Before describing the relatively new set of measures of biological processes designed to find and quantitate what are assumed to be relatively sample size insensitive, distributionally nonconvergent and multiply correlated processes that are without a single time or space scale, we should remind ourselves that there is already much more apparent “order” in a generically random situation than our intuitions would lead us to believe. For example, if we keep cumulative scores in a competition between heads and tails and determine the distribution of trials between those in which the number of heads and tails are even, we will get periods between zero crossings of many lengths with very short ones and very (very) long ones being most statistically prominent. The distribution of these wavelengths is shaped like a symmetrically fat-tailed, bowl (Feller, 1968). As another illustration, expected runs of heads or tails in this Gaussian random task are longer and more frequent than we might suspect. It has been proven that the expected run length grows with n coin flips (as an order of magnitude estimate) like the logarithm (for a fair coin, base 1/p = 1/0.5 = 2) of n. For example, in 512 ( e.g. 2°) tosses, we cannot report a run of 9 heads as a evidence for a biased coin or the sign of some deterministic coin tossing mechanism (Erdos and Renyi, 1970). If we had a 0.6 head biased coin, then the observation of a run of 13 heads couldn’t dissuade us from a random
mechanism!
Unlike our random coin task, the variances of many, perhaps most, time series of biologically-relevant events, do not tend to converge onto a limiting value as sample size, n, grows, but rather continue to increase (or decrease) with n ina scale invariant manner. Instead of “regressing to the mean” with increasing sample length or number, the likelinood of a larger deviation than previously observed increases with n. Analyses of inter-event intervals reveals a multiplicity of characteristic times. One interpretation of these finding might be that this represents evidence for the inherent “nonstationarity” of biological mechanisms as reflected in, for examples, the frequency of saccades concomitant with ceaselessly shifting foci of visual attention (Steriade and McCarley, 1990), or our inability to not think of “white bear’ when so instructed (Wegner, 1994). Hermann Haken, the father of
laser-inspired “synergetics,” has said that biological mechanisms are not in a steady
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