Nonlinear Dynamics

, Volume 94, Issue 1, pp 679–692 | Cite as

Elimination of spiral waves in excitable media by magnetic induction

  • Zahra Rostami
  • Sajad Jafari
  • Matjaž Perc
  • Mitja Slavinec
Original Paper


The formation of spiral waves in excitable media is a fascinating example of the beauty of nonlinear dynamics in spatiotemporal systems. Apart from the beauty of the patterns, the subject also has many practical application. For example, the emergence of spiral waves in cardiac tissue can lead to arrhythmias. Cortical spiral waves are also involved in epileptic seizures. Motivated by this, we here study the effects of magnetic induction on the formation of spiral waves in excitable media. An external sinusoidal magnetic induction with different amplitudes and angular frequencies is applied in order to study whether spiral waves could be eliminated. We use a network of coupled neurons as a model for the excitable medium. The four-variable magnetic Hindmarsh–Rose model is used for the local dynamics of each isolated neuron. The distribution of the cell membrane potential over time, affected by magnetic induction, is determined and the results are depicted as snapshots of the 2D network. Our research reveals that the continuance of rotating spiral seeds is impaired by high-amplitude magnetic induction. Moreover, we show that low-frequency induction is not capable of breaking the reorganizing rhythm of the spiral seeds, while much higher frequencies can be too fast to overcome this special rhythm.


Spiral wave Spatiotemporal pattern Magnetic flux Neuronal network Magnetic Hindmarsh–Rose model 



