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Why Biological Systems Suddenly Change State: An Intuitive Guide to Freidlin–Wentzell Theory

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  Stochasticity is ubiquitous in biology and neuroscience, manifesting in various forms, including ion channel noise, synaptic variability, gene regulatory fluctuations, noisy population dynamics, and more. Many biological systems spend long periods in a stable “state” and only rarely transition to another state due to noise. For instance, a neuron typically remains inactive but may occasionally trigger a spontaneous spike. Similarly, a gene can switch from the OFF state to the ON state due to rare bursts of transcription factors. Cells can also transition out of metabolic or epigenetic states, populations might shift between different ecological equilibria, and a viral infection can fluctuate between phases of control and uncontrollability. Freidlin–Wentzell theory provides a mathematically rigorous framework to study these phenomena when noise is small but nonzero . It tells you, firstly, h ow likely rare transitions are,    secondly,   h ow fast they occ...
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The Relationship Between Weighted Geometric Mean and Shannon's Entropy

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The geometric mean is a type of average that is useful for comparing quantities that have different units or vary over a wide range. It is defined as the n-th root of the product of n numbers G M = ( ∏ i = 1 n x i ) 1 / n The geometric mean has some interesting properties that make it useful for certain applications. For instance, the geometric mean is always less than or equal to the arithmetic mean (the ordinary average), and it is always greater than or equal to the harmonic mean (another type of average that is useful for rates and ratios). The geometric mean also preserves the ratios of the data values, meaning that multiplying or dividing all the data values by a constant does not change their geometric mean, and it tends to be closer to the smaller values in the data set. This makes it more robust to outliers and skewed distributions. However, the geometric mean has a limitation: it assumes that all the values in the...
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R function Kullback_Leibler

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The  Kullback_Leibler()  function calculates the  Kullback-Leibler divergence  between two probability distributions  p  and  q . The function takes two arguments  p  and  q , which can be either discrete or continuous probability distributions. The type of distribution is specified using the  type  argument. The function first checks the type of distributions and their lengths. If the distributions are discrete, it checks that they have non-negative values and sum to one. If the distributions are continuous, it checks that they have positive values. The Kullback-Leibler divergence is then calculated using the formula  sum(p * log(p / q)) . The function also calculates a normalized version of the divergence by dividing it by the entropy of  p . The asymmetry of the divergence is checked by swapping  p  and  q . Finally, the function returns a list of results that includes the Kullback-Leibler divergence...
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The Kullback-Leibler Divergence: a bridge between Probability Theory and Information Theory

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1. Introduction Evaluating the difference between two probability distributions is an important task in many fields of statistics, machine learning, and data science. For example, we may want to compare the performance of different models, test hypotheses about the underlying data-generating process, or measure the divergence or similarity between two populations or groups. There are different ways to evaluate the difference between two probability distributions, depending on the type and nature of the data, the assumptions we make about the distributions, and the goal of the analysis. Some common methods are: ·          The Kolmogorov-Smirnov test : This is a non-parametric test that compares the empirical cumulative distribution functions (CDFs) of two samples and measures the maximum vertical distance between them. The test can be used to check if two samples come from the same distribution or if one distribution is stochastically larger th...
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Understanding Anaerobic Threshold (VT2) and VO2 Max in Endurance Training

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  Introduction: The Science Behind Ventilatory Thresholds VT1 vs. VT2: Defining Energy Transitions p (Aerobic Threshold) : The transition from lipid metawwwpp (fat-burning) to a mixed metabolism, where both carbohydrates and fats fuel muscle activity. V0T2 (Anaerobic Threshold) : The point where the  shifts to a predominantly carbohydrate-based wmetabolism , leading to rapid lactate waccumpulation and increased reliance on anaerobic energy pathways. www w In p VT1 and VT2 Wwww ewAgep & Sex Younger athletes generally exhibit higher VT2 values , while aging naturally reduces aerobicw capacity. Men tend to have a higher VO2 Max due to greater lung capacity and muscle mass, but women can achieve similar endurance levels through optimized training. Body Composition & Health Status High body fat percentage may reduce VT2 efficiency, as excess weight increases ...
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Owen's Function: A Simple Solution to Complex Problems

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Definition   If you are interested in statistics or probability, you may have encountered the Owen's function , a special function that arises in various applications involving bivariate normal distributions. Let’s explain what Owen's function is, how it is defined and notated, and why it is useful.  The Owen's function is defined as follows: for any real numbers h and a, Owen's function T(h,a) is given by T ( h , a ) = 1 2 π · ∫ 0 a exp ( - h 2 2 · ( 1 ...
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Cell Count Analysis with cycleTrendR

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  1. Introduction Long‑term monitoring of adherent cell cultures (e.g., Vero, MDCK) is central to vaccine development, viral propagation, and QC workflows in CRO environments. Standard analyses typically focus on short‑term growth metrics such as doubling time, confluence progression, and endpoint viability. While informative, these metrics overlook slow drifts, cyclic environmental influences, and structural transitions that emerge over multi‑day or multi‑week monitoring. Operational routines (e.g., weekly medium changes), environmental oscillations (e.g., daily temperature cycles), and metabolic rhythms can introduce periodic components into cell‑density trajectories. These patterns are rarely analyzed explicitly, despite their potential impact on viral yield, infection kinetics, and batch‑to‑batch reproducibility. This post demonstrates how cycleTrendR can extract long‑term trends, dominant cycles, and structural transitions from a realistically simulated adherent cell‑co...
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