Bifurcations and Multistability in Periodically Stimulated Cardiac Cells

  • Elena SurovyatkinaEmail author


Application of fractal dimensions, Lyapunov exponents, and other measures from dynamical systems theory to characterize the function of the heart has led to extended and quite vivid discussions about how to interpret the observed irregularity of the human heart beat [10]. It is generally accepted that the heart rhythm tends to become more regular with age, but it is also clear that certain conditions that predispose a person for heart failure are reflected in particular patterns of irregular heart beat.


Bifurcation Diagram Period Doubling Bifurcation Restitution Protocol Ionic Property Coupling Interval 
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. 1.
    Adam DR, Akselrod S, Cohen RJ (1981) Estimation of ventricular vulnerability to fibrillation through T wave time series analysis. Comput Cardiol 8:307–310Google Scholar
  2. 2.
    Arnold VI, Afrajmovich VS, Il’yashenko YS, Shil’nikov LP (1994) Bifurcation theory and catastophe theory, Series: Encyclopaedia of Mathematical Sciens, Vol. 5. Springer, New YorkGoogle Scholar
  3. 3.
    Balanov A, Janson N, Postnov D and Sosnovtseva O (2009) Synchronization: From Simple to Complex. Springer, BerlinGoogle Scholar
  4. 4.
    CellML Model Repository (2010):
  5. 5.
    Cherry EM, Fenton FH (2004) Suppression of alternans and conduction blocks despite steep APD restitution: electrotonic, memory and conduction velocity effects. Am J Physiol 286:2332–2341Google Scholar
  6. 6.
    Cherry EM, Fenton FH (2007) A tale of two dogs: analyzing two models of canine ventricular electrophysiology. Am J Physiol Heart Circ Physiol 292:43–55CrossRefGoogle Scholar
  7. 7.
    Chialvo DR, Jalife J (1987) Non-linear dynamics of cardiac excitation and impulse propagation. Nature 330:749–752PubMedCrossRefGoogle Scholar
  8. 8.
    Chialvo DR, Michaels D, Jalife J (1990) Supernormal excitability as a mechanism of chaotic dynamics of activation in cardiac Purkinje fibers. Circ Res 66:525–545PubMedCrossRefGoogle Scholar
  9. 9.
    Cytrynbaum EN (2004) Periodic stimulus and the single cardiac cell - getting more out of 1D maps. J Theor Biol 229:69–83PubMedCrossRefGoogle Scholar
  10. 10.
    Glass L (2009) Introduction to controversial topics in nonlinear science: is the normal heart rate chaotic? Chaos 19:028501–028504PubMedCrossRefGoogle Scholar
  11. 11.
    Goldberger MD, Bahargava V, West BJ, Mandell AJ (1986) Some observations on the question: is ventricular fibrillation ”chaos”? Physica D 19:282–289CrossRefGoogle Scholar
  12. 12.
    Guevara MR (1984) Chaotic Cardiac Dynamics. Doctoral thesis, McGill University, MontrealGoogle Scholar
  13. 13.
    Guevara MR, Alonso F, Jeandupeux D, Ginneken ACG (1989) Alternans in periodically stimulated isolated ventricular myocytes: Experiment and model. In: Goldbeter A (ed) Cell to Cell Signalling: From Experiments to Theoretical Models. Harcourt Brace Jovanovich, LondonGoogle Scholar
  14. 14.
    Guevara MR, Glass L, Shrier A (1981) Phase Locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells. Science 214:1350–1352PubMedCrossRefGoogle Scholar
  15. 15.
    Hall GM, Bahar S, Gauthier DJ (1999) Prevalence of rate dependent behaviors in cardiac muscle. Phys Rev Lett 82:2995–2998CrossRefGoogle Scholar
  16. 16.
    Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544PubMedGoogle Scholar
  17. 17.
    Hutter OF, Noble D (1960) Rectifying properties of heart muscle. Nature 188:495Google Scholar
  18. 18.
