  We study two families of excitable cellular automata known as the
Greenberg-Hastings Model (GHM) and the Cyclic Cellular Automaton (CCA). Each
family consists of local deterministic oscillating lattice dynamics, with
parallel discrete-time updating, parametrized by the range of interaction, the
"shape" of its neighbor set, threshold value for contact updating, and number
of possible states per site. GHM and CCA are mathematically tractable
prototypes for the spatially distributed periodic wave activity of so-called
excitable media observed in diverse disciplines of experimental science.
Earlier work by Fisch, Gravner, and Griffeath studied the ergodic behavior of
these excitable cellular automata on Z^2, and identified two distinct (but
closely-related) elaborate phase portraits as the parameters vary. In
particular, they noted the emergence of asymptotic phase diagrams (and
Euclidean dynamics) in a well-defined threshold-range scaling limit. In this
study we present several rigorous results and some experimental findings
concerning various phase transitions in the asymptotic diagrams, focusing on
evaluating the limiting threshold cutoff for existence of the spirals that
characterize many excitable media. For mathematical expediency our main results
are formulated in terms of spo(p), the cutoff for existence of stable periodic
objects that arise as spiral cores.
