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A correspondence between the models of Hodgkin-Huxley and FitzHugh-Nagumo revisited

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Abstract.

We present the discussion on the possibility to scale the classical dimensionless FitzHugh-Nagumo model of neuronal self-sustained oscillations to the range of variables corresponding to the results, which are provided by the biophysically relevant reduced two-dimensional Hodgkin-Huxley equations (the Rinzel model). It is shown that there exists a relatively simple choice of affine transformation, which results in time-dependent solutions, which reproduce with a high accuracy the time course of the recovery variable and the sharp onsets (intervals of fast motions on a phase trajectories) of the voltage spikes. As for the latter, the reasons for unavoidable difference are discussed as well as a necessity of taking into account applied current values during such a scaling procedure.

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References

  1. R. FitzHugh, Biophys. J. 1, 445 (1961)

    Article  ADS  Google Scholar 

  2. J. Nagumo, S. Arimoto, S. Yoshizawa, Proc. IRE 50, 2061 (1962)

    Article  Google Scholar 

  3. E.M. Izhikevich, R. FitzHugh, Scholarpedia 1, 1349 (2006)

    Article  ADS  Google Scholar 

  4. A.L. Hodgkin, A.F. Huxley, J. Physiol. 117, 500 (1952)

    Article  Google Scholar 

  5. A.C. Scott, Rev. Mod. Phys. 47, 487 (1975)

    Article  ADS  Google Scholar 

  6. R.G. Casten, H. Cohen, P.A. Lagerstrom, Q. Appl. Math. 32, 365 (1975)

    MathSciNet  Google Scholar 

  7. T.B. Kepler, L.F. Abbott, E. Marder, Biol. Cybern. 66, 381 (1992)

    Article  Google Scholar 

  8. J. Rinzel, Fed. Proc. 44, 2944 (1985)

    Google Scholar 

  9. B. Lindner, J. Garca-Ojalvo, A. Neiman, L. Schimansky-Geier, Phys. Rep. 392, 321 (2004)

    Article  ADS  Google Scholar 

  10. P.E. Phillipson, P. Schuster, Int. J. Bifurcat. Chaos 15, 3851 (2005)

    Article  MathSciNet  Google Scholar 

  11. Y. Che et al., Chaos 22, 023139 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  12. A. Concha, R. Garrido, J. Comput. Nonlinear Dyn. 10, 021023 (2015)

    Article  Google Scholar 

  13. A.C. Ventura, G.B. Mindlin, S.P. Dawson, Phys. Rev. E 65, 046231 (2002)

    Article  ADS  Google Scholar 

  14. C. Iadecola, Nat. Rev. Neurosci. 5, 347 (2004)

    Article  Google Scholar 

  15. M.R. Metea, E.A. Newman, J. Neurosci. 26, 2862 (2006)

    Article  Google Scholar 

  16. D.E. Postnov, A.Y. Neganova, J.C.B. Jacobsen, N.-H. Holstein-Rathlou, O. Sosnovtseva, Eur. Phys. J. ST 222, 2667 (2013)

    Article  Google Scholar 

  17. R.C. Sotero, R. Martinez-Cancino, Neural Comp. 22, 969 (2010)

    Article  Google Scholar 

  18. R. Jolivet, J.S. Coggan, T. Allaman, P.J. Magistretti, PLoS Comput. Biol. 11, e1004036 (2016)

    Article  Google Scholar 

  19. K. Chhabria, V.S. Chakravarthy, Front Neurol. 7, 24 (2016)

    Article  Google Scholar 

  20. D.E. Postnov, L.S. Ryazanova, O.V. Sosnovtseva, Biosystems 89, 84 (2007)

    Article  Google Scholar 

  21. M. Hayati, M. Nouri, S. Haghiri, D. Abbott, IEEE Trans. Biomed. Circ. Syst. 10, 518 (2016)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Theoretical Physics, Kursk State University, Radishcheva st. 33, 305000, Kursk, Russia

    Eugene B. Postnikov

  2. Department of Mathematical Analysis and Applied Mathematics, Kursk State University, Radishcheva st. 33, 305000, Kursk, Russia

    Olga V. Titkova

Authors
  1. Eugene B. Postnikov
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  2. Olga V. Titkova
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Correspondence to Eugene B. Postnikov.

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Postnikov, E.B., Titkova, O.V. A correspondence between the models of Hodgkin-Huxley and FitzHugh-Nagumo revisited. Eur. Phys. J. Plus 131, 411 (2016). https://doi.org/10.1140/epjp/i2016-16411-1

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  • DOI: https://doi.org/10.1140/epjp/i2016-16411-1

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