topological dimension as that of a line equal to one. If each time step had the

topological dimension as that of a line equal to one. If each time step had the largest up or down amplitude as possible, its fractal dimension would approach (but not reach) that of the embedding plane, Euclidean d = 2. The Do of the one dimensional Richardson technique (Mandelbrot, 1967) can be computed by covering the one dimensional surface of a time series with a number, #, of line segments of several orders of magnitude range of lengths, / -Graphing log(l) along the x-axis and log #(I) along the y-axis yields a negative linear slope, -s. As defined, 1- s = Do noting that (-(-s)+s) such that 1 < Do = 1+s < 2. Strain differences and peptide and psychotropic drug-induced changes in Do computed in this way were found in time series of fluctuations in rat brainstem tyrosine and tryptophan hydroxylase activities under far-from-equilibrium co- reactant concentrations (Mandell and Russo, 1981; Knapp et al, 1981; Knapp and Mandell, 1983; 1984). Systematic influences of stimulant drug dose on Do were found as well in these systems (Mandell et al, 1982). This simple measure, made directly on the “roughness” of the graph of a one dimensional time series rather than on its orbital reconstruction, has been used to discriminate the pattern of fluctuations in daily mood scales in normal subjects and mood disordered patients (Woyshville et al, 1999). These findings confirmed dimensional scaling exponents on higher dimensional embeddings of similar time series in mood disordered patients (Gottschalk et al, 1995; Pezard et al, 1996). Due to the ease and rapidity of its computation, techniques involving Do on one dimensional time series are currently in development as possible real time epilepsy predictors when analyzing the output of a large number of EEG leads simultaneously. If M(e) is the minimum number of d-dimensional cubes of side ¢ required to cover the d-dimensionally embedded attractor, plotting a logarithmic range of rulers of length ¢ (as e—0) along the x axis and a logarithmic range of number of cubes, M(e), each of corresponding «-edge size, along the y axis, results in a negative (more smaller M(e) ‘s and fewer bigger M(e) ‘s) power law slope Do. Here the numbered covering cubes, M(e), are those in which the probability of containing at least one point (its “probability density measure,” often called uw) is not zero. We 227 HOUSE_OVERSIGHT_013727