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

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References | Similar Articles | Additional Information

**[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*, 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) 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]**J. D. Cole,*Perturbation methods in applied mathematics*, Blaisdell, Waltham, Mass., 1968 MR**0246537****[16]**S. P. Hastings,*On a third order differential equation from biology*, informal report, State Univ. of N. Y., Buffalo, N. Y., 1972 MR**0324145****[17]**J. M. Greenberg,*A note on the Nagumo equation*, informal report, State Univ. of N. Y., Buffalo, N. Y., 1972 MR**0402301****[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 Mathematics*4*, 209-223 (1970) MR**0260438****[20]**J. Rinzel and J. B. Keller,*Traveling wave solutions of a nerve conduction equation*(to be published)

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DOI:
https://doi.org/10.1090/qam/445095

Article copyright:
© Copyright 1975
American Mathematical Society