    Jalife J, Delmar M, Anumonwo J, Berenfeld O, Kalifa J (2009) Bioelectricity, in Basic Cardiac Electrophysiology for the Clinician, 2nd edn. Wiley-Blackwell, OxfordCrossRefGoogle Scholar
  19. 19.
    Kalb SS, Dobrovolny HM, Tolkacheva EG, Idriss SF, Krassowska W, Gauthier DJ (2004) The restitution portrait: A new method for investigation rate-dependent restitution. J Cardiovasc Electrophysiol 15:698–709PubMedCrossRefGoogle Scholar
  20. 20.
    Kléber AG, Rudy Y (2004) Basic mechanisms of cardiac impulse propagation and associated arrhythmias. Physiol Rev 84:431–488PubMedCrossRefGoogle Scholar
  21. 21.
    Koller ML, Riccio MK, Gilmour RF Jr (1998) Dynamic restitution of action potential duration during electrical alternans and ventricular fibrillation. Am J Physiol Heart Circ Physiol 275:1635–1642Google Scholar
  22. 22.
    Kurata Y, Hisatome I, Matsuda H, Shibamoto T (2005) Dynamical mechanisms of pacemaker generation in I K1-downregulated human ventricular myocytes: Insights from bifurcation analyses of a mathematical model. Biophys J 89:2865–2887PubMedCrossRefGoogle Scholar
  23. 23.
    Lewis TJ, Guevara MR (1990) Chaotic dynamics in an ionic model of the propagated cardiac action potential. J Theor Biol 146:407–432PubMedCrossRefGoogle Scholar
  24. 24.
    Luo CH, Rudy Y (1991) A model of the ventricular cardiac action potential. Depolarization, repolarization and their interaction. Circulation 68:1501–1526CrossRefGoogle Scholar
  25. 25.
    Luo CH, Rudy Y (1994) A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res 74:1071–1096Google Scholar
  26. 26.
    Mines GR (1913) On dynamic equilibrium in the heart. J Physiol 46:349–383PubMedGoogle Scholar
  27. 27.
    Mosekilde M, Maistrenko Yu, Postnov D (2002) Chaotic Synchronization: Applications to Living Systems. World Scientific, SingaporeGoogle Scholar
  28. 28.
    Munteanu A, Kondratyev AA, Kucera JP (2008) Analysis of damped oscillations during reentry: A new approach to evaluate cardiac restitution. Biophys J 94:1094–1109PubMedCrossRefGoogle Scholar
  29. 29.
    Nash MP, Bradley CP, Sutton P, Hayward M, Paterson DJ, Taggart P (2004) Human hearts possess large regions of steep and flat APD restitution. Europace 6:187Google Scholar
  30. 30.
    Nash MP, Bradley CP, Sutton PM, Hayward M, Paterson DJ, Taggart P (2005) Spatial heterogeneity of action potential duration restitution in humans. Heart Rhythm 2:216–217CrossRefGoogle Scholar
  31. 31.
    Nash MP, Bradley CP, Sutton PM, Clayton RH, Kallis P, Hayward M, Paterson DJ, Taggart P (2006) Whole heart APD restitution properties in cardiac patients: A combined clinical and modeling study. Exp Physiol 91:339–354PubMedCrossRefGoogle Scholar
  32. 32.
    Noble D (1962) A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pace-maker potentials. J Physiol 160:317–352PubMedGoogle Scholar
  33. 33.
    Noble D (2002) Modelling the heart: from genes to cells to the whole organ. Science 295:1678–1682PubMedCrossRefGoogle Scholar
  34. 34.
    Noble D (2006) The Music of Life. Oxford University Press, OxfordGoogle Scholar
  35. 35.
    Noble D (2007) From the Hodgkin-Huxley axon to the virtual heart. J Physiol 580:15–22PubMedCrossRefGoogle Scholar
  36. 36.
    Noble D (2008) Computational models of the heart and their use in assessing the actions of drugs. J Pharmacol Sci 107:107–117PubMedCrossRefGoogle Scholar
  37. 37.
    Nolasco JB, Dahlen RW (1968) A graphic method for the study of alternation in cardiac action potentials. J Appl Physiol 25:191–196PubMedCrossRefGoogle Scholar
  38. 38.
