Proportional-integral control of propagating wave segments in excitable media

Numerical simulations are performed to demonstrate that proportional-integral control, one of the most commonly used feedback schemes in control engineering, can stabilize propagating wave segments in excitable media to a desired size. The proportional-integral controller measures the size of a wave segment and applies a spatially uniform signal to the medium. This controller has the following features: difficult trial-and-error adjustment is not necessary, wave segments can be stabilized to different sizes without readjusting the controller, and the wave segment size can be maintained even in media having position-dependent parameters.

Figure 1

Snapshots of propagating wave segments [i.e., excited area $\mathrm{\Omega }\left(t\right)$] in the Bär model and the Oregonator model without feedback control: (a) $U\equiv 0.142$ (light-colored segments) and 0.144 (dark-colored segments) for the Bär model with $L_1=80$ and $L_2=40$, (b) $U\equiv 0.090$ (light-colored segments) and 0.094 (dark-colored segments) for the Oregonator model with $L_1=12$ and $L_2=8$. Snapshots were taken for increment of (a) $t=4$ and (b) $t=0.5$.

Figure 2

Controlled object with input $U\left(t\right)$ and output $w\left(t\right)$.

Figure 3

Block diagram of the feedback control system consisting of excitable medium $\mathrm{\Sigma }$ and the P controller (3).

Figure 4

Snapshots and time series data of propagating wave segments with P control in (a) the Bär model ($L_1=200$ and $L_2=40$) and (b) the Oregonator model ($L_1=30$ and $L_2=8$). (a) Bär model with $K_{\mathrm{P}}=-3\times {10}^{-3}$, $I=0.15$, and $r\left(t\right)\equiv 20$. (b) Oregonator model with $K_{\mathrm{P}}=-3\times {10}^{-3}$, $I=0.093$, and $r\left(t\right)\equiv 4$.

Figure 5

Block diagram of the feedback control system consisting of excitable medium $\mathrm{\Sigma }$ and PI controller (5).

Figure 6

Snapshots and time series data of propagating wave segments with PI control in (a) the Bär model ($L_1=200$ and $L_2=40$) and (b) the Oregonator model ($L_1=30$ and $L_2=8$). (a) Bär model with $K_{\mathrm{P}}=-3\times {10}^{-3}$, $K_{\mathrm{I}}=-2\times {10}^{-4}$, $r\left(t\right)\equiv 20$, and $z\left(0\right)=0.15$. (b) Oregonator model with $K_{\mathrm{P}}=-3\times {10}^{-3}$, $K_{\mathrm{I}}=-2\times {10}^{-3}$, $r\left(t\right)\equiv 4$, and $z\left(0\right)=0.093$.

Figure 7

Snapshots and time series data of propagating wave segments with P and PI control in the Bär model ($L_1=700$ and $L_2=40$) for the size change: $r\left(t\right)\equiv w_0=10$ to $r\left(t\right)\equiv \overline{r}=20$ at $t=200$. P control with $K_{\mathrm{P}}=-1\times {10}^{-3}$ and $I=U_0=0.1413$ and PI control with $K_{\mathrm{P}}=-1\times {10}^{-3}$, $K_{\mathrm{I}}=-1\times {10}^{-4}$, and $z\left(0\right)=0.14$. Snapshots were taken every $t=8$.

Figure 8

Snapshots and time series data of propagating wave segments with P and PI control in the Oregonator model ($L_1=60$ and $L_2=8$), the parameter $\varepsilon$ of which depends on position: $\varepsilon =0.010$ in $x_1\in [0,20]\cup [40,60]$ and $\varepsilon =0.012$ in $x_1\in (20,40)$. P control with $K_{\mathrm{P}}=-4\times {10}^{-3}$ and $I=U_0=0.0913$ and PI control with $K_{\mathrm{P}}=-4\times {10}^{-3}$, $K_{\mathrm{I}}=-2\times {10}^{-3}$, and $z\left(0\right)=0.091$. Snapshots were taken every $t=1$.

Figure 9

Stability regions on $K_{\mathrm{P}}\text{−}{K}_{\mathrm{I}}$ space for (a) the Bär model ($L_1=300$ and $L_2=40$) and (b) the Oregonator model ($L_1=40$ and $L_2=8$). Open circles indicate that stabilization without steady-state error is achieved, whereas crosses indicate that stabilization without steady-state error is not achieved.

Figure 10

Schematic diagram of an excitable medium consisting of region ${\mathbf{\Pi }}_0$ without feedback control and region $\mathbf{\Pi }$ with feedback control for the numerical simulations of the present study.