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


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.


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



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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  1. Alfonsi, A., Cancès E., Turinici, G., Di Ventura, B., Huisinga, W. (2005). Adaptive simulation of hybrid stochastic and deterministic models for biochemical systems. In ESAIM: Proceedings (Vol. 14, pp. 1–13). EDP Sciences Google Scholar
  2. Anderson, D.F. (2007). A modified next reaction method for simulating chemical systems with time dependent propensities and delays. Journal of Chemical Physics, 127(21), 214107.PubMedCrossRefGoogle Scholar
  3. Anderson, D.F. (2012). An efficient finite difference method for parameter sensitivities of continuous time Markov chains. SIAM Journal on Numerical Analysis, 50(5), 2237–2258.CrossRefGoogle Scholar
  4. Anderson, D.F., Ganguly, A., Kurtz, T.G. (2011). Error analysis of tau-leap simulation methods. Annals of Applied Probability, 21(6), 2226–2262.CrossRefGoogle Scholar
  5. Anderson, D.F., & Higham, D.J. (2012). Multi-level Monte Carlo for continuous time Markov chains, with applications in biochemical kinetics. SIAM: Multiscale Modeling and Simulation, 10(1), 146–179.Google Scholar
  6. Anderson, D.F., Higham, D.J., Sun, Y. (2014). Complexity analysis of multilevel monte carlo tau-leaping. Submitted.Google Scholar
  7. Anderson, D.F., & Koyama, M. (2014). An asymptotic relationship between coupling methods for stochastically modeled population processes. Accepted for publication to IMA Journal of Numerical Analysis.Google Scholar
  8. Anderson, D.F., & Kurtz, T.G. (2011). Design and analysis of biomolecular circuits, chapter 1. Continuous Time Markov chain models for chemical reaction networks. Springer.Google Scholar
  9. Anderson, D.F., & Wolf, E.S. (2012). A finite difference method for estimating second order parameter sensitivities of discrete stochastic chemical reaction networks. Journal of Chemical Physics, 137(22), 224112.PubMedCrossRefGoogle Scholar
  10. Ball, K., Kurtz, T.G., Popovic, L., Rempala, G. (2006). Asymptotic analysis of multiscale approximations to reaction networks. Annals of Applied Probability, 16(4), 1925–1961.CrossRefGoogle Scholar
  11. Bressloff, P.C. (2009). Stochastic neural field theory and the system-size expansion. SIAM Journal on Applied Mathematics, 70(5), 1488–1521.CrossRefGoogle Scholar
  12. Bressloff, P.C., & Newby, J.M. (2013). Metastability in a stochastic neural network modeled as a velocity jump Markov process. SIAM Journal on Applied Dynamical Systems, 12(3), 1394–1435.CrossRefGoogle Scholar
  13. Bressloff, P.C., & Newby, J.M. (2014a). Stochastic hybrid model of spontaneous dendritic NMDA spikes. Physical Biology, 11(1), 016006.PubMedCrossRefGoogle Scholar
  14. Bressloff, P.C., & Newby, J.M. (2014b). Path integrals and large deviations in stochastic hybrid systems. Physical Review E, 89(4), 042701.CrossRefGoogle Scholar
  15. Buckwar, E., & Riedler, M.G. (2011). An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution. Journal of Mathematical Biology, 63, 1051–1093.PubMedCrossRefGoogle Scholar
  16. Buice, M.A., Cowan, J.D., Chow, C.C. (2010). Systematic fluctuation expansion for neural network activity equations. Neural Computation, 22(2), 377–426.PubMedCentralPubMedCrossRefGoogle Scholar
  17. Cao, Y., Gillespie, D.T., Petzold, L.R. (2006). Efficient step size selection for the tau-leaping simulation method. Journal of Chemical Physics, 124(4), 044109.PubMedCrossRefGoogle Scholar
  18. Clay, J.R., & DeFelice, L.J. (1983). Relationship between membrane excitability and single channel open-close kinetics. Biophysical Journal, 42(2), 151–7.PubMedCentralPubMedCrossRefGoogle Scholar
  19. Colquhoun, D., & Hawkes, A.G. (1983). Single-channel recording, chapter the principles of the stochastic interpretation of ion-channel mechanisms. New York: Plenum Press.Google Scholar
  20. Davis, M.H.A. (1984). Piecewise-deterministic markov processes: a general class of non-diffusion stochastic models. Journal of the Royal Statistical Society. Series B, 46(3), 353–388.Google Scholar
  21. Dorval, Jr., A.D., & White, J.A. (2005). Channel noise is essential for perithreshold oscillations in entorhinal stellate neurons. The Journal of Neuroscience, 25(43), 10025–10028.PubMedCrossRefGoogle Scholar
  22. Earnshaw, B.A., & Keener, J.P. (2010). Invariant manifolds of binomial-like nonautonomous master equations. SIAM Journal Applied Dynamical Systems, 9(2), 568–588.CrossRefGoogle Scholar
  23. Ermentrout, G.B., & Terman, D.H. (2010). Foundations of mathematical neuroscience. Springer.Google Scholar
