Hodgkin–Huxley-Type Models of Cardiac Muscle Cells

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


Following the HH formalism introduced in Chap. 2, various kinds of HH-type models of neurons and other excitable cells are proposed (Canavier et al. 1991; Chay and Keizer 1983; Cronin 1987; Gerber and Jakobsson 1993; Hayashi and Ishizuka 1992; Keener and Sneyd 1998; Noble 1995; Rinzel 1990; Traub et al. 1991), and are analyzed (Alexander and Cai 1991; Av-Ron 1994; Bertram 1994; Bertram et al. 1995; Butera 1998; Canavier et al. 1993; Chay and Rinzel 1985; Doi and Kumagai 2005; Guckenheimer et al. 1993; Maeda et al. 1998; Rush and Rinzel 1994; Schweighofer et al. 1999; Terman 1991; Tsumoto et al. 2003, 2006; Yoshinaga et al. 1999). The HH-type equations include many variables depending on the number of different ionic currents and their gating variables considered in the equations, whereas the original HH equations possess only four variables (a membrane voltage, activation and inactivation variables of Na+ current and an activation variable of K+ current). Among the diverse family of HH-type equations, this chapter explores the dynamics and the bifurcation structure of the HH-type equations of heart muscle cells (cardiac myocytes).


Periodic Solution Hopf Bifurcation Bifurcation Diagram Bifurcation Point Bifurcation Curve 
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.


