Alternans and Spiral Breakup in an Excitable Reaction-Diffusion System: A Simulation Study

Abstract

The determination of the mechanisms of spiral breakup in excitable media is still an open problem for researchers. In the context of cardiac electrophysiological activities, spiral breakup exhibits complex spatiotemporal pattern known as ventricular fibrillation. The latter is the major cause of sudden cardiac deaths all over the world. In this paper, we numerically study the instability of periodic planar traveling wave solution in two dimensions. The emergence of stable spiral pattern is observed in the considered model. This pattern occurs when the heart is malfunctioning (i.e., ventricular tachycardia). We show that the spiral wave breakup is a consequence of the transverse instability of the planar traveling wave solutions. The alternans, that is, the oscillation of pulse widths, is observed in our simulation results. Moreover, we calculate the widths of spiral pulses numerically and observe that the stable spiral pattern bifurcates to an oscillatory wave pattern in a one-parameter family of solutions. The spiral breakup occurs far below the bifurcation when the maximum and the minimum excited states become more distinct, and hence the alternans becomes more pronounced.

Figure 1

(a) The nullclines of model (1). Solid line is the u-nullcline and dashed lines are the v-nullcline for different values of the parameter b. (b) The action potential of (1) corresponding to the three different values of b in (a), that is, b = 1.2, 1.1, and 1.05. The parameter settings are the same in Table 1.

Figure 2

Breakup of planar pulses in model (1) by transverse instability. The snapshots are at time (a) t = 0, (b) t = 187.62, (c) t = 199.32, and (d) t = 220.12. Numerical integration with space step dx = dy = 0.2 and time step dt = 0.01 on the grid of 150 × 150 elements. The red area represents excited state and the blue area represents the resting state of the tissue.

Figure 3

Time sequence illustrating the dynamical mechanism of spiral breakup in (1). The parameter settings are same as in Table 1 with b = 1.015. The pictures are at time (a) t = 0, (b) t = 546, (c) t = 2270, and (d) t = 4909. Numerical integration with space step dx = dy = 0.25 and time step dt = 0.05 on the grid of 960 × 960 elements. The red area represents excited state and the blue area represents the resting state of the tissue.

Figure 4

The spiral breakup in (1), when b = 1.015. The pictures are at time (a) t = 0, (b) t = 1266, (c) t = 2356, and (d) t = 3657. Numerical integration with space step dx = dy = 0.25 and time step dt = 0.05 on the grid of 960 × 960 elements. The red area represents excited state and the blue area represents the resting state of the tissue.

Figure 5

The spiral dynamics of the tissue as a function of parameter b in (1). The panels are at (a) b = 1.05, (b) b = 1.02, (c) b = 1.018, (d) b = 1.015. Numerical integration with space step dx = dy = 0.25 and time step dt = 0.05 on the grid of 960 × 960 elements.

Figure 6

(a) The two-dimensional plot u(x) for b = 1.05 of (1). (b) The line AB in (a) gives a one-dimensional plot of u(x) by two-dimensional interpolation. (c) A one-dimensional cross-sectional plot of u(x), when b = 1.025. (d) The maximum and minimum widths of spiral pulses as a function of the parameter b of (1). The values of b on the x-axis are 1.06, 1.05, 1.045, 1.04, 1.038, 1.037, 1.035, 1.032, 1.03, and 1.025 (from left to right).

Figure 7

Time sequence illustrating the dynamics of spiral waves in two dimensions. The plots in (a) and (b) are cross-sectional plots in one-dimensional form. (a) A stable spiral traveling wave for b = 1.05 and (b) an oscillatory spiral wave pattern for b = 1.025. (c) The alternans or oscillation of different spiral pulses for b = 1.032 as the development of time. The parameter values used in the simulation are taken from Table 1.

Figure 8

Space-time plots of integrations of (1) in one dimension. Time sequence illustrating the dynamics of (a) a stable periodic traveling wave solution for b = 1.05 and (b) an oscillatory pattern of solution for b = 1.025 in one dimension. The parameter settings are same as those in Table 1.

Figure 9

The spiral wave dynamics of the tissue as a function of the parameter b in (5). The other parameter values are in Table 1. The panels are at (a) b = 1.2, (b) b = 1.1, (c) b = 1.05, (d) b = 1.04, (e) b = 1.03, and (f) b = 1.02. Numerical integration with space step dx = dy = 0.25 and time step dt = 0.05 on the grid of 960 × 960 elements.

Figure 10

Space-time plots of integrations of (5) in one dimension. Time sequence illustrating the dynamics of (a) a stable PTW solution for b = 1.055 and (b) an oscillatory pattern of solution for b = 1.03. The parameter settings are same as in Figure 9.

Figure 11

The APD restitution curves in model (5). The first one is for b = 1.5, having slope <1, and the second one is for b = 1.05, having slope > 1. The parameter settings are the same as in Figure 9.

Algorithm 1

Calculate widths of different spiral pulses.