HOUSE_OVERSIGHT_013700.jpg

21...) (Shenker and Kadanoff, 1982). Similar quantitative scaling properties were also discovered in the parametric period adding route (Kaneko, 1983).

All of these scaling numbers have been found in experiments and in remarkable agreement with theory. Examples have been discovered in electronic circuits, hydrodynamic and mercury flows, acoustic systems, laser dynamics and oscillating chemical reactions (see Cvitanovic', 1989, for representative list of references). Whereas qualitative evidence for all of these bifurcation scenarios have been found in brain relevant experiments, there is yet to be a bifurcating experimental biological system with adequate precision across a sufficient range of magnitudes such that quantitative universality could be demonstrated across a sufficient range of values to be convincing. We remind ourselves that in order to establish a Fiegenbaum number, each period doubling bifurcation of the several required necessitates about a five-fold improvement in the experimenter’s ability to specify the control parameter.

Using Invariant Measures of Dynamical Neurobiological Systems

Before the modern era of dissipatively forced (energy utilizing) dynamical systems research, the known attactors of an experiment’s initial values resulted from their convergence onto either a fixed point or a limit cycle. An attractor can be regarded as a set which remains in bounded space and to which all orbits in this neighborhood converge (Milnor, 1985). Since by the rules of differential equations, orbits are required to be both smooth (graphable without lifting the pencil) and unique (different trajectories don’t intersect since the point of intersection would no longer be unique), the foundational Poincare-Bendixon theorem says that any such orbit confined to a two dimensional phase space that doesn’t converge to a fixed point must, no matter how long it irregularly wanders, must, eventually intersect with itself and then go around the same route again in a (perhaps very long) cycle. In most neuroscience research as well, we have generally regarded our data as manifesting either tolerable (or intolerable) fluctuations around mean values (fixed

200

HOUSE_OVERSIGHT_013700