Self-organized pacemakers and bistability of pulses in an excitable medium

Pattern formation in an excitable medium described by a three-component reaction-diffusion system is investigated. Our focus is on stable self-organized pacemakers which give rise to spatially extended target patterns. Bistability of pulse solutions in the excitable regime is also reported, and interactions of the different pulses with each other and the pacemaker are studied. Self-organized pacemakers are created by a suitable perturbation from the steady state or through interaction of pulses. Bound states of one-dimensional pacemakers and phase flips are also observed.

Figure 1

Formation of pulse pairs and pacemakers. Characteristic stages in the development of a pulse pair (a) and a pacemaker (b) are displayed. The initial conditions for $u$ are shown as thin solid lines. Space-time diagrams for the formation of a pulse pair and a pacemaker are displayed in (c) and (d), respectively. There, the variable $u$ is shown in gray scale. The gray scale maps are always scaled to the minimum (white) and maximum (black) values of the variable in the displayed time interval. Here, the gray scale map is furthermore nonlinear to emphasize the increased values of $u$ in the tail of the waves. The parameters are $\alpha =1$, $\beta =0.2$, $\kappa =1$, ${\tau }_u=0.1$, ${\tau }_v=1$, ${\tau }_s=0.1$, $l_u=0.001$, and $l_s=0.017$. The simulated (displayed) system size is $L=1.0$ $(\Delta L=0.3)$, and the shown time interval in (c), (d) is $\Delta t=5$. In all space-time diagrams, space is displayed vertically and time horizontally.

Figure 2

Stable self-organized pacemaker in one spatial dimension. Space-time diagrams for (a) $s$ and (b) $u$ are shown. The parameters are $\alpha =1$, $\beta =0.2$, $\kappa =1$, ${\tau }_u=0.05$, ${\tau }_v=1$, ${\tau }_s=0.1$, $l_u=0.00071$, and $l_s=0.017$. The displayed time interval is $\Delta t=50$, and the system size is $L=1$. Periodic boundary conditions are used.

Figure 3

Stable self-organized pacemaker in two spatial dimensions. Snapshots of the distributions of (a) $s$ and (b) $u$ are displayed. The parameters are $\alpha =1$, $\beta =0.2$, $\kappa =1$, ${\tau }_u=0.1$, ${\tau }_v=1$, ${\tau }_s=0.1$, $l_u=0.05$, $l_s=1$, $L_x=L_y=15$, and $t=500$. No-flux boundary conditions are used.

Figure 4

Pacemaker and pulse solutions. The profiles of the variables are shown for a pacemaker (a) and the two pulse solutions (b). The parameters are $\alpha =1$, $\beta =0.2$, $\kappa =1$, ${\tau }_u=0.1$, ${\tau }_v=1$, ${\tau }_s=0.1$, $l_u=0.05$, $l_s=1$, and $L=30$. The solid, dotted, and dashed lines correspond to $u$, $v$, and $s$, respectively. No-flux boundary conditions are present in (a), being responsible for the increase of $s$ at the boundary. The arrows indicate the direction of wave propagation.

Figure 5

Two colliding small pulses lead to a bound state of two pacemakers and a wave sink. Space-time diagrams for (a) $s$ and (b) $u$. The parameters are $\alpha =1$, $\beta =0.2$, $\kappa =1$, ${\tau }_u=0.1$, ${\tau }_v=1$, ${\tau }_s=0.1$, $l_u=0.05$, and $l_s=1$. The displayed time interval is $\Delta t=40$, and the system size is $L=20$. No-flux boundary conditions are used.

Figure 6

Unstable bound state of pacemakers. Space-time diagrams for (a) $s$ and (b) $u$ are shown. The parameters are like in Fig. 5. The displayed time interval is $\Delta t=50$, and the system size is $L=25$. No-flux boundary conditions are imposed.

Figure 7

Interaction of a small pulse (coming from above) with a large pulse (coming from below). A stable pacemaker is formed. Space-time diagrams for (a) $s$ and (b) $u$ for the initial evolution of the system are shown. The parameters of the system are as in Fig. 5.

Figure 8

Interaction of a small pulse with a pacemaker. Space-time diagrams for the initial evolution of (a) $s$ and (b) $u$ are displayed. The parameters of the system are as in Fig. 5, except the system size, which is $L=50$.

Figure 9

Traveling phase flip. Space-time diagrams for (a) $s$ and (b) $u$ are shown. The parameters of the system are as in Fig. 5, except the system size $(L=22.625)$ and the displayed time interval $(\Delta t=50)$. Periodic boundary conditions are used.