computer experimental findings had already been anticipated in a remarkable

computer experimental findings had already been anticipated in a remarkable mathematical proof by Sharkovskii (1964). The dynamical richness of these simple, single maximum, one dimensional maps was computationally explored in the context of ecological and epidemiological issues in the classical studies of Robert May (1976). It has been possible to relate the individually characteristic L, R sequence behavior of human subjects on a computer task to a unique parameter of a tent map generating those sequences which predicted age and discriminated subclinical obsessive compulsive from borderline syndromes (Selz and Mandell, 1993). The dynamical entropy of unstructured L,R behavior also discriminated a population of schizophrenic patients from normals (Paulus et al, 1996). More generally, parameter dependent dynamical coding, built into the universal behavior of its constitutive equations, is a mechanism with which a nonlinear dynamical system, such as nerve membrane equations as above, or in the aggregate, the middle layer of a completely connected neural network, can encode, Morse code-like, messages (Paulus et al, 1989). Bifurcations in Biologically Relevant Dynamical Systems Bifurcations, “splitting into (two) branches,” are observed over a smooth change in control parameter(s) (independent variables), as a discontinuous and qualitative change in the dynamical (time-dependent) pattern of the observable (Guckenheimer and Holmes, 1993; Wiggens, 1990; see Strogatz, 1994, for a particularly intuitive description). Qualitative here means how the dynamics of the trajectory appear as a geometric-topological (relative shaped not necessarily sized) pattern in phase space. In such a space, the orbital points are located along the x- axis by their value, x at time f, and along the y-axis by their time rate of change at that ¢, = To visualize a representative phase portrait in the plane, start by imagining the pattern made by mass hanging on a linear spring at rest as dx represented by a point centered at x=0,y= a 0. When perturbed from rest, the 194 HOUSE_OVERSIGHT_013694