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Nonlinear dynamics of two-dimensional cardiac action potential duration mapping model with memory

  • M. Kesmia
  • S. Boughaba
  • S. JacquirEmail author
Article
  • 47 Downloads

Abstract

The aim of this work is the analysis of the nonlinear dynamics of two-dimensional mapping model of cardiac action potential duration (2D-map APD) with memory derived from one dimensional map (1D-map). Action potential duration (APD) restitution, which relates APD to the preceding diastolic interval (DI), is a useful tool for predicting cardiac arrhythmias. For a constant rate of stimulation the short action potential during alternans is followed by a longer DI and inversely. It has been suggested that these differences in DI are responsible for the occurrence and maintenance of APD alternans. We focus our attention on the observed bifurcations produced by a change in the stimulation period and a fixed value of a particular parameter in the model. This parameter provides new information about the dynamics of the APD with memory, such as the occurrence of bistabilities not previously described in the literature, as well as the fact that synchronization rhythms occur in different ways and in a new fashion as the stimulation frequency increases. Moreover, we show that this model is flexible enough as to accurately reflect the chaotic dynamics properties of the APD: we have highlighted the fractal structure of the strange attractor of the 2D-map APD, and we have characterized chaos by tools such as the calculation of the Lyapunov exponents, the fractal dimension and the Kolmogorov entropy, with the next objective of refining the study of the nonlinear dynamics of the duration of the action potential and to apply methods of controlling chaos.

Keywords

Cardiac action potential duration Memory Bifurcation Periodic dynamics Fixed point Strange attractor Fractal structure Chaos 

Mathematics Subject Classification

37D45 37N25 92C50 92B25 93C10 93C55 

Notes

Acknowledgements

We would like to thank Professor Michael R. Guevara (Department of Physiology and Centre for Nonlinear Dynamics in Physiology and Medicine, McGill University, Montreal, Quebec) for his insightful comments and helpful suggestions, which have helped improve the quality and presentation of the paper significantly.

