Problems of measurement precision, amplified by the expansive actions of systems that are sensitive to initial conditions, yield parameter sensitive entropies of two (mathematically) fundamental kinds called topological and metric entropies, hr and hw, proven to be the upper and lower bounds of any estimate of the entropy ina uniformly expanding and/or equidistributed system (Adler and Weiss, 1965). Measures of entropy, as “missing information related to the number of alternatives which remain possible to a physical system” (Boltzmann, 1909), “index of probability” (Gibbs, 1902) or the “amount of uncertainty associated with a finite scheme” (Khinchin,1957) are obviously sensitive to the partition rules and its fineness of the grain. The most theoretically defensible partition is called the “generating partition” in which no box contains more than one point. Comparisons of control and experimental data can be differentially sensitive to partition construction, so that if a generating partition is not practicable due to sample length or dense curdling in the point distribution, some arbitrary choices have to be made. These have included naturally renormalized variational partitions, such that in one dimension the boxes are defined by +1, +2, +3,...standard deviations, or quartiles or quintile, above and below the mean and in n dimensions. Partitions have also been constructed and used to described drug effects on rat exploratory behavior by sequential partitioning along the dimension of the highest remaining variation (after the previous partition) called the “KD” partition (Paulus et al, 1991). Partition strategies to capture entropic measures on serial ordering (Klemm and Sherry, 1981; Strong et al, 1998) can grow from knowledge or hypotheses about the physiological sources of temporal irregularities and discontinuities in brain dynamics including characteristic interval(s) of refractoriness, relaxation times of the inhibitory surround, correlation time in dendritic tree summation, the time course of reciprocal inhibition and its decay and chemical influences such as the synaptic half-life and
time of action of inhibitory influences such as GABA on cell firing.
The logarithmic growth rates of occupancy of new symbolically indexed boxes or, equivalently, the growth rates of visitations to old ones generating unstable periodic orbits, are called topological entropies, hy . They record new
happenings, the growth rate of the diversity of orbits, and not how likely with respect
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