period doubling bifurcations accumulate (Grassberger, 1981). This is a dust-like region, which when endlessly dilated looks like the same dust. Some mathematicians call these objects “Lebesgue points” because even though at low magnifications when they look rather solid, they are not. Composed of points, they have topological measure zero (a line has measure one) and non-integer fractal dimension. These 4=0, D = Integer, period doubling accumulation points can be found in a wide variety of attractors, though in each case the parameter space in which they are located is so small (in point set topology also called “Lebesgue measure zero”) that they are very difficult to locate and therefore have little chance of being physiologically significant. This constrasts with a relatively new category of dynamical systems which promises to be important in studies of the nervous system. These are ones that are driven by two or more independent frequencies (called quasiperiodic driving). We found them to be relevant to brain stem, thalamocortical neurophysiology of perceptual processes and states of consciousness. They have the properties, A=0, Do and D; # integer and a characteristic scaling “spectral distribution function” (see below). They have been named “strange nonchaotic attractors” (Grebogi et al, 1984; Romeiras et al, 1987; Ding et al, 1989). In addition, the strange nonchaotic behavior of these quasiperiodically-driven, nonlinear oscillators has positive (>0) measure in parameter space and thus is of potential physiological significance. A good demonstration of a multiple frequency driven strange nonchaotic attractor can be found and manipulated in the software package of Nusse and Yorke (1991). The neurobiological substrate for this system is the brain stem neuronal modulatory driving of on- going thalamocortcal oscillatory brain waves (once called “recruitment waves” in the 7-14 Hz, 8 to a, day dreaming to quiet alert range) and as perturbed by multifrequency drivi