Mestivier et al, 1998); and predicting defects in visual learning functions from decreases in the A(+)of the cardiac interbeat interval attractor in patients with
multiple sclerosis (Ganz and Faustman, 1996).
We recall that on theoretical grounds (Guckenheimer and Holmes, 1983), a decrease in the positivity of 4(+)—>/(0)in a delay coordinate, geometric
reconstruction of a time series of observables may auger an incipient global bifurcation in the system’s dynamics. As reviewed above, this has turned out to be the case in several studies of the EEG and electrocorticogram in epileptic patients. Futher research will be required to see if this idea has substance more generally for
predicting “catastrophic” changes in other brain-related systems.
Power Law Scaling of Orbital Geometries in Time Series Reconstructions
Benoit Mandelbrot’s book in its first incarnation was derived from his lectures at College de France in 1973 and 1974 and was called Les Objets Fractals: Forme, Hasard et Dimension (Mandelbrot, 1975). This essay was translated into English as Fractals, Form Chance and Dimension (Mandelbrot, 1977). Later expanded and reworked editions displayed another title, The Fractal Geometry of Nature (Mandelbrot, 1982) but the deep conceptual, sometimes poetic fusion and confusion generated by the apparent identity among the objects of his first title remains. “Fractal,” along with “chaos” and “strange attractor” are among the most widely familiar new words in modern dynamical systems research. Fractal is the most difficult to rigorously define and is commonly misunderstood due to the evocative yet dream-like cognitive condensations provoked by the first title and its reflections in Mandelbrot’s prose. A common conceptual confusion is exemplified by the assumed relation between “fractal time event distributions” of the cardiac interbeat interval and the “fractal like” anatomy of the purkinje network of the cardiac conduction system. Data from both contexts are often shown juxtaposed in the same illustration as though their relationships were obvious (Goldberger et al, 1990; Goldberger, 1996; Liebovitch and Todorov, 1996). “Fractal times” and “fractal
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