Hysteresis phenomena between periodic and stationary solutions in a model of pacemaker and nonpacemaker coupled cardiac cells

We were interested in investigating the behaviour of a cardiac electrophysiological model including coupled pacemaker (PM) and nonpacemaker (NPM) cells. To this aim, a modified version of the model of Van Capelle and Durrer was used. First, few discrete values were assigned to coupling resistance (CR) and respective cell sizes and numerical simulations versus time showed three possible kinds of response pattern: sustained rhythmic activity, subthreshold oscillations, and complete inhibition. Then, after setting a fixed value to PM cell size, we undertake a thorough study of the system by using bifurcation-continuation techniques and CR was chosen as the continuation parameter. On the maximum action potential — CR plane representation, we could describe five behavioural zones: complete inhibition, coexistence of complete inhibition and NPM large oscillations, NPM large oscillations, coexistence of NPM large oscillations and subthreshold oscillations, subthreshold oscillations. Within the zones of qualitatively different coexisting solutions, a detailed exploration clearly demonstrated the presence of hysteresis cycles. Indeed, the status of the system depended on its immediate previous story within narrow ranges of CR values. Such a coexistence of stable solutions for identical values of CR may suggest an explanation of the intermittant activity elicited from abnormal ectopic foci observed in certain ventricular rhythm disturbances. In addition, a Hopf bifurcation point, from which emerged stationary and periodic solutions, was followed on the PM cell size — CR plane and from this representation we could deduce that the smaller the PM cell, the higher the CR must be for the PM cell to escape from the NPM cell inhibition.