Skip to main content
SpringerLink
Log in

Exact solutions for Fitzhugh–Nagumo model of nerve excitation via Kudryashov method

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

This paper presents a number of new solutions obtained for solving the Fitzhugh–Nagumo model via the Kudryashov method. The merit of the presented method is finding the further solutions of the considering problems including soliton, periodic, kink, kink-singular wave solutions. Comparing our new results with other results show that our results give the further solutions. The proposed method also allow us to establish many new types of exact solutions. By utilizing the Maple software package, we show that all obtained solutions satisfy the conditions of the studied model. More importantly, the solutions found in this work can have significant applications in engineering and physics sciences.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Abbasbandy, S.: Soliton solutions for the Fitzhugh–Nagumo equation with the homotopy analysis method. Appl. Math. Model 32, 2706–2714 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Abdelrahman, M.A.E., Khater, M.M.A.: Exact traveling wave solutions for Fitzhugh–Nagumo equation and modified Liouville equation. Int. J. Comput. Appl. 113, 1–7 (2015)

    Google Scholar 

  • Baskonus, H.M.: New acoustic wave behaviors to the Davey–Stewartson equation with power-law nonlinearity arising in fluid dynamics. Nonlinear Dyn. 86, 177–183 (2016)

    Article  MathSciNet  Google Scholar 

  • Baskonus, H.M.: New complex and hyperbolic function solutions to the generalized double combined Sinh–Cosh–Gordon equation. AIP Conf. Proc. 1798, 020018 (2017). doi:10.1063/1.4972610

    Article  Google Scholar 

  • Baskonus, H.M., Bulut, H.: New hyperbolic function solutions for some nonlinear partial differential equation arising in mathematical physics. Entropy 17, 4255–4270 (2015)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Baskonus, H.M., Bulut, H.: Exponential prototype structures for (2+1)-dimensional Boiti–Leon–Pempinelli systems in mathematical physics. Waves Random Complex Media 26, 201–208 (2016)

    MathSciNet  Google Scholar 

  • Biswas, A.: 1-soliton solution of the generalized Zakharov–Kuznetsov modified equal width equation. Appl. Math. Lett. 22, 1775–1777 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Bulut, H., Baskonus, H.M., Cüvelek, E.: On the prototype solutions of symmetric regularized long wave equation by generalized Kudryashov method. Math. Lett. 1, 10–16 (2015)

    Google Scholar 

  • Dehghan, M., Manafian, J.: The solution of the variable coefficients fourth–order parabolic partial differential equations by homotopy perturbation method. Zeitschrift fr Naturforschung A 64a, 420–430 (2009)

    ADS  Google Scholar 

  • Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differ. Equ. J. 26, 448–479 (2010a)

    MATH  MathSciNet  Google Scholar 

  • Dehghan, M., Manafian, J., Saadatmandi, A.: Application of semi-analytic methods for the Fitzhugh–Nagumo equation, which models the transmission of nerve impulses. Math. Methods Appl. Sci. 33, 1384–1398 (2010b)

    MATH  MathSciNet  Google Scholar 

  • Dehghan, M., Manafian, J., Saadatmandi, A.: Application of the exp-function method for solving a partial differential equation arising in biology and population genetics. Int. J. Numer. Methods Heat Fluid Flow 21, 736–53 (2011)

    Article  MathSciNet  Google Scholar 

  • Dehghan, M., Manafian, J., Saadatmandi, A.: Analytical treatment of some partial differential equations arising in mathematical physics by using the exp-function method. Int. J. Mod. Phys. B 25, 2965–2981 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Fitzhugh, R.: Impulse and physiological states in models of nerve membrane. Biophys. J. 1, 445–466 (1961)

    Article  ADS  Google Scholar 

  • Gaiko, V.A.: Multiple limit cycle bifurcations of the FitzHugh–Nagumo neuronal model. Nonlinear Anal. 74, 7532–7542 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Hasseine, A., Barhoum, Z., Attarakih, M., Bart, H.J.: Analytical solutions of the particle breakage equation by the Adomian decomposition and the variational iteration methods. Adv. Powder Technol. 24, 252–256 (2013)

    Article  Google Scholar 

  • Islam, MdS, Khan, K., Arnous, A.H.: Generalized Kudryashov method for solving some \((3+1)\)-dimensional nonlinear evolution equations. New Trends Math. Sci. 3, 46–57 (2015)

    MathSciNet  Google Scholar 

  • Izhikevich, E.M.: Dynamical Systems in Neuro Science: The Geometry of Excitability and Bursting. The MIP Press, Cambridge (2007)

