Conditions for propagation and block of excitation in an asymptotic model of atrial tissue

Abstract

Detailed ionic models of cardiac cells are difficult for numerical simulations because they consist of a large number of equations and contain small parameters. The presence of small parameters, however, may be used for asymptotic reduction of the models. Earlier results have shown that the asymptotics of cardiac equations are nonstandard. Here we apply such a novel asymptotic method to an ionic model of human atrial tissue to obtain a reduced but accurate model for the description of excitation fronts. Numerical simulations of spiral waves in atrial tissue show that wave fronts of propagating action potentials break up and self-terminate. Our model, in particular, yields a simple analytical criterion of propagation block, which is similar in purpose but completely different in nature to the "Maxwell rule" in the FitzHugh-Nagumo type models. Our new criterion agrees with direct numerical simulations of breakup of reentrant waves.

FIGURE 1

Asymptotic properties of the atrial model of Courtemanche et al. (6). (A) Timescale functions of dynamical variables versus time. (B) Quasi-stationary values of the gating variables and (C) Transmembrane voltage V as a function of time. (D) Main ionic currents versus time. I_in = I_b,Na + I_NaK + I_Ca,L + I_b,Ca + I_NaCa and I_out = I_p,Ca + I_K1 + I_to + I_Kur + I_Kr + I_Ks + I_b,K are the sums of all inward and outward currents, respectively, and the individual currents are described in Courtemanche et al. (6). The results are obtained for a space-clamped version of the model at values of the parameters as given in Courtemanche et al. (6). In C and D, a typical AP is triggered by initializing the transmembrane voltage to a nonequilibrium value of V = −20 mV.

FIGURE 2

Response to a temporary local block of excitability (ℬ) in the models of (A) Courtemanche et al. (6), (B) FitzHugh-Nagumo Eqs. 1, and (C) in Eqs. 4. The border of the blocked region is shown by broken lines. Solutions are represented by shades of gray: black is the smallest, and white is the largest value of V within the solution. The parameters of the FitzHugh-Nagumo model are β = 0.75, γ = 0.5, and ε_g = 0.03, whereas for the two other models the same parameter values as described in Courtemanche et al. (6) are used; the block is described in the plots. The value of j = 0.28 in the block in C is just below the propagation threshold (see Fig. 8). The time and space ranges (in dimensionless units) are 70 × 70 in B and 80 × 50 in A and C.

FIGURE 3

(A) AP and (B) the gating variables h and m as functions of the traveling wave coordinate The solution of the model of Courtemanche et al. (6) is given by circles, of the full three-variable model of Eqs. 4 by thin lines, and the analytical solution given by Eqs. 10 for τ_h = τ_h(V_m) = 1.077, τ_m = τ_m(V_m) = 0.131, V_α = −81.18 mV, and j = 0.956 by thick lines. The gates h and m are indicated in the plot. The position of the internal boundary point is indicated by a dash-dotted line.

FIGURE 4

Wave speed C as a function of the excitation parameter j. (Thick lines) The numerical solution of Eqs. 7. (Thin lines) Solution Eq. 14 for values of τ_h and τ_m corresponding to a selected voltage V = V₀ in Eqs. 5. From right to left, V₀ = −28, −30, V_m, – 34, −36, and −38 (mV). In both cases, V_α = −81.18 mV and mm⁻¹.

FIGURE 5

Wave speed C as a function of the timescale ratio τ_h/τ_m in the caricature model Eqs. 7 and 8. The values of τ_h and are fixed to the values of the corresponding functions in Eqs. 5 at a selected voltage V = V₀, the prefront voltage is V_α = −81.18 mV, and curvature is K = 0 mm⁻¹. (Left plot) Left to right, V₀ = −38, −36, −34 and −32.7 = V_m (mV), and j = 0.9775. (Right plot) Right to left, j = 0.2 to 1.0 and V₀ = V_m.

FIGURE 6

Threshold value j_min above which propagation is possible, as a function of the prefront voltage V_α for the same values of the parameters as in Fig. 3, i.e., τ_h = 1.077 and τ_m = 0.131. Shown are different approximations to the perturbation expansion given by Eq. 16. (Solid line) Zeroth order, Eq. 17. (Dashed line) First order, Eq. 18. (Dotted line) Second order.

FIGURE 7

Wave speed C as a function of j and V_α, for the model of Eqs. 7. Rapid changes are indicated by a higher density of curves. The thick dotted line on the base represents the threshold value j_min and may be compared to the results in Fig. 6.

FIGURE 8

Thick solid line represents the threshold value j_min for excitation failure as a function of V_α for the model given by Eqs. 7. The dotted lines represent projections of AP trajectories in the space-clamped detailed model of Courtemanche et al. (6).

FIGURE 9

Local propagation block, dissipation, and breakup of the front of a reentrant excitation wave. The density plots represent the distribution of the transmembrane voltage V (red component) in regions of superthreshold (white) and of subthreshold (blue) excitability j. The white arrow indicates the time and place the propagation block begins. The time increases from A to F with Δt = 20 ms; size of the simulation domain is 75 mm × 75 mm.

FIGURE 10

(A and B) Wave speed C for the model of Eqs. 7 and 8 as a function of the curvature for values of j = 1…0.4 (from top to bottom). Results for the detailed model (6) are denoted by thick solid lines. (C) The wave speed C in the model given by Eqs. 7 as a function of j for 𝒦 = 0.1, 0, and −0.1 mm⁻¹ (from top to bottom).