Controlling the dynamical behavior of a circle map model of the human heart

One-dimensional circle maps are good models for describing the nonlinear dynamical behavior of two interacting oscillators. They have been employed to characterize the interaction between a periodic external forcing stimulus and an in vitro preparation of chick embryonic cardiac cells. They have also been used to model some human cardiac arrythmias such as modulated ventricular parasystole. In this paper, we describe several techniques involving engineering feedback control theory applied to a circle map model of human heart parasystole. Through simulations of the mathematical model, we demonstrate that a desired target phase relationship between the normal sinus rhythm and an abnormal ectopic pacemaker can be achieved rapidly with low-level external stimulation applied to the system. Specifically, we elucidate the linear, self-tuning, and nonlinear feedback approaches to control. The nonlinear methods are the fastest and most accurate, yet the most complex and computationally expensive to implement of the three types. The linear approach is the easiest to implement but may not be accurate enough in real applications, and the self-tuning methods are a compromise between the other two. The latter was successful in tracking a variety of period-1, period-2, and period-3 target phase trajectories of the heart model.