Une équation simple au comportement complexe
According to WJ Freeman experiments and simulations, arousal level can be simulated by the "complexity parameter" in his numerical model. When arousal or the complexity parameter increase, we saw that electrophysiologic responses (real or simulated) pass by simple attractors (sleep), then limit cycle (waking rest), then strange attractors (motivated state), and finally by limit cycle (seizures).
Writing a code for freeman equations is rather difficult, but another (obviously) very simple equation perfectly illustrates such a behavior:
in which, xt may be a potential recorded at time t, and r the complexity parameter (or arousal level).
We must choose 0<x<1, and 0<r<4
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References
May RM. 1991. Le chaos en biologie. La Recherche. 232(22); 588:598.
This paper cites the following articles (but I did not read them):
May. 1974. Science. 186; 645.
York & Tien Yien. 1975. Amer. Mathematical Monthly. 82; 985.
May. 1976. Nature. 261; 459.
May & Oster. 1976. The Amer. Naturalist. 110; 573.
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A small R code to explore the behavior of the equation :
# Now, let explore the behavior of the equation depending on r value
rep = readline(prompt="Valeur de r ? (0<r<4)")
r = as.numeric(rep)
# If you want to explore the behavior of the equation depending on initial x value, just unlock the lines 7 and 8 (remove their first "#")
x[1]= 0.7
#xep = readline(prompt="Valeur de depart de x? (0<x<1)") # line 7
#x[1] = as.numeric(xep) # line 8
# Computation and plotting
plot( x)
for (t in 2:1000){
coord = t - 1
x[t] = r * x[coord] * (1 - x[coord]) # "The" equation
points(t, x[t], pch=20)
}
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