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A geometric understanding of how fast activating potassium channels promote bursting in pituitary cells

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Abstract

The electrical activity of endocrine pituitary cells is mediated by a plethora of ionic currents and establishing the role of a single channel type is difficult. Experimental observations have shown however that fast-activating voltage- and calcium-dependent potassium (BK) current tends to promote bursting in pituitary cells. This burst promoting effect requires fast activation of the BK current, otherwise it is inhibitory to bursting. In this work, we analyze a pituitary cell model in order to answer the question of why the BK activation must be fast to promote bursting. We also examine how the interplay between the activation rate and conductance of the BK current shapes the bursting activity. We use the multiple timescale structure of the model to our advantage and employ geometric singular perturbation theory to demonstrate the origin of the bursting behaviour. In particular, we show that the bursting can arise from either canard dynamics or slow passage through a dynamic Hopf bifurcation. We then compare our theoretical predictions with experimental data using the dynamic clamp technique and find that the data is consistent with a burst mechanism due to a slow passage through a Hopf.

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Notes

  1. We do not draw the bursting/plateau boundary since in AUTO, there seems to be no way of distinguishing between these two types of trajectories.

  2. We point out that for our model equations, det J| S and tr J| S are both independent of the calcium variable c.

  3. Folded nodes have a sector of canards (the funnel). Folded saddles have precisely two canards (tangent to the eigendirections of the folded saddle) and folded foci have no canards.

  4. In the classic slow-fast approach to bursting, the criticality of the fast subsystem Hopf differentiates between plateau and pseudo-plateau bursting. Plateau (pseudo-plateau) bursts are associated with supercritical (subcritical) Hopf bifurcations of the layer problem. For further details, we refer to Stern et al. (2008), Osinga and Tsaneva-Atanasova (2010), Tsaneva-Atanasova et al. (2010), Teka et al. (2011).

  5. Variations in r have no effect on the singular canards since they are associated with the slow subsystem and r is only present in the fast subsystem.

  6. In our case, the homoclinic associated with the BT bifurcation has no influence on the full system dynamics.

  7. The BT bifurcation is a codimension 2 bifurcation. The reason we are able to draw a curve of BT bifurcations is that detJ| S and tr J| S are both independent of c

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Acknowledgments

TV was partially supported by an A.E. and F.A.Q. Stephens Scholarship and a Philipp Hofflin International Research Scholarship (University of Sydney). TV is also grateful to Florida State University for its hospitality, where this work was carried out. RB and JT were supported by NSF grant DMS 1220063. MW was partially supported by the Australian Research Council and the Marsden Fund, New Zealand.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia

    Theodore Vo & Martin Wechselberger

  2. Department of Mathematics, Florida State University, Tallahassee, FL, USA

    Joël Tabak

  3. Department of Mathematics and Programs in Neuroscienceand Molecular Biophysics, Florida State University, Tallahassee, FL, USA

    Richard Bertram

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  1. Theodore Vo
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Correspondence to Theodore Vo.

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Vo, T., Tabak, J., Bertram, R. et al. A geometric understanding of how fast activating potassium channels promote bursting in pituitary cells. J Comput Neurosci 36, 259–278 (2014). https://doi.org/10.1007/s10827-013-0470-8

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