Sajad Jafari was supported by the Iran National Science Foundation (Grant No. 96000815). Matjaž Perc was supported by the Slovenian Research Agency (Grants Nos. J1-7009 and P5-0027).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Goldenfeld, N., Kadanoff, L.P.: Simple lessons from complexity. Science 284, 87–89 (1999)CrossRefGoogle Scholar
  2. 2.
    Gell-Mann, M.: Simplicity and complexity in the description of nature. Eng. Sci. 51, 2–9 (1988)Google Scholar
  3. 3.
    Perc, M.: Stability of subsystem solutions in agent-based models. Eur. J. Phys. 39, 014001 (2017)CrossRefGoogle Scholar
  4. 4.
    Holovatch, Y., Kenna, R., Thurner, S.: Complex systems: physics beyond physics. Eur. J. Phys. 38(2), 023002 (2017)CrossRefGoogle Scholar
  5. 5.
    Guo, S., Xu, Y., Wang, C., Jin, W., Hobiny, A., Ma, J.: Collective response, synapse coupling and field coupling in neuronal network. Chaos Solitons Fractals 105, 120–127 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wu, F., Wang, Y., Ma, J., Jin, W., Hobiny, A.: Multi-channels coupling-induced pattern transition in a tri-layer neuronal network. Physica A 493, 54–68 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Yilmaz, E., Ozer, M., Baysal, V., Perc, M.: Autapse-induced multiple coherence resonance in single neurons and neuronal networks. Sci. Rep. 6, 30914 (2016)CrossRefGoogle Scholar
  8. 8.
    Li, X., Rakkiyappan, R., Sakthivel, N.: Non-fragile synchronization control for markovian jumping complex dynamical networks with probabilistic time-varying coupling delays. Asian J. Control 17, 1678–1695 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Li, X., Fu, X.: Synchronization of chaotic delayed neural networks with impulsive and stochastic perturbations. Commun. Nonlinear Sci. Numer. Simul. 16, 885–894 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ma, J., Tang, J.: A review for dynamics in neuron and neuronal network. Nonlinear Dyn. 89, 1569–1578 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gosak, M., Markovič, R., Dolenšek, J., Rupnik, M.S., Marhl, M., Stožer, A., Perc, M.: Network science of biological systems at different scales: a review. Phys. Life Rev. 24, 118–135 (2018)CrossRefGoogle Scholar
  12. 12.
    Gosak, M., Markovič, R., Dolenšek, J., Rupnik, M.S., Marhl, M., Stožer, A., Perc, M.: Loosening the shackles of scientific disciplines with network science: reply to comments on network science of biological systems at different scales: a review. Phys. Life Rev. 24, 162–167 (2018)CrossRefGoogle Scholar
  13. 13.
    Milton, J., Jung, P.: Epilepsy as a Dynamic Disease. Springer, Berlin (2013)zbMATHGoogle Scholar
  14. 14.
    Xu, Y., Jia, Y., Ma, J., Hayat, T., Alsaedi, A.: Collective responses in electrical activities of neurons under field coupling. Sci. Rep. 8, 1349 (2018)CrossRefGoogle Scholar
  15. 15.
    Wang, C., Ma, J.: A review and guidance for pattern selection in spatiotemporal system. Int. J. Mod. Phys. B 32, 1830003 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mvogo, A., Takembo, C.N., Fouda, H.P.E., Kofané, T.C.: Pattern formation in diffusive excitable systems under magnetic flow effects. Phys. Lett. A 381, 2264–2271 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Takembo, C.N., Mvogo, A., Ekobena Fouda, H.P., Kofané, T.C.: Modulated wave formation in myocardial cells under electromagnetic radiation. Int. J. Mod. Phys. B 32, 1850165 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Xiang, W., Huangpu, Y.: Second-order terminal sliding mode controller for a class of chaotic systems with unmatched uncertainties. Commun. Nonlinear Sci. Numer. Simul. 15, 3241–3247 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sinha, S., Sridhar, S.: Patterns in Excitable Media: Genesis, Dynamics, and Control. CRC Press, Boca Raton (2014)CrossRefGoogle Scholar
  20. 20.
    Zhang, J., Tang, J., Ma, J., Luo, J.M., Yang, X.Q.: The dynamics of spiral tip adjacent to inhomogeneity in cardiac tissue. Physica A 491, 340–346 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Banerjee, M., Ghorai, S., Mukherjee, N.: Approximated spiral and target patterns in Bazykins prey-predator model: Multiscale perturbation analysis. Int. J. Bifurc. Chaos 27, 1750038 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Woo, S.-J., Lee, J., Lee, K.J.: Spiral waves in a coupled network of sine-circle maps. Phys. Rev. E 68, 016208 (2003)CrossRefGoogle Scholar
  23. 23.
    Hu, B., Ma, J., Tang, J.: Selection of multiarmed spiral waves in a regular network of neurons. PLoS ONE 8, e69251 (2013)CrossRefGoogle Scholar
  24. 24.
    Li, F., Ma, J.: Pattern selection in network of coupled multi-scroll attractors. PLoS ONE 11(4), e0154282 (2016)CrossRefGoogle Scholar
  25. 25.
    Perc, M.: Effects of small-world connectivity on noise-induced temporal and spatial order in neural media. Chaos Solitons Fractals 31, 280–291 (2007)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Panfilov, A.V., Müller, S.C., Zykov, V.S., Keener, J.P.: Elimination of spiral waves in cardiac tissue by multiple electrical shocks. Phys. Rev. E 61, 4644–4647 (2000)CrossRefGoogle Scholar
