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idealized systems, topological entropy has been proven be equivalent to the product
of an index of expansion and the dimension of the support such that an increase in
expansiveness , λ(+), is compensated by a decrease in Do leaving hT invariant
(Manning, 1981). This relationship has also been found in the behavior of some
nonuniformly expansive neuroendocrine, neuronal and human behavioral systems
(Mandell and Selz, 1995; Smotherman et al, 1996; Mandell and Selz, 1997a;).

Is Randomness Versus Determinism a Productive Question for the Biological
Sciences? Are There Better Ones?

Measures made on realistically nonuniformly expansive behavior of
dynamical systems emerging from nonlinear differential equations and that arising
from a variety of non-classical random walk models overlap such that making what
may be more a metaphysical discrimination at this point is labor intensive,
contentious and unproductive for generating new experimental work in the
neurosciences. It is important to note that random walk theory and computation has
matured to such an extent that almost any “nonlinear dynamical behavior” can, with
respect to statistical measure, be modeled using one of many varieties. For
examples, power law distributions in continuous time random walks (times of
movement are also randomly chosen) , random walks with traps (temporarily
immobilizing the trajectory like unstable fixed points), random walks in random
environments, time of passage of ants in a labyrinth and Levy leaps and local
diffusive exploration (looking for a wallet) among many others can represent much
of the irregular behavior we observe in the brain (Shlesinger et al, 1982; Montroll
and Shlesinger, 1984; Hughes, 1995; Klafter et al, 1996). On the other hand,
(Markoff) partition of the sequence and a probabilistic style of analysis of nonlinear
dynamical systems has been a major strategy for description and quantification
from the field’s beginnings (Parry, 1964; Adler and Weiss, 1967; Bowen, 1970;
Lasota and Yorke, 1973). The issue of randomness versus determinism remains
current although many if not most properties of deterministic dynamical systems can

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