Rhythms from Two Competing Periodic Sources Embedded in an Excitable Medium

In an excitable medium, a stimulus generates a wave that propagates in space until it reaches the boundary or collides with another wave and annihilates. We study the dynamics generated by two periodic sources with different frequencies in excitable cardiac tissue culture using optogenetic techniques. The observed rhythms, which can be modeled using cellular automata and studied analytically, show unexpected regularities related to classic results in number theory. We apply the results to identify cardiac arrhythmias in people that are due to a putative mechanism of two competing pacemakers.

Figure 1

Space-time plot of an experimental monolayer, which is periodically stimulated with period $T_N=1.00\mathrm{s}$ at $N$ (blue arrows) and $T_V=1.69\mathrm{s}$ at $V$ (red arrows). Waves coming from $N$ are in blue, and those coming from $V$ are in red. The numbers on the right are the number of intervening $N$ beats predicted in a given region of the monolayer. These data are shown at specific locations in Figs. 2 and 2.

Figure 2

Interbeat interval plots for the experimental (left) and cellular automaton (right) models at a specific location ($x$). The dots represent IBIs of type NN (blue), NV (red), and VN (green), where the two letters denote the type of beat at the start and end of the interval. Inset text gives NIBs observed. (a)–(b) 0.79 cm from V at $R=1.69$ (c)–(d) 0.06 cm from $V$ at $R=1.69$. (e)–(f) 0.34 cm from V at $R=2.13$.

Figure 3

Space-time schematic illustrating the window $w(x)$ where $V$ beats may occur at a distance $x$ from the $V$ pacemaker. Red dashed lines show the earliest and latest $V$ waves that may arrive at position $x$. Solid blue lines show wavefronts of $N$ waves. The refractory period ($\theta$) is shaded blue. Full propagation of the second $N$ wave is for illustrative purposes only—a $V$ wave could annihilate the $N$ wave, or block it at the source.

Figure 4

(a) Allowed values for the number of intervening $N$ beats (NIB) between $V$ beats in the $(R,\alpha )$ plane, where $R=T_V/T_N$ and $\alpha$ is defined in Eq. (5). Shaded areas can be continually divided into smaller regions and their NIBs determined [8]. The cross-hatched region $\alpha \ge 1$ is the range of parameters for which no $V$ beats occur. Red crosses indicate parameter values used in experiments and cellular automaton simulations and red rectangles show parameters observed in the clinical record. In the experimental model and the cellular automaton, $(R,\alpha )$ values are (1.69,0.75), (1.69,0.56), and (2.13,0.84), for Figs. 2. In the clinical record, $(R,\alpha )$ ranges are given in Fig. 5. (b) Observed and predicted NIBs in space with $R=1.69$. Continuous lines represent theoretical NIB probabilities. Cross mark symbols represent NIB probabilities across selected experimental monolayers where $R=1.69$. The NIB values 1, 2, 4, 6, and 11 are labeled in red, green, blue, purple, and orange, respectively.

Figure 5

Interbeat interval plots for a clinical record hypothesized to show a pure parasystole mechanism (left) and the cellular automaton model (right). Inset text gives NIB values observed. (a) Section of clinical record where $R$ and $\alpha$ fluctuate in the range 2.10–2.38 and 0.62–0.70, respectively. (b) Cellular automaton simulation with $T_N=1.08$, $T_V=2.35$, $\theta =0.68\mathrm{s}$, and $x=0$. (c) Section of clinical record where $R$ and $\alpha$ fluctuate in the range 1.70–1.85 and 0.54–0.59, respectively. (d) Simulation with $T_N=1.28$, $T_V=2.26$, $\theta =0.72\mathrm{s}$, and $x=0$.