Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Length dependence of solutions of FitzHugh-Nagumo equations


Author: Clyde Collins
Journal: Trans. Amer. Math. Soc. 280 (1983), 809-832
MSC: Primary 35K57; Secondary 35B99, 92A09
MathSciNet review: 716852
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the behavior of the solutions of the problem

\begin{displaymath}\begin{array}{*{20}{c}} {{u_t} = {u_{xx}} - \alpha u - \upsil... ... h(t),} & {{u_x}(L,t) = {\upsilon _x}(L,t) = 0} \\ \end{array} \end{displaymath}

where $ t \geqslant 0$ and $ 0 < x < L \leqslant \infty $.

Solutions of the above equations are considered a qualitative model of conduction of nerve axon impulses. Using explicit constructions and semigroup methods, we obtain decay results on the norms of differences between the solution for $ L$ infinite and the solutions when $ L$ is large but finite. We conclude that nerve impulses for long finite nerves become uniformly close to those of the semi-infinite nerves away from the right endpoint of the finite nerve.


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

  • [1] Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR 0069338 (16,1022b)
  • [2] C. Collins, Length dependence of solutions of generalized FitzHugh-Nagumo equations, Ph.D. Thesis, Indiana Univ., Bloomington, 1981.
  • [3] R. FitzHugh, Mathematical models of excitation and propagation in nerve, Biological Engineering (H. Schwan, ed.), McGraw-Hill, New York, 1969, pp. 1-85.
  • [4] Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088 (56 #3433)
  • [5] S. P. Hastings, Some mathematical problems from neurobiology, Amer. Math. Monthly 82 (1975), no. 9, 881–895. MR 0381744 (52 #2633)
  • [6] Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244 (83j:35084)
  • [7] Fritz John, Partial differential equations, 3rd ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York-Berlin, 1978. MR 514404 (80f:35001)
  • [8] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1980.
  • [9] R. Keynes, Ion channels in the nerve-cell membrane, Sci. Amer. 240 (1979), 126-135.
  • [10] S. Ochs, Elements of neurobiology, Wiley, New York, 1965.
  • [11] Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861 (36 #2935)
  • [12] Jeffrey Rauch and Joel Smoller, Qualitative theory of the FitzHugh-Nagumo equations, Advances in Math. 27 (1978), no. 1, 12–44. MR 0487094 (58 #6759)
  • [13] M. Schonbek, Technical Reports 1739 and 1740, Math. Res. Center, Madison, Wis., 1977.
  • [14] Kôsaku Yosida, Functional analysis, 5th ed., Springer-Verlag, Berlin-New York, 1978. Grundlehren der Mathematischen Wissenschaften, Band 123. MR 0500055 (58 #17765)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35K57, 35B99, 92A09

Retrieve articles in all journals with MSC: 35K57, 35B99, 92A09


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0716852-6
PII: S 0002-9947(1983)0716852-6
Article copyright: © Copyright 1983 American Mathematical Society