  From genetic regulatory networks to nervous systems, the interactions between
elements in biological networks often take a sigmoidal or S-shaped form. This
paper develops a probabilistic characterization of the parameter space of
continuous-time sigmoidal networks (CTSNs), a simple but dynamically-universal
model of such interactions. We describe an efficient and accurate method for
calculating the probability of observing effectively M-dimensional dynamics in
an N-element CTSN, as well as a closed-form but approximate method. We then
study the dependence of this probability on N, M, and the parameter ranges over
which sampling occurs. This analysis provides insight into the overall
structure of CTSN parameter space.
