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Bifurcation Analysis

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Encyclopedia of Computational Neuroscience

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References

  • Barnett W, Cymbalyuk G (2014) A codimension-2 bifurcation controlling endogenous bursting activity and pulse-triggered responses of a neuron model. PLoS One 9:e85451

    Article  PubMed Central  PubMed  Google Scholar 

  • Booth V, Carr TW, Erneux T (1997) Near threshold bursting is delayed by a slow passage near a limit point. SIAM J Appl Math 57:1406–1420

    Article  Google Scholar 

  • Doiron B, Laing C, Longtin A, Maler L (2003) Ghostbursting: a novel neuronal burst mechanism. J Comput Neurosci 12:5–25

    Article  Google Scholar 

  • Ermentrout G, Terman D (2010) Mathematical foundations of neuroscience, vol 35, Interdisciplinary applied mathematics. Springer, New York

    Google Scholar 

  • Ghigliazza R, Holmes P (2004) Minimal models of bursting neurons: how multiple currents, conductances, and timescales affect bifurcation diagrams. SIAM J Appl Dyn Syst 3:636–670

    Article  Google Scholar 

  • Guckenheimer J, Gueron S, Harris-Warrick R (1993) Mapping the dynamics of a bursting neuron. Phil Trans R Soc Lond 341:345–359

    Article  CAS  Google Scholar 

  • Guckenheimer J, Harris-Warrick R, Peck J, Willms A (1997) Bifurcation, bursting, and spike frequency adaptation. J Comput Neurosci 4:257–277

    Article  CAS  PubMed  Google Scholar 

  • Gutkin BS, Ermentrout GB (1998) Dynamics of membrane excitability determine interspike interval variability: a link between spike generation mechanisms and cortical spike train statistics. Neural Comput 10:1047–1065

    Article  CAS  PubMed  Google Scholar 

  • Izhikevich E (2007) Dynamical systems in neuroscience. MIT Press, Cambridge, MA

    Google Scholar 

  • Lukyanov V, Shilnikov L (1978) On some bifurcations of dynamical systems with homoclinic structures. Soviet Math Dokl 19:1314

    Google Scholar 

  • Medvedev GS (2006) Transition to bursting via deterministic chaos. Phys Rev Lett 97:048102

    Article  PubMed  Google Scholar 

  • Rabinovich M, Huerta R, Laurent G (2008) Transient dynamics for neural processing. Science 321:48–50

    Article  CAS  PubMed  Google Scholar 

  • Rinzel J (1978) On repetitive activity in nerve. Fed Proc 37:2793–2802

    CAS  PubMed  Google Scholar 

  • Rinzel J (1987) Mathematical topics in population biology, morphogenesis, and neuroscience, vol 71, Lecture notes in biomathematics. Springer, Berlin

    Google Scholar 

  • Rinzel J, Ermentrout B (1998) Analysis of neural excitability and oscillations. In: Koch C, Segev I (eds) Methods in neural modeling. The MIT Press, Cambridge, MA, pp 251–292

    Google Scholar 

  • Roa MAD, Copelli M, Kinouchi O, Caticha N (2007) Scaling law for the transient behavior of type-II neuron models. Phys Rev E 75:021911

    Article  CAS  Google Scholar 

  • Shilnikov AL, Cymbalyuk GS (2005) Transition between tonic-spiking and bursting in a neuron model via the blue-sky catastrophe. Phys Rev Lett 94:048101

    Article  PubMed  Google Scholar 

  • Shilnikov L, Turaev D (2000) A new simple bifurcation of a periodic orbit of blue sky catastrophe type. Am Math Soc Transl II Ser 200:165–188

    Google Scholar 

  • Shilnikov AL, Calabrese RL, Cymbalyuk GS (2005) Mechanism of bistability: tonic spiking and bursting in a neuron model. Phys Rev E 71:056214

    Article  Google Scholar 

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Acknowledgment

This work was supported by National Science Foundation grant PHY-0750456.

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Correspondence to Gennady Cymbalyuk .

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Barnett, W., Cymbalyuk, G. (2014). Bifurcation Analysis. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7320-6_156-1

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  • DOI: https://doi.org/10.1007/978-1-4614-7320-6_156-1

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  • Online ISBN: 978-1-4614-7320-6

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