decrease in entropy in brain-relevant dynamical systems. As we shall see, certain pathophysiological processes appear to manifest themselves as reductions in background or “resting” state entropy which then limits its supply with respect to information gain and/or transport. Relationships between “physical” thermodynamic observables, such as changes in heat capacity or temperature dependence of kinetic constants, and information-transport driven, neurotransmitter evoked conformational changes in neural membrane proteins may someday come together in an experimentally productive way (Hitzemann et al, 1985; Zeman et al, 1987;
Borea et al, 1988), but they are beyond the scope of this paper.
The idea of taming the orbit of an expanding flow (with at least one 2(+) ) by
partitioning the geometric space supporting its actions, its “manifold,” and then labeling each box so that its trajectory is representable by a symbol string of box indices is the way “symbolic dynamics” are applied to dynamical systems. Symbolic dynamics arose in pure mathematics in the context of obtaining a one-to-one, topological (sequence not distance preserving ) representation of a difficult to characterize system of “geodesics on surfaces of negative curvature” (Hadamard, 1898; Morse, 1917; Morse and Hedlund, 1938). Geodesics here are the shortest lines in this curved, non-Euclidean space in which nearby lines spread apart and far
away ones came together with (in Euclidian space) parallel lines meeting at infinity.
Remarkably, symbolic dynamic encoding of the motions on this abstract manifold of negative curvature also capture how uniformly divergent (and convergent), “hyperbolic” chaotic systems, such as brain systems, behave in Euclidean space, an intuitive similarity about which Poincare experienced his famous vacation bus trip epiphany (Stillwell, 1985). It should also be noted that encoding neural spike trains in one dimension for symbolic dynamical comparisons of sequence structure and recurrances, “favored patterns” has been developed independently of orbital dynamics on manifolds (Dayhoff, 1984; Dayhoff and Gerstein, 1983a; 1983b). A similar approach has been used to characterize firing patterns and their response to acupuncture in dopamine neurons in the substantia
nigra and hypothalamic neurons (Chen and Ku, 1992).
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