Unstable cardiac multi-spiral waves in a FitzHugh–Nagumo soliton model under magnetic flow effect

Abstract

This work deals with the stimulation of cardiac spiral waves in a two-dimensional FitzHugh–Nagumo model through modulational instability phenomenon in the presence of intracellular magnetic flux. The nonlinear generic model is firstly transformed into a two-dimensional complex Ginzburg–Landau equation with a small real damping term. Then, the latter is explored to perform the linear stability analysis and some modulational instability properties are derived and found to be modified with the change of either the magnetic coupling strength or the diffusion coefficient which are set as the control parameters. The bifurcation theory analysis is numerically performed, and the range of values of the stimulation parameter for which firing patterns emerge in cardiac media is well estimated. The recording of membrane potential as well as the magnetic flux as functions of time or spatial coordinates allows to picture some features such as time series, phase portraits and spatial patterns of membrane potential. As a result, diamond-shaped, squared-shaped and spiral-shaped waves are obtained. An elimination process of multi-spiral waves is proposed by exposing such patterns to an external magnetic flux. Thus, for suited values of parameters, the cardiac spiral waves are transmuted into target waves which obviously protect heart against harmful attacks such as ventricular tachycardia and fibrillation. Our results suggest that uncontrolled electromagnetic induction within cardiac tissue may be considered as the principal source of many heart injuries which are the first cause of mortality in the industrial world.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  1. 1.

    Glass, L.: Synchronization and rhythmic processes in physiology. Nature (London) 410, 277 (2001)

    Google Scholar 

  2. 2.

    Qu, Z., Hu, G., Garfinkel, A., Weiss, J.N.: Nonlinear and stochastic dynamics in the heart. Phys. Rep. 543, 61 (2014)

    MathSciNet  Google Scholar 

  3. 3.

    Scardigli, M., Crocini, C., Ferrantini, C., Gabbrielli, T., Silvestri, L., Coppini, R., Tesi, C., Rog-Zielinska, E.A., Kohl, P., Cerbai, E., Poggesi, C., Pavone, F.S., Sacconi, L.: Quantitative assessment of passive electrical properties of the cardiac T-tubular system by FRAP microscopy. Proc. Natl. Acad. Sci. USA 114, 5737 (2017)

    Google Scholar 

  4. 4.

    Nash, M.P., Panfilov, A.V.: Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias. Prog. Biophys. Mol. Biol. 85, 501 (2004)

    Google Scholar 

  5. 5.

    Tusscher, K.T., Noble, D., Noble, P.J., Panfilov, A.V.: A model for human ventricular tissue. Am. J. Physsiol. 286, H1573–H1589 (2004)

    Google Scholar 

  6. 6.

    Dadivenko, J.M., Pertsov, A., Salomonsz, R., Baxter, W., Jalife, J.: Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature 355, 349 (1992)

    Google Scholar 

  7. 7.

    Winfree, A.T.: Electrical turbulence in three-dimensional heart muscle. Science 266, 1003 (1994)

    Google Scholar 

  8. 8.

    Witkowski, F.X., Leon, L.J., Penkoske, P.A., Giles, W.R., Spano, M.L., Ditto, W.L., Winfree, A.T.: Spatiotemporal evolution of ventricular fibrillation. Nature 392, 78 (1998)

    Google Scholar 

  9. 9.

    Abildskov, J.A., Lux, R.L.: The mechanism of simulated torsade de pointes in a computer model of propagated excitation. J. Cardiovasc. Electrophysiol. 2, 224–237 (1991)

    Google Scholar 

  10. 10.

    Gray, R.A., Jalife, J., Panfilov, A.V., Baxter, W.T., Cabo, C., Pertsov, A.M.: Nonstationary vortexlike reentrant activity as a mechanism of polymorphic ventricular tachycardia in the isolated rabbit heart. Circulation 91, 2454–2469 (1995)

    Google Scholar 

  11. 11.

    Garfinkel, A., Qu, Z.: Nonlinear dynamics of excitation and propagation in cardiac tissue. In: Zipes, D.P., Jalife, J. (eds.) Cardiac electrophysiology. From cell to bedside, 3rd edn, pp. 315–320. W.B. Saunders Company, Philadelphia (1999)

    Google Scholar 

  12. 12.

