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Model Neurons of Bifurcation Type 2

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An Introduction to Modeling Neuronal Dynamics

Part of the book series: Texts in Applied Mathematics ((TAM,volume 66))

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Abstract

A neuron is said to be of bifurcation type 2 if the transition that occurs as I crosses I c is a Hopf bifurcation [47, 75, 129]. Examples of model neurons of bifurcation type 2 include the classical Hodgkin-Huxley model, and the Erisir model described in Section 5.3 The Hopf bifurcation in the classical Hodgkin-Huxley model is analyzed in great detail in [67]. For numerical evidence that the transition from rest to firing in the Erisir model involves a subcritical Hopf bifurcation, see [17], and also Fig. 17.9.

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Notes

  1. 1.

    There are general methods for computing unstable periodic orbits in higher dimensions (see, for instance, [144]), but we won’t use them in this book.

  2. 2.

    See, however, also Section 15.1, where parameters a and τ n are given that yield a supercritical Hopf bifurcation.

Bibliography

  1. C. Börgers and B. Walker, Toggling between gamma-frequency activity and suppression of cell assemblies, Frontiers in Computational Neuroscience, 7, doi: 10.3389/fncom.2013.00033 (2013).

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  2. G. B. Ermentrout, Type I membranes, phase resetting curves, and synchrony, Neural Comp., 8 (1996), pp. 879–1001.

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  3. J. Guckenheimer and M. Myers, Computing Hopf bifurcations. II: Three examples from Neurophysiology, SIAM J. Sci. Comput., 17 (1996), pp. 1275–1301.

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  4. A. L. Hodgkin, The local changes associated with repetitive action in a non-medullated axon, J. Physiol. (London), 107 (1948), pp. 165–181.

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  5. J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations, in Methods in Neuronal Modeling, C. Koch and I. Segev, eds., Cambridge, MA, 1998, MIT Press, pp. 251–292.

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Authors and Affiliations

  1. Department of Mathematics, Tufts University, Medford, MA, USA

    Christoph Börgers

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  1. Christoph Börgers
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Börgers, C. (2017). Model Neurons of Bifurcation Type 2. In: An Introduction to Modeling Neuronal Dynamics. Texts in Applied Mathematics, vol 66. Springer, Cham. https://doi.org/10.1007/978-3-319-51171-9_14

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