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.
Similar content being viewed by others
References
Amsallem, D., Nordstrom, J.: High-order accurate difference schemes for the Hodgkin–Huxley equations. J. Comput. Phys. 252, 573–590 (2013)
FitzHugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophys. 17(4), 257–278 (1955)
Gao, H., Zhao, R.X.: New exact solutions to the generalized Burgers–Huxley equation. Appl. Math. Comput. 217, 1598–1603 (2010)
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)
Heimburg, T., Jackson, A.D.: On soliton propagation in biomembranes and nerves. Proc. Nat. Acad. Sci. 102(28), 9790–9795 (2005)
Hines, M.: Efficient computation of branched nerve equations. J. BioMed. Comput. 15, 69–76 (1984)
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)
Johnston, D., Wu, S.M.S.: Foundations of Cellular Neurophysiology. MIT press, Cambridge (1994)
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)
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)
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)
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)
Lazar, A.A.: Information representation with an ensemble of Hodgkin–Huxley neurons. Neurocomputing 70(10–12), 1764–1771 (2007)
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)
Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50(10), 2061–2070 (1962)
Nagy, A.N., Sweilan, N.H.: An efficient method for solving fractional Hodgkin–Huxley model. Phys. Lett. A 378, 1980–1984 (2014)
Navickas, Z., Bikulciene, L.: Expressions of solutions of ordinary differential equations by standard functions. Math. Model Anal. 11, 399–412 (2006)
Navickas, Z., Bikulciene, L., Ragulskis, M.: Generalization of Exp-function and other standard function methods. Appl. Math. Comput. 216, 2380–2393 (2010)
Navickas, Z., Ragulskis, M., Bikulciene, L.: Be careful with the Exp-function method—additional remarks. Commun. Nonlinear Sci. Numer. Simul. 15, 3874–3886 (2010)
Navickas, Z., Ragulskis, M., Bikulciene, L.: Special solutions of Huxley differential equation. Math. Model Anal. 16, 248–259 (2011)
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)
Navickas, Z., Telksnys, T., Ragulskis, M.: Comments on “the exp-function method and generalized solitary solutions”. Comput. Math. Appl. 69(8), 798–803 (2015)
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)
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)
Sakyte, E., Ragulskis, M.: Self-calming of a random network of dendritic neurons. Neurocomputing 74(18), 3912–3920 (2011)
Scott, A. (ed.): Encyclopedia of Nonlinear Science. Routledge, New York (2004)
Tass, P.: Effective desynchronization with a resetting pulse train followed by a single pulse. EPL Europhy. Lett. 55(2), 171 (2001)
Wojcik, G.M., Kaminski, W.A.: Liquid state machine built of Hodgkin–Huxley neurons and pattern recognition. Neurocomputing 58, 245–251 (2004)
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)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Apppendix A: Details of the application of the inverse balancing technique to the Hodgkin–Huxley equation
Setting the solitary solution order to \(m = 2\) transforms (26) into:
Taking \({\widehat{x}} = {\widehat{c}} x_1\), \({\widehat{x}} = {\widehat{c}} x_2\), \({\widehat{x}} = {\widehat{c}} y_1\), \({\widehat{x}} = {\widehat{c}} y_2\), \({\widehat{x}} = {\widehat{c}} (x_1+x_2)/2\),\({\widehat{x}} = {\widehat{c}} (y_1+y_2)/2\) and \({\widehat{x}} = 0\) results in seven linear equations. The solutions to these equations with respect to \(b, {\widehat{a}}_0, \dots , {\widehat{a}}_3\) are given as follows:
where
where
Appendix B: Construction of solitary solutions to (41)
An independent variable transformation (9) is applied to (41):
Initial conditions (37) are transformed to:
The generalized differential operator with respect to (56) reads:
Computing Hankel determinants (16) for the sequence \({\widehat{p}}_j := \frac{1}{j!} \varvec{\mathrm {D}}^j u; \, j = 0,1,\dots \), yields the following relation:
where \(f\left( \eta , u\right) \) is a polynomial in \(\eta , u\). The above equation yields that \(d_4 = 0\) only if:
Equation (56) can admit closed-form solutions only if conditions (19) hold true. To verify (19), the characteristic roots \({\widehat{\rho }}_k, k = 1,2,3\), are computed from the characteristic polynomial:
Solution to (61) reads:
Formula (18) together with (62)–(64) yields:
Note that \(0^0 := 1\). Definition of the generalized differential operator \(\varvec{\mathrm {D}}\) and sequence \({\widehat{p}}_j\) for \(j = 0,1,2\) results in:
Using (62)–(64) and (66)–(68), it is verified that conditions (19) hold true; thus, the solution to (56) reads:
Using the inverse of transformation (9) yields the solution to (41):
where
and
The functions \(K(u), L(u)\) have the following expressions:
Rights and permissions
About this article
Cite this article
Telksnys, T., Navickas, Z., Timofejeva, I. et al. Symmetry breaking in solitary solutions to the Hodgkin–Huxley model. Nonlinear Dyn 97, 571–582 (2019). https://doi.org/10.1007/s11071-019-04998-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-019-04998-4