    Difrancesco, D.: Pacemaker mechanisms in cardiac. Annu. Rev. Physiol. 55, 455–472 (1993)

    Google Scholar 

  13. 13.

    Seemann, G., Höper, C., Sachse, F.B., Dössel, O., Holden, A.V., Zhang, H.: Heterogeneous three-dimensional anatomical and electrophysiological model of human atria. Philos. Trans. A 364, 1465–1481 (2006)

    Google Scholar 

  14. 14.

    Fox, K.: Resting heart rate in cardiovascular disease. J. Am. Coll. Cardiol. 50, 823–830 (2007)

    Google Scholar 

  15. 15.

    Fenton, F.H., Karma, A.: Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: filament instability and fibrillation. Chaos 8, 20–47 (1998)

    MATH  Google Scholar 

  16. 16.

    Fenton, F.H., Cherry, E.M., Hastings, H.M., Evans, S.G.: Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. Chaos 12, 852–892 (2002)

    Google Scholar 

  17. 17.

    Karma, A.: Universal limit of spiral wave propagation in excitable media. Phys. Rev. Lett. 66, 2274 (1991)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Karma, A.: Scaling regime of spiral wave propagation in single-diffusive media. Phys. Rev. Lett. 68, 397 (1992)

    Google Scholar 

  19. 19.

    Krinsky, V., Pumir, A.: Models of defibrillation of cardiac tissue. Chaos 8, 188 (1998)

    Google Scholar 

  20. 20.

    Winfree, A.T.: When Time Breaks down. Princeton University Press, Princeton (1987)

    Google Scholar 

  21. 21.

    Fenton, F.H., Cherry, E.M., Karma, A., Rappel, W.J.: Modeling wave propagation in realistic heart geometries using the phase-field method. Chaos 15, 013502 (2005)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Luther, S., Fenton, F.H., Kornreich, B.G., Squires, A., Bittihn, P., Hornung, D., Zabel, M., Flanders, J., Gladuli, A., Campoy, L., Cherry, E.M., Luther, G., Hasenfuss, G., Krinsky, V.I., Pumir, A., Gilmour Jr., R.F., Bodenschatz, E.: Low-energy control of electrical turbulence in the heart. Nat. Lett. 475, 235–239 (2011)

    Google Scholar 

  23. 23.

    Fenton, F.H., Cherry, E.M.: Models of cardiac cell. Scholarpedia 3, 1868 (2008)

    Google Scholar 

  24. 24.

    Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)

    Google Scholar 

  25. 25.

    FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)

    Google Scholar 

  26. 26.

    Nagumo, J.S., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061 (1962)

    Google Scholar 

  27. 27.

    Aliev, R.R., Panfilov, A.V.: A simple two-variable model of cardiac excitation. Chaos Solitons Fractals 7, 293–301 (1996)

    Google Scholar 

  28. 28.

    Hindmarsh, J.L., Rose, R.M.: A model of the nerve impulse using two first-order differential equations. Nature 296, 162 (1982)

    Google Scholar 

  29. 29.

    Hindmarsh, J.L., Rose, R.M.: A model of neuronal bursting using three coupled first order differential equations. Proc. Royal Soc. Lond. B Biol. Sci. 221, 87 (1984)

    Google Scholar 

  30. 30.

    Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193 (1981)

    Google Scholar 

  31. 31.

    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.: Nonlinear diffusion and thermo-electric coupling in a two-variable model of cardiac action potential. Chaos 27, 093919 (2017)

    MathSciNet  Google Scholar 

  32. 32.

    Fenton, F.H., Gizzi, A., Cherubini, C., Pomella, N., Filippi, S.: Role of temperature on nonlinear cardiac dynamics. Phys. Rev. E 87, 042717 (2013)

    Google Scholar 

  33. 33.

    Karma, A.: Spiral breakup in model equations of action potential propagation in cardiac tissue. Phys. Rev. Lett. 71, 1103 (1993)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Karma, A.: Electrical alternans and spiral wave breakup in cardiac tissue. Chaos 4, 461–472 (1994)

    Google Scholar 

  35. 35.

