Empirical Gramian Based Controllability of Alternans in a Cardiac Map Model

Electrical alternans, a beat-to-beat alternation in action potential duration, sometimes precedes the formation of dangerous cardiac arrhythmias. Controllability analysis answers questions about what types of interventions can suppress alternans or other unwanted phenomena in a dynamical model. Previously, we used a conventional linearization approach to determine controllability properties of the $Qu$,, Shiferaw, and Weiss nonlinear map model of alternans dynamics. The model was of interest due to its ability to represent different drivers of alternans, including instabilities in voltage or calcium dynamics. Our computational methods had the disadvantage of requiring numerical evaluation of Jacobian matrices, a process that becomes computationally intensive for higher-dimensional systems. In the present study, we used a Jacobian-free “empirical” method to compute a measure of controllability based on minimum singular values of Gramian matrices and showed that empirical values matched closely with conventional ones. We compared the singular value measure with a modal controllability measure and demonstrated that the former predicts energy usage for certain scenarios, while the latter indicates strategies that lead to smaller sizes of alternans-suppressing perturbations.