  1. Alexander JC, Cai DY (1991) On the dynamics of bursting systems. J Math Biol 29:405–423MathSciNetMATHCrossRefGoogle Scholar
  2. Av-Ron E (1994) The role of a transient potassium current in a bursting neuron model. J Math Biol 33:71–87MATHCrossRefGoogle Scholar
  3. Beeler GW, Reuter H (1977) Reconstruction of the action potential of ventricular myocardial fibres. J Physiol (London) 268:177–210Google Scholar
  4. Bertram R (1994) Reduced-system analysis of the effects of serotonin on a molluscan burster neuron. Biol Cybern 70:359–368MATHCrossRefGoogle Scholar
  5. Bertram R, Butte MJ, Kiemel T, Sherman A (1995) Topological and phenomenological classification of bursting oscillations. Bull Math Biol 57:413–439MATHGoogle Scholar
  6. Butera RJ Jr (1998) Multirhythmic bursting. Chaos 8:274–284MathSciNetMATHCrossRefGoogle Scholar
  7. Canavier CC, Clark JW, Byrne JH (1991) Simulation of the bursting activity of neuron R15 in Aplisia: role of ionic currents, calcium balance, and modulatory transmitters. J Neurophysiol 66:2107–2124Google Scholar
  8. Canavier CC, Baxter DA, Clark JW, Byrne JH (1993) Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of postsynaptic activity. J Neurophysiol 69:2252–2257Google Scholar
  9. Chay TR, Keizer J (1983) Minimal model for membrane oscillations in the pancreatic β-cell. Biophys J 42:181–190CrossRefGoogle Scholar
  10. Chay TR, Rinzel J (1985) Bursting, beating, and chaos in an excitable membrane model. Biophys J 47:357–366CrossRefGoogle Scholar
  11. Cronin J (1987) Mathematical aspects of Hodgkin–Huxley neural theory. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  12. DiFrancesco D, Noble D (1985) A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Philos Trans R Soc Lond [Biol] 307:353–398CrossRefGoogle Scholar
  13. Doi S, Kumagai S (2005) Generation of very slow neuronal rhythms and chaos near the Hopf bifurcation in single neuron models. J Comp Neurosci 19:325–356MathSciNetCrossRefGoogle Scholar
  14. Gerber B, Jakobsson E (1993) Functional significance of the A-current. Biol Cybern 70:109–114CrossRefGoogle Scholar
  15. Guckenheimer J, Gueron S, Harris-Warrick RM (1993) Mapping the dynamics of a bursting neuron. Phil Trans R Soc Lond B 341:345–359CrossRefGoogle Scholar
  16. Hayashi H, Ishizuka S (1992) Chaotic nature of bursting discharges in the Onchidium pacemaker neuron. J Theor Biol 156:269–291CrossRefGoogle Scholar
  17. Hilgemann DW, Noble D (1987) Excitation-contraction coupling and extracellular calcium transients in rabbit atrium: reconstruction of the basic cellular mechanisms. Proc R Soc Lond B Biol Sci 230:163–205CrossRefGoogle Scholar
  18. Keener JP, Sneyd J (1998) Mathematical physiology. Springer, BerlinMATHGoogle Scholar
  19. Luo CH, Rudy Y (1991) A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ Res 68:1501–1526CrossRefGoogle Scholar
  20. Luo CH, Rudy Y (1994) A dynamic model of the ventricular cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res 74:1071–1096Google Scholar
  21. Maeda Y, Pakdaman K, Nomura T, Doi S, Sato S (1998) Reduction of a model for an Onchidium pacemaker neuron. Biol Cybern 78:265–276MATHCrossRefGoogle Scholar
  22. McAllister RE, Noble D, Tsien RW (1975) Reconstruction of the electrical activity of cardiac Purkinje fibres. J Physiol (London) 251:1–59Google Scholar
  23. Noble D (1962) Modification of Hodgkin–Huxley equations applicable to purkinje fibre action and pace-maker potentials. J Physiol (London) 160:317–352Google Scholar
  24. Noble D (1975) The initiation of the heartbeat. Oxford University Press, OxfordGoogle Scholar
  25. Noble D (1995) The development of mathematical models of the heart. Chaos Solitons Fractals 5:321–333MATHCrossRefGoogle Scholar
  26. Noble D, Noble SJ (1984) A model of sino-atrial node electrical activity based on a modification of the DiFrancesco–Noble (1984) equations. Proc R Soc Lond B Biol Sci 222:295–304CrossRefGoogle Scholar
  27. Priebe L, Beuckelmann DJ (1998) Simulation study of cellular electric properties in heart gailure. Circ Res 82:1206–1223CrossRefGoogle Scholar
  28. Ramirez RJ, Nattel S (2000) Courtemanche M: Mathematical analysis of canine atrial action potentials: rate, regional factors, and electrical remodeling. Am J Physiol Heart Circ Physiol 279:H1767–H1785Google Scholar
  29. Rinzel J (1990) Discussion: electrical excitability of cells, theory and experiment: review of the Hodgkin–Huxley foundation and update. Bull Math Biol 52:5–23CrossRefGoogle Scholar
  30. Rush ME, Rinzel J (1994) Analysis of bursting in a thalamic neuron model. Biol Cybern 71:281–291MATHCrossRefGoogle Scholar
  31. Sarai N, Matsuoka S, Kuratomi S, Ono K, Noma A (2003) Role of individual ionic current systems in the SA node hypothesized by a model study. Jpn J Physiol 53:125–134CrossRefGoogle Scholar
  32. Schweighofer N, Doya K, Kawato M (1999) Electrophysiological properties of inferior olive neurons: a compartmental model. J Neurophysiol 82:804–817Google Scholar
  33. ten Tusscher KHW, Panfilov AV (2006a) Alternans and spiral breakup in a human ventricular tissue model. Am J Physiol Heart Circ Physiol 291:H1088–H1100CrossRefGoogle Scholar
  34. ten Tusscher KHW, Noble D, Noble PJ, Panfilov AV (2004) A model for human ventricular tissue. Am J Physiol Heart Circ Physiol 286:H1573–H1589CrossRefGoogle Scholar
  35. Terman D (1991) Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J Appl Math 51:1418–1450MathSciNetMATHCrossRefGoogle Scholar
  36. Traub RD, Wong RKS, Miles R, Michelson H (1991) A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances. J Neurophysiol 66:635–650Google Scholar
  37. Tsumoto K, Yoshinaga T, Aihara K, Kawakami H (2003) Bifurcations in synaptically coupled Hodgkin–Huxley neurons with a periodic input. Int J Bifurcat Chaos 13:653–666MathSciNetMATHCrossRefGoogle Scholar
  38. Yamaguchi R, Doi S, Kumagai S (2007) Bifurcation analysis of a detailed cardiac cell model and drug sensitivity of ionic channels. In: Proc. 15th IEEE international workshop on Nonlinear Dynamics of Electronic Systems 2007, pp 205–208Google Scholar
  39. Yanagihara K, Noma A, Irisawa H (1980) Reconstruction of sinoatrial node pacemaker potential based on the voltage clamp experiments. Jpn J Physiol 30:841–857CrossRefGoogle Scholar
  40. Yoshinaga T, Sano Y, Kawakami H (1999) A method to calculate bifurcations in synaptically coupled Hodgkin–Huxley equations. Int J Bifurcat Chaos 9:1451–1458MATHCrossRefGoogle Scholar
  41. Zhang H, Holden AV, Kodama I, Honjo H, Lei M, Varghese T, Boyett MR (2000) Mathematical models of action potential in the periphery and center of the rabbit sinoatrial node. Am J Physiol 279:H397–H421Google 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

Personalised recommendations