References

  1. Babloyantz A, Destexhe A (1988) Is the normal heart a periodic oscillator. Biol Cybern 58:203–211MathSciNetCrossRefGoogle Scholar
  2. Bergé P, Pomeau Y, Vidal C (1984) L’ordre dans le chaos. Hermann, PariszbMATHGoogle Scholar
  3. Bittihn P, Berg S, Parlitz U, Luther S (2017) Emergent dynamics of spatio-temporal chaos in a heterogeneous excitable medium. Chaos Interdiscip J Nonlinear Sci 27(9):093,931.  https://doi.org/10.1063/1.4999604 MathSciNetCrossRefGoogle Scholar
  4. Cain JW (2007) Criterion for stable reentry in a ring of cardiac tissue. J Math Biol 55Google Scholar
  5. Chatterjee K, Harris A, Davies J et al (1969) T-wave changes after artificial pacing. Lancet 1:759–760CrossRefGoogle Scholar
  6. Cherubini C et al (2017) A note on stress-driven anisotropic diffusion and its role in active deformable media. J Theor Biol 430:221–228CrossRefGoogle Scholar
  7. Chialvo D, Jalife J (1987) Low dimensional chaos in cardiac tissue. Nature 330:749–752CrossRefGoogle Scholar
  8. Ding M, Grebogi C, Ott E, Sauer T, Yorke J (1993) Estimating correlation dimension from a chaotic time series: When does plateau onset occur? Physica D 69:404–424MathSciNetCrossRefGoogle Scholar
  9. Ditto WL, Rauseo SN, Spano ML (1990) Experimental control of chaos. Phys Rev Lett 65:3211–3214.  https://doi.org/10.1103/PhysRevLett.65.3211 CrossRefGoogle Scholar
  10. Eckmann J (1981) Roads to turbulence in dissipative dynamical systems. Rev Mod Phys 53:643–654.  https://doi.org/10.1103/RevModPhys.53.643 MathSciNetCrossRefzbMATHGoogle Scholar
  11. Eyebe Fouda J, Koepf W, Jacquir S (2017) The ordinal Kolmogorov–Sinai entropy: a generalized approximation. Commun Nonlinear Sci Numer Simul 46:103–115MathSciNetCrossRefGoogle Scholar
  12. Fenton F, Karma A (1998) Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: filament instability and fibrillation. Chaos Interdiscip J Nonlinear Sci 8(1):20–47.  https://doi.org/10.1063/1.166311 CrossRefzbMATHGoogle Scholar
  13. Fenton F, Gizzi A, Cherubini C, Pomella N, Filippi S (2013) Role of temperature on nonlinear cardiac dynamics. Phys Rev E 87(042):717.  https://doi.org/10.1103/PhysRevE.87.042717 CrossRefGoogle Scholar
  14. Fleetwood D, Masden J, Giordano N (1983) 1/f noise in platinum films and ultrathin platinum wires: evidence for a common bulk origin. Phys Rev Lett 50:450–453CrossRefGoogle Scholar
  15. Frigg R (2004) Kolmogorov–Sinai, entropy a measure for chaotic behaviour? Bridging the gap between dynamical systems theory and communication theory. Br J Philos Sci 55:411–434MathSciNetCrossRefGoogle Scholar
  16. Garfinkel A, Spano M, Ditto W, Weiss J (1992) Controlling cardiac chaos. Science 257:1230–1235CrossRefGoogle Scholar
  17. Garzón A, Grigoriev R (2017) Memory effects, transient growth, and wave breakup in a model of paced atrium. Chaos Interdiscip J Nonlinear Sci 27(9):093,917.  https://doi.org/10.1063/1.4999601
  18. Garzón A, Grigoriev R, Fenton F (2009) Model-based control of cardiac alternans on a ring. Phys Rev E 80(021):932.  https://doi.org/10.1103/PhysRevE.80.021932 CrossRefGoogle Scholar
  19. Garzón A, Grigoriev R, Fenton F (2011) Model-based control of cardiac alternans in purkinje fibers. Phys Rev E 84(041):927.  https://doi.org/10.1103/PhysRevE.84.041927 CrossRefGoogle Scholar
  20. Garzón A, Grigoriev R, Fenton F (2014) Continuous-time control of alternans in long purkinje fibers. Chaos Interdiscip J Nonlinear Sci 24(3):033,124.  https://doi.org/10.1063/1.4893295
  21. Gizzi A, Cherry EM, Gilmour R, Luther S, Filippi S, Fenton F (2013) Effects of pacing site and stimulation history on alternans dynamics and the development of complex spatiotemporal patterns in cardiac tissue. Front Physiol 4(71)Google Scholar
  22. Gizzi A, Loppini A, Ruiz-Baier R, Ippolito A, Camassa A, La Camera A, Emmi E, Di Perna L, Garofalo V, Cherubini C, Filippi S (2017) Nonlinear diffusion and thermo-electric coupling in a two-variable model of cardiac action potential. Chaos Interdiscip J Nonlinear Sci 27(9):093,919.  https://doi.org/10.1063/1.4999610
  23. Glass L, Guevara M, Shrier A, Perez R (1983) Bifurcation and chaos in a periodically stimulated cardiac oscillator. Physica D 7(1):89–101CrossRefGoogle Scholar
  24. Grebogi C, Ott E, Yorke J (1987) Chaos, strange attractors and fractal basin boundaries in nonlinear dynamics. Science 238:632–638MathSciNetCrossRefGoogle Scholar
  25. Guevara M (1997) Concepts and techniques in bioelectric measurements: Is the medium carrying the message? edited by J Billette and AR Leblanc Editions de l’Ecole polythenique de Montreal 7:67–87Google Scholar
