Induced spiral motion in cardiac tissue due to alternans

Spiral wave meander is a typical feature observed in cardiac tissue and in excitable media in general. Here, we show for a simple model of excitable cardiac tissue that a transition to alternans—a beat-to-beat temporal alternation in the duration of cardiac excitation—can also induce a transition in the spiral core motion that is related to the presence of synchronization defect lines (SDLs) or nodal lines. While this is similar to what has been predicted and indeed observed for complex-oscillatory media close to onset, we find important qualitative differences. For example, single straight SDLs rotate and induce an additional nonresonant frequency characterizing the core motion of the attached spiral. We analyze this behavior quantitatively as a function of the steepness of the restitution curve and show that the velocity and the directionality of the core motion vary monotonically with the control parameter. Our findings agree with recent observations in rat heart tissue cultures indicating that the described behavior is of rather general nature. In particular, it could play an important role in the context of potentially life-threatening cardiac arrhythmias such as fibrillation for which alternans and spiral waves are known precursors.

Figure 1

Snapshots of the voltage field for spirals in three regimes relevant to this study: (a) $\mathrm{Re}<1.05$, (b) $1.05<\mathrm{Re}<1.125,$ and (c) $\mathrm{Re}>1.15$. These images were taken of circular domains with a radius of 2.62 cm and a no-flux boundary. (d) shows a space-time plot along the indicated radial cut (thick line) from the spiral core in (b), clearly exhibiting alternans.

Figure 2

(a) Average spiral rotation period, and (b) the average action potential duration${}^{\mathrm{a}}$, as functions of $\mathrm{Re}$. After the period doubling, both plots contain two branches, indicating the period-two behavior of the spiral. The dashed line in the middle is an average of the two branches. For the spiral rotation period in (a), this average corresponds to half of the new period. For both spiral rotation period and APD, each data point corresponds to a numerical estimate of the asymptotic long-time average sufficiently far away from the spiral core.${}^{\mathrm{a}}$For this study, APD is defined as the length of time $E\ge 2$

Figure 3

(a) Dimensionless voltage, (b) the associated SDL detected using the method described in Ref. [28], (c) the method used in Ref. [10], and (d) the method used in this study.${}^{\mathrm{a}}$ The images were taken on a circular tissue with a radius of 5.24 cm and a no-flux boundary.${}^{\mathrm{a}}$

Figure 4

Conduction velocity of a single stable pulse or action potential traveling on a ring with $\mathrm{Re}=1.08$ as a function of ring length. The behavior is qualitatively the same for all $\mathrm{Re}$ we considered ($1.02\le \mathrm{Re}\le 1.16$).

Figure 5

Period of rotation of a single SDL attached to a free spiral in a circular domain with diameter 10.48 cm as a function of $\mathrm{Re}$. For low $\mathrm{Re}$ values, no alternans are present. After the appearance of alternans, increasing $\mathrm{Re}$ increases the period of rotation. This continues until a point where SDLs start to appear and disappear constantly (turbulent SDLs). For even larger $\mathrm{Re}$, the spiral simply breaks up. The variation with the spatial resolution, $▵x$, used for the numerical integration gives a rough estimate of the uncertainties.

Figure 6

Spiral core trajectories for (a) $\mathrm{Re}$ over 493 ms, (b) $\mathrm{Re}=1.08$ over 13493 ms, and (c) a zoomed in portion of (b). Both of the core trajectories have had running averages applied over 6.25 ms in order to smooth out the trajectories, while (c) has had different running averages applied. The red (light gray) trajectory seen in (c) is the result of a running average of 25 ms (one spiral rotation), while the blue (dark gray) trajectory is the result of a running average of 200 ms (eight spiral rotations). For $\mathrm{Re}=1.02$, alternans have not yet set in, the only motion present is a circular orbit. In the alternans regime for $\mathrm{Re}=1.08$, a slowly rotating SDL is present introducing a second frequency and causing the core to exhibit inward petals.

Figure 7

Drift speed (solid triangles) and the angle between the direction of the drift and the orientation of the SDL, $\theta$ (open circles), as functions of $\mathrm{Re}$. For $\mathrm{Re}<1.07$ numerical accuracy is insufficient to obtain a clear drift trajectory, and subsequently neither a drift velocity or a drift-SDL angle could be measured for $\mathrm{Re}=1.06$.

Figure 8

Images of the voltage field with SDLs overlaid for two spirals in a periodic tissue of size 20.96 cm by 10.48 cm and $\mathrm{Re}=1.08$. Images were taken at (a) 3450 ms, (b) 5750 ms, (c) 7950 ms, (d) 10550 ms, (e) 11100 ms, and (f) 13650 ms, respectively. The SDLs of each spiral both rotate, and are connected to each other at all times. Notice also how another SDL disconnects and lies on the shock line for part of the rotation.

Figure 9

Snapshots the voltage field with SDLs overlaid from a simulation of many spirals with Re $=$ 1.08. The simulation was run on a square periodic tissue of length 20.96 cm. Snapshots show (1) nonrotation of some SDLs, (2) rotation of others, and (3) reconnection between SDLs. (4) Four tiny spirals can be seen inside the shock lines in the upper right corners of the images. Snapshots were taken at times (a) 20 s, (b) 22 s, (c) 24 s, and (d) 26 s.