For real neurobiological data, a time series and its n time delays are first reconstructed as a trajectory in an n+1 dimensional geometric embedding space and, following partition of that geometric space into n+1 dimensional lettered boxes (the choice of partition being a sensitive step), what was once an orbit has become a sequence of symbols. Dynamical systems in geometric space become symbolic dynamics in sequence space. It was Kolmogoroff (1958) who first applied Shannon’s ideas of entropy and information (Shannon and Weaver, 1949; Khinchin, 1957) to the quantification of these dynamical system’s telegraphic messages as
discrete, “stochastic” (random, probabilistic) output. Kolmogoroff turned to Shannon
entropy, -S'p, log p, (where p = 1/n and n = number of possibilities) to decide the
question whether a dynamical system that naturally partitioned into a two or three box system per unit time had the same entropy. His answer was no, that —3(1/3 In (1/3)) = 1.098 > -2(1/2 In (1/2) = 0.6931 loge and in computer relevant logz, 1.5850 > 1.0 (Kolmogorov, 1959). Entropy increases with possibility.
Nonlinear differential equations representing brain-relevant expanding dynamical systems replace Shannon’s linguistically weighted and serially ordered, Markoff-dependent random number generator of probabilistic language. As noted above, in the case of the Sharkovskii sequences (Sharkovskii, 1964; Metropolis et al, 1973; Misiurewicz, 1995), a small change in the single parameter of an entire class of single maximum maps generating motions that are coded from their position at the left or right of center of the unit interval, alters and determines precisely the periodic output such as {1,0,0,1,0,1,1,0,0,1,0,1...) of its binary
message. In higher dimensional examples such as the Rossler and Lorenz systems, one can visualize the joint actions of 24(+) and A(-) moving the trajectory so as to both enter, “create,” new boxes and generate new letters as well as visit old ones, unstable fixed points, thus forming unstable periodic orbits. The latter, one of three diagnostic features of chaotic attractors (see above), can also be seen as resulting from the “coarse-grained” imprecision of real world neurobiological measurement such that two points that are brought close to attractive-repelling
points are, within measurement error, recorded as having the same value.
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