Mechanism of spiral formation in heterogeneous discretized excitable media

Spiral waves on excitable media strongly influence the functions of living systems in both a positive and negative way. The spiral formation mechanism has thus been one of the major themes in the field of reaction-diffusion systems. Although the widely believed origin of spiral waves is the interaction of traveling waves, the heterogeneity of an excitable medium has recently been suggested as a probable cause. We suggest one possible origin of spiral waves using a Belousov-Zhabotinsky reaction and a discretized FitzHugh-Nagumo model. The heterogeneity of the reaction field is shown to stochastically generate unidirectional sites, which can induce spiral waves. Furthermore, we found that the spiral wave vanished with only a small reduction in the excitability of the reaction field. These results reveal a gentle approach for controlling the appearance of a spiral wave on an excitable medium.

Figure 1

(a–c) Space-time diagrams and (d–f) superimposed images, in which the color changes from blue (dark gray) to yellow (light gray) as time progresses, showing three types of wave propagations. (a, d) The wave vanished before it reached the left-hand side of the excitable region (block state). (b, e) A spiral wave was generated spontaneously (spiral state), and spiral cores were formed at the positions indicated by white broken circles in (e). See Ref. [13] for movie M1. (c, f) The wave propagated and reached the left-hand side of the excitable region (propagating state).

Figure 2

Snapshots of the wave propagation around a spiral core. The wave did not pass the gap in the center of the images while propagating from the top to the bottom (15–30 s). However, the wave moved through the same gap from the bottom to the top (60–90 s), and the spiral core survived. The gap acts as a chemical diode [16], and unidirectional wave propagation appeared.

Figure 3

Snapshots of three types of dynamics in the D-FHN model with heterogeneous reaction terms connections ($D=0.65$). Each panel shows an array 200 cells wide and 100 cells high. A planar wave propagates from the right-hand side of the field, and time progresses as indicated by the red (dark gray) arrow. Yellow (light gray) cells are excitable states, and black cells are refractory, rest states. (a) For a low value of $R_{re}=0.70$, the planar wave is blocked from reaching the left-hand side. (b) For an intermediate value of $R_{re}=0.89$ (see Ref. [13] for movie M2), multiple spiral waves emerge. (c) For a large value of $R_{re}=0.95$, the planar wave propagates across the field.

Figure 4

Snapshots of a typical time series of spiral formation in a heterogeneous excitable medium with different excitabilities. In (a) and (b), each panel shows the time sequence of spiral formation, and the insets are enlarged figures of the wave propagation near a spiral core. Orange (dark gray) cells indicate an additional inhibitor $A_2=0.01$. (a) The site indicated by the cross at $t=13.25$ prevents the excitable wave from propagating to the neighboring cell above, and at $t=27.25$, the excitable wave is able to propagate from the bottom to the top through this same cell. (b) At $t=100.00$, we change the conditions of the excitable medium to $A_1=5.0\times {10}^{-5}$ from $A_1=0.0$. The site indicated by the cross at $t=103.00$ prevents the excitable wave from propagating to the neighboring cell above. At $t=200.00$, we stimulate the right-hand side column of cells again. The last three panels show the behavior of the planar wave after this stimulation. (c) A schematic of the conditions of the additional inhibitor term (see Ref. [13] for movie M3).