Predicting effective quenching of stable pulses in slow-fast excitable media

Abstract

We develop a linear theory for the prediction of excitation wave quenching -- the construction of minimal perturbations which return stable excitations to quiescence -- for localized pulse solutions in models of excitable media. The theory accounts for an additional equivariance compared to the homogeneous ignition problem, and thus requires a reconsideration of heuristics for choosing optimal reference states from their group representation. We compare predictions made with the linear theory to direct numerical simulations across a family of perturbations and assess their accuracy for several models with distinct stable excitation structures. We find that the theory achieves qualitative predictive power with only the effort of continuing a scalar root, and achieves quantitative predictive power in many circumstances. Finally, we compare the computational cost of our prediction technique to other numerical methods for the determination of transitions in extended excitable systems.

Figures (11)

• Figure 1: (a) Supercritical, (b) subcritical, and (c) unperturbed dynamics for the FitzHugh-Nagumo model stable pulse in a co-moving frame with speed $\check{c}$. For (a,b), the quenching perturbation is a rectangular envelope in the $u_1$ channel, centered at $x=1$, with width substantially smaller than the pulse ($\sim5\%$), and amplitudes ${U_{q}^\pm} = -18.8314 \mp 10^{-4}$ for the super- and sub-critical perturbations, respectively.
• Figure 2: Space-time dynamics of the fast variable $u_1(t,x)$ for (a) Super-critical and (b) sub-critical perturbations to the stable pulse of the FitzHugh-Nagumo model \ref{['eq:fhn']} corresponding to figure \ref{['fig:1']}(a,b), and (c) their diagnostic function $\psi(t)$ for the (dots) super-critical and (dash) sub-critical dynamics.
• Figure 3: Nullclines of the FitzHugh-Nagumo model (black; $f_1=0$ solid, $f_2=0$ dashed) and the stable pulse solutions for $\gamma \in \{0.025, 0.020, 0.010, 0.001\}$.
• Figure 4: Linear theory ingredients for the FitzHugh-Nagumo model. (Top row) Unstable pulse solution $\hat{\mathbf{u}}$ for $\gamma \in \{0.001, 0.010, 0.020, 0.025\}$, with (second row) $\hat{\mathbf{v}}_1$, (third row) $\hat{\mathbf{v}}_2$, (fourth row) $\hat{\mathbf{w}}_1$, and (fifth row) $\hat{\mathbf{w}}_2$. The first component of each variable is the solid (blue) curve, and the second component is shown as dashed (orange).
• Figure 5: Critical quenching perturbations sampled over the $(x_s,\theta)$-plane, for (a) $\gamma=0.001$, (b) $\gamma=0.010$, (c) $\gamma=0.020$, and (d) $\gamma=0.025$. In all cases, the perturbation quenching amplitude $U_q$ determines the red color and blue denotes regions in which \ref{['eq:rootfinding']} has no solution for all bounding samples (c.f. black dots in figure \ref{['fig:6']}(a-d)). The $|\beta-1|$ asymptotic estimate contour is denoted by a solid black curve and the linear bounding region between successful and unsuccessful quenching searches is denoted by black dashed lines.