    Google Scholar 

  • Jing, J., Li, B.: Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrödinger equation with variable coefficients. Chin. Phys. B 22(1), 010303 (2013)

    Article  ADS  Google Scholar 

  • Kabir, M.M., Khajeh, A., Aghdam, E.A., YousefiKoma, A.: Modified Kudryashov method for finding exact solitary wave solutions of higher-order nonlinear equations. Math. Methods Appl. Sci. 34(2), 213–219 (2011). doi:10.1002/mma.1349

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Li, H., Guo, Y.: New exact solutions to the Fitzhugh–Nagumo equation. Appl. Math. Comput. 180, 524–528 (2006)

    MATH  MathSciNet  Google Scholar 

  • Manafian, J.: On the complex structures of the Biswas–Milovic equation for power, parabolic and dual parabolic law nonlinearities. Eur. Phys. J. Plus 130, 1–20 (2015)

    Article  Google Scholar 

  • Manafian, J., Lakestani, M.: Optical solitons with Biswas–Milovic equation for Kerr law nonlinearity. Eur. Phys. J. Plus 130, 1–12 (2015)

    Article  Google Scholar 

  • Manafian, J., Lakestani, M.: Solitary wave and periodic wave solutions for Burgers, Fisher, Huxley and combined forms of these equations by the (G’/G)-expansion method. Pramana 130, 31–52 (2015)

    Article  ADS  Google Scholar 

  • Mohyud-Din, S.T., Khan, Y., Faraz, N., Yildirim, A.: Exp-function method for solitary and periodic solutions of Fitzhugh–Nagumo equation. Int. J. Numer. Methods Heat Fluid Flow 22, 335–341 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Nagumo, J.S., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962)

    Article  Google Scholar 

  • Nawaz, T., Yildirim, A., Mohyud-Din, S.T.: Analytical solutions Zakharov–Kuznetsov equations. Adv. Powder Technol. 24, 252–256 (2013)

    Article  Google Scholar 

  • Nucci, M.C., Clarkson, P.A.: The nonclassical method is more general than the direct method for symmetry reductions: an example of the Fitzhugh–Nagumo equation. Phys. Lett. A 164, 49–56 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  • Rashidi, M.M., Hayat, T., Keimanesh, T., Yousefian, H.: A study on heat transfer in a second-grade fluid through a porous medium with the modified differential transform method. Heat Transf. Asian Res. 42, 31–45 (2013)

    Article  Google Scholar 

  • Ringkvist, M., Zhou, Y.: On the dynamical behaviour of FitzHugh–Nagumo systems: revisited. Nonlinear Anal. 71, 2667–2687 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Rocsoreanu, C., Georgescu, A., Giurgiteanu, N.: The FitzHugh–Nagumo Model: Bifurcation and Dynamics. Kluwer, Boston (2000)

    Book  MATH  Google Scholar 

  • Ryabov, P.N., Sinelshchikov, D.I., Kochanov, M.B.: Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations. Appl. Math. Comput. 218, 3965–3972 (2011)

    MATH  MathSciNet  Google Scholar 

  • Taghipour-Farshia, H., Frounchi, J., Ahmadiasl, N., Shahabi, P., Salekzamani, Y.: Assessment on selectivity of multi-contact cuff electrode for recording peripheral nerve signals using Fitzhugh–Nagumo model of nerve excitation. J. Back Musculoskelet. Rehabil. 29, 749–756 (2016)

    Article  Google Scholar 

  • Wang, H., Yu, Y., Zhao, R., Wang, S.: Two-parameter bifurcation in a two-dimensional simplified Hodgkin–Huxley model. Commun. Nonlinear Sci. Numer. Simul. 18, 184–193 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their valuable suggestions and comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran

    Mohammadreza Foroutan

  2. Young Researchers and Elite Club, Ilkhchi Branch, Islamic Azad University, Ilkhchi, Iran

    Jalil Manafian

  3. Department of Electrical Engineering, Ilkhchi Branch, Islamic Azad University, Ilkhchi, Iran

    Hamed Taghipour-Farshi

Authors
  1. Mohammadreza Foroutan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jalil Manafian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hamed Taghipour-Farshi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jalil Manafian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Foroutan, M., Manafian, J. & Taghipour-Farshi, H. Exact solutions for Fitzhugh–Nagumo model of nerve excitation via Kudryashov method. Opt Quant Electron 49, 352 (2017). https://doi.org/10.1007/s11082-017-1197-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-017-1197-y

Keywords

Search

Navigation