brain enzyme kinetics (Mandell, 1984) and single neuron firing patterns (Selz and Mandell, 1991) to human psychomotor and cognitive behavior (Selz, 1992; Selz and Mandell, 1993). The Leading 4(+) of Some Biologically Relevant Time Series An early application of a simplified form of leading Lyapounov exponent to brain data involved the computation of the one dimensional averaged slope of in vitro studies of psychopharmacological drug and peptide effects on time series of catecholamine and indoleamine biosynthetic enzyme activities studied at physiological, far-from-equilibrium reactant concentrations (Russo and Mandell, 1984b; Knapp and Mandell, 1984). A contemporaneous study also suggested the influence of differences in initial conditions for pharmacokinetic equilibrium times in drug binding kinetics by proteins (Bayne and Hwang, 1985). The most extensive applications to the clincial neurosciences of the Lyapounov measure of the exponential divergence of orbital points has involved reconstructed brain wave attractors from the intracranial or scalp recordings of the EEG (Duke and Pritchard, 1991; Dvorak and Holden, 1991; Jansen and Brandt, 1993). Space prevents us from surveying more than a small representative set of the studies (Jansen, 1996). It should be noted, however, that this is an area in which “state of the art” research has grown quite complicated and somewhat controversial with respect to technical issues. The choices of the digitizing frequency of the smooth record, the dimension of the embedding space and time delays continue to be debated in the context of numerical computations of 2 and dimension measures (Mayer-Kress, 1986; Ott et al, 1994). Controls for the implicitly required statistical discrimination between “randomness” and “deterministic chaos” consist of sequence and (Fourier) phase randomization generating “surrogate data” which conserve the probability distributions and destroy the correlation properties and attractor geometries (Sauer