    Tanskanen, A.J., Alvarez, L.H.R.: Voltage noise influences action potential duration in cardiac myocytes. Math. Biosci. 208, 125–146 (2007)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Rostami, Z., Jafari, S., Perc, M., Slavinec, M.: Elimination of spiral waves in excitable media by magnetic induction. Nonlinear Dyn. 94, 679–692 (2018)

    Google Scholar 

  37. 37.

    Wu, F., Wang, C., Xu, Y., Ma, J.: Model of electrical activity in cardiac tissue under electromagnetic induction. Sci. Rep. 6, 28 (2016)

    Google Scholar 

  38. 38.

    Ma, J., Wu, F., Hayat, T., Zhou, P., Tang, J.: Electromagnetic induction and radiation-induced abnormality of wave propagation in excitable media. Physica A 486, 508 (2017)

    MathSciNet  Google Scholar 

  39. 39.

    Fu, Y.-X., Kang, Y.-M., Xie, Y.: Subcritical hopf bifurcation and stochastic resonance of electrical activities in neuron under electromagnetic induction. Front. Comput. Neurosci. 12, 6 (2018)

    Google Scholar 

  40. 40.

    Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, Berlin (1984)

    Google Scholar 

  41. 41.

    Aranson, I.S., Kramer, L.: The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99 (2002)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    GarcÍa-Morales, V., Krischer, K.: The complex Ginzburg–Landau equation: an introduction. Contemp. Phys. 53, 79–95 (2012)

    Google Scholar 

  43. 43.

    Zykov, V.S.: Spiral waves and dissipative solitons in weakly excitable media. Lect. Notes Phys. 751, 453–473 (2008)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Tabi, C.B.: Fractional unstable patterns of energy in \(\alpha \)-helix proteins with long-range interactions. Chaos Solit. Fract. 116, 386–391 (2018)

    MathSciNet  Google Scholar 

  45. 45.

    Mimshe, J.C.F., Tabi, C.B., Edongue, H., Ekobena, H.P.F., Mohamadou, A., Kofané, T.C.: Wave patterns in \(\alpha \)-helix proteins with interspine coupling. Phys. Scr. 87, 025801 (2013)

    MATH  Google Scholar 

  46. 46.

    Tabi, C.B., Ondoua, R.Y., Ekobena, H.P., Mohamadou, A., Kofané, T.C.: Energy patterns in coupled \(\alpha \)-helix protein chains with diagonal and off-diagonal couplings. Phys. Lett. A 380, 2374–2381 (2016)

    Google Scholar 

  47. 47.

    Zanga, D., Fewo, S.I., Tabi, C.B., Kofané, T.C.: Modulational instability in weak nonlocal nonlinear media with competing Kerr and non-Kerr nonlinearities. Commun. Nonlinear Sci. Numer. Simul. 80, 104993 (2020)

    MathSciNet  Google Scholar 

  48. 48.

    Panguetna, C.S., Tabi, C.B., Kofané, T.C.: Electronegative nonlinear oscillating modes in plasmas. Commun. Nonlinear Sci. Numer. Simul. 55, 326–337 (2018)

    Google Scholar 

  49. 49.

    Panguetna, C.S., Tabi, C.B., Kofané, T.C.: Low relativistic effects on the modulational instability of rogue waves in electronegative plasmas. J. Theor. Appl. Phys. 13, 237–249 (2019)

    Google Scholar 

  50. 50.

    Tabi, C.B., Etémé, A.S., Mohamadou, A., Kofané, T.C.: Oscillating two-dimensional \(\text{ Ca }^{2+}\) waves in cell networks with bidirectional paracrine signaling. Waves Rand. Comp. Med. (2019). https://doi.org/10.1080/17455030.2019.1644465

    Article  Google Scholar 

  51. 51.

    Xu, D., Roth, B.J.: The magnetic field produced by the heart and its influence on MRI. Math. Prob. Eng. 2017, 3035479 (2017)

    MathSciNet  Google Scholar 

  52. 52.

    Raghavachari, S., Glazier, J.A.: Waves in diffusively coupled bursting cells. Phys. Rev. Lett. 82, 2991 (1999)

    Google Scholar 

  53. 53.

    Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851 (1993)

    MATH  Google Scholar 

  54. 54.

    Malomed, B.A.: Complex Ginzburg–Landau Equation, in Encyclopedia of Nonlinear Science, pp. 157–160. Routledge, New York (2005)

    Google Scholar 

  55. 55.

