Empirical Gramian Based Controllability of Alternans in a Cardiac Map Model

Laura Munoz1, Mark Ampofo1, Elizabeth Cherry2
1Rochester Institute of Technology, 2Georgia Institute of Technology


Motivation: Electrical alternans, a beat-to-beat alternation in action potential (AP) dynamics, 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 (QSW) nonlinear map model of alternans dynamics. Our approach had the disadvantage of requiring numerical evaluation of Jacobian matrices, a process that becomes computationally intensive for higher-dimensional systems such as spatially-distributed geometries. An alternative method reported by other groups is to calculate controllability properties using an “empirical” Jacobian-free approach.

Aims: We aimed to determine whether controllability measures based on Gramian matrices were similar for both empirical and conventional calculation methods. We compared the two methods in tests of a cell model under a range of different control strategies, pacing cycle lengths, and model parameters.

Methods: We computed controllability Gramians for the QSW model using a Jacobian-based method and an empirical approach that relies on collecting state variable values from perturbed simulations of the model. Minimum singular values (MSVs) of Gramians were calculated, where larger MSVs indicate less energy required to control the system. Model parameters were adjusted to produce alternans driven by instabilities in either voltage or calcium dynamics.

Results: For appropriate choices of perturbation sizes (approximately three to four orders of magnitude smaller than reference values of the system state), empirical and conventional Gramian MSVs agreed to within 10^(-8) average absolute error. Controllability was maximized when we perturbed intracellular calcium to suppress voltage-driven alternans and perturbed AP duration to suppress calcium-driven alternans.

Conclusions: An empirical approach accurately reproduced Jacobian-based controllability measures for a map model of alternans. These results serve as a useful basis for extending matrix-free methods to higher-dimensional cardiac models.