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Transient spike adding in the presence of equilibria

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  • Numerical Continuation in Self-sustained Oscillators
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Abstract

Many models of neuronal activity exhibit complex oscillations in response to an input from other neurons in a network or to an input from a stimulus. We consider the effect of a single short stimulus on a simple model designed to mimic some features of neuronal dynamics. We focus on the transient response induced by the stimulus, particularly on the spike-adding behaviour of the response. Our main goal is to explain how the transient response is affected by the presence of unstable equilibria. We also investigate the dependence of the number of spikes on the amplitude and duration of the stimulus. In our analysis, we use numerical continuation methods and exploit the presence of different time scales in the model.

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References

  1. L.A. Belyakov, Math. Notes Acad. Sci. USSR 36, 838 (1984)

    Article  MathSciNet  Google Scholar 

  2. J. Burke, M. Desroches, A.M. Barry, T.J. Kaper, M.A. Kramer, J. Math. Neurosci. 2, 3 (2012)

    Article  MathSciNet  Google Scholar 

  3. E.J. Doedel, Congr. Numer. 30, 265 (1981)

    MathSciNet  Google Scholar 

  4. E.J. Doedel, AUTO-07P: Continuation and bifurcation software for ordinary differential equations, with major contributions from A.R. Champneys, T.F. Fairgrieve, Yu.A. Kuznetsov, B.E. Oldeman, R.C. Paffenroth, B. Sandstede, X.J. Wang and C. Zhang (http://cmvl.cs.concordia.ca/auto/) (2007)

  5. N. Fenichel, J. Differ. Equations 31, 53 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  6. G. Hek, J. Math. Biol. 60, 347 (2010)

    Article  MathSciNet  Google Scholar 

  7. E.M. Izhikevich, Int. J. Bifurc. Chaos 10, 1655 (2000)

    Article  MathSciNet  Google Scholar 

  8. B. Krauskopf, T. Rieß, Nonlinearity 21, 1655 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  9. B. Krauskopf, K. Schneider, J. Sieber, S. Wieczorek, M. Wolfrum, Opt. Commun. 215, 367 (2003)

    Article  ADS  Google Scholar 

  10. J. Nowacki, H.M. Osinga, K.T. Tsaneva-Atanasova, J. Math. Neurosci. 2, 7 (2012)

    Article  MathSciNet  Google Scholar 

  11. J. Nowacki, H.M. Osinga, K.T. Tsaneva-Atanasova, Neural Comput. 25, 877 (2013)

    Article  MathSciNet  Google Scholar 

  12. H.M. Osinga, K.T. Tsaneva-Atanasova, Neuroendocrinology 264, 1301 (2010)

    Article  MathSciNet  Google Scholar 

  13. H.M. Osinga, K.T. Tsaneva-Atanasova, Chaos 23, 046107 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  14. J. Rinzel, Proc. Int. Congress Math. 1 & 2, 1578 (1986)

    Google Scholar 

  15. K.T. Tsaneva-Atanasova, H.M. Osinga, T. Rieß, A. Sherman, J. Theor. Biol. 264, 1133 (2010)

    Article  Google Scholar 

  16. S. Wieczorek, B. Krauskopf, D. Lenstra, Phys. Rev. Lett. 88, 063901 (2002)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    Saeed Farjami, Vivien Kirk & Hinke M. Osinga

Authors
  1. Saeed Farjami
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  2. Vivien Kirk
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  3. Hinke M. Osinga
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Correspondence to Saeed Farjami, Vivien Kirk or Hinke M. Osinga.

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Farjami, S., Kirk, V. & Osinga, H.M. Transient spike adding in the presence of equilibria. Eur. Phys. J. Spec. Top. 225, 2601–2612 (2016). https://doi.org/10.1140/epjst/e2015-50332-9

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  • DOI: https://doi.org/10.1140/epjst/e2015-50332-9

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