Periodic sequence of stabilized wave segments in an excitable medium

Numerical computations show that a stabilization of a periodic sequence of wave segments propagating through an excitable medium is possible only in a restricted domain within the parameter space. By application of a free-boundary approach, we demonstrate that at the boundary of this domain the parameter $H$ introduced in our Rapid Communication is constant. We show also that the discovered parameter predetermines the propagation velocity and the shape of the wave segments. The predictions of the free-boundary approach are in good quantitative agreement with results from numerical reaction-diffusion simulations performed on the modified FitzHugh-Nagumo model.

Figure 1

Kinematical parameters of a propagating wave segment train. (a) A sketch of two neighboring segments. (b) The selected shape of a wave segment obtained as a solution of the free-boundary problem for $c_t=0.142,\epsilon =0.001,\delta =-1.682,\lambda =127$, and $k_{\epsilon }=14$. The thick solid line represents the wave front. The thin solid line corresponds to the wave back. The dashed (dotted) line corresponds to $c_t<0.142(c_t>0.142)$.

Figure 2

Wave segment shapes computed using free-boundary Eqs. (12, 13, 14, 15) for different values of (a) $\epsilon$ and (b) $\lambda$. The parameters $\delta =-1.682$ and $k_{\epsilon }=1$ are fixed.

Figure 3

The existence domain of a stabilized wave segment train (the dark region) computed using the free-boundary model with $\delta =-1.682$ and $k_{\epsilon }=14$. The dashed and dotted lines show the existence boundaries computed in accordance to Eqs. (22) and (23) for $k_{\epsilon }=14$ and $k_{\epsilon }=1$. Pluses and stars indicate the results of the reaction-diffusion computations corresponding to successful and unsuccessful stabilizations, respectively.

Figure 4

Kinematical parameters of the wave segment train vs the dimensionless parameter $H$. (a) The normalized translational velocity. (b) Half-width of the segment. Free-boundary computations have been performed at $\epsilon =0.00001$ (the pluses), $\epsilon =0.0001$ (the asterisks), $\epsilon =0.0005$