugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysics J. 1, 445-466 (1961)
  • [6] J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve, axon, Proc. IRE 50 ,2061-2070 (1972)
  • [7] B. I. Khodarov, Ye. N. Timin, S. Ya. Vilenkin, and F. B. Gul'ko, Theoretical analysis of the mechanisms of conduction of a nerve pulse over an inhomogeneous axon. I. Conduction through a portion with increased diameter, Biofizika 14, 304-315 (1969); II. Conduction of a single impulse across a region of the fibre with modified functional properties, Biofizika, 15, 140-146 (1970). References are to the English translation version.
  • [8] F. A. Dodge and J. Cooley, Action potential of the motor neuron, to appear in IBM J. Res. Devel.
  • [9] A. Mauro, A. Freeman, J. Cooley, and A. Cass, Propagated subthreshold oscillatory response and classical electrotonic response of squid giant axon, Biophysik 8, 118-132 (1972)
  • [10] N. H. Sabah and K. N. Leibovic, Subthreshold oscillatory responses of the Hodgkin-Huxley cable model for the giant squid axon, Biophys. J. 9, 1206 (1969)
  • [11] R. Fitz Hugh, Computation of impulse initiation and saltatory conduction in a myelinated nerve fiber, Biophys. J. 2, 11-21 (1962)
  • [12] W. L. Hardy, Propagation speed in myelinated nerve. Dependence on external sodium, Ph.D. dissertation, U. of Washington, Seattle, 1969
  • [13] G. F. Carrier, Boundary layer problems in applied mechanics, Advances in Applied Mechanics, vol. 3, Academic Press Inc., New York, N. Y., 1953, pp. 1–19. MR 0062315
  • [14] A. A. Dorodnitsyn, Asymptotic solution of the Van der Pol equations, Prik. Mat. i Mekh 11, 313-328 (1970) (in Russian)
  • [15] Julian D. Cole, Perturbation methods in applied mathematics, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1968. MR 0246537
  • [16] S. P. Hastings, On a third order differential equation from biology, Quart. J. Math. Oxford Ser. (2) 23 (1972), 435–448. MR 0324145, https://doi.org/10.1093/qmath/23.4.435
  • [17] J. M. Greenberg, A note on the Nagumo equation, Quart. J. Math. Oxford Ser. (2) 24 (1973), 307–314. MR 0402301, https://doi.org/10.1093/qmath/24.1.307
  • [18] C. Conley, On the existence of bounded progressive wave solutions of Nagumo equation (in preparation)
  • [19] H. P. McKean Jr., Nagumo’s equation, Advances in Math. 4 (1970), 209–223 (1970). MR 0260438, https://doi.org/10.1016/0001-8708(70)90023-X
  • [20] J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation (to be published)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35B99, 92A05

Retrieve articles in all journals with MSC: 35B99, 92A05

Additional Information

DOI: https://doi.org/10.1090/qam/445095
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society