Analytical Approach to Initiation of Propagating Fronts

We consider the problem of initiation of a propagating wave in a one-dimensional bistable or excitable fiber. In the Zeldovich-Frank-Kamenetskii equation, also known as the Nagumo equation and Schlögl model, the key role is played by the “critical nucleus” solution whose stable manifold is the threshold surface separating initial conditions leading to the initiation of propagation and decay. An approximation of this manifold by its tangent linear space yields an analytical criterion of initiation which is in good agreement with direct numerical simulations.

Figure 1

(a),(b) Response to a below- and above-threshold initial perturbation in ZFK equations (1, 2, 8). Parameter values: $\theta =0.13$, $I_s=0$, and $x_s=2.10$ for both (a) subthreshold $u_s=0.3304831$ and (b) superthreshold $u_s=0.3304833$ cases, numerics using central difference centered in space with step $h_x=0.15$ and forward Euler in time with step $h_t=0.01$. Dashed-dotted black lines: initial conditions; bold solid black lines: the critical nuclei. (c) The corresponding critical strength-extent curve, separating initiation initial conditions from decay initial conditions. (d) The sketch of a stable manifold of the critical solution for the ZFK equation. The critical nucleus is represented by the black dot; the critical trajectories, constituting the stable manifold, are shown in black. The family of initial conditions is represented by the dashed-dotted line. The bold black line is the critical trajectory with initial condition in that family. The subthreshold trajectories are represented by the blue line, while the red lines represent superthreshold trajectories. Note that the point where the initial condition intersects the stable manifold is shown as the empty circle.

Figure 2

Comparison of analytical predictions with numerical simulations. (a) Strength-extent curves for rectangular initial conditions. (b) Strength-duration curves for point stimulation. Red solid lines: Analytical approximations, (13) for (a) and (14, 15, 16) for (b). Blue stars (“cub”): Numerical results for cubic kinetics (2). Magenta diamonds (“quad”): Numerical results for quadratic kinetics (12).