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 Free Access
References |
Similar Articles |
Additional Information
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)
A. F. Huxley, Can a nerve propagate a sub-threshold disturbance? Proc. Phys. Soc. 10–11, July 1959, J. Physiol. 148: 80–81P
J. Cooley and F. Dodge, Digital computer solutions for excitation and propagation of the nerve impulse, Biophysics J. 6, 583 (1966)
J. Evans and N. Shenk, Solutions to axon equations, Biophysics J. 10, 1090–1101 (1970)
R. Fitz Hugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysics J. 1, 445–466 (1961)
J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve, axon, Proc. IRE 50 ,2061–2070 (1972)
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.
F. A. Dodge and J. Cooley, Action potential of the motor neuron, to appear in IBM J. Res. Devel.
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)
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)
R. Fitz Hugh, Computation of impulse initiation and saltatory conduction in a myelinated nerve fiber, Biophys. J. 2, 11–21 (1962)
W. L. Hardy, Propagation speed in myelinated nerve. Dependence on external sodium, Ph.D. dissertation, U. of Washington, Seattle, 1969
- 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
A. A. Dorodnitsyn, Asymptotic solution of the Van der Pol equations, Prik. Mat. i Mekh 11, 313–328 (1970) (in Russian)
- Julian D. Cole, Perturbation methods in applied mathematics, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1968. MR 0246537
- S. P. Hastings, On a third order differential equation from biology, Quart. J. Math. Oxford Ser. (2) 23 (1972), 435–448. MR 324145, DOI https://doi.org/10.1093/qmath/23.4.435
- J. M. Greenberg, A note on the Nagumo equation, Quart. J. Math. Oxford Ser. (2) 24 (1973), 307–314. MR 402301, DOI https://doi.org/10.1093/qmath/24.1.307
C. Conley, On the existence of bounded progressive wave solutions of Nagumo equation (in preparation)
- H. P. McKean Jr., Nagumo’s equation, Advances in Math. 4 (1970), 209–223 (1970). MR 260438, DOI https://doi.org/10.1016/0001-8708%2870%2990023-X
J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation (to be published)
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)
A. F. Huxley, Can a nerve propagate a sub-threshold disturbance? Proc. Phys. Soc. 10–11, July 1959, J. Physiol. 148: 80–81P
J. Cooley and F. Dodge, Digital computer solutions for excitation and propagation of the nerve impulse, Biophysics J. 6, 583 (1966)
J. Evans and N. Shenk, Solutions to axon equations, Biophysics J. 10, 1090–1101 (1970)
R. Fitz Hugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysics J. 1, 445–466 (1961)
J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve, axon, Proc. IRE 50 ,2061–2070 (1972)
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.
F. A. Dodge and J. Cooley, Action potential of the motor neuron, to appear in IBM J. Res. Devel.
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)
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)
R. Fitz Hugh, Computation of impulse initiation and saltatory conduction in a myelinated nerve fiber, Biophys. J. 2, 11–21 (1962)
W. L. Hardy, Propagation speed in myelinated nerve. Dependence on external sodium, Ph.D. dissertation, U. of Washington, Seattle, 1969
G. F. Carrier, Boundary layer problems in applied mechanics, Adv. Appl. Mech., Academic Press, N. Y., 3, 1–19 (1953). Also see W. L. Miranker, Singular perturbation analyses of the differential equations of a tunnel diode circuit, Quant, of Appl. Math. 20, 279–299 (1962)
A. A. Dorodnitsyn, Asymptotic solution of the Van der Pol equations, Prik. Mat. i Mekh 11, 313–328 (1970) (in Russian)
J. D. Cole, Perturbation methods in applied mathematics, Blaisdell, Waltham, Mass., 1968
S. P. Hastings, On a third order differential equation from biology, informal report, State Univ. of N. Y., Buffalo, N. Y., 1972
J. M. Greenberg, A note on the Nagumo equation, informal report, State Univ. of N. Y., Buffalo, N. Y., 1972
C. Conley, On the existence of bounded progressive wave solutions of Nagumo equation (in preparation)
H. P. McKean, Jr., Nagumo’s equation, Advances in Mathematics 4, 209–223 (1970)
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
Article copyright:
© Copyright 1975
American Mathematical Society
