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: calculate the anaerobic threshold

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RunTime <- function(Time,Distance,Distance_reference,bpm) {

## INPUT

#Time: run time measured [min, s]

#Distance: run distance [mt]

#Distance_reference: reference distance [mt] of which we want to evaluate the expected time 

#bpm: cardiac frequency (beats per minute) 

## OUTPUT

# T2 = tempo corsa atteso su distanza di riferimento (s)

# V2 = velocità attesa su distanza di riferimento (km / h)

# V1 = velocità media singole corse (km / h)

# Vm = velocità media su distanze e tempi rilevati cumulati (km / h)

# FC = Frequenza cardiaca per minuto

# SAN = Soglia Anaerobica stimata

# SAN2 = proiezione Soglia Anaerobica su tempi attesi e distanza di riferimento

# D1 = distanze percorse

# P1 = passo corsa per km [min, s]  

# P2 = passo corsa atteso per km [min, s]  


T1 <- Time

D1 <- Distance

D2 <- Distance_reference

T = T1[,2] + T1[,1]*60 # run time (seconds)

T2 = T* (D2/D1)^(1.06) # expected time (seconds)

V2 = 3.6 * (D2/T2)

V1 = 3.6 * (D1/T);

p = (1000*T/D1)/60 # run pace per km 

P1 = data.frame(floor(p), ((p - floor(p))*60)) # run pace per km [min, s]  

p2 = (1000*T2/D2)/60 # expected run pace per km 

P2 = data.frame(floor(p2), ((p2 - floor(p2))*60)) # expected run pace per km [min, s]  

Vm = 3.6 * (sum(D1)/sum(T));

SAN = 35000/(T*(10000/D1)) #Anaerobic Threshold

SAN2 = 35000/(T2*(10000/D2)) #projection of the Anaerobic Threshold over expected time and reference distance

FC <- bpm

## summary statistics

stats=NULL

stats$V1 = c(mean(V1), sd(V1))

stats$D1 = c(mean(D1), sd(D1))

stats$V2 = c(mean(V2), sd(V2))

stats$T1 = c(mean(T), sd(T))

stats$T2 = c(mean(T2), sd(T2))

stats$SAN = c(mean(SAN), sd(SAN))

stats$SAN2 = c(mean(SAN2), sd(SAN2))

stats$FC = c(mean(FC), sd(FC))

stats$P1 = c(mean(p), sd(p))

stats$P2 = c(mean(p2), sd(p2))

DF <- data.frame(

    Run = seq_along(Distance),

    Distance = Distance,

    Velocity = V1,

    Expected_Velocity = V2,

    Time = T,

    Expected_Time = T2,

    bpm = bpm,

    Anaerobic_Threshold = SAN

  )

##Plots

par(mfrow=c(2,2))

plot(V1,main="Velocity(km / h)",xlab="Run",ylab="km / h",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(D1,main="Distance",xlab="Run",ylab="mt",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(T,main="Run time",xlab="Run",ylab="s",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(P1[,2] + P1[,1]*60,main="Run pace per km", xlab="Run",ylab="s",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)


par(mfrow=c(2,2))

plot(T2,main="Projection of times",xlab="Run",ylab="s",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(V2,main=paste0("expected velocity\n on reference distance of",D2," mt",sep=""),xlab="Run",ylab="km / h",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(P2[,2] + P2[,1]*60,main="Expected run pace per km (s)", xlab="Run", ylab="s",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)


par(mfrow=c(1,2))

plot(T, SAN,main="Anaerobic Threshold vs. Time",xlab="Time (s)",ylab="Anaerobic Threshold (km / h)",pch=16)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(SAN,main="Anaerobic Threshold",xlab="Run",ylab="Anaerobic Threshold (km / h)",pch=16)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)


plot(D1,V1,main="Velocity vs. Distance",xlab="Distance",ylab="Velocity (km / h)",pch=16)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)



par(mfrow=c(2,1))

plot(D1,FC,main="bpm vs. distance",ylab="bpm",xlab="distance (mt)",pch=16)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(V1,FC,main="bpm vs. velocity",xlab="velocity (km / h)",ylab="bpm",pch=16)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)


return(DF)

}


This function evaluates the performance of running training sessions based on distance, time, and heart rate data. 

It calculates expected time, velocity, anaerobic threshold, and other metrics to provide insights into a runner's progress.



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