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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...

Adding gaussian noise to break the tied values

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  In statistics, the " rank " represents the indexed position of the numerical values in ascending (or descending) order measured on a random sample. So, by ranking, we transform the data and get another random variable. Numerically, ranking a random sample of size n corresponds to making a random permutation of the integers from 1 to n. Importantly, since each of the n! possible permutations have the same probability of occurrence, we can calculate the means of these random variables which are invariant with respect to any monotonic transformation of the original data. There are many statistics based on ranks, among which: Wilcoxon-Mann-Whitney test (also known as Wilcoxon Rank Sum Test ), Wilcoxon signed-rank test, Kruskal-Wallis test, Friedman test, Spearman’s rank correlation coefficient. The rank-based tests are generally more robust compared to their homologous parametric tests, enable to deal easily with ordinal data, and the asymptotic approximations of the rank sta...

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