A two-current model for the dynamics of cardiac membrane

Abstract

In this paper we introduce and study a model for electrical activity of cardiac membrane which incorporates only an inward and an outward current. This model is useful for three reasons: (1) Its simplicity, comparable to the FitzHugh-Nagumo model, makes it useful in numerical simulations, especially in two or three spatial dimensions where numerical efficiency is so important. (2) It can be understood analytically without recourse to numerical simulations. This allows us to determine rather completely how the parameters in the model affect its behavior which in turn provides insight into the effects of the many parameters in more realistic models. (3) It naturally gives rise to a one-dimensional map which specifies the action potential duration as a function of the previous diastolic interval. For certain parameter values, this map exhibits a new phenomenon-subcritical alternansthat does not occur for the commonly used exponential map.

An Ionically Based Mapping Model with Memory for Cardiac Restitution

“…For the map (26), which is derived from the ODE's (21,22), it is shown in Mitchell and Schaeffer (2003) that this threshold behavior follows from asymptotic analysis of the 14 Incidentally, the APD's in Figure 7 are on the order of 5% longer than the values graphed in Figure 3. This discrepancy provides an estimate for the degree of accuracy of the aymptotic approximation.…”

“…Specifically, suppose the stimulus current raises the voltage by an amount v stim in a time that is short compared to τ in ; then the threshold DI for (26) If D n < D thr , the heart is assumed to ignore the stimulus and wait for a later stimulus. (For the mapping (26), this behavior is derived from asymptotics in Mitchell and Schaeffer (2003).) Thus, if the heart is paced at a constant, very short, basic cycle length B, the effective diastolic interval is kB − A n , where k is the smallest integer such that kB − A n > D thr .…”

“…The present work builds on the two-current ionic model of Karma (1993) and of Mitchell and Schaeffer (2003), which we now summarize. The two-current model contains two functions of time, the transmembrane potential v(t) and a gating variable h(t), both of which are dimensionless and scaled to lie in the interval (0, 1).…”

Asymptotic approximation of an ionic model for cardiac restitution

“…In particular, it does not capture any memory effects [6,10,14]. To illustrate this, consider tissue that, after repeated pacing with period B 0 , has achieved a steady-state response.…”

“…For a single cell or a small piece of tissue, an ionic model consists of a system of ODEs that specifies how the voltage across the cell membrane changes in response to currents of ions that flow through the cell walls. Both idealized ionic models [14,18]-low-dimensional systems aimed at qualitative understanding-and realistic models [8,13]-complicated systems intended to describe all currents in the cell-have been proposed. As in the Hodgkin-Huxley equations, given these ODEs, one may introduce an appropriate diffusive term in the voltage equation (or equations, if the bidomain model is used) to obtain PDEs that describe propagation in extended tissue.…”