  27. 27.
    Pertsov, A.M., Davidenko, J.M., Salomonsz, R., Baxter, W.T., Jalife, J.: Spiral waves of excitation underlie reentrant activity in isolated cardiac muscle. Circ. Res. 72, 631–650 (1993)CrossRefGoogle Scholar
  28. 28.
    Cherry, E.M., Fenton, F.H., Krogh-Madsen, T., Luther, S., Parlitz, U.: Introduction to focus issue complex cardiac dynamics (2017)Google Scholar
  29. 29.
    Cherry, E.M., Fenton, F.H.: Visualization of spiral and scroll waves in simulated and experimental cardiac tissue. New J. Phys. 10, 125016 (2008)CrossRefGoogle Scholar
  30. 30.
    Christini, D.J., Glass, L.: Introduction: mapping and control of complex cardiac arrhythmias. Chaos 12, 732–739 (2002)CrossRefGoogle Scholar
  31. 31.
    Gray, R.A., Pertsov, A.M., Jalife, J.: Spatial and temporal organization during cardiac fibrillation. Nature 392, 75 (1998)CrossRefGoogle Scholar
  32. 32.
    Takagaki, K., Zhang, C., Wu, J.-Y., Ohl, F.W.: Flow detection of propagating waves with temporospatial correlation of activity. J. Neurosci. Methods 200, 207–218 (2011)CrossRefGoogle Scholar
  33. 33.
    Schiff, S.J., Huang, X., Wu, J.-Y.: Dynamical evolution of spatiotemporal patterns in mammalian middle cortex. BMC Neurosci. 8, P61 (2007)CrossRefGoogle Scholar
  34. 34.
    Li, Y., Oku, M., He, G., Aihara, K.: Elimination of spiral waves in a locally connected chaotic neural network by a dynamic phase space constraint. Neural Netw. 88, 9–21 (2017)CrossRefGoogle Scholar
  35. 35.
    Ge, M., Jia, Y., Xu, Y., Yang, L.: Mode transition in electrical activities of neuron driven by high and low frequency stimulus in the presence of electromagnetic induction and radiation. Nonlinear Dyn. 91, 515–523 (2018)CrossRefGoogle Scholar
  36. 36.
    Li, J., Liu, S., Liu, W., Yu, Y., Wu, Y.: Suppression of firing activities in neuron and neurons of network induced by electromagnetic radiation. Nonlinear Dyn. 83, 801–810 (2016)MathSciNetCrossRefGoogle 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)CrossRefGoogle Scholar
  38. 38.
    Lv, M., Ma, J.: Multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Neurocomputing 205, 375–381 (2016)CrossRefGoogle Scholar
  39. 39.
    Ma, J., Mi, L., Zhou, P., Xu, Y., Hayat, T.: Phase synchronization between two neurons induced by coupling of electromagnetic field. Appl. Math. Comput. 307, 321–328 (2017)MathSciNetGoogle Scholar
  40. 40.
    Perez-Olivas, H., Cordova-Fraga, T., Gómez-Aguilar, F., Rosas-Padilla, E., Lopez-Briones, S., Espinoza-García, A., Villagómez-Castro, J., Bernal-Alvarado, J., Sosa-Aquino, M.: Magnetic exposure system to stimulate human lymphocytes proliferation. In: AIP Conference Proceedings, Volume 1494, pp. 146–148. AIP (2012)Google Scholar
  41. 41.
    Rastogi, P., Lee, E., Hadimani, R.L., Jiles, D.C.: Transcranial magnetic stimulation-coil design with improved focality. AIP Adv. 7, 056705 (2017)CrossRefGoogle Scholar
  42. 42.
    Hindmarsh, J., Rose, R.: A model of the nerve impulse using two first-order differential equations. Nature 296, 162 (1982)CrossRefGoogle Scholar
  43. 43.
    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)CrossRefGoogle Scholar
  44. 44.
    Wu, J., Xu, Y., Ma, J.: Lévy noise improves the electrical activity in a neuron under electromagnetic radiation. PLoS ONE 12, e0174330 (2017)CrossRefGoogle Scholar
  45. 45.
    Lv, M., Wang, C., Ren, G., Ma, J., Song, X.: Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn. 85, 1479–1490 (2016)CrossRefGoogle Scholar
  46. 46.
    Chua, L.: Memristor-the missing circuit element. IEEE Trans. Circuit Theory 18, 507–519 (1971)CrossRefGoogle Scholar
  47. 47.
    Li, X., Rakkiyappan, R., Velmurugan, G.: Dissipativity analysis of memristor-based complex-valued neural networks with time-varying delays. Inf. Sci. 294, 645–665 (2015)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Strukov, D.B., Snider, G.S., Stewart, D.R., Williams, R.S.: The missing memristor found. Nature 453, 80 (2008)CrossRefGoogle Scholar
  49. 49.
    Rakkiyappan, R., Sivasamy, R., Li, X.: Synchronization of identical and nonidentical memristor-based chaotic systems via active backstepping control technique. Circuits Syst. Signal Process. 34, 763–778 (2015)CrossRefGoogle Scholar
  50. 50.
    Kandel, E.R., Schwartz, J.H., Jessell, T.M., Siegelbaum, S.A., Hudspeth, A.J., et al.: Principles of Neural Science, vol. 4. McGraw-Hill, New York (2000)Google Scholar
  51. 51.
    Shu, Y., Duque, A., Yu, Y., Haider, B., McCormick, D.A.: Properties of action-potential initiation in neocortical pyramidal cells: evidence from whole cell axon recordings. J. Neurophysiol. 97, 746–760 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Biomedical Engineering DepartmentAmirkabir University of TechnologyTehranIran
  2. 2.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  3. 3.CAMTP – Center for Applied Mathematics and Theoretical PhysicsUniversity of MariborMariborSlovenia
  4. 4.Complexity Science HubViennaAustria

Personalised recommendations