Criteria for the Convergence, Oscillation, and Bistability of Pulse Circulation in a Ring of Excitable Media

A discrete model based on a nonlinear difference equation (equivalent to a coupled map lattice of high dimension) is used to study the dynamics of a circulating pulse in a ring of excitable media, such as cardiac cells. Based on the global and local properties of monotonic restitution and dispersion curves, criteria are obtained for the asymptotic stability of the unique steady state (pulse circulating at constant frequency) as well as for non-convergent oscillatory behavior of all non-equilibrium trajectories (pulse circulating at variable frequency). We also demonstrate that in certain cases the system is bistable, where an asymptotically stable equilibrium coexists with stable oscillatory solutions.