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A dynamical system analysis of the adaptive spike threshold

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Recent in vivo experiments have revealed that the action potential threshold depends on the rate of depolarization just preceding the spike. This phenomenon can be reproduced in the Hodgkin-Huxley model. We analyzed spike initiation in the (V, h) phase space, where h is the sodium inactivation variable, and found that the dynamical system exhibits a saddle equilibrium, whose stable manifold is the curve of the threshold. We derived an equation of this manifold, which relates the threshold to the sodium inactivation variable. It leads to a differential equation of the threshold depending on the membrane potential, which translates into an integrate-and-fire model with an adaptive threshold. The model accounts well for the variability of threshold and the slope-threshold relationship. See figure 1.

Figure 1
figure1

Sample trace of a noise-driven integrate-and-fire model with adaptive threshold. Blue: membrane potential, red: spike threshold.

Acknowledgements

This work was partially supported by the EC IP project FP6-015879, FACETS, and the EADS Corporate Research Foundation.

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Correspondence to Jonathan Platkiewicz.

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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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Platkiewicz, J., Brette, R. A dynamical system analysis of the adaptive spike threshold. BMC Neurosci 8, P119 (2007) doi:10.1186/1471-2202-8-S2-P119

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Keywords

  • Differential Equation
  • Animal Model
  • Dynamical System
  • Phase Space
  • Membrane Potential