We study two families of excitable cellular automata known as the Greenberg-Hastings model and the cyclic cellular automaton. Each family consists of local deterministic oscillating lattice dynamics, with parallel discrete-time updating, parametrized by the range $\rho$ of interaction, $\ell^p$ shape of its neighbor set, threshold $\theta$ for contact updating, and number $\kappa$ of possible states per site. These models are mathematically tractable prototypes for the spatially distributed periodic wave activity of so-called excitable media observed in diverse disciplines of experimental science.
¶ Fisch, Gravner and Griffeath studied experimentally the ergodic behavior of
these models on $\Bbb Z^2$, started from random initial states. Among other phenomena, they noted the emergence of asymptotic phase diagrams (and dynamics on $\Bbb R^2$) in the threshold-range scaling limit as $\rho,\theta\to
\infty$ with $\theta/\rho^2$ constant.
¶ Here we present several rigorous results and some experimental findings concerning various phase transitions in the asymptotic diagrams. Our efforts focus on evaluating bend$(p)$, the limiting threshold cutoff for existence of the spirals that characterize many excitable media. Our main results are formulated in terms of spo$(p)$, the cutoff for existence of stable periodic objects that arise as spiral cores. Some subtle consequences of
anisotropic neighbor sets $(p\neq2)$ are also discussed; the case of box neighborhoods $(p=\infty)$ is examined in detail.