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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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R function slopeRatio

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  The slopeRatio function calculates the relative potency of two samples based on their linear regression slopes. It also returns the confidence interval for the relative potency The function slopeRatio takes four arguments: x, y0, y1, and alpha x is the independent variable (e.g., dose, time, ...) y0 is the standard sample y1 is the test sample alpha is the significance level for the confidence interval (default is 0.05) slopeRatio uses the R function lm to fit linear models for y0 and y1 as functions of x.  It then extracts the slopes and standard errors of the models and  computes the ratio of the slopes (R) and its standard error using a formula from Finney (1978). It also checks if the denominator slope (B0) is significantly different from zero using a g-value. If g > 1, then the use of the ratio is not sensible for the data. It then calculates the lower and upper bounds of the confidence interval for R.  It returns a list with four eleme...
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What is the Slope Ratio Test and Why You Should Use It for Bioassay

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Introduction Bioassay is a method of measuring the potency or concentration of a substance by its effect on living cells or tissues. Bioassays are often used in drug development, vaccine production, and environmental monitoring. Potency assays are used to determine the concentration of a substance that produces a specific biological effect. The potency of a substance is expressed as the amount of substance required to produce a specific biological effect. One of the challenges of bioassay is to ensure that the dose-response curves of the test and reference samples are parallel, meaning that they have the same shape and slope. We have already discussed the importance of  parallelism as a prerequisite for the determination of relative potency in bioactivity assays   because it indicates that the test and reference samples have the same mechanism of action and that the results are comparable. To test for parallelism, one of the commonly used methods is the slope ratio test ....
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Parallelism Test: A Quality Control Measure for Bioassay Standardization

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Bioassays are experiments that measure the biological activity or effect of a substance, such as a drug, vaccine, or hormone. Bioassay validation is the process of ensuring that the bioassay method is reliable, accurate, and reproducible. Bioassay analysis is the process of interpreting the bioassay data and drawing conclusions about the substance's potency, efficacy, or safety.  One of the key aspects of bioassay validation and analysis is parallelism . Parallelism refers to the assumption that the test sample and the reference sample have the same response function, meaning that they produce similar results when diluted at different concentrations. This assumption allows us to compare the test sample and the reference sample on a common scale and estimate the relative potency of the test sample.  To test for parallelism, we need to perform a statistical test that compares the slopes and intercepts of the test sample and the reference sample curves. There are different metho...
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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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