The Construction of Arbitrary Stable Dynamics in Non-Linear Neural Networks

In this paper, two methods for constructing systems of ordinary differential equations realizing any fixed finite set of equilibria in any fixed finite dimension are introduced; no spurious equilibria are possible for either method. By using the first method, one can construct a system with the fewest number of equilibria, given a fixed set of attractors. Using a strict Lyapunov function for each of these differential equations, a large class of systems with the same set of equilibria is constructed. A method of fitting these nonlinear systems to trajectories is proposed. In addition, a general method which will produce an arbitrary number of periodic orbits of shapes of arbitrary complexity is also discussed. A more general second method is given to construct a differential equation which converges to a fixed given finite set of equilibria. This technique is much more general in that it allows this set of equilibria to have any of a large class of indices which are consistent with the Morse Inequalities. It is clear that this class is not universal, because there is a large class of additional vector fields with convergent dynamics which cannot be constructed by the above method. The easiest way to see this is to enumerate the set of Morse indices which can be obtained by the above method and compare this class with the class of Morse indices of arbitrary differential equations with convergent dynamics. The former set of indices are a proper subclass of the latter, therefore, the above construction cannot be universal. In general, it is a difficult open problem to construct a specific example of a differential equation with a given fixed set of equilibria, permissible Morse indices, and permissible connections between stable and unstable manifolds. A strict Lyapunov function is given for this second case as well. This strict Lyapunov function as above enables construction of a large class of examples consistent with these more complicated dynamics and indices. The determination of all the basins of attraction in the general case for these systems is also difficult and open.

An unrecognized source of PCB contamination in schools and other buildings

An investigation of 24 buildings in the Greater Boston Area revealed that one-third (8 of 24) contained caulking materials with polychlorinated biphenyl (PCB) content exceeding 50 ppm by weight, which is the U.S. Environmental Protection Agency (U.S. EPA) specified limit above which this material is considered to be PCB bulk product waste. These buildings included schools and other public buildings. In a university building where similar levels of PCB were found in caulking material, PCB levels in indoor air ranged from 111 to 393 ng/m3; and in dust taken from the building ventilation system, < 1 ppm to 81 ppm. In this building, the U.S. EPA mandated requirements for the removal and disposal of the PCB bulk product waste as well as for confirmatory sampling to ensure that the interior and exterior of the building were decontaminated. Although U.S. EPA regulations under the Toxic Substances Control Act stipulate procedures by which PCB-contaminated materials must be handled and disposed, the regulations apparently do not require that materials such as caulking be tested to determine its PCB content. This limited investigation strongly suggests that were this testing done, many buildings would be found to contain high levels of PCBs in the building materials and potentially in the building environment. The presence of PCBs in schools is of particular concern given evidence suggesting that PCBs are developmental toxins.