    Oliver RA, Henriquez CS, Krassowska W (2000) Bistability and correlation with arrhythmogenesis in a model of the right atrium. Ann Biomed Eng 33:577–589CrossRefGoogle Scholar
  39. 39.
    Oliver RA, Krassowska W (2005) Reproducing cardiac restitution properties using the Fenton-Karma membrane model. Ann Biomed Eng 33:907–911PubMedCrossRefGoogle Scholar
  40. 40.
    Pikovsky A, Rosenblum M, Kurths J (2001) Synchronization – A Universal Concept in Nonlinear Sciences. Cambridge University Press, UKCrossRefGoogle Scholar
  41. 41.
    Qu Z (2004) Dynamical effects of diffusive cell coupling on cardiac excitation and propagation: a simulation study. Am J Physiol Heart Circ Physiol 287:2803–2812CrossRefGoogle Scholar
  42. 42.
    Ritzenberg AL, Adam DR, Cohen RJ (1984) Period multypling evidence for nonlinear behaviour of the canine heart. Nature 307:159–161PubMedCrossRefGoogle Scholar
  43. 43.
    Rubart M, Zipes D (2005) Mechanisms of sudden cardiac death. J Clin Investig 115:2305–2315PubMedCrossRefGoogle Scholar
  44. 44.
    Samie FH, Mandapati R, Gray RA, Watanabe Y, Zuur C, Beaumont J, Jalife J (2000) A mechanism of transition from ventricular fibrillation to tachycardia. Effect of calcium channel blockade on the dynamics of rotating waves. Circ Res 86:684–691Google Scholar
  45. 45.
    Savino GV, Romanelli L, González DL, Piro O, Valentinuzzi ME (1989) Evidence for chaotic behavior in driven ventricles. Biophys J 56:273–280PubMedCrossRefGoogle Scholar
  46. 46.
    Smith NP, Mulquiney PJ, Nash MP, Bradley CP, Nickerson DP, Hunter PJ (2001) Mathematical modelling of the heart: cell to organ. Chaos Solitons Fractals 13:1613–1621CrossRefGoogle Scholar
  47. 47.
    Surovyatkina E, Egorchenkov R, Ivanov G (2007) Multistability as intrinsic property of a single cardiac cell: a simulation study. Conf Proc IEEE Eng Med Biol Soc 927–930Google Scholar
  48. 48.
    Surovyatkina E, Noble D, Gavaghan D, Sher A (2010) Multistability Phenomenon in Ionic Models of Mammalian and Human Cardiac Ventricular Cells. Progr Biophys Mol Biol 103:131–141CrossRefGoogle Scholar
  49. 49.
    Ten Tusscher KHWJ, Noble D, Noble PJ, Panfilov AV (2004) A model for human ventricular tissue. Am J Physiol Heart Circ Physiol 286:1573–1589CrossRefGoogle Scholar
  50. 50.
    Ten Tusscher KHWJ, Panfilov AV (2006) Alternans and spiral breakup in a human ventricular tissue model. Am J Physiol Heart Circ Physiol 291:1088–1100CrossRefGoogle Scholar
  51. 51.
    Wu R, Parwardhan A (2004) Asymmetry in dynamics of action potential duration transition between steady states: a simulation study. Conf Proc IEEE Eng Med Biol Soc 6:3979–3982PubMedGoogle Scholar
  52. 52.
    Yehia AR, Jeandupeux D, Alonso F, Guevara MR (1999) Hysteresis and Bistability in the Direct Transition from 1:1 to 2:2 Rhythm in Periodically Driven Single Ventricular Cells. Chaos 9:916–931PubMedCrossRefGoogle Scholar
  53. 53.
    Yue AM, Franz MR, Roberts PR, Morgan JM (2005) Global endocardial electrical restitution in human right and left ventricles determined by noncontact mapping. J Am Coll Cardiol 46:1067–1075PubMedCrossRefGoogle Scholar
  54. 54.
    Zemlin Ch, Storch E, Herzel H (2002) Alternans and 2:1 rhythms in an ionic model of heart cells. BioSystems 66:1–10PubMedCrossRefGoogle Scholar
  55. 55.
    Zipes DP, Jalife J, Zorab R (2009) Cardiac Electrophysiology: From Cell to Bedside. Elseivier Saunders, PhiladelphiaGoogle Scholar

Copyright information

© Springer-Verlag/Wien 2011

Authors and Affiliations

  1. 1.Space Dynamics and Data Analysis Department, Space Research InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations