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and Mumolo, 1998). Spontaneous changes in the apparent syllabic sound made by
regularly presented, word-like auditory stimuli emerge irregularly, the duration of
perceived sameness demonstrating a power law distribution of “dwell” times (Tuller
et al, 1998). The same kind of power law distribution of characteristic “brain times”
can be found in studies of gait cycle durations in normal walking (Hausdorff et al,
1996) with a decrease in this locally detrended, α-like index compared with controls
(0.91±0.05) in patients with the basal ganglia disorders of Parkinson’s (0.82±0.06)
and Huntington’s (0.60±0.04) Diseases (Hausdorff et al, 1998). Hurst > 0.5 has
been speculated to more accurately quantitate the fundamental time structure of
cells that was previously called circahoralian (ultradian) intracellular rhythms
(Brodski, 1998).

Reconstructions of Time Series as Orbital Geometries

Rene Thom (1972), extending the ideas of Poincaré and D’Arcy Thompson
(1942), argued that experimentally useful, intuitive connections between the
qualities of biological processes and the quantities of an explicit (equations known)
or implicit (equations unknown) dynamical system could be best achieved through
the use of graphic representations of their geometric and topological forms. Notably
successful examples can be found in the work of Thom, Arnold (1984) and Zeeman
(1977), who were inspired by “caustics” (the shapes made on surfaces by the
coincidence of reflected or refracted light rays) and Whitney’s representation of
parametric manifolds (surfaces) by the shadows they would make on a plane when
back lit (Whitney, 1955). This led to a small number of qualitatively predictive,
number-of-independent-parameters dependent shapes, such as “folds” “cusps” and
“wavefronts.” Experimentally crossing the values of these independent variable
forms at their singular boundaries successfully predicted discontinuities in the
otherwise smooth alterations in the dependent variable; i.e. bifurcations
(“catastrophes”) in the behavior of the observable. This approach was best suited to
the study of systems with many independent variables and one dependent variable
that could be mapped on the axis of the latter to represent a continuum of

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