Pu Shusen, Thomas Peter J
Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, OH, USA.
Department of Biomedical Engineering, Vanderbilt University, Nashville, TN, USA.
Biol Cybern. 2021 Jun;115(3):267-302. doi: 10.1007/s00422-021-00877-7. Epub 2021 May 22.
Molecular fluctuations can lead to macroscopically observable effects. The random gating of ion channels in the membrane of a nerve cell provides an important example. The contributions of independent noise sources to the variability of action potential timing have not previously been studied at the level of molecular transitions within a conductance-based model ion-state graph. Here we study a stochastic Langevin model for the Hodgkin-Huxley (HH) system based on a detailed representation of the underlying channel state Markov process, the "[Formula: see text]D model" introduced in (Pu and Thomas in Neural Computation 32(10):1775-1835, 2020). We show how to resolve the individual contributions that each transition in the ion channel graph makes to the variance of the interspike interval (ISI). We extend the mean return time (MRT) phase reduction developed in (Cao et al. in SIAM J Appl Math 80(1):422-447, 2020) to the second moment of the return time from an MRT isochron to itself. Because fixed-voltage spike detection triggers do not correspond to MRT isochrons, the inter-phase interval (IPI) variance only approximates the ISI variance. We find the IPI variance and ISI variance agree to within a few percent when both can be computed. Moreover, we prove rigorously, and show numerically, that our expression for the IPI variance is accurate in the small noise (large system size) regime; our theory is exact in the limit of small noise. By selectively including the noise associated with only those few transitions responsible for most of the ISI variance, our analysis extends the stochastic shielding (SS) paradigm (Schmandt and Galán in Phys Rev Lett 109(11):118101, 2012) from the stationary voltage clamp case to the current clamp case. We show numerically that the SS approximation has a high degree of accuracy even for larger, physiologically relevant noise levels. Finally, we demonstrate that the ISI variance is not an unambiguously defined quantity, but depends on the choice of voltage level set as the spike detection threshold. We find a small but significant increase in ISI variance, the higher the spike detection voltage, both for simulated stochastic HH data and for voltage traces recorded in in vitro experiments. In contrast, the IPI variance is invariant with respect to the choice of isochron used as a trigger for counting "spikes."
分子涨落可导致宏观上可观测的效应。神经细胞膜中离子通道的随机门控就是一个重要例子。此前尚未在基于电导的模型离子态图的分子跃迁层面研究独立噪声源对动作电位时间变异性的贡献。在此，我们基于对基础通道状态马尔可夫过程的详细表示，研究霍奇金 - 赫胥黎（HH）系统的随机朗之万模型，即（Pu和Thomas在《神经计算》32(10):1775 - 1835, 2020中引入的）“[公式：见原文]D模型”。我们展示了如何解析离子通道图中每个跃迁对峰间期（ISI）方差的个体贡献。我们将（Cao等人在《SIAM应用数学杂志》80(1):422 - 447, 2020中提出的）平均返回时间（MRT）相位约简扩展到从MRT等时线到其自身的返回时间的二阶矩。由于固定电压尖峰检测触发与MRT等时线不对应，相间间隔（IPI）方差仅近似于ISI方差。我们发现当两者都可计算时，IPI方差和ISI方差在百分之几的范围内一致。此外，我们严格证明并通过数值表明，我们的IPI方差表达式在小噪声（大系统规模） regime中是准确的；我们的理论在小噪声极限下是精确的。通过选择性地仅包含与导致大部分ISI方差的少数跃迁相关的噪声，我们的分析将随机屏蔽（SS）范式（Schmandt和Galán在《物理评论快报》109(11):118101, 2012中提出的）从稳态电压钳情况扩展到电流钳情况。我们通过数值表明，即使对于更大的、生理相关的噪声水平，SS近似也具有高度准确性。最后，我们证明ISI方差不是一个明确界定的量，而是取决于作为尖峰检测阈值设置的电压水平的选择。我们发现，对于模拟的随机HH数据和体外实验中记录的电压迹线，尖峰检测电压越高，ISI方差有小幅但显著的增加。相比之下，IPI方差对于用作计数“尖峰”触发的等时线的选择是不变的。