  24. Ethier, S.N., & Kurtz, T.G. (1986). Markov processes: characterization and convergence. New York: John Wiley.CrossRefGoogle Scholar
  25. Fisch, K., Schwalger, T., Lindner, B., Herz, A.V.M., Benda, J. (2012). Channel noise from both slow adaptation currents and fast currents is required to explain spike-response variability in a sensory neuron. Journal of Neuroscience, 32(48), 17332–44.PubMedCrossRefGoogle Scholar
  26. Fox, R.F., & Yan-nan, L. (1994). Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels. Physical Review E Statistical Physics Plasmas Fluids Related Interdisciplinary Topics, 49(4), 3421– 3431.Google Scholar
  27. Giles, M.B. (2008). Multilevel Monte Carlo path simulation. Operations Research, 56, 607–617.CrossRefGoogle Scholar
  28. Gillespie, D.T. (1977). Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry, 81, 2340–2361.CrossRefGoogle Scholar
  29. Gillespie, D.T. (2007). Stochastic simulation of chemical kinetics. Annual Review of Physical Chemistry, 58, 35–55.PubMedCrossRefGoogle Scholar
  30. Glynn, P.W. (1989). A GSMP formalism for discrete event systems. Proceedings of the IEEE, 77(1), 14–23.CrossRefGoogle Scholar
  31. Goldwyn, J.H., Imennov, N.S., Famulare, M., Shea-Brown, E. (2011). Stochastic differential equation models for ion channel noise in hodgkin-huxley neurons. Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics, 83(4 Pt 1), 041908.PubMedCentralPubMedCrossRefGoogle Scholar
  32. 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.PubMedCentralPubMedCrossRefGoogle Scholar
  33. Groff, J.R., DeRemigio, H., Smith G.D. (2009). Markov chain models of ion channels and calcium release sites. In Stochastic Methods in neuroscience (pp. 29–64).Google Scholar
  34. Haas, P.J. (2002). Stochastic petri nets: modelling stability, simulation, 1st edn. New York: Springer.CrossRefGoogle Scholar
  35. Hodgkin, A.L., & Huxley, A.F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117, 500–544.PubMedCentralPubMedCrossRefGoogle Scholar
  36. Kang, H.W., & Kurtz, T.G. (2013). Separation of time-scales and model reduction for stochastic reaction models. Annals of Applied Probability, 23, 529–583.CrossRefGoogle Scholar
  37. Keener, J.P., & Newby, J.M. (2011). Perturbation analysis of spontaneous action potential initiation by stochastic ion channels. Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics, 84(1-1), 011918.PubMedCrossRefGoogle Scholar
  38. Kispersky, T., & White, J.A. (2008). Stochastic models of ion channel gating. Scholarpedia, 3(1), 1327.CrossRefGoogle Scholar
  39. Kurtz, T.G. (1980). Representations of markov processes as multiparameter time changes. Annals of Probability, 8(4), 682–715.CrossRefGoogle Scholar
  40. Kurtz, T.G. (1981). Approximation of population processes, CBMS-NSF Reg. Conf. Series in Appl. Math.: 36, SIAM.Google Scholar
  41. Laing, C., & Lord, G.J. (Eds.) 2010. Stochastic methods in neuroscience. Oxford University Press.Google Scholar
  42. Lee, C., & Othmer, H. (2010). A multi-time-scale analysis of chemical reaction networks: I. deterministic systems. Journal of Mathematical Biology, 60, 387–450. doi: 10.1007/s00285-009-0269-4.PubMedCrossRefGoogle Scholar
  43. Milescu, L.S., Yamanishi, T., Ptak, K., Smith, J.C. (2010). Kinetic properties and functional dynamics of sodium channels during repetitive spiking in a slow pacemaker neuron. Journal of Neuroscience, 30(36), 12113–27.PubMedCentralPubMedCrossRefGoogle Scholar
  44. Mino, H., Rubinstein, J.T., White, J.A. (2002). Comparison of algorithms for the simulation of action potentials with stochastic sodium channels. Annals of Biomedical Engineering, 30(4), 578–87.PubMedCrossRefGoogle Scholar
  45. Morris, C., & Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal, 35(1), 193–213.PubMedCentralPubMedCrossRefGoogle Scholar
  46. Newby, J.M., Bressloff, P.C., Keener, J.P. (2013). Breakdown of fast-slow analysis in an excitable system with channel noise. Physical Review Letters, 111(12), 128101.PubMedCrossRefGoogle Scholar
  47. Pakdaman, K., Thieullen, M., Wainrib, G. (2010). Fluid limit theorems for stochastic hybrid systems with application to neuron models. Advances in Applied Probability, 42(3), 761–794.CrossRefGoogle Scholar
  48. Pakdaman, K., Thieullen, M., Wainrib, G. (2012). Asymptotic expansion and central limit theorem for multiscale piecewise-deterministic Markov processes. Stochastic Proceedings of Applied, 122, 2292–2318.CrossRefGoogle Scholar
  49. Rathinam, M., Sheppard, P.W., Khammash, M. (2010). Efficient computation of parameter sensitivities of discrete stochastic chemical reaction networks. Journal of Chemical Physics, 132, 034103.PubMedCentralPubMedCrossRefGoogle Scholar
  50. Riedler, M., & Notarangelo, G. (2013). Strong Error Analysis for the Θ-Method for Stochastic Hybrid Systems arXiv preprint. arXiv:1310.0392.
