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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 D₀ 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, l
.Graphing log(l) along the x-axis and log #(l) along the y-axis yields a negative linear
slope, -s. As defined, 1- s = D₀ noting that (-(-s)→+s) such that 1 < D₀ = 1+s < 2.
Strain differences and peptide and psychotropic drug-induced changes in D₀
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 D₀ 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 D₀ 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(ε) is the minimum number of d-dimensional cubes of side ε required to
cover the d-dimensionally embedded attractor, plotting a logarithmic range of rulers
of length ε (as ε→0) along the x axis and a logarithmic range of number of cubes,
M(ε), each of corresponding ε-edge size, along the y axis, results in a negative
(more smaller M(ε) ‘s and fewer bigger M(ε) ‘s) power law slope D₀. Here the
numbered covering cubes, M(ε), are those in which the probability of containing at
least one point (its "probability density measure,” often called μ) is not zero. We

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