  We study the stable attractors of a class of continuous dynamical systems
that may be idealized as networks of Boolean elements, with the goal of
determining which Boolean attractors, if any, are good approximations of the
attractors of generic continuous systems. We investigate the dynamics in simple
rings and rings with one additional self-input. An analysis of switching
characteristics and pulse propagation explains the relation between attractors
of the continuous systems and their Boolean approximations. For simple rings,
"reliable" Boolean attractors correspond to stable continuous attractors. For
networks with more complex logic, the qualitative features of continuous
attractors are influenced by inherently non-Boolean characteristics of
switching events.
