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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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ECONOMICS' DIARY IN THE EURO AREA BETWEEN EVIDENCE AND MYTHS

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1. THE SYLLOGISM ( Aristoteles, 384 BC - 322 BC ) Major premise: All humans are mortal. Minor premise: All Greeks are humans.         Conclusion: All Greeks are mortal . For years citizens in the Euro area have been informed that only by pursuing policies of rigour the economies of the countries with large public debts would have  increased.   Although their crushing effects,  on the base of this reasoning  still now the austerity policies are  widely implemented in the euro area (EA) .  We can see in Figures 1-4 the ratio of public debt to GDP and the GDP growth rate for some countries compared to the averaged series of the EA countries in the period 2000-2013.   Figure 1. Ratio public debt to GDP: EA vs. rest of the world (2000-2013) . Figure 2.  Ratio public debt to GDP within EA (2000-2013) . Figure 3. GDP growth rate:   EA vs. rest of the world (2000-2013) . Figure 4.  ...
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ANALYSIS OF THE CURRENT ACCOUNT BALANCES: GREECE, SPAIN AND ITALY

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We have already seen [ here ] that the difference between national savings and investment  corresponds to the  current account balance . A country   gets into debt with foreign countries if it saves too little (i.e., if it consumes too much) or if it makes too many capital expenditures.  Let's see now the time course of the current account under its possible different perspecitives for some euro area (EA) countries whose financial status and outlook are suffering (Greece, Spain,  Italy ). Data have been collected from the IMF :  World Economic Outlook  October 2012  [ here ] . The synthetic matrix of the elaborated data for all the countries of the EA is downloadable here . 1. GREECE The first country that we consider is Greece, which has been the earliest to be exposed to the financial crisis within the EA. Greece, often is represented as the bad model. Figure1-3. The c urrent account and its components (1991-2013): Greece ....
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EURO CURRENCY DOES NOT CIRCULATE IN AN OPTIMAL CURRENCY AREA

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1. PRIVATE VICE, PUBLIC VICE In the " Fable of the Bees " (1705)  Bernard Mandeville represented the paradox " private vices, public benefits " [1] . He showed that actions which may be considered vicious from the individual perspective, give benefits for the collectivity. By rewording this sentence,  one would say that prosperity relies on expenditure (vice) rather than on saving (virtue). And going a step further, if that  assessment were applied in terms of the whole private and public sectors, then one would expect that deficits of the private sector yields benefits to the public sector.    Instead, this is not always the case. Especially when the imbalance in the private sector becomes too large.  In a previous post [ here ] we introduced the fundamental identity of macroeconomics which allows  to assess the contribution of both the private sector and the public sector to the formation of the current account balance :  10)...
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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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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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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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