  A broad range of nonlinear processes over networks are governed by threshold
dynamics. So far, existing mathematical theory characterizing the behavior of
such systems has largely been concerned with the case where the thresholds are
static. In this paper we extend current theory of finite dynamical systems to
cover dynamic thresholds. Three classes of parallel and sequential dynamic
threshold systems are introduced and analyzed. Our main result, which is a
complete characterization of their attractor structures, show that sequential
systems may only have fixed points as limit sets whereas parallel systems may
only have period orbits of size at most two as limit sets. The attractor states
are characterized for general graphs and enumerated in the special case of
paths and cycle graphs; a computational algorithm is outlined for determining
the number of fixed points over a tree. We expect our results to be relevant
for modeling a broad class of biological, behavioral and socio-technical
systems where adaptive behavior is central.
