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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 , 4(+), is compensated by a decrease in Do leaving hr 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 248 HOUSE_OVERSIGHT_013748