Figure 1

Schematic illustration of the experimental system for the photosensitive BZ reaction.Reuse & Permissions

Figure 2

Experimental results. (a) Top view of the reaction tracks at $t=0$ on R1 ($t=-110$ on R2). The white region was illuminated at $66\mathrm{klx}$, and showed complete inhibition. The two stripes R1 and R2 were initially affected by a slight suppression effect at $0.5\mathrm{klx}$. At $t=0$ the light intensity started to increase to $17\mathrm{klx}$ (largely suppressed state), and the rates of change were different between R1 and R2. (b) Spatiotemporal plots showing the different nature of pulse propagation between the tracks. The pulse propagated throughout the track under a slow change (R1), whereas it failed to propagate under an abrupt change in light intensity (R2).Reuse & Permissions

Figure 3

Numerical results based on the Oregonator model, Eqs. (1, 2, 3, 4, 5). The parameter for light intensity was varied: (a) $T=2.8$ and (b) 0. (i) Time trace of light intensity. We set $t=0$ at the moment that $\varphi$ begins to change. (ii) Spatiotemporal plots of single-pulse propagation in one-dimensional excitable media for different $T$. The propagating pulse disappears in (b) and survives in (a).Reuse & Permissions

Figure 4

Lifetime $\tau$ of the traveling pulse for different $T$ as deduced from the numerical calculation. A pulse fails to propagate when the duration of light growth $T$ is smaller than $T_c$, and survives when $T$ is larger than $T_c$. Calculations were performed with the modified Oregonator model [Eqs. (1, 2, 3, 4, 5)], where the parameters are ${\varphi }_I=0.015$, ${\varphi }_{II}=0.0974$, $f=3$, $q=0.002$, ${ϵ}_u=0.004$, ${ϵ}_v=1$, $D_u=1$, and $D_v=0.6$.Reuse & Permissions

Figure 5

Schematic representation of the response of the system to slow and abrupt changes in inhibitory stimuli. (a) Value of the slow variable in the vicinity of the pulse front $v_f$ can be considered to be equal to that in the homogeneous area $\left(H\right)$. (b) Nullclines of Eqs. (8, 9) are shown for $\varphi ={\varphi }_I$ and ${\varphi }_{II}$ [$f_I=f(u,v,{\varphi }_I)$, $f_{II}=f(u,v,{\varphi }_{II})$]. The points $P$ and $Q$ give stable fixed points under $\varphi ={\varphi }_I$ and ${\varphi }_{II}$, respectively. Under a slow change, the resting state changes by a quasistatic process $\left(PQ\right)$. On the other hand, with an abrupt change, it changes by a two-step process $\left(PRQ\right)$. (c) Excitation threshold $E_{\mathrm{th}}(v,\varphi )$, which can be estimated as $v-v_{\min }$ [22], where $v_{\min }$ is the value of $v$ at the bottom of the $u$ nullcline $f(u,v,\varphi )=0$. Under a slow change in $\varphi$, the value of the excitation threshold in the vicinity of the pulse front is equal to $E_{\mathrm{th}}(v_Q,{\varphi }_{II})$. On the other hand, under an abrupt change, it changes from $E_{\mathrm{th}}(v_R,{\varphi }_{II})$ to $E_{\mathrm{th}}(v_Q,{\varphi }_{II})$, where $E_{\mathrm{th}}(v_R,{\varphi }_{II})>E_{\mathrm{th}}(v_Q,{\varphi }_{II})$.Reuse & Permissions

Figure 6

Diagram of the transition of the traveling pulse under abrupt and slow changes in the environmental parameter, obtained from numerical results using the modified FHN model [Eqs. (1, 2, 6, 7)]. (a) Diagram of the transition in three dimensions. (b) Projection figures of (a) to (i) $\varphi \text{−}S$ and (ii) $\varphi \text{−}{E}_{\mathrm{th}}^f$ planes. The parameter $S$ is defined as the size of the traveling pulse in space units. The value of the excitation threshold at the pulse front $E_{\mathrm{th}}^f$ and the environmental parameter $\varphi$ are plotted. The black $\left[P\right(\varphi \left)\right]$ and gray $\left[U\right(\varphi \left)\right]$ dots indicate the stable traveling pulse and the trivial uniform solutions for various $\varphi$, respectively. As $T$ increases, the trajectory more closely approaches the set of stable pulse solutions. The black dashed trajectory $(T=40)$ is very close to the set of stable pulse solutions. Under a slow change $(T=6)$, the stable pulse solution under $\varphi ={\varphi }_I$ $\left[P\right({\varphi }_I\left)\right]$ changes to the stable pulse solution under $\varphi ={\varphi }_{II}$ $\left[P\right({\varphi }_{II}\left)\right]$. Under an abrupt change $(T=4)$, the stable pulse solution under $\varphi ={\varphi }_I$ $\left[P\right({\varphi }_I\left)\right]$ changes to the trivial uniform solution under $\varphi ={\varphi }_{II}$ $\left[U\right({\varphi }_{II}\left)\right]$.Reuse & Permissions