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Metastability and Eye Movements: A Dynamical‑Systems Interpretation

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  1. Why metastability is a natural language for gaze behavior The idea that perception, attention, and neural activity evolve through transiently stable states has deep roots in cognitive science and neuroscience. Concepts such as attractors , basins , noise‑induced transitions , and escape times appear in work on perceptual switching, decision‑making, and neural population dynamics (Kelso, 1995; Rabinovich et al., 2008; Deco & Jirsa, 2012). Eye movements, especially the alternation between fixations and saccades ,   fit remarkably well into this metastable picture. The Freidlin–Wentzell theory of rare events (Freidlin & Wentzell, 2012) provides a rigorous mathematical language for these intuitions. 2. Fixations as metastable states  Consider the gaze position X t as the state of a stochastic dynamical system. During a fixation the  gaze remains confined to a small region, microsaccades and noise generate small fluctuations, and the gaze tends to return...
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Sanov’s Theorem in Living Systems: Quantifying Rare Events in Biology and Neuroscience

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Biological and neural systems operate under substantial intrinsic stochasticity. Despite this, they maintain stable distributions of molecular, cellular, and population-level states. Occasionally, however, these systems exhibit rare, high-impact deviations. Sanov’s theorem provides a unifying framework for quantifying the improbability of such events through the geometry of empirical distributions. 1. Introduction Rare events in living systems often correspond not to isolated fluctuations but to atypical empirical distributions emerging from many interacting components. Sanov’s theorem characterizes the exponential decay of the probability of such deviations using the Kullback–Leibler divergence. This post surveys applications across molecular biology, cell dynamics, and neuroscience. Notation The following symbols and variables are used throughout this post: \(X_1, \dots, X_n\) : observations or molecular states sampled from a biological system. \(P\) : bas...
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Diagnosing Singular Fits in Linear Mixed Models: Practical Examples, Code, and Alternatives

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  Introduction This post is the practical follow‑up to When Linear Mixed Models Don’t Converge , where I discussed the theoretical reasons why convergence issues often arise in linear mixed models. Here, we move from theory to practice: simulated examples, diagnostic tools, interpretation of singular fits, and practical modeling alternatives. If you haven’t read the first part yet, you may want to start there for the conceptual foundations. Figure. Conceptual map showing how model complexity and available information interact Legend of Figure 1. Green = identifiable models; yellow = risk of singularity; red = non‑identifiable models. A Concrete Example Using R: When a Random-Slope Model Becomes Singular Let’s simulate a simple dataset with only 8 groups and a very small random‑slope variance. This is a classic scenario where the model converges numerically but is statistically non‑identifiable. ```r set.seed(123) n_groups <- 8 n_per_group <- 20 group <- factor(rep(1:n_gro...
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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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Unit Root Testing in Practice: A Tutorial on ADF–KPSS, Specification, and Diagnostics

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  Stationary Signals   A stationary signal is defined by the stability of its statistical properties over time. Specifically, this implies that the signal’s mean, variance, and autocorrelation function remain consistent, regardless of the time in which the signal is evaluated. There are two main types of stationarity: ·        Strict-sense stationarity refers to the property of a signal in which all statistical characteristics remain unchanged over time. This encompasses all moments of the distribution, indicating that the entire probability distribution remains constant throughout the observed period. ·        Wide-sense stationarity requires that the first two moments, specifically the mean and variance, remain constant over time. The autocorrelation function is dependent only on the time difference between two points, not on the exact time points themselves.   Importance in Signal...
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