The Hodgkin–Huxley Theory of Neuronal Excitation

  • Shinji Doi
  • Junko Inoue
  • Zhenxing Pan
Part of the A First Course in “In Silico Medicine” book series (FCISM, volume 2)


Hodgkin and Huxley (1952) proposed the famous Hodgkin–Huxley (hereinafter referred to as HH) equations which quantitatively describe the generation of action potential of squid giant axon, although there are still arguments against it (Connor et al. 1977; Strassberg and DeFelice 1993; Rush and Rinzel 1995; Clay 1998). The HH equations are important not only in that it is one of the most successful mathematical model in quantitatively describing biological phenomena but also in that the method (the HH formalism or the HH theory) used in deriving the model of a squid is directly applicable to many kinds of neurons and other excitable cells. The equations derived following this HH formalism are called the HH-type equations.


Periodic Solution Hopf Bifurcation Phase Plane Repetitive Firing Squid Giant Axon 
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  1. Abbott LF, Kepler TB (1990) Model neurons: from Hodgkin–Huxley to Hopfield. In: Garrido L (ed) Statistical mechanics of neural networks. Springer, BerlinGoogle Scholar
  2. Adams P (1982) Voltage-dependent conductances of vertebrate neurones. Trends Neurosci 5:116–119CrossRefGoogle Scholar
  3. Bedrov YA, Akoev GN, Dick OE (1992) Partition of the Hodgkin–Huxley type model parameter space into the regions of qualitatively different solutions. Biol Cybern 66:413–418MATHCrossRefGoogle Scholar
  4. Carpenter GA (1977) A geometric approach to singular perturbation problems with applications to nerve impulse equations. J Diff Eqns 23:335–367MathSciNetMATHCrossRefGoogle Scholar
  5. Clay JR (1998) Excitability of the squid giant axon revisited. J Neurophysiol 80:903–913Google Scholar
  6. Connor JA, Walter D, McKown R (1977) Neural repetitive firing: modifications of the Hodgkin–Huxley axon suggested by experimental results from crustacean axons. Biophys J 18:81–102CrossRefGoogle Scholar
  7. Crill WE, Schwindt PC (1983) Active currents in mammalian central neurons. Trends Neurosci 6:236–240CrossRefGoogle Scholar
  8. Doedel E, Wang X, Fairgrieve T (1995) AUTO94 – software for continuation and bifurcation problems in ordinary differential equations. CRPC-95-2, California Institute of TechnologyGoogle Scholar
  9. FitzHugh R (1960) Thresholds and plateaus in the Hodgkin–Huxley nerve equations. J Gen Physiol 43:867–896CrossRefGoogle Scholar
  10. FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophy J 1:445–466CrossRefGoogle Scholar
  11. Fukai H, Doi S, Nomura T, Sato S (2000a) Hopf bifurcations in multiple parameter space of the Hodgkin–Huxley equations. I. Global organization of bistable periodic solutions. Biol Cybern 82:215–222MATHGoogle Scholar
  12. Golomb D, Guckenheimer J, Gueron S (1993) Reduction of a channel-based model for a stomatogastric ganglion LP neuron. Biol Cybern 69:129–137MATHCrossRefGoogle Scholar
  13. Guckenheimer J, Labouriau IS (1993) Bifurcation of the Hodgkin and Huxley equations: a new twist. Bull Math Biol 55:937–952MATHGoogle Scholar
  14. Guttman R, Lewis S, Rinzel J (1980) Control of repetitive firing in squid axon membrane as model for neuroneoscillator. J Physiol 305:377–395Google Scholar
  15. Hassard B (1978) Bifurcation of periodic solutions of the Hodgkin–Huxley model for the squid giant axon. J Theor Biol 71:401–420MathSciNetCrossRefGoogle Scholar
  16. Hassard BD, Shiau LJ (1989) Isolated periodic solutions of the Hodgkin–Huxley equations. J Theor Biol 136:267–280MathSciNetCrossRefGoogle Scholar
  17. Hille B (1992) Ionic channels of excitable membranes, 2nd edn. Sinauer, Sunderland, MAGoogle Scholar
