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Journal of Computational Neuroscience

, Volume 38, Issue 1, pp 67–82 | Cite as

Stochastic representations of ion channel kinetics and exact stochastic simulation of neuronal dynamics

  • David F. Anderson
  • Bard Ermentrout
  • Peter J. Thomas
Article

Abstract

In this paper we provide two representations for stochastic ion channel kinetics, and compare the performance of exact simulation with a commonly used numerical approximation strategy. The first representation we present is a random time change representation, popularized by Thomas Kurtz, with the second being analogous to a “Gillespie” representation. Exact stochastic algorithms are provided for the different representations, which are preferable to either (a) fixed time step or (b) piecewise constant propensity algorithms, which still appear in the literature. As examples, we provide versions of the exact algorithms for the Morris-Lecar conductance based model, and detail the error induced, both in a weak and a strong sense, by the use of approximate algorithms on this model. We include ready-to-use implementations of the random time change algorithm in both XPP and Matlab. Finally, through the consideration of parametric sensitivity analysis, we show how the representations presented here are useful in the development of further computational methods. The general representations and simulation strategies provided here are known in other parts of the sciences, but less so in the present setting.

Keywords

Markov process Conductance based model Exact stochastic simulation Morris-Lecar model 

Notes

Acknowledgments

Anderson was supported by NSF grant DMS-1318832 and Army Research Office grant W911NF-14-1-0401. Ermentrout was supported by NSF grant DMS-1219754. Thomas was supported by NSF grants EF-1038677, DMS-1010434, and DMS-1413770, by a grant from the Simons Foundation (#259837), and by the Council for the International Exchange of Scholars (CIES). We gratefully acknowledge the Mathematical Biosciences Institute (MBI, supported by NSF grant DMS 0931642) at The Ohio State University for hosting a workshop at which this research was initiated. The authors thank David Friel and Casey Bennett for helpful discussions and testing of the algorithms.

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • David F. Anderson
    • 1
  • Bard Ermentrout
    • 2
  • Peter J. Thomas
    • 3
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA
  2. 2.Department of MathematicsUniversity of PittsburghPittsburghUSA
  3. 3.Department of Mathematics, Applied Mathematics, and StatisticsCase Western Reserve UniversityClevelandUSA

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