  26. Guevara MR (1988) Spatiotemporal patterns of block in an ionic model of cardiac purkinje fibre. In: Markus M, Müller SC, Nicolis G (eds) From chemical to biological organization. Springer, Heidelberg, pp 273–281CrossRefGoogle Scholar
  27. Guevara M, Glass L, Shrier A (1981) Phaselocking, period-doubling bifurcations, and irregular dynamics inperiodically stimulated cardiac cells. Science 214(4527):1350–1353.  https://doi.org/10.1126/science.7313693. http://science.sciencemag.org/content/214/4527/1350. http://science.sciencemag.org/content/214/4527/1350.full.pdf
  28. Guevara M, Ward G, Shrier A, Glass L (1984) Electrical alternans and period doubling bifurcations. In: IEEE Computers in Cardiology pp 167–170Google Scholar
  29. Guevara M, Alonso F, Jeandupeux D, Vanginneken A (1989) Alternans in periodically stimulated isolated ventricular myocytes: experiment and model. In: Cell to cell signalling: from experiment to theoretical model pp 551–563Google Scholar
  30. Guevara M, Shrier A, Orlowski J, Glass L (2016) George Ralph mines (1886–1914): the dawn of cardiac nonlinear dynamics. J Physiol 594(9):2361–2371CrossRefGoogle Scholar
  31. Hescheler J, Speicher R (1989) Regular and chaotic behaviour of cardiac cells stimulated at frequencies between 2 and 20 Hz. Eur Biophys J 17:273–280CrossRefGoogle Scholar
  32. Hilborn RC (2000) Chaos and nonlinear dynamics: an introduction for scientists and engineersGoogle Scholar
  33. Hoekstra B, Diks C, Allessie M, Goedb J (1995) Nonlinear analysis of epicardial atrial electrograms of electrically induced atrial fibrillation in man. J Cardiovasc Electrophysiol 6(6):419–440.  https://doi.org/10.1111/j.1540-8167.1995.tb00416.x CrossRefGoogle Scholar
  34. Jensen J et al (1984) Chaos in the Beeler-Reuter system for the action potential of ventricular myocardial fibres. Physica D 13(1):269–277.  https://doi.org/10.1016/0167-2789(84)90283-5 MathSciNetCrossRefzbMATHGoogle Scholar
  35. Karmakar C, Udhayakumar R, Palaniswami M (2015) Distribution entropy (disten): a complexity measure to detect arrhythmia from short length rr interval time series. In: 37th Annual international conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp 5207–5210Google Scholar
  36. Kesmia M, Boughaba S, Jacquir S (2016) Predictive chaos control for the 1d-map of action potential duration. Chaotic Model Simul J 3:387–398Google Scholar
  37. Kesmia M, Boughaba S, Jacquir S (2018a) Control of irregular cardiac rhythm. Chaotic Model Simul J 1:91–99zbMATHGoogle Scholar
  38. Kesmia M, Boughaba S, Jacquir S (2018b) New approach of controlling cardiac alternans. Discrete Contin Dyn Syst B 23(2):975–989MathSciNetCrossRefGoogle Scholar
  39. King C (1991) Fractal and chaotic dynamics in nervous systems. Prog Neurobiol 36(4):279–308CrossRefGoogle Scholar
  40. Kolmogorov A (1959) Entropy per unit time as a metric invariant of automorphism. Doklady Russ Acad Sci 124:754–755MathSciNetzbMATHGoogle Scholar
  41. Krogh-Madsen T, Kold Taylor L, Skriver A, Schaffer P, Guevara M (2017) Regularity of beating of small clusters of embryonic chick ventricular heart-cells: experiment versus stochastic single-channel population model. Chaos Interdiscip J Nonlinear Sci 27(9):093,929.  https://doi.org/10.1063/1.5001200
  42. Landaw J, Garfinkel A, Weiss J, Qu Z (2017) Memory-induced chaos in cardiac excitation. Phys Rev Lett 118(138):101.  https://doi.org/10.1103/PhysRevLett.118.138101 CrossRefGoogle Scholar
  43. Lee K et al (2009) Effect of short-term cardiac memory on ventricular electrical restitution and qt intervals in humans. Kaohsiung J Med Sci 25(2):53–61CrossRefGoogle Scholar
  44. Lee K, Chu C, Lin T et al (2008) Effect of sodium and calcium channel blockers on short-term cardiac memory in humans. Int J Cardiol 123(2):94–101CrossRefGoogle Scholar
  45. Lewis T, Guevara M (1990) Chaotic dynamics in an ionic model of the propagated cardiac action potential. J Theor Biol 146(3):407–432CrossRefGoogle Scholar
  46. Luther ea (2011) Low-energy control of electrical turbulence in the heart. Nature 475:235Google Scholar
  47. Mayordomo E, Kolmogorov A (2002) Complexity characterization of constructive Hausdorff dimension. Inf Process Lett 84:1–3MathSciNetCrossRefGoogle Scholar
  48. Morgan SW, Biktasheva IV, Biktashev VN (2008) Control of scroll-wave turbulence using resonant perturbations. Phys Rev E 78(046):207.  https://doi.org/10.1103/PhysRevE.78.046207 MathSciNetCrossRefGoogle Scholar
  49. Nagaiah C, Kunisch K, Plank G (2013) Optimal control approach to termination of re-entry waves in cardiac electrophysiology. J Math Biol 67:359–388MathSciNetCrossRefGoogle Scholar
  50. Ohnishi M, Inaba N (1994) A singular bifurcation into instant chaos in a piecewise-linear circuit. IEEE Trans Circuits Syst 41:433–442MathSciNetCrossRefGoogle Scholar
