to box occupancy densities they are likely to occur ( Adler et al, 1964; Alexeev and Jacobson, 1981; Cornfield et al, 1982; Ornstein, 1989; Ruelle, 1990). The close
relationships in real brain observables between the appearance rate of new symbols or new unstable periodic orbits, hr , and log 4(+), reflecting the rate of
divergence from the next expected value generating a new, unexpected value, is not surprising. In fact, a maximal estimate of the entropy of a dynamical system, hr = log 4(+) whereas the largest value that hy can attain is log(#of states). A great deal of substantial mathematics has gone into proofs that similarities (“equivalence
relations”) and differences between dynamical patterns are robustly indicated by differences in hy and hy (Adler et al, 1977; Adler and Marcus, 1979).
lf the sum of the densities in each | box were normalized so as to sum to 1.0, such that each is a probability, pj , then - X pj log p; represents the metric entropy, hu. hy was first described in the dynamical context by Kolmogorov (1958;1959). The sum having a —1 prefactor converts the negative log of < 1 to a meaningful positive value in the expression. hy is maximal for the equidistributed, uniformly expansive, C or Axiom A systems (see above). As noted above, generally hy = the maximum estimate of the entropy and hy the minimum estimate (Adler and Weiss, 1965). ht = hw in uniformly hyperbolic systems (Bowen, 1975) and the difference, [hz — hy] is an index of non-uniformity found useful in discriminating among classes of single neurons from their discharge patterns (Mandell, 1987; Selz and Mandell, 1992: Mandell and Selz, 1993; Mandell and Selz, 1997a). These measures applied to temporal and spatial patterns of rat exploratory behavior have been used to discriminate among stimulant drug effects (Paulus et al, 1990; Paulus and Geyer, 1992). Similar computations involving the symbolic dynamics and disallowed transitions have been used to study the complexity of the the EEG (Xu, 1994) in which both extremely low (fixed point, periodic) and high (Gaussian random) entropies are seen as manifesting low “complexity as a function of the diversity of
the available patterns of behavior (Crutchfield and Young, 1989a). Before describing the simple but definitional matrix operations for ht and hy
below which might seem forbidding to those “not up on their linear algebra,” we note
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