Analytically Solvable Asymptotic Model of Atrial Excitability

We report a three-variable simplified model of excitation fronts in human atrial tissue. The model is derived by novel asymptotic techniques \new{from the biophysically realistic model of Courtemanche et al (1998) in extension of our previous similar models. An iterative analytical solution of the model is presented which is in excellent quantitative agreement with the realistic model. It opens new possibilities for analytical studies as well as for efficient numerical simulation of this and other cardiac models of similar structure.

💡 Research Summary

The paper presents a rigorously derived three‑variable asymptotic model that captures the essential dynamics of excitation fronts in human atrial tissue. Starting from the detailed Courtemanche et al. (1998) model, which contains 21 state variables and dozens of ionic currents, the authors apply a novel asymptotic expansion that separates fast sodium influx from slower repolarizing currents. By introducing a small parameter ε that scales the rapid upstroke dynamics, they systematically reduce the full system to three coupled ordinary differential equations governing the transmembrane voltage V, the fast activation gate m, and the slower inactivation gate h.

The reduction proceeds in two stages. First, the total ionic current is split into a fast component I_Na_fast(V,m) that dominates during the steep voltage rise, and a slow component I_K_slow(V,h)+I_leak that governs the recovery phase. The fast component is treated in the singular limit ε→0, which forces m to instantaneously follow its quasi‑steady‑state m∞(V) while retaining a finite but small time constant τ_m(V). The slower gate h retains its full dynamics with a time constant τ_h(V) on the order of tens of milliseconds. The resulting reduced system is:

1. dV/dt = –