  51. Oida E, Moritani T, Yamori Y (1997) Tone-entropy analysis on cardiac recovery after dynamic exercise. J Appl Physiol 82(6):1794–1801.  https://doi.org/10.1152/jappl.1997.82.6.1794 (pMID: 9173943)CrossRefGoogle Scholar
  52. Otani N (2017) Theory of the development of alternans in the heart during controlled diastolic interval pacing. Chaos Interdiscip J Nonlinear Sci 27(9):093,935.  https://doi.org/10.1063/1.5003250
  53. Otani N, Gilmour R (1997) Memory models for the properties of local cardiac systems. J Theor Biol 187:409–436CrossRefGoogle Scholar
  54. Pincus S, Viscarello R (1992) Approximate entropy: a regularity measure for fetal heart rate analysis. Obstet Gynecol 79:249–255Google Scholar
  55. Pumir A, Nikolski V, Hörning M, Isomura A, Agladze K, Yoshikawa K, Gilmour R, Bodenschatz E, Krinsky V (2007) Wave emission from heterogeneities opens a way to controlling chaos in the heart. Phys Rev Lett 99(208):101.  https://doi.org/10.1103/PhysRevLett.99.208101 CrossRefGoogle Scholar
  56. Rappel W, Fenton F, Karma A (1999) Spatiotemporal control of wave instabilities in cardiac tissue. Phys Rev Lett 83Google Scholar
  57. Rosenbaum M, Blanco H, Elizari M et al (1982) Electrotonic modulation of the t wave and cardiac memory. Am J Cardiol 50(2):213–222CrossRefGoogle Scholar
  58. Ryan S et al (1994) Gender- and age-related differences in heart rate dynamics: Are women more complex than men? J Am Coll Cardiol 24(7):1700–1707CrossRefGoogle Scholar
  59. Schuster HG (1988) Deterministic chaos: an introduction (2nd revised edition)Google Scholar
  60. Shilnikov L (1981) The bifurcation theory and quasi-hyperbiloc attractors. Uspehi Mat Nauk 36:240–241Google Scholar
  61. Sinai Y (1959) On the notion of entropy of a dynamical system. Doklady Russ Acad Sci 124:768–771zbMATHGoogle Scholar
  62. Smith J, Cohen R (1984) Simple finite-element model accounts for wide range of cardiac dysrhythmias. Proceedings of the National Academy of Sciences 81(1):233–237.  https://doi.org/10.1073/pnas.81.1.233. http://www.pnas.org/content/81/1/233. http://www.pnas.org/content/81/1/233.full.pdf
  63. Sprott J, Rowlands G (1995) Chaos data analyzer: the professional version. Physics Academic Software, RaleighGoogle Scholar
  64. Tolkacheva E, Schaeffer D, Gauthier D, Mitchell C (2002) Analysis of the fentonkarma model through an approximation by a one-dimensional map. Chaos Interdiscip J Nonlinear Sci 12(4):1034–1042.  https://doi.org/10.1063/1.1515170 CrossRefzbMATHGoogle Scholar
  65. Voss A, Schulz S, Schroeder R, Baumert M, Caminal P(2009) Methods derived from nonlinear dynamics for analysing heart rate variability. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 367(1887):277–296.  https://doi.org/10.1098/rsta.2008.0232.http://rsta.royalsocietypublishing.org/content/367/1887/277. http://rsta.royalsocietypublishing.org/content/367/1887/277.full.pdf
  66. Wei N, Mori Y, Tolkacheva E (2015) The role of short term memory and conduction velocity restitution in alternans formation. J Theor Biol 367:21–28MathSciNetCrossRefGoogle Scholar
  67. Wessel N, Schumann A, Schirdewan A, Voss A, Kurths J (2000) Entropy measures in heart rate variability data. In: Brause RW, Hanisch E (eds) Medical data analysis. Springer, Heidelberg, pp 78–87CrossRefGoogle Scholar
  68. Wu R, Patwardhan A (2004) Restitution of action potential duration during sequential changes in diastolic intervals shows multimodal behavior. Circ Res 94(5):634–641.  https://doi.org/10.1161/01.RES.0000119322.87051.A9 CrossRefGoogle Scholar
  69. Xu B, Jacquir S, Laurent G, Bilbault J, Binczak S (2011) A hybrid stimulation strategy for suppression of spiral waves in cardiac tissue. Chaos Solitons Fractals 44(8):633–639CrossRefGoogle Scholar
  70. Xu B, Jacquir S, Laurent G, Binczak S, Pont O, Yahia H (2015) In vitro arrhythmia generation by mild hypothermia: a pitchfork bifurcation type process. Physiol Meas 36(3):579. http://stacks.iop.org/0967-3334/36/i=3/a=579
  71. Yehia A, Jeandupeux D, Alonso F, Guevara M (1999) Hysteresis and bistability in the direct transition from 1:1 to 2:1 rhythm in periodically driven single ventricular cells. Chaos Interdiscip J Nonlinear Sci 9(4):916–931.  https://doi.org/10.1063/1.166465 CrossRefzbMATHGoogle Scholar
  72. Yentes J, Hunt N, Schmid K, Kaipust J, McGrath D, Stergiou N (2013) The appropriate use of approximate entropy and sample entropy with short data sets. Ann Biomed Eng 41(2):349–365CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Constantine IConstantineAlgeria
  2. 2.Le2i, FRE CNRS 2005, Arts et MétiersUniversité Bourgogne Franche-ComtéDijonFrance

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