  18. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its applications to conduction and excitation in nerve. J Physiol 117:500–544Google Scholar
  19. Horikawa Y (1994) Period-doubling bifurcations and chaos in the decremental propagation of a spike train in excitable media. Phys Rev E 50:1708–1710CrossRefGoogle Scholar
  20. Kepler TB, Marder E (1993) Spike initiation and propagation on axons with slow inward currents. Biol Cybern 68:209–214CrossRefGoogle Scholar
  21. Kepler TB, Abbott LF, Marder E (1992) Reduction of conductance-based neuron models. Biol Cybern 66:381–387MATHCrossRefGoogle Scholar
  22. Kokoz YuM, Krinskii VI (1973) Analysis of the equations of excitable membranes. II. Method of analysing the electrophysiological characteristics of the Hodgkin–Huxley membrane from the graphs of the zero-isoclines of a second order system. Biofizika 18:878–885Google Scholar
  23. Krinskii VI, Kokoz YuM (1973) Analysis of the equations of excitable membranes. I. Reduction of the Hodgkin–Huxley equations to a second order system. Biofizika 18:506–511Google Scholar
  24. Labouriau IS (1985) Degenerate Hopf bifurcation and nerve impulse. SIAM J Math Anal 16:1121–1133MathSciNetMATHCrossRefGoogle Scholar
  25. Labouriau IS, Ruas MAS (1996) Singularities of equations of Hodgkin–Huxley type. Dyn Stab Syst 11:91–108MathSciNetMATHCrossRefGoogle Scholar
  26. Llinas RR (1988) The intrinsic electrophysiological properties of mammalian neurons: insights into central nervous system function. Science 242:1654–1664CrossRefGoogle Scholar
  27. Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line stimulating nerve axon. Proc Inst Radio Eng 50:2061–2070Google Scholar
  28. Plant RE (1976) The geometry of the Hodgkin–Huxley model. Comp Prog Biomed 6:85–91CrossRefGoogle Scholar
  29. Poznanski RR (1998) Electrophysiology of a leaky cable model for coupled neurons. J Austral Math Soc B 40:59–71MathSciNetMATHCrossRefGoogle Scholar
  30. Rinzel J (1978) On repetitive activity in nerve. Fed Proc 37:2793–2802Google Scholar
  31. Rinzel J (1985) Excitation dynamics: insights from simplified membrane models. Fed Proc 44:2944–2946Google Scholar
  32. Rinzel J, Keener JP (1983) Hopf bifurcation to repetitive activity in nerve. SIAM J Appl Math 43:907–922MathSciNetMATHCrossRefGoogle Scholar
  33. Rinzel J, Miller RN (1980) Numerical calculation of stable and unstable periodic solutions to the Hodgkin–Huxley equations. Math Biosci 49:27–59MathSciNetMATHCrossRefGoogle Scholar
  34. Rush ME, Rinzel J (1995) The potassium A-current, low firing rates and rebound excitation in Hodgkin–Huxley models. Bull Math Biol 57:899–929MATHGoogle Scholar
  35. Shiau LJ, Hassard BD (1991) Degenerate Hopf bifurcation and isolated periodic solutions of the Hodgkin–Huxley model with varying sodium ion concentration. J Theor Biol 148:157–173CrossRefGoogle Scholar
  36. Strassberg AF, DeFelice LJ (1993) Limitations of the Hodgkin–Huxley formalism: effects of single channel kinetics upon transmembrane voltage dynamics. Neural Comput 5:843–855CrossRefGoogle Scholar
  37. Troy WC (1978) The bifurcation of periodic solutions in the Hodgkin–Huxley equations. Q Appl Math 36:73–83MathSciNetMATHGoogle Scholar
  38. Yanagida E (1985) Stability of fast traveling pulse solutions of the FitzHugh–Nagumo equations. J Math Biol 22:81–104MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  • Shinji Doi
    • 1
  • Junko Inoue
    • 2
  • Zhenxing Pan
    • 3
  1. 1.Graduate School of EngineeringKyoto UniversityKyoto-Daigaku Katsura, Nishikyo-kuJapan
  2. 2.Faculty of Human ScienceKyoto Koka Women’s UniversityJapan
  3. 3.Graduate School of EngineeringOsaka UniversityJapan

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