Skip to main content
SpringerLink
Log in

Reliability of signal transmission in stochastic nerve axon equations

  • Published:
Journal of Computational Neuroscience Aims and scope Submit manuscript

Abstract

We introduce a method for computing probabilities for spontaneous activity and propagation failure of the action potential in spatially extended, conductance-based neuronal models subject to noise, based on statistical properties of the membrane potential. We compare different estimators with respect to the quality of detection, computational costs and robustness and propose the integral of the membrane potential along the axon as an appropriate estimator to detect both spontaneous activity and propagation failure. Performing a model reduction we achieve a simplified analytical expression based on the linearization at the resting potential (resp. the traveling action potential). This allows to approximate the probabilities for spontaneous activity and propagation failure in terms of (classical) hitting probabilities of one-dimensional linear stochastic differential equations. The quality of the approximation with respect to the noise amplitude is discussed and illustrated with numerical results for the spatially extended Hodgkin-Huxley equations. Python simulation code is supplied on GitHub under the link https://github.com/deristnochda/Hodgkin-Huxley-SPDE.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Alili, L., Patie, P., & Pedersen, J. L. (2005). Representations of the first hitting time density of an Ornstein-Uhlenbeck process 1. Stochastic Models, 21(4), 967–980.

    Article  Google Scholar 

  • Faisal, A. A., & Laughlin, S. B. (2007). Stochastic simulations on the reliability of action potential propagation in thin axons. PLoS Computational Biology, 3(5), e79.

    Article  PubMed Central  PubMed  Google Scholar 

  • Faisal, A. A., Selen, L. P., & Wolpert, D. M. (2008). Noise in the nervous system. Nature Reviews Neuroscience, 9(4), 292–303.

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  • Goldwyn, J. H., & Shea-Brown, E. (2011). The what and where of adding channel noise to the Hodgkin-Huxley equations. PLoS Computational Biology, 7(11), e1002247.

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  • Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4), 500–544.

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  • Horikawa, Y. (1991). Noise effects on spike propagation in the stochastic Hodgkin-Huxley models. Biological Cybernetics, 66(1), 19–25.

    Article  CAS  PubMed  Google Scholar 

  • Linaro, D., Storace, M., & Giugliano, M. (2010). Accurate and fast simulation of channel noise in conductance-based model neurons by diffusion approximation. PLoS Computational Biology, 7(3), e1001102.

    Article  Google Scholar 

  • Sacerdote, L., & Giraudo, M. T. (2013). Stochastic Integrate and Fire models: a review on mathematical methods and their applications. In: Stochastic biomathematical models (pp. 99–148). Springer.

  • Sauer, M., & Stannat, W. (2014). Analysis and approximation of stochastic nerve axon equations. arXiv:1402.4791, accepted for publication in Mathematics of Computation.

  • Sauer, M., & Stannat, W. (2015). Lattice approximation for stochastic reaction diffusion equations with one-sided lipschitz condition. Mathematics of Computation, 84(292), 743– 766.

    Article  Google Scholar 

  • Stannat, W. (2014). Stability of travelling waves in stochastic bistable reaction-diffusion equations. arXiv:1404.3853.

  • Tuckwell, H. C. (2008). Analytical and simulation results for the stochastic spatial Fitzhugh-Nagumo model neuron. Neural Computation, 20(12), 3003–3033.

    Article  PubMed  Google Scholar 

  • Tuckwell, H. C., & Jost, J (2010). Weak noise in neurons may powerfully inhibit the generation of repetitive spiking but not its propagation. PLoS Computational Biology, 6(5), e1000794.

    Article  PubMed Central  PubMed  Google Scholar 

  • Tuckwell, H. C., & Jost, J. (2011). The effects of various spatial distributions of weak noise on rhythmic spiking. Journal of Computational Neuroscience, 30(2), 361–371.

    Article  PubMed  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Straße des 17, Juni 136, 10623, Berlin, Germany

    Martin Sauer & Wilhelm Stannat

  2. Bernstein Center for Computational Neuroscience, Philippstr. 13, 10115, Berlin, Germany

    Martin Sauer & Wilhelm Stannat

Authors
  1. Martin Sauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wilhelm Stannat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wilhelm Stannat.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Action Editor: Brent Doiron

This work is supported by the BMBF, FKZ 01GQ1001B

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sauer, M., Stannat, W. Reliability of signal transmission in stochastic nerve axon equations. J Comput Neurosci 40, 103–111 (2016). https://doi.org/10.1007/s10827-015-0586-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10827-015-0586-0

Keywords

Search

Navigation