Advertisement

Propagating Action Potentials

  • G. Bard Ermentrout
  • David H. Terman
Chapter
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 35)

Abstract

Neurons need to communicate over long distances. This is accomplished by electrical signals, or action potentials, that propagate along the axon. We have seen that linear cables cannot transmit information very far; neural signals are able to reach long distances because there exist voltage-gated channels in the cell membrane. The combination of ions diffusing along the axon together with the nonlinear flow of ions across the membrane allows for the existence of an action potential that propagates along the axon with a constant shape and velocity.

Keywords

Periodic Orbit Wave Speed Travel Wave Solution Unstable Manifold Homoclinic Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 6.
    D. G. Aronson and H. F. Weinberger. Nonlinear diffusion in population genetics, combustion and nerve pulse propagation. In J. Goldstein, editor, Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, pages 5–49. Springer, New York, 1975.CrossRefGoogle Scholar
  2. 34.
    G. Carpenter. A geometric approach to singular perturbation problems with applications to nerve impulse equations. J. Differ. Equat., 23:335–367, 1977.MathSciNetMATHCrossRefGoogle Scholar
  3. 35.
    A. Carpio, S. J. Chapman, S. Hastings, and J. B. McLeod. Wave solutions for a discrete reaction-diffusion equation. Eur. J. Appl. Math., 11(4):399–412, 2000.MathSciNetMATHCrossRefGoogle Scholar
  4. 44.
    C. Conley. Isolated invariant Sets and the Morse Index. CBMS Lecture Notes in Math, volume 38. AMS Press, Providence, RI, 1978.Google Scholar
  5. 48.
    S. Coombes and M. R. Owen. Evans functions for integral neural field equations with heaviside firing rate function. SIAM J. Appl. Dyn. Syst., 3:574–600, 2004.MathSciNetMATHCrossRefGoogle Scholar
  6. 54.
    P. Dayan and L. F. Abbott. Theoretical Neuroscience. MIT, Cambridge, MA; London, England, 2001.MATHGoogle Scholar
  7. 75.
    G. B. Ermentrout and J. Rinzel. Waves in a simple, excitable or oscillatory, reaction-diffusion model. J. Math. Biol., 11(3):269–294, 1981.MathSciNetMATHCrossRefGoogle Scholar
  8. 79.
    G. B. Ermentrout, R. F. Galán, and N. N. Urban. Reliability, synchrony and noise. Trends Neurosci., 31:428–434, 2008.CrossRefGoogle Scholar
  9. 80.
    J. W. Evans. Nerve axon equations. iv. Indiana Univ. Math. J., 24:1169–1190, 1975.MATHCrossRefGoogle Scholar
  10. 85.
    O. Feinerman, M. Segal, and E. Moses. Signal propagation along unidimensional neuronal networks. J. Neurophysiol., 94:3406–3416, 2005.CrossRefGoogle Scholar
  11. 86.
    N. Fenichel. Geometric singular perturbation theory. J. Diff. Equat., 31:53–91, 1979.MathSciNetMATHCrossRefGoogle Scholar
  12. 87.
    J. A. Feroe. Existence and stability of multiple impulse solutions of a nerve axon equation. SIAM J. Appl. Math., 42:235–246, 1982.MathSciNetMATHCrossRefGoogle Scholar
  13. 116.
    B. J. Hall and K. R. Delaney. Contribution of a calcium-activated non-specific conductance to NMDA receptor-mediated synaptic potentials in granule cells of the frog olfactory bulb. J. Physiol. (Lond.), 543:819–834, 2002.CrossRefGoogle Scholar
  14. 117.
    F. Han, N. Caporale, and Y. Dan. Reverberation of recent visual experience in spontaneous cortical waves. Neuron, 60:321–327, 2008.CrossRefGoogle Scholar
  15. 139.
    D. Johnston and S. M. Wu. Foundations of Cellular Neurophysiology. MIT, Cambridge, MA, 1995.Google Scholar
  16. 140.
    D. Johnston and S. Wu. Foundations of Cellular Neurophysiology. MIT, Cambridge, MA, 1999.Google Scholar
  17. 141.
    D. Johnston, J. C. Magee, C. M. Colbert, and B. R. Cristie. Active properties of neuronal dendrites. Annu. Rev. Neurosci., 19:165–186, 1996.CrossRefGoogle Scholar
  18. 144.
    E. Kandel, J. Schwartz, and T. Jessell. Principles of Neural Science. Appleton & Lange, Norwalk, CT, 1991.Google Scholar
  19. 146.
    J. P. Keener. Proagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math., 47:556–572, 1987.MathSciNetMATHCrossRefGoogle Scholar
  20. 184.
    Y. Loewenstein and H. Sompolinsky. Temporal integration by calcium dynamics in a model neuron. Nat. Neurosci., 6:961–967, 2003.CrossRefGoogle Scholar
  21. 185.
    M. London and M. Hausser. Dendritic computation. Annu. Rev. Neurosci., 28:503–532, 2005.CrossRefGoogle Scholar
  22. 186.
    G. Maccaferri and C. J. McBain. The hyperpolarization-activated current (Ih) and its contribution to pacemaker activity in rat CA1 hippocampal stratum oriens-alveus interneurones. J. Physiol. (Lond.), 497( Pt 1):119–130, 1996.Google Scholar
  23. 197.
    W. S. McCulloch and W. Pitts. The statistical organization of nervous activity. Biometrics, 4:91–99, 1948.CrossRefGoogle Scholar
  24. 198.
    G. Medvedev. Reduction of a model of an excitable cell to a one-dimensional map. Phys. D, 202:37–59, 2005.MathSciNetMATHCrossRefGoogle Scholar
  25. 226.
    J. Rinzel. On repetitive activity in nerve. Fed. Proc., 37:2793–2802, 1978.Google Scholar
  26. 288.
    H. R. Wilson and J. D. Cowan. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik, 13:55–80, 1973.MATHCrossRefGoogle Scholar
  27. 289.
    H. R. Wilson, R. Blake, and S. H. Lee. Dynamics of travelling waves in visual perception. Nature, 412:907–910, 2001.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. MathematicsUniversity of PittsburghPittsburghUSA
  2. 2.Dept. MathematicsOhio State UniversityColumbusUSA

Personalised recommendations