Advertisement

Symmetry breaking in solitary solutions to the Hodgkin–Huxley model

  • Tadas TelksnysEmail author
  • Zenonas Navickas
  • Inga Timofejeva
  • Romas Marcinkevicius
  • Minvydas Ragulskis
Original Paper
  • 63 Downloads

Abstract

This paper presents necessary and sufficient conditions for the existence of bright/dark solitary solutions in the Hodgkin–Huxley model. The second-order analytic solitary solutions are derived using the generalized differential operator technique. It is shown that the heteroclinic bifurcation in the Hodgkin–Huxley model yields a symmetry breaking effect. Trajectories of solitary solutions before the bifurcation lie on manifolds of one of the saddle points and the separatrix between periodic and non-periodic solutions. A new separatrix emerges after the heteroclinic bifurcation—but solitary solutions do not lie on this trajectory. This symmetry breaking effect is demonstrated using analytic and computational experiments.

Keywords

Solitary solution Hodgkin–Huxley model Generalized differential operator Heteroclinic bifurcation 

Mathematics Subject Classification

35B32 35C08 

Notes

Compliance with ethical standards

Conflicts of interest

The authors declare no conflict of interest.

References

  1. 1.
    Amsallem, D., Nordstrom, J.: High-order accurate difference schemes for the Hodgkin–Huxley equations. J. Comput. Phys. 252, 573–590 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    FitzHugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophys. 17(4), 257–278 (1955)CrossRefGoogle Scholar
  3. 3.
    Gao, H., Zhao, R.X.: New exact solutions to the generalized Burgers–Huxley equation. Appl. Math. Comput. 217, 1598–1603 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Hanslien, M., Karlsen, K.H., Tveito, A.: A maximum principle for an explicit finite difference scheme approximating the Hodgkin–Huxley model. BIT Numer. Math. 45, 725–741 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Heimburg, T., Jackson, A.D.: On soliton propagation in biomembranes and nerves. Proc. Nat. Acad. Sci. 102(28), 9790–9795 (2005)CrossRefGoogle Scholar
  6. 6.
    Hines, M.: Efficient computation of branched nerve equations. J. BioMed. Comput. 15, 69–76 (1984)CrossRefGoogle Scholar
  7. 7.
    Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500–544 (1952)CrossRefGoogle Scholar
  8. 8.
    Johnston, D., Wu, S.M.S.: Foundations of Cellular Neurophysiology. MIT press, Cambridge (1994)Google Scholar
  9. 9.
    Jun, M., Li-Jian, Y., Ying, W., Cai-Rong, Z.: Spiral wave in small-world networks of Hodgkin–Huxley neurons. Commun. Theor. Phys. 54(3), 583 (2010)CrossRefGoogle Scholar
  10. 10.
    Krisnangkura, M., Chinviriyasit, S., Chinviriyasit, W.: Analytic study of the generalized Burgers–Huxley equation by hyperbolic tangent method. Appl. Math. Comput. 218, 10843–10847 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kurakin, V.L., Kuzmin, A.S., Mikhalev, A.V., Nechaev, A.A.: Linear recurring sequences over rings and modules. J. Math. Sci. 76, 2793–2915 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lautrup, B., Appali, R., Jackson, A.D., Heimburg, T.: The stability of solitons in biomembranes and nerves. Eur. Phys. J. E 34(6), 57 (2011)CrossRefGoogle Scholar
  13. 13.
    Lazar, A.A.: Information representation with an ensemble of Hodgkin–Huxley neurons. Neurocomputing 70(10–12), 1764–1771 (2007)CrossRefGoogle Scholar
  14. 14.
    Majtanik, M., Dolan, K., Tass, P.A.: Desynchronization in networks of globally coupled neurons with dendritic dynamics. J. Biol. Phys. 32(3–4), 307–333 (2006)CrossRefGoogle Scholar
  15. 15.
    Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50(10), 2061–2070 (1962)CrossRefGoogle Scholar
  16. 16.
    Nagy, A.N., Sweilan, N.H.: An efficient method for solving fractional Hodgkin–Huxley model. Phys. Lett. A 378, 1980–1984 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Navickas, Z., Bikulciene, L.: Expressions of solutions of ordinary differential equations by standard functions. Math. Model Anal. 11, 399–412 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Navickas, Z., Bikulciene, L., Ragulskis, M.: Generalization of Exp-function and other standard function methods. Appl. Math. Comput. 216, 2380–2393 (2010)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Navickas, Z., Ragulskis, M., Bikulciene, L.: Be careful with the Exp-function method—additional remarks. Commun. Nonlinear Sci. Numer. Simul. 15, 3874–3886 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Navickas, Z., Ragulskis, M., Bikulciene, L.: Special solutions of Huxley differential equation. Math. Model Anal. 16, 248–259 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Navickas, Z., Ragulskis, M., Telksnys, T.: Existence of solitary solutions in a class of nonlinear differential equations with polynomial nonlinearity. Appl. Math. Comput. 283, 333–338 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Navickas, Z., Telksnys, T., Ragulskis, M.: Comments on “the exp-function method and generalized solitary solutions”. Comput. Math. Appl. 69(8), 798–803 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pang, J.C.S., Monterola, C.P., Bantang, J.Y.: Noise-induced synchronization in a lattice Hodgkin–Huxley neural network. Phys. A Stat. Mech. Appl. 393, 638–645 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Parand, K., Rad, J.A.: Exp-function method for some nonlinear PDE’s and a nonlinear ODE’s. J. King Saud Univ. Sci. 24, 1–10 (2012)CrossRefGoogle Scholar
  25. 25.
    Sakyte, E., Ragulskis, M.: Self-calming of a random network of dendritic neurons. Neurocomputing 74(18), 3912–3920 (2011)CrossRefGoogle Scholar
  26. 26.
    Scott, A. (ed.): Encyclopedia of Nonlinear Science. Routledge, New York (2004)zbMATHGoogle Scholar
  27. 27.
    Tass, P.: Effective desynchronization with a resetting pulse train followed by a single pulse. EPL Europhy. Lett. 55(2), 171 (2001)CrossRefGoogle Scholar
  28. 28.
    Wojcik, G.M., Kaminski, W.A.: Liquid state machine built of Hodgkin–Huxley neurons and pattern recognition. Neurocomputing 58, 245–251 (2004)CrossRefzbMATHGoogle Scholar
  29. 29.
    Ganji, Z.Z., Ganji, D.D., Asgari, A.: Finding general and explicit solutions of high nonlinear equations by the Exp-Function method. Comput. Math. Appl. 58, 2124–2130 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Center for Nonlinear SystemsKaunas University of TechnologyKaunasLithuania
  2. 2.Department of Software EngineeringKaunas University of TechnologyKaunasLithuania

Personalised recommendations