Criteria for the Convergence, Oscillation and Bistability of Pulse Circulation in a Ring of Excitable Media H. SEDAGHAT 1 , C.M. KENT 1 , M.A. WOOD 2 Abstract. 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 mono- tonic 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 oscil- latory 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. 1 Introduction. The periodic contractions of muscles that result in the beating of our hearts are caused by elec- trochemical signals or excitations called action potentials that propagate through chains of cardiac cells. Normally, cardiac cells generate and conduct action potentials in response to excitation by the self-oscillatory pacemaker cells in the heart’s sinoatrial and atrioventricular nodes. However, in certain circumstances a closed loop or ring of tissue is formed within the heart that unidirec- tionally recycles a previously generated action potential. Such a reentrant circuit is capable of blocking the much slower pacemaker signals by transmitting its rapid pulses outward through adja- cent cell layers, thus taking over the beating of the heart and leading to potentially life-threatening arrhythmias. Reentry of an action potential pulse in a ring of cardiac cells or other excitable media and the resulting self-sustained propagation is relatively easy to model mathematically because of the simple one dimensional geometry. The study of such models contributes to our understanding of cardiac arrhythmias and the results of the study find concrete applications to experimental models of reentrant electrical activity in cardiac muscle. Nevertheless, the mathematical expressions of the manner in which a reentrant pulse propagates in a loop involve complex nonlinear equations whose study requires the application of a variety of different methods from the dynamical systems theory. In Ito and Glass [14] a discrete model of a reentrant circuit is developed based on the restitution and dispersion properties of cardiac cells. Mathematically, the centerpiece of this model is a coupled- map lattice whose dimension is equal to the number of cells (or excitable units) in the loop. Close 01 Department of Mathematics, Virginia Commonwealth University, Richmond, VA 23284-2014, USA; 02 Department of Internal Medicine, Virginia Commonweath University Health Systems, Richmond, Virginia, 23298-0053, USA 1