Although counter-intuitive when expressed in words, the trajectories that one

Although counter-intuitive when expressed in words, the trajectories that one sees in the graphics of chaotic attractors result from the actions along the “unstable” directions of the stretching distortion; the actions in the otherwise invisible stable directions “iron down” the points onto this unstable manifold (n dimensional abstract surface). As might be expected from this set of characteristic motions, the diagnostic triad of chaotic dynamical systems are: (1) Sensitivity to initial conditions—tiny distances between starting points are magnified and large distances between starting points are reduced under the stretching and folding actions of the system; (2) The presence of a theoretically infinite but countable number of unstable periodic orbits of theoretically all period lengths—points in phase space can be viewed an attractive-repellers, visited and left by the orbits recursively as the dynamics proceed; and (3) Indecomposability—the attractor is not separable into isolated regions and no points escape (see Devaney, 1989, for one of the clearest definitions). Of particular relevance to information encoding and transport by brain mechanisms, it is important to visualize that new information in the form of unstable periodic orbits is being created as well as destroyed by the dynamics. The logarithmic rate of formation of these new orbits is computed as the system’s topological entropy (see below). Assuming the real neurobiological system under study is behaving in these ways (and often much has to be done to help justify such a claim), the observables take the form of an irregular and/or episodic time series of amplitudes, as in repeated sample, neuroendocrine studies of plasma hormone levels (Veldhuis and Johnson, 1992) or a sequence of times between events as in neuronal interspike intervals (Katz, 1966; Perkel et al, 1967). These time or time-sequence series are generally studied from three relatively distinct yet complementary quantitative perspectives: (1) As stochastic (“random”) processes with various amounts of sequential dependency (autocorrelations) and scale (sample length) dependencies; (2) As “deterministic” smooth or discrete, vectorial geometries in phase space following reconstruction and/or embedding of the series as phase portraits or return maps; (3) As information generating and transporting, topological (about relative 205 HOUSE_OVERSIGHT_013705