Two-Term Asymptotic Approximation of a Cardiac Restitution Curve

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 APD(next) = f(B - APD(previous)). 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.