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The What and Where of Adding Channel Noise to the Hodgkin-Huxley Equations

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  • Joshua H Goldwyn
  • Eric Shea-Brown

Abstract

Conductance-based equations for electrically active cells form one of the most widely studied mathematical frameworks in computational biology. This framework, as expressed through a set of differential equations by Hodgkin and Huxley, synthesizes the impact of ionic currents on a cell's voltage—and the highly nonlinear impact of that voltage back on the currents themselves—into the rapid push and pull of the action potential. Later studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations or their counterparts. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the equations of Hodgkin-Huxley type. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic equations of Hodgkin-Huxley type as well as to more modern models of ion channel dynamics. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly MATLAB simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html.

Suggested Citation

  • Joshua H Goldwyn & Eric Shea-Brown, 2011. "The What and Where of Adding Channel Noise to the Hodgkin-Huxley Equations," PLOS Computational Biology, Public Library of Science, vol. 7(11), pages 1-9, November.
  • Handle: RePEc:plo:pcbi00:1002247
    DOI: 10.1371/journal.pcbi.1002247
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    References listed on IDEAS

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    1. Antti Saarinen & Marja-Leena Linne & Olli Yli-Harja, 2008. "Stochastic Differential Equation Model for Cerebellar Granule Cell Excitability," PLOS Computational Biology, Public Library of Science, vol. 4(2), pages 1-11, February.
    2. Daniele Linaro & Marco Storace & Michele Giugliano, 2011. "Accurate and Fast Simulation of Channel Noise in Conductance-Based Model Neurons by Diffusion Approximation," PLOS Computational Biology, Public Library of Science, vol. 7(3), pages 1-17, March.
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    1. I.B., Tagne nkounga & F.M., Moukam kakmeni & R., Yamapi, 2022. "Birhythmic oscillations and global stability analysis of a conductance-based neuronal model under ion channel fluctuations," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    2. Jae Kyoung Kim & Eduardo D Sontag, 2017. "Reduction of multiscale stochastic biochemical reaction networks using exact moment derivation," PLOS Computational Biology, Public Library of Science, vol. 13(6), pages 1-24, June.
    3. Tuckwell, Henry C. & Jost, Jürgen, 2012. "Analysis of inverse stochastic resonance and the long-term firing of Hodgkin–Huxley neurons with Gaussian white noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(22), pages 5311-5325.
    4. Wang, Jia-Zeng & Ma, Shu & Ji, Yu & Sun, Qi, 2023. "Response to multiplicative noise: The cross-spectrum of membrane voltage fluctuation and voltage-independent conductance noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 622(C).
    5. Quentin Clairon & Adeline Samson, 2022. "Optimal control for parameter estimation in partially observed hypoelliptic stochastic differential equations," Computational Statistics, Springer, vol. 37(5), pages 2471-2491, November.
    6. Gu Huaguang & Zhao Zhiguo & Jia Bing & Chen Shenggen, 2015. "Dynamics of On-Off Neural Firing Patterns and Stochastic Effects near a Sub-Critical Hopf Bifurcation," PLOS ONE, Public Library of Science, vol. 10(4), pages 1-29, April.
    7. Ramírez-Piscina, L. & Sancho, J.M., 2018. "Periodic spiking by a pair of ionic channels," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 345-354.
    8. Susanne Ditlevsen & Adeline Samson, 2019. "Hypoelliptic diffusions: filtering and inference from complete and partial observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(2), pages 361-384, April.
    9. Bendahmane, M. & Karlsen, K.H., 2019. "Stochastically forced cardiac bidomain model," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5312-5363.
    10. Yu, Haitao & Galán, Roberto F. & Wang, Jiang & Cao, Yibin & Liu, Jing, 2017. "Stochastic resonance, coherence resonance, and spike timing reliability of Hodgkin–Huxley neurons with ion-channel noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 263-275.
    11. Nkounga, I.B. Tagne & Xia, Yibo & Yanchuk, Serhiy & Yamapi, R. & Kurths, Jürgen, 2023. "Generalized FitzHugh–Nagumo model with tristable dynamics: Deterministic and stochastic bifurcations," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).

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