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

Thalamocortical (TC) neurons 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 (< 1 Hz) oscillation whose rhythm is largely determined by the steady state component of IT and its interaction with a leak current [1]. This steady state, or window current as it is so often called, has received relatively little theoretical attention despite its importance in determining the electro-responsiveness and input-output relationship of TC neurons.

In this paper, we introduce an integrate-and-fire spiking neuron model that includes a biophysically realistic model of I T . We briefly review the sub-threshold 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 non-spiking 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 [2].

References

  1. 1.

    Hughes SW, Cope DW, Blethyn KL, Crunelli V: Cellular mechanisms of the slow (<1 Hz) oscillation in thalamocortical neurons in vitro. Neuron. 2002, 33: 947-958. 10.1016/S0896-6273(02)00623-2.

  2. 2.

    Nordmark AB: Universal limit mapping in grazing bifurcations. Phys Rev E. 1997, 55: 266-270. 10.1103/PhysRevE.55.266.

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Acknowledgements

J.Laudanski was supported by a Marie Curie Early Stage Researcher Training Fellowship from the European Commission (EC Contract No: MEST-CT-2005-020723).

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Correspondence to Jonathan Laudanski.

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Open Access This article is published under license to BioMed Central Ltd. This is an Open Access article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Keywords

  • Periodic Orbit
  • Direct Numerical Simulation
  • Bifurcation Diagram
  • Chaotic Behavior
  • Spike Neuron