Extending Phase Reduction to Excitable Media: Theory and Applications

Phase reduction methods have been tremendously useful for understanding the dynamics of nonlinear oscillators, but have been difficult to extend to systems with a stable fixed point, such as an excitable system. Using the notion of isostables, which measure the time it takes for a given initial condition in phase space to approach a stable fixed point, we present a general method for isostable reduction for excitable systems. We also devise an adjoint method for calculating infinitesimal isostable response curves, which are analogous to infinitesimal phase response curves for oscillatory systems. Through isostable reduction, we are able to implement sophisticated control strategies in a high-dimensional model of cardiac activity for the termination of alternans, a precursor to cardiac fibrillation.