Spiral formation and degeneration in heterogeneous excitable media

Spontaneous spiral formation occurs when an excitation wave is input to a heterogeneous network of low- and high-light-intensity cells projected onto a light-sensitive Belousov-Zhabotinsky reaction. The range of network conditions where spirals form is increased if two waves are input at critical time intervals. Spirals degenerate to form multiple spirals and spirals trapped within excitable cells. Spiral formation and degeneration is dependent on network excitability, cell size, and network size. Results exhibit parallels with spiral formation in excitable biological systems such as the heart.

Figure 1

Spiral formation resulting from one wave input to the network (a),(f) (experiment, ${\Phi }_1=1.3$, ${\Phi }_2=8.6\mathrm{mW}{\mathrm{cm}}^{-2}$; model, ${\Phi }_1=0.0285$, ${\Phi }_2=0.0409$). Behavior when two waves are input to the network with time interval $\left(T\right)$. (Experiment, ${\Phi }_1=0.394$, ${\Phi }_2=9.97\mathrm{mW}{\mathrm{cm}}^{-2}$; model, ${\Phi }_1=0.027$, ${\Phi }_2=0.042$). In the model waves with different time intervals are initiated by altering the stoichiometric factor $f$ of the entire medium. Two stable waves (b),(g), and single (c),(h), double (d),(i), and multiple spirals (e),(j). Projected checkerboard pattern (k). Wavefronts are shown in white [high concentration of oxidized catalyst $\mathrm{Ru}\left(\mathrm{bpy}\right)_3{}^{3+}$]. Arrows show the location and direction of initiation, numbers show time after initiation [seconds or dimensionless time units (dtu)]. See also movies S1–S3 in [10].

Figure 2

(Color online) Numerical results showing spiral degeneration in $11\times 11$ (a),(b) and $10\times 10$ (c)–(f) networks, cell size 18. Orthogonal (a)–(d) and nonorthogonal initiation (e),(f) (shown by arrows) $(f=1.1)$. (a), (c), (e) ${\Phi }_1=0.0245$ and ${\Phi }_2=0.0465$; (b), (d), (f) ${\Phi }_1=0.028$ and ${\Phi }_2=0.039$. Images captured at 70, 143, and 843 dtu. Circles indicate spirals trapped in excitable cells (a),(c),(e) and two opposed fragments rotating around a nonexcitable cell (b),(d),(f). See also movies S4 and S5 in [10].

Figure 3

(Color online) Phase diagram of sampled combinations of ${\Phi }_1$ and ${\Phi }_2$ in numerical simulation $(f=1.1)$. ×, fragmented wave propagation $\left(F\right)$; △, wave does not fragment $\left(W\right)$; ◇, wave does not propagate $\left(N\right)$; at the boundary between $F$ and $N$ waves fragment but propagate only diagonally $\left(D\right)$; ◻, spiral formation $\left(S\right)$; +, unbounded wave fragments (SE). Images illustrate typical wave behaviors; numbers are ${\Phi }_1,{\Phi }_2$.

Figure 4

(Color online) (a) Effects of cell size and $\Delta \Phi$ on wave behavior. ◼, spiral $\left(S\right)$; ▲, diagonal propagation $\left(D\right)$, ∗, fragmented wave $\left(F\right)$; ●, continuous wave $\left(W\right)$; ⊠, no propagation $\left(N\right)$; ◻, diagonal propagation of first wave, second wave annihilated; ◆, both waves fail after crossing first junction. (b) Wave behavior with decreasing cell size. ${\Phi }_1=0.027$, ${\Phi }_2=0.042$, $\Delta \Phi =0.015$. Numbers on images show cell size and time after initiation (dtu).