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The Onset and Extinction of Neural Spiking: A Numerical Bifurcation Approach

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Abstract

We study the onset of neural spiking when the equilibrium rest state loses stability by the change of a critical parameter, the applied current. In the case of the well-known Morris-Lecar model, we start from a complete numerical study of the bifurcation diagram in the most relevant two-parameter range. This diagram includes all equilibrium and limit cycle bifurcations, thus correcting and completing earlier studies.

We discuss and classify the behavior of the spiking orbits, when increasing or decreasing the applied current. A complete classification can be extracted from the complete bifurcation diagram. It is based on three components: bifurcation type of the equilibrium at the loss of stability, subcritical behavior in the limit of decreasing the applied current and supercritical behavior in the limit of increasing the applied current.

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References

  • Bazykin AD (1985) Mathematical Biophysics of Interacting Populations. Nauka, Moscow (in Russian).

  • Bazykin AD (1998) Nonlinear Dynamics of Interacting Populations. World Scientific, Singapore.

    Google Scholar 

  • Crook SM, Ermentrout GB, Bower JM (1998) Spike frequency adaptation affects the synchronization properties of networks of cortical oscillators. Neural Comput. 10: 837–854.

    Google Scholar 

  • Dhooge A, Govaerts W, Kuznetsov Yu A (2003) matcont: A matlab package for numerical bifurcation analysis of ODEs. ACM TOMS 29(2): 141–164.

    Google Scholar 

  • Dhooge A, Govaerts W, Kuznetsov Yu A, Mestrom W, Riet AM (2003) A Continuation Toolbox in matlab. Manual, http://allserv.UGent.be/~ajdhooge/doc_cl_matcont.zip.

  • Doedel EJ, Champneys AR, Fairgrieve TF, Kuznetsov Yu A, Sandstede B, Wang XJ (1997–2000) auto97-auto2000: Continuation and Bifurcation Software for Ordinary Di.erential Equations (with HomCont). User’s Guide, (Concordia University, Montreal, Canada) http://sourceforge.net/projects/auto2000

  • Ermentrout GB (1996) Type I membranes, phase resetting curves and synchrony. Neural Comput. 8(5): 979–1001.

    Google Scholar 

  • Ermentrout GB, Pascal M, Gutkin B (2001) The Effects of Spike Frequency Adaptation and Negative Feedback on the Synchronization of Neural Oscillators. Neural Comput. 13(6): 1285–1310.

    Google Scholar 

  • Fall CP, Marland ES, Wagner JM and Tyson JJ (2002) Computational Cell Biology. Springer-Verlag, New York, NY.

    Google Scholar 

  • Govaerts W, Sautois B (2005) Bifurcation software in Matlab with applications in neuronal modeling. Comput. Meth. Prog. Bio. 77(2): 141–153.

    Google Scholar 

  • Hodgkin AL (1948) The local electric changes associated with repetitive action in a non-medullated axon. J. Physiol. 107: 165–181.

    Google Scholar 

  • Hos C, Champneys A, Kullmann L (2003) Bifurcation analysis of surge and rotating stall in the Moore-Greitzer compression system. IMA J. App. Math. 68: 205–228.

    Google Scholar 

  • Izhikevich EM (2000) Neural excitability, spiking and bursting. Int. J. Bifurcat. Chaos 10(6): 1171–1266.

    Google Scholar 

  • Khibnik A, Krauskopf B, Rousseau C (1998) Global study of a family of cubic Lie’nard equations. Nonlinearity 11: 1505–1519.

    Google Scholar 

  • Kuznetsov Yu, A (1998) Elements of Applied Bifurcation Theory. Springer-Verlag, New York, NY.

    Google Scholar 

  • Kuznetsov Yu, A Levitin VV (1997) content: Integrated Environment for Analysis of Dynamical Systems. www.math.uu.nl/people/kuznet/CONTENT (CWI, Amsterdam).

  • Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35: 193–213.

    Google Scholar 

  • Rinzel J, Ermentrout B (1989) Analysis of Neural Excitability and Oscillations. In: Koch C, Segev I, eds., Methods in Neuronal Modeling: From Synapses to Networks, MIT Press, Cambridge, MA, pp. 135–169.

    Google Scholar 

  • Tsumoto K, Yoshinaga T, Aihara K, Kawakami H (2002) Bifurcations in Morris-Lecar neuron model. Proceedings of the Int. Symp. on Nonlinear Theory and its Appls., Xi’an, China, pp. 363–366.

  • van Vreeswijk C, Hansel, D (2001) Patterns of synchrony in neural networks with spike adaptation. Neural Comput. 13: 959–992.

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000, Ghent, Belgium

    W. Govaerts & B. Sautois

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  1. W. Govaerts
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  2. B. Sautois
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Correspondence to W. Govaerts.

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Govaerts, W., Sautois, B. The Onset and Extinction of Neural Spiking: A Numerical Bifurcation Approach. J Comput Neurosci 18, 265–274 (2005). https://doi.org/10.1007/s10827-005-0328-9

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  • DOI: https://doi.org/10.1007/s10827-005-0328-9

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