## Length dependence of solutions of FitzHugh-Nagumo equations

HTML articles powered by AMS MathViewer

- by Clyde Collins
- Trans. Amer. Math. Soc.
**280**(1983), 809-832 - DOI: https://doi.org/10.1090/S0002-9947-1983-0716852-6
- PDF | Request permission

## Abstract:

We investigate the behavior of the solutions of the problem \[ \begin {array}{*{20}{c}} {{u_t} = {u_{xx}} - \alpha u - \upsilon + {u^2}(1 + \alpha - u),} & {{\upsilon _t} = \eta {\upsilon _{xx}} + \sigma u - \gamma \upsilon ,} \\ {u(0,t) = g(t),\quad \upsilon (0,t) = h(t),} & {{u_x}(L,t) = {\upsilon _x}(L,t) = 0} \\ \end {array} \] 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

- Earl A. Coddington and Norman Levinson,
*Theory of ordinary differential equations*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR**0069338**
C. Collins, - Avner Friedman,
*Partial differential equations*, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR**0445088** - S. P. Hastings,
*Some mathematical problems from neurobiology*, Amer. Math. Monthly**82**(1975), no. 9, 881–895. MR**381744**, DOI 10.2307/2318490 - Daniel Henry,
*Geometric theory of semilinear parabolic equations*, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR**610244** - Fritz John,
*Partial differential equations*, 3rd ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York-Berlin, 1978. MR**514404**
T. Kato, - Murray H. Protter and Hans F. Weinberger,
*Maximum principles in differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR**0219861** - Jeffrey Rauch and Joel Smoller,
*Qualitative theory of the FitzHugh-Nagumo equations*, Advances in Math.**27**(1978), no. 1, 12–44. MR**487094**, DOI 10.1016/0001-8708(78)90075-0
M. Schonbek, Technical Reports 1739 and 1740, Math. Res. Center, Madison, Wis., 1977.
- Kôsaku Yosida,
*Functional analysis*, 5th ed., Grundlehren der Mathematischen Wissenschaften, Band 123, Springer-Verlag, Berlin-New York, 1978. MR**0500055**

*Length dependence of solutions of generalized FitzHugh-Nagumo equations*, Ph.D. Thesis, Indiana Univ., Bloomington, 1981. R. FitzHugh,

*Mathematical models of excitation and propagation in nerve*, Biological Engineering (H. Schwan, ed.), McGraw-Hill, New York, 1969, pp. 1-85.

*Perturbation theory for linear operators*, Springer-Verlag, New York, 1980. R. Keynes,

*Ion channels in the nerve-cell membrane*, Sci. Amer.

**240**(1979), 126-135. S. Ochs,

*Elements of neurobiology*, Wiley, New York, 1965.

## Bibliographic Information

- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**280**(1983), 809-832 - MSC: Primary 35K57; Secondary 35B99, 92A09
- DOI: https://doi.org/10.1090/S0002-9947-1983-0716852-6
- MathSciNet review: 716852