The European Physical Journal Special Topics

, Volume 225, Issue 13–14, pp 2601–2612 | Cite as

Transient spike adding in the presence of equilibria

Regular Article Numerical Continuation in Self-sustained Oscillators
Part of the following topical collections:
  1. Temporal and Spatio-Temporal Dynamic Instabilities: Novel Computational and Experimental Approaches

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. 1.
    L.A. Belyakov, Math. Notes Acad. Sci. USSR 36, 838 (1984)MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Burke, M. Desroches, A.M. Barry, T.J. Kaper, M.A. Kramer, J. Math. Neurosci. 2, 3 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    E.J. Doedel, Congr. Numer. 30, 265 (1981)MathSciNetGoogle Scholar
  4. 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. 5.
    N. Fenichel, J. Differ. Equations 31, 53 (1979)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    G. Hek, J. Math. Biol. 60, 347 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    E.M. Izhikevich, Int. J. Bifurc. Chaos 10, 1655 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    B. Krauskopf, T. Rieß, Nonlinearity 21, 1655 (2008)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    B. Krauskopf, K. Schneider, J. Sieber, S. Wieczorek, M. Wolfrum, Opt. Commun. 215, 367 (2003)ADSCrossRefGoogle Scholar
  10. 10.
    J. Nowacki, H.M. Osinga, K.T. Tsaneva-Atanasova, J. Math. Neurosci. 2, 7 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J. Nowacki, H.M. Osinga, K.T. Tsaneva-Atanasova, Neural Comput. 25, 877 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    H.M. Osinga, K.T. Tsaneva-Atanasova, Neuroendocrinology 264, 1301 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    H.M. Osinga, K.T. Tsaneva-Atanasova, Chaos 23, 046107 (2013)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    J. Rinzel, Proc. Int. Congress Math. 1 & 2, 1578 (1986)Google Scholar
  15. 15.
    K.T. Tsaneva-Atanasova, H.M. Osinga, T. Rieß, A. Sherman, J. Theor. Biol. 264, 1133 (2010)CrossRefGoogle Scholar
  16. 16.
    S. Wieczorek, B. Krauskopf, D. Lenstra, Phys. Rev. Lett. 88, 063901 (2002)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2016

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand

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