Front interaction induces excitable behavior

Spatially extended systems can support local transient excitations in which just a part of the system is excited. The mechanisms reported so far are local excitability and excitation of a localized structure. Here we introduce an alternative mechanism based on the coexistence of two homogeneous stable states and spatial coupling. We show the existence of a threshold for perturbations of the homogeneous state. Subthreshold perturbations decay exponentially. Superthreshold perturbations induce the emergence of a long-lived structure formed by two back to back fronts that join the two homogeneous states. While in typical excitability the trajectory follows the remnants of a limit cycle, here reinjection is provided by front interaction, such that fronts slowly approach each other until eventually annihilating. This front-mediated mechanism shows that extended systems with no oscillatory regimes can display excitability.

Figure 1

Evolution of $u(x,t)$ after a perturbation on $u_{-}$ with $\mathrm{\Gamma }=15,t_0=50$, and $\mathrm{\Delta }t=2$. (a) and (c) correspond to a subthreshold perturbation with $G=0.79$. (b) and (d) correspond to a superthreshold perturbation with $G=0.81$. The vertical dashed lines indicate the switching on and off of the perturbation. In panels (a) and (b) the range of $x$ is $[-22,22]$ and the range of $t$ is $[0,200]$.

Figure 2

Projection of the dynamics into the $(h,d)$ phase space. Red lines correspond to the analytical $h$-nullclines (5). Lines with arrows show trajectories obtained integrating Eq. (1) starting from an initial condition (6) with different $d_0$ and $H_0$. The arrows correspond to the velocity field, allowing one to identify fast and slow time scales. Trajectories in color correspond to $d_0=4.98$ and different $H_0$, whose temporal evolution is plotted in Fig. 4.

Figure 3

Phase portrait as in Fig. 2 in the $(h,A)$ plane.

Figure 4

Time evolution of $h$ for the trajectories plotted in color in Figs. 2 and 3 obtained integrating Eq. (1) starting from an initial condition (6) with $d_0=4.98$ and, from bottom to top, $H_0=0.6,0.8,1.0,1.1,1.2,1.25,1.3$.

Figure 5

Dynamics generated by the repetitive addition of Gaussian signals (2) with a noisy amplitude (8) (see text). The range of $x$ plotted is $[-22,22]$ and the range of $t$ is $[0,5000]$ (left panel) and $[5000,10000]$ (right panel).