• Open Access

Calcium window currents, periodic forcing, and chaos: Understanding single neuron response with a discontinuous one-dimensional map

J. Laudanski, C. Sumner, and S. Coombes
Phys. Rev. E 82, 011924 – Published 26 July 2010

Abstract

Thalamocortical (TC) neurones are known to express the low-voltage activated, inactivating Ca2+ current IT. The triggering of this current underlies the generation of low threshold Ca2+ potentials that may evoke single or bursts of action potentials. Moreover, this current can contribute to an intrinsic slow (<1Hz) oscillation whose rhythm is partly determined by the steady state component of IT and its interaction with a leak current. This steady state, or window current as it is so often called, has received relatively little theoretical attention despite its importance in determining the electroresponsiveness and input-output relationship of TC neurones. In this paper, we introduce an integrate-and-fire spiking neuron model that includes a biophysically realistic model of IT. We briefly review the subthreshold bifurcation diagram of this model with constant current injection before moving on to consider its response to periodic forcing. Direct numerical simulations show that as well as the expected mode-locked responses there are regions of parameter space that support chaotic behavior. To reveal the mechanism by which the window current generates a chaotic response to periodic forcing we consider a piecewise linear caricature of the dynamics for the gating variables in the model of IT. This model can be analyzed in closed form and is shown to support an unstable set of periodic orbits. Trajectories are repelled from these organizing centers until they reach the threshold for firing. By determining the condition for a grazing bifurcation (at the border between a spiking and nonspiking event) we show how knowledge of the unstable periodic orbits (existence and stability) can be combined with the grazing condition to determine an effective one-dimensional map that captures the essentials of the chaotic behavior. This map is discontinuous and has strong similarities with the universal limit mapping in grazing bifurcations derived in the context of impacting mechanical systems [A. B. Nordmark, Phys. Rev. E 55, 266 (1997)].

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  • Received 26 April 2010

DOI:https://doi.org/10.1103/PhysRevE.82.011924

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Authors & Affiliations

J. Laudanski1,2, C. Sumner2, and S. Coombes1

  • 1School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
  • 2MRC Institute of Hearing Research, Science Road, Nottingham NG7 2RD, United Kingdom

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Vol. 82, Iss. 1 — July 2010

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