  51. Riedler, M.G., Thieullen, M., Wainrib, G. (2012). Limit theorems for infinite-dimensional piecewise deterministic Markov processes. Applications to stochastic excitable membrane models. Electronic Journal of Probability, 17(55), 1–48.Google Scholar
  52. Rinzel, J., & Ermentrout, G.B. (1989). Analysis of neural excitability and oscillations. In: C. Koch, & I. Segev (Eds.) Methods in Neuronal Modeling, 2nd ed. MIT Press.Google Scholar
  53. Terman, D., & Rubin, J. (2002). Geometric singular pertubation analysis of neuronal dynamics. In B. Fiedler (Ed.), Handbook of dynamical systems, vol. 2: towards applications (pp. 93–146). Elsevier.Google Scholar
  54. Schmandt, N.T., & Galán, R.F. (2012). Stochastic-shielding approximation of Markov chains and its application to efficiently simulate random ion-channel gating. Physical Review Letters, 109(11), 118101.PubMedCrossRefGoogle Scholar
  55. Schmidt, D.R., & Thomas, P.J. (2014). Measuring edge importance: a quantitative analysis of the stochastic shielding approximation for random processes on graphs. The Journal of Mathematical Neuroscience, 4(1), 6.CrossRefGoogle Scholar
  56. Schwalger, T., Fisch, K., Benda, J., Lindner, B. (2010). How noisy adaptation of neurons shapes interspike interval histograms and correlations. PLoS Computers in Biology, 6(12), e1001026.CrossRefGoogle Scholar
  57. Shingai, R., Quandt, F.N. (1986). Single inward rectifier channels in horizontal cells. Brain Research, 369(1-2), 65–74.PubMedCrossRefGoogle Scholar
  58. Skaugen, E., & Walløse, L. (1979). Firing behaviour in a stochastic nerve membrane model based upon the Hodgkin-Huxley equations. Acta Physiologica Scandinavica, 107(4), 343–63.PubMedCrossRefGoogle Scholar
  59. Smith, G.D., Keizer, J. (2002). Modeling the stochastic gating of ion channels. In Computational cell biology (pp. 285–319). New York: Springer.Google Scholar
  60. Srivastava, R., Anderson, D.F., Rawlings, J.B. (2013). Comparison of finite difference based methods to obtain sensitivities of stochastic chemical kinetic models. Journal of Chemical Physics, 138(7), 074110.PubMedCentralPubMedCrossRefGoogle Scholar
  61. Strassberg, A.F., & DeFelice, L.J. (1993). Limitations of the Hodgkin-Huxley formalism: Effects of single channel kinetics on transmebrane voltage dynamics. Neural Computation, 5, 843–855.CrossRefGoogle Scholar
  62. Wainrib, G., Thieullen, M., Pakdaman, K. (2012). Reduction of stochastic conductance-based neuron models with time-scales separation. Journal of Computational Neuroscience, 32, 327–346.CrossRefGoogle Scholar
  63. White, J.A., Chow, C.C., Ritt, J., Soto-Trevino, C., Kopell, N. (1998). Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons. Journal of Computational Neuroscience, 5, 5–16.PubMedCrossRefGoogle Scholar
  64. White, J.A., Rubinstein, J.T., Kay, A.R. (2000). Channel noise in neurons. Trends in Neurosciences, 23, 131–137.PubMedCrossRefGoogle Scholar
  65. Wilkinson, D.J. (2011). Stochastic modelling for systems biology. Chapman & Hall/CRC.Google Scholar

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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