Using Local Feedback Control to Stabilize Global Behavior in Excitable Media
Abstract
If one end of a one-dimensional excitable medium is forced periodically via impulsive stimuli, the usual response is a periodic wavetrain of propagating pulses. When the forcing period is large, the pulses are uniformly spaced and have identical propagation speed. If the forcing period B becomes critically small, the periodic wavetrain may lose stability via a period-doubling bifurcation that occurs at the stimulus site. In certain contexts (e.g., if the excitable medium is cardiac tissue), it is desirable to stabilize the periodic wavetrain solution by perturbing B, adjusting the timing of the nth stimulus by some small amount ϵn. Previous studies have suggested that if the stimuli are delivered at a single point, then stabilization is possible only in some small radius of the stimulus site. In this article, we explain why controlling global spatiotemporal dynamics via locally applied feedback control (i.e., perturbations to B at the stimulus site) is so difficult. Not only does our analysis reveal why traditional feedback control typically fails, it leads to a constructive algorithm for selecting the perturbations ϵn in such a way that stabilization of the periodic wavetrain can succeed.