Nonlinear Dynamics

, Volume 95, Issue 1, pp 433–444 | Cite as

Electromagnetic induction on a map-based action potential model

  • G. S. Bortolotto
  • R. V. Stenzinger
  • M. H. R. TragtenbergEmail author
Original Paper


Neurons and cardiac cells are known to be susceptible to electromagnetic radiation. Although many mathematical models exist to represent these cells, only recently there was an effort to include the electromagnetic induction on the membrane potential equations. In this paper, we investigate the effects of the induction on the logistic KTz, a computationally efficient map-based action potential model, and compare them to the more widely used Hindmarsh–Rose model. We study the effects of a self-induced current on a single cell and the synchronization of cells coupled through an induction current caused by the magnetic flux of the neighbor. We also study the emergence of aperiodic behaviors and the presence of chaos, as an effect of the inclusion of the induction. Besides, we use a simple network of KTz elements to show that the electromagnetic induction is relevant for the study of pattern formation. Additionally, we report for the first time the presence of cardiac spikes in the Hindmarsh–Rose model. Our results demonstrate the importance of implementing the induction current on different models and we provide a computationally efficient alternative to better understand how the induction acts on neuronal and cardiac cells.


Electromagnetic induction Synchronization Lyapunov exponents Computational efficiency Map-based neuron model Spiral waves 



Author G.S.B. thank financial support from FAPESC. Author R.V.S. thank financial support from CNPq, FAPESC and CAPES. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Akiyama, H., Shimizu, Y., Miyakawa, H., Inoue, M.: Extracellular DC electric fields induce nonuniform membrane polarization in rat hippocampal CA1 pyramidal neurons. Brain Res. 1383, 22–35 (2011)CrossRefGoogle Scholar
  2. 2.
    Alonso, S., Bär, M., Echebarria, B.: Nonlinear physics of electrical wave propagation in the heart: a review. Rep. Prog. Phys. 79(9), 096601 (2016)CrossRefGoogle Scholar
  3. 3.
    Eckmann, J., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57(3), 617–656 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Fenton, F., Cherry, E.: Models of cardiac cell. Scholarpedia 3(8), 1868 (2008)CrossRefGoogle Scholar
  5. 5.
    Fenton, F.H., Cherry, E.M., Hastings, H.M., Evans, S.J.: Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. Chaos 12(3), 852–892 (2002)CrossRefGoogle Scholar
  6. 6.
    Frederickson, P., Kaplan, J.L., Yorke, E.D., Yorke, J.A.: The Liapunov dimension of strange attractors. J. Differ. Equ. 49(2), 185–207 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Girardi-Schappo, M., Bortolotto, G.S., Stenzinger, R.V., Gonsalves, J.J., Tragtenberg, M.H.R.: Phase diagrams and dynamics of a computationally efficient map-based neuron model. PLoS ONE 12(3), e0174621 (2017)CrossRefGoogle Scholar
  8. 8.
    Hindmarsh, J.L., Rose, R.M.: A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B Biol. Sci. 221(1222), 87–102 (1984)CrossRefGoogle Scholar
  9. 9.
    Huang, X., Troy, W., Yang, Q., Ma, H., Laing, C., Schiff, S., Wu, J.Y.: Spiral waves in mammalian neocortex. J. Neurosci. 24(44), 9897–9902 (2004)CrossRefGoogle Scholar
  10. 10.
    Huang, X., Xu, W., Liang, J., Takagaki, K., Gao, X., Wu, J.Y.: Spiral wave dynamics in neocortex. Neuron 68(5), 978–990 (2010)CrossRefGoogle Scholar
  11. 11.
    Itoh, M., Chua, L.O.: Memristor oscillators. Int. J. Bifurc. Chaos 18(11), 3183–3206 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Izhikevich, E.: Which model to use for cortical spiking neurons? IEEE Trans. Neural Netw. 15(5), 1063–1070 (2004)CrossRefGoogle Scholar
  13. 13.
    Krasteva, V.T., Papazov, S.P., Daskalov, I.K.: Peripheral nerve magnetic stimulation: influence of tissue non-homogeneity. BioMed Eng Online 2(1), 19 (2003)CrossRefGoogle Scholar
  14. 14.
    Kuva, S.M., Lima, G.F., Kinouchi, O., Tragtenberg, M.H.R., Roque, A.C.: A minimal model for excitable and bursting elements. Neurocomputing 38–40, 255–261 (2001)CrossRefGoogle Scholar
  15. 15.
    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(1–2), 801–810 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Luo, C.H., Rudy, Y.: Original contributions a model of the ventricular cardiac action potential. Circ. Res. 68(6), 1501–1526 (1991)CrossRefGoogle Scholar
  17. 17.
    Lv, M., Ma, J.: Multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Neurocomputing 205, 375–381 (2016)CrossRefGoogle Scholar
  18. 18.
    Lv, M., Wang, C., Ren, G., Ma, J., Song, X.: Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn. 85(3), 1479–1490 (2016)CrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    Ma, J., Tang, J.: A review for dynamics in neuron and neuronal network. Nonlinear Dyn. 89(3), 1569–1578 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ma, J., Wang, Y., Wang, C., Xu, Y., Ren, G.: Mode selection in electrical activities of myocardial cell exposed to electromagnetic radiation. Chaos Solitons Fractals 99, 219–225 (2017)CrossRefGoogle Scholar
  22. 22.
    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, 508–516 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Reato, D., Rahman, A., Bikson, M., Parra, L.C.: Low-intensity electrical stimulation affects network dynamics by modulating population rate and spike timing. J. Neurosci. 30(45), 15067–15079 (2010)CrossRefGoogle Scholar
  24. 24.
    Rosenblum, M., Pikovsky, A., Kurths, J.: From phase to lag synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 78(22), 4193–4196 (1997)zbMATHCrossRefGoogle Scholar
  25. 25.
    Roth, A., van Rossum, M.C.W.: Modeling synapses. In: De Schutter, E. (ed.) Computational Modeling Methods for Neuroscientists, pp. 139–160. MIT Press, Cambridge (2009)CrossRefGoogle Scholar
  26. 26.
    Storace, M., Linaro, D., De Lange, E.: The Hindmarsh–Rose neuron model: bifurcation analysis and piecewise-linear approximations. Chaos 18(3), 033128 (2008)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, Boulder (2014)zbMATHGoogle Scholar
  28. 28.
    Wu, F., Wang, C., Xu, Y., Ma, J.: Model of electrical activity in cardiac tissue under electromagnetic induction. Sci. Rep. 6(1), 28 (2016)CrossRefGoogle Scholar
  29. 29.
    Wu, J., Xu, Y., Ma, J.: Lévy noise improves the electrical activity in a neuron under electromagnetic radiation. PLoS ONE 12(3), e0174330 (2017)CrossRefGoogle Scholar
  30. 30.
    Xu, Y., Ying, H., Jia, Y., Ma, J., Hayat, T.: Autaptic regulation of electrical activities in neuron under electromagnetic induction. Sci. Rep. 7(1), 43452 (2017)CrossRefGoogle Scholar
  31. 31.
    Ye, H., Steiger, A.: Neuron matters: electric activation of neuronal tissue is dependent on the interaction between the neuron and the electric field. J. Neuroeng. Rehabil. 12(1), 65 (2015)CrossRefGoogle Scholar
  32. 32.
    Zhao, Y., Sun, X., Liu, Y., Kurths, J.: Phase synchronization dynamics of coupled neurons with coupling phase in the electromagnetic field. Nonlinear Dyn. 93, 1315–1324 (2018)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidade Federal de Santa CatarinaFlorianópolisBrazil

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