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This paper presents a new methodology for generating continuous statistical distributions, integrating the exponentiated odds ratio within the framework of survival analysis. This new method enhances the flexibility and adaptability of distribution models to effectively address the complexities inherent in contemporary datasets. The core of this advancement is illustrated by introducing a particular subfamily, the "Type-2 Gumbel Weibull-G Family of Distributions." We provide a comprehensive analysis of the mathematical properties of these distributions, encompassing statistical properties such as density functions, moments, hazard rate and quantile functions, R\'enyi entropy, order statistics, and the concept of stochastic ordering. To establish the robustness of our approach, we apply five distinct methods for parameter estimation. The practical applicability of the Type-2 Gumbel Weibull-G distributions is further supported through the analysis of three real-world datasets. These empirical applications illustrate the exceptional statistical precision of our distributions compared to existing models, thereby reinforcing their significant value in both theoretical and practical statistical applications.

The contributions of independent noise sources to the variability of action potential timing has not previously been studied at the level of individual directed molecular transitions within a conductance-based model ion-state graph. The underlying connection provides an important example of how mathematics can be applied to study the effects of unobservable microscopic fluctuations to macroscopically observable quantities. We study a stochastic Langevin model and 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. 2020, SIAM J. Appl. Math) 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 et al. 2012, Phys. Rev. Lett.) 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. We show that the ISI variance is not an unambiguously defined quantity, but depends on the choice of voltage level set as the spike-detection threshold, both in vitro and in silico., Comment: 51 pages, 14 Figures

Fox and Lu introduced a Langevin framework for discrete-time stochastic models of randomly gated ion channels such as the Hodgkin-Huxley (HH) system. They derived a Fokker-Planck equation with state-dependent diffusion tensor $D$ and suggested a Langevin formulation with noise coefficient matrix $S$ such that $SS^\intercal=D$. Subsequently, several authors introduced a variety of Langevin equations for the HH system. In this paper, we present a natural 14-dimensional dynamics for the HH system in which each \emph{directed} edge in the ion channel state transition graph acts as an independent noise source, leading to a $14\times 28$ noise coefficient matrix $S$. We show that (i) the corresponding 14D system of ordinary differential \rev{equations} is consistent with the classical 4D representation of the HH system; (ii) the 14D representation leads to a noise coefficient matrix $S$ that can be obtained cheaply on each timestep, without requiring a matrix decomposition; (iii) sample trajectories of the 14D representation are pathwise equivalent to trajectories of Fox and Lu's system, as well as trajectories of several existing Langevin models; (iv) our 14D representation (and those equivalent to it) give the most accurate interspike-interval distribution, not only with respect to moments but under both the $L_1$ and $L_\infty$ metric-space norms; and (v) the 14D representation gives an approximation to exact Markov chain simulations that are as fast and as efficient as all equivalent models. Our approach goes beyond existing models, in that it supports a stochastic shielding decomposition that dramatically simplifies $S$ with minimal loss of accuracy under both voltage- and current-clamp conditions., Comment: 55 pages, 9 figures