Global Organization of Dynamics in Oscillatory Heterogeneous Excitable Media

An oscillatory heterogeneous excitable medium undergoes a transition from periodic target patterns to a bursting rhythm driven by the spontaneous initiation and termination of spiral waves as coupling or density is reduced. We illustrate these phenomena in monolayers of chick embryonic heart cells using calcium-sensitive fluorescent dyes. These results are modeled in a heterogeneous cellular automaton in which the neighborhood of interaction and cell density is modified. Parameters that give rise to bursting rhythms are organized in distinct zones in parameter space, leading to a global organization that should be applicable to the dynamics in a large class of excitable media.

Figure 1

The activity of embryonic chick heart monolayers plated at three different densities under different concentrations of gap junction blocker $\alpha \mathrm{G}\mathrm{A}$ visualized with calcium-sensitive fluorescent dye. Each box is approximately $0.8{\mathrm{c}\mathrm{m}}^2$. A trace above each panel shows the activity for 50 sec of the four center pixels. The images are presented as activation plots which show the overlayed location of active cells at 100 msec intervals.

Figure 2

The dynamics of the model for different values of density and neighborhood size. The position of the wave front at different times is shown for the first propagated pulse. For this simulation $F_{\mathrm{s}\mathrm{u}\mathrm{b}}=0.999$. The images are represented as activation plots showing the overlayed location of active cells at ten iterate intervals.

Figure 3

The dynamics of the experimental preparation and the model for different values of density and cell-cell coupling. (a) Entropy values calculated for 500 consecutive frames for the experimental data shown in Fig. 1. (b) Entropy values calculated for parameters of the model, shown as a contour plot. Since the size of large clusters is constrained by the boundary of the simulation, we consider clusters of greater than 10 000 pixels to be in the same size class in order to minimize boundary effects. (c) Model parameters that give rise to bursting dynamics are indicated with black crosses. The results for the model were obtained by simulating three different geometries at each parameter point.

Figure 4

Cluster size distributions for 500 frames from the experimental data under intermediate gap junction block ($\alpha \mathrm{G}\mathrm{A}=5\mu \mathrm{M}$) for different densities (panels a and b) and 6000 iterates of the model (panels c and d) for $r=2.5$ for different density values. (a) Cluster size distribution for a high density ($20\times {10}^3\mathrm{\text{cells}}/{\mathrm{c}\mathrm{m}}^2$) monolayer. (b) Cluster size distribution for a low density ($5\times {10}^3\mathrm{\text{cells}}/{\mathrm{c}\mathrm{m}}^2$) monolayer. (c) Cluster size distribution for the model for high density $\sigma =1$. (d) Cluster size distribution for the model for low density $\sigma =0.5$. Cluster sizes for the model were divided by a scaling factor of 3 to fit the data.

Figure 5

Schematic global organization in the cellular automaton based on computations at 2000 values evenly spaced on the $F_{\mathrm{s}\mathrm{u}\mathrm{b}}$, $r$ parameter space. Each parameter point was tested for three randomly generated arrays.