Two-Term Asymptotic Approximation of a Cardiac Restitution Curve

John W. Cain, University of Richmond
David G. Schaeffer


If spatial extent is neglected, ionic models of cardiac cells consist of systems of ordinary differential equations (ODEs) which have the property of excitability, i.e., a brief stimulus produces a prolonged evolution (called an action potential in the cardiac context) before the eventual return to equilibrium. Under repeated stimulation, or pacing, cardiac tissue exhibits electrical restitution: the steady-state action potential duration (APD) at a given pacing period B shortens as B is decreased. Independent of ionic models, restitution is often modeled phenomenologically by a one-dimensional mapping of the form APDnext = f(B - APDprevious). Under some circumstances, a restitution function f can be derived as an asymptotic approximation to the behavior of an ionic model.

In this paper, extending previous work, we derive the next term in such an asymptotic approximation for a particular ionic model consisting of two ODEs. The two-term approximation exhibits excellent quantitative agreement with the actual restitution curve, whereas the leading-order approximation significantly underestimates actual APD values.