    Mohamadou, A., Ayissi, B.E., Kofané, T.C.: Instability criteria and pattern formation in the complex Ginzburg–Landau equation with higher-order terms. Phys. Rev. E 74, 046604 (2006)

    Google Scholar 

  56. 56.

    Tiofack, C.G.L., Mohamadou, A., Kofané, T.C., Moubissi, A.B.: Generation of pulse trains in nonlinear optical fibers through the generalized complex Ginzburg–Landau equation. Phys. Rev. E 80, 066604 (2009)

    Google Scholar 

  57. 57.

    Ndzana II, F., Mohamadou, A., Kofané, T.C., English, L.Q.: Modulated waves and pattern formation in coupled discrete nonlinear LC transmission lines. Phys. Rev. E 78, 016606 (2008)

    MathSciNet  Google Scholar 

  58. 58.

    Kengne, E., Chui, S.T., Liu, W.M.: Modulational instability criteria for coupled nonlinear transmission lines with dispersive elements. Phys. Rev. E 74, 036614 (2006)

    Google Scholar 

  59. 59.

    Wamba, E., Mohamadou, A., Kofané, T.C.: Modulational instability of a trapped Bose–Einstein condensate with two- and three-body interactions. Phys. Rev. E 77, 046216 (2008)

    Google Scholar 

  60. 60.

    Kevrekidis, P.G., Carretero-Gonzalez, R., Theocharis, G., Frantzeskakis, D.J., Malomed, B.A.: Stability of dark solitons in a Bose–Einstein condensate trapped in an optical lattice. Phys. Rev. A 68, 035602 (2003)

    Google Scholar 

  61. 61.

    Akers, B.F.: Modulational instabilities of periodic traveling waves in deep water. Physica D 300, 26–33 (2015)

    MathSciNet  MATH  Google Scholar 

  62. 62.

    Göktepe, S., Wong, J., Kuhl, E.: Atrial and ventricular fibrillation: computational simulation of spiral waves in cardiac tissue. Arch. Appl. Mech. 80, 569–580 (2010)

    MATH  Google Scholar 

  63. 63.

    Clayton, R., Zhuchkova, E., Panfilov, A.: Phase singularities and filaments: simplifying complexity in computational models of ventricular fibrillation. Prog. Biophys. Mol. Biol. 90, 378 (2006)

    Google Scholar 

  64. 64.

    Koplan, B.A., Stevenson, W.G.: Ventricular tachycardia and sudden cardiac death. Mayo Clin. Proc. 84, 289 (2009)

    Google Scholar 

  65. 65.

    Ma, J., Jia, Y., Yi, M., Tang, J., Xia, Y.-F.: Suppression of spiral wave and turbulence by using amplitude restriction of variable in a local square area. Chaos Solitons Fract. 41, 1331–1339 (2009)

    Google Scholar 

  66. 66.

    Etémé, A.S., Tabi, C.B., Mohamadou, A., Kofané, T.C.: Elimination of spiral waves in a two-dimensional Hindmarsh–Rose neural network under long-range interaction effect and frequency excitation. Physica A 533, 122037 (2019)

    MathSciNet  Google Scholar 

  67. 67.

    Baysal, V., Yilmaz, E.: Effects of electromagnetic induction on vibrational resonance in single neurons and neuronal networks. Physica A 537, 122733 (2020)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The work by CBT is supported by the Botswana International University of Science and Technology under the Grant DVC/RDI/2/1/16I (25). I thank the Kavli Institute for Theoretical Physics (KITP), University of California Santa Barbara (USA), where this work was supported in part by the National Science Foundation Grant No. NSF PHY-1748958, NIH Grant No. R25GM067110, and the Gordon and Betty Moore Foundation Grant No. 2919.01.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Conrad B. Tabi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tabi, C.B., Etémé, A.S. & Kofané, T.C. Unstable cardiac multi-spiral waves in a FitzHugh–Nagumo soliton model under magnetic flow effect. Nonlinear Dyn 100, 3799–3814 (2020). https://doi.org/10.1007/s11071-020-05750-z

Download citation

Keywords

  • Spiral wave
  • Cardiac tissue
  • Wave pattern
  • Magnetic FitzHugh–Nagumo model
  • Modulational instability