Skip to main content

Travelling Wave Solutions of a Simple Nerve Conduction Equation for Inhomogeneous Axons

  • Chapter
Nonlinear Wave Processes in Excitable Media

Part of the book series: NATO ASI Series ((NSSB,volume 244))

Abstract

Helmholtz’s measurement of impulse conduction velocity in frog sciatic nerve which he accomplished in 1850 marked the beginning of the exact, quantitative description of excitation phenomena in nerves (Fig. 1). Currently, nerve impulse conduction represents one of the simplest, and because of this, most fully studied nonlinear wave phenomena in excitable media.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Babitsch, W.M. et al. (1972). Lineare Differential- gleichungen der mathematischen Physik. Akademie- Verlag: Berlin.

    Google Scholar 

  2. Bernstein, J.L. (1900). Lehrbuch der Physiologie. Verlag F. Enke: Stuttgart.

    Google Scholar 

  3. Dodge, F.A. & Cooley, J.W. (1973). Action potential of the motoneuron. IBM J. Res. Develop. 17, 219–229.

    Article  Google Scholar 

  4. FitzHugh, R. (1969). Mathematical models of excitation and propagation in nerve. In Biological Engineering, pp. 1–85, Schwan, H.P. (ed.). McGraw-Hill: New York.

    Google Scholar 

  5. FitzHugh, R. (1973). Dimensional analysis of nerve models. J. Theor. Biol. 40, 517–541.

    Article  Google Scholar 

  6. Goldman, L. & Albus, J.S. (1968). Computation of impulse conduction in myelinated fibres: theoretical basis of the velocity-diameter relation. Biophys. J. 8, 596–607.

    Article  ADS  Google Scholar 

  7. Goldstein, S.S. & Rall, W. (1974). Changes of action potential shape and velocity for changing core conductor geometry. Biophys. J. 14, 731–757.

    Article  Google Scholar 

  8. Hodgkin, A.L. & Huxley, A.F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.) 117, 500–544.

    Google Scholar 

  9. Hunter, P.J., McNaughton, P.A. & Noble, D. (1975). Analytic models of propagation in excitable cells. Prog. Biophys. Violec. Biol. 30, 99–144.

    Article  Google Scholar 

  10. Kamke, E. (1967). Differentialgleichungen Bd. I. Geest & Portig: Leipzig.

    Google Scholar 

  11. Kelly, J.J. & Ghausi, M.S. (1965). Tapered distribution RC networks with similar immittances. IEEE Trans. Circuit Theory CT-12, 554–558.

    Google Scholar 

  12. Khodorov, B.I. & Timin, E.N. (1975). Nerve impulse propagation along non-uniform fibres (investigations using mathematical models). Prog. Biophys. Molec. Biol. 30, 145–184.

    Article  Google Scholar 

  13. Nagumo, J., Arimoto, S. & Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070.

    Article  Google Scholar 

  14. Parnas, I., Hochstein, S. & Parnas, H. (1976). Theoretical analysis of parameters leading to frequency modulation along an inhomogeneous axon. J. Neurophysiol. 39, 909–922.

    Google Scholar 

  15. Pawelussen, J.P. (1980). Heteroclinic waves of the Fitzhugh-Nagumo equations. Amsterdam, preprint.

    Google Scholar 

  16. Rinzel, J. & Keller, J.B. (1973). Traveling wave solutions of a nerve conduction equation. Biophys. J. 13, 1313–1337.

    Article  ADS  Google Scholar 

  17. Scott, A.C. (1975). The electrophysics of a nerve fiber. Rev. Mod. Phys. 47, 487–533.

    Article  ADS  Google Scholar 

  18. Schierwagen, A.K. A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree. J. Theor. Biol., in press.

    Google Scholar 

  19. Waxman, S.G. (ed.). Physiology and Pathobiology of Axons. Raven Press: New York.

    Google Scholar 

  20. Zykov, V.S. (1987). Simulation of Wave Processes in Excitable Media. Manchester University Press: Manchester.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer Science+Business Media New York

About this chapter

Cite this chapter

Schierwagen, A.K. (1991). Travelling Wave Solutions of a Simple Nerve Conduction Equation for Inhomogeneous Axons. In: Holden, A.V., Markus, M., Othmer, H.G. (eds) Nonlinear Wave Processes in Excitable Media. NATO ASI Series, vol 244. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-3683-7_11

Download citation

  • DOI: https://doi.org/10.1007/978-1-4899-3683-7_11

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4899-3685-1

  • Online ISBN: 978-1-4899-3683-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics