The Hodgkin–Huxley Equations

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


All living cells have an electrical voltage, or potential difference, between their inside and outside. Since the cell’s membrane is what separates the inside from the outside, this potential difference is referred to as the membrane potential. In mathematical terms, the membrane potential V M is defined as
$${V }_{\mathrm{M}} = {V }_{\mathrm{in}} - {V }_{\mathrm{out}},$$
where V in is the potential on the inside of the cell and V out is the potential on the outside. This will change during an action potential, for example.


Membrane Potential Planck Equation Nernst Equation Cable Equation Squid Axon 
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  1. 17.
    L. J. Borg-Graham. Modeling the nonlinear conductances of excitable membranes. In H. Wheal and J. Chad, editors, Cellular and Molecular Neurobiology: A Practical Approach, pages 247–275. Oxford University Press, Oxford, 1991.Google Scholar
  2. 54.
    P. Dayan and L. F. Abbott. Theoretical Neuroscience. MIT, Cambridge, MA; London, England, 2001.zbMATHGoogle Scholar
  3. 82.
    A. A. Faisal, L. P. Selen, and D. M. Wolpert. Noise in the nervous system. Nat. Rev. Neurosci., 9:292–303, 2008.CrossRefGoogle Scholar
  4. 120.
    F. Helmchen, K. Imoto, and B. Sakmann. Ca2 + buffering and action potential-evoked Ca2 + signaling in dendrites of pyramidal neurons. Biophys. J., 70:1069–1081, 1996.CrossRefGoogle Scholar
  5. 122.
    B. Hille. Ion Channels of Excitable Membranes. Sinauer, Sunderland, MA, second edition, 2001.Google Scholar
  6. 134.
    E. M. Izhikevich. Simple model of spiking neurons. IEEE Trans Neural Netw., 14:1569–1572, 2003.CrossRefGoogle Scholar
  7. 135.
    E. M. Izhikevich. Dynamical Systems in Neuroscience. MIT, Cambridge, MA, 2007.Google Scholar
  8. 137.
    J. J. B. Jack, D. Noble, and R. W. Tsien. Electrical Current Flow in Excitable Cells. Clarendon Press, Oxford, 1975.Google Scholar
  9. 142.
    D. Johnston, D. A. Hoffman, C. M. Colbert, and J. C. Magee. Regulation of back-propagating action potentials in hippocampal neurons. Curr. Opin. Neurobiol., 9:288–292, 1999.CrossRefGoogle Scholar
  10. 146.
    J. P. Keener. Proagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math., 47:556–572, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 154.
    P. E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations, volume 23 of Applications of Mathematics (New York). Springer, Berlin, 1992.zbMATHGoogle Scholar
  12. 188.
    J. Magee, D. Hoffman, C. Colbert, and D. Johnston. Electrical and calcium signaling in dendrites of hippocampal pyramidal neurons. Annu. Rev. Physiol., 60:327–346, 1998.CrossRefGoogle Scholar
  13. 224.
    A. D. Reyes. Synchrony-dependent propagation of firing rate in iteratively constructed networks in vitro. Nat. Neurosci., 6:593–599, 2003.CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

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

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