The effects of the intensity of coupling and the time delay on the properties of circulating spatiotemporal firing patterns in several simple cell arrays have been studied for the Belousov-Zhabotinskii (BZ) and chlorine dioxide-iodide reaction kinetics models. Single firing and finite cascade of firings have been found for both systems, but permanent firing has been observed only in the Oregonator model of the BZ reaction. The existence of regimes with permanent firing have been explained on the basis of the difference in the course of concentration trajectories of the studied systems.