Topological constraints on spiral wave dynamics in spherical geometries with inhomogeneous excitability

We analyze the way topological constraints and inhomogeneity in the excitability influence the dynamics of spiral waves on spheres and punctured spheres of excitable media. We generalize the definition of an index such that it characterizes not only each spiral but also each hole in punctured, oriented, compact, two-dimensional differentiable manifolds and show that the sum of the indices is conserved and zero. We also show that heterogeneity and geometry are responsible for the formation of various spiral-wave attractors, in particular pairs of spirals in which one spiral acts as a source and a second as a sink—the latter similar to an antispiral. The results provide a basis for the analysis of the propagation of waves in heterogeneous excitable media in physical and biological systems.

Figure 1

(Color online) Left: spiral waves of excitation (light fronts) on the sphere for constant $\beta ={\beta }_{\mathit{ex}}$ emanating from spiral cores close to the poles on an equator projection. The white arrows show the direction of propagation. One annihilation front along the equator can be identified. Right: sketch of the constant gradient in the inhomogeneous case. The dashed lines are the equi-$\beta$ lines and we choose ${\beta }_{\mathit{max}}=1.0$ and ${\beta }_{\mathit{min}}={\beta }_{\mathit{ex}}$. The angle $\varphi$ describes the orientation with respect to the axis from pole to pole. The results described in the text do not depend qualitatively on the choice of $\varphi$ or ${\beta }_{\mathit{min}}$ and ${\beta }_{\mathit{max}}$ as long as they yield stable spirals.

Figure 2

(Color online) The local excitability $\beta \left(\mathbf{r}\left(t\right)\right)$ at the spiral cores versus time. Gradient-induced motion of the two spiral cores leads to a change in the local excitability at the cores with time. The spiral period in the final state is $T_0=13.2\pm 0.1$. Four different regimes can be identified (see text). Inset: the final bound pair of counterrotating spirals in regime III for $\varphi =51.0°$ is shown on an equator projection such that the point of lowest excitability lies on the central longitude. The spiral closer to the equator has index $-1$ while the other one has index $+1$.

Figure 3

(Color online) The distance between the two spiral cores $d\left(t\right)$ versus $t$. Gradient-induced motion of the two spiral cores leads to a change in the distance $d$ between the cores with time. The lower and upper curves correspond to the distance in ${\mathbb{R}}^3$ and ${\mathbb{S}}^2$, respectively. The dashed lines are the respective upper bounds given by the size of the sphere.

Figure 4

(Color online) Waves of excitation on the sphere in regime I of the gradient-induced dynamics shown in Figs. 2, 3. A source-sink pair has formed. For random spatial variations of excitability with a correlation length comparable to the diameter of the spiral core meander, the final state is very similar to the one shown here [22]. Upper panel from left to right: view centered at the north pole, south pole, and the equator. The source at the south pole has index $-1$ and the sink at the north pole index $+1$. The black circles show possible choices of $C$. The white arrow shows the direction of wave propagation. Lower panel: dynamics at the north pole. Time increases from left to right with $\Delta t=2.5$ between snapshots.

Figure 5

(Color online) Waves of excitation on the punctured sphere propagating towards the hole. A source-sink spiral wave pair has formed in the final state. Upper panel from left to right: view centered at the north pole, south pole, and the equator. Lower panel: dynamics at the south pole. Time increases from left to right with $\Delta t=2.5$ between snapshots.