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note that changing the ratios of the numbers of cubes that are dense in point probability to those that are sparse would not influence the value of D₀. This helps differentiate D₀ from other dimensions and, as noted above, D₀ as a maximal estimate of the fractal dimension, is called the capacity dimension and by convention the scaling law is written M(ε) ≈ ε⁻ᴰ⁰. More specifically, D₀ is calculated by repeatedly dividing the d-dimensionally embedded phase space into equal d-dimensional hypercubes and plotting the log of the fraction of the hypercubes containing data points versus the log of the (normalized) linear dimension (“length scale”) of the hypercubes. The slope fitted to the most linear part of the slope (usually the middle 50%) indicates the capacity dimension. D₀ is computed for increasing embedding (and cube) dimension, d, until it achieves an asymptotic plateau, it “saturates”. This is but one of a range of geometric scaling exponents, “dimensions,” that are currently being computed (Farmer et al, 1983; Grassberger and Procaccia, 1983; Meyer-Kress, 1986; Theiler, J. (1990); Gershenfeld, 1992; Ott et al,, 1994).

Although still subject to debate, convention has it that the sample length required to determine this most primitive of dimension computations goes like 10ᴰ⁰ (e.g. a dimension of 2.45 requires a sample length of at least 282 points). Assuming robust findings using D₀ as indicated by non-parametric tests of significance in test-retest, before and after, drug treatment designs, this arbitrary criteria sounds more like ritual than meaningful help for the clinical neuroscientist with (say) 100 spinal fluid hormone and metabolite samples painfully and laboriously collected from a patient’s indwelling catheter over 48 hours. In the context of real data (and not numerical studies of differential equations), we are dealing with empirical findings that must find their meaning (or lack of) in the context of questions about issues in the neurosciences, not in abstract questions such as those about the number of dimensions that an unknown differential equation would require to represent the data (Broomhead and King, 1986). In a similar arbitrary spirit, a system manifesting a D₀ > 5 is considered not discriminable from a random process; e.g. the difference between D₀ = 5 versus D₀ = 7 (though perhaps statistically significant) is thought to be without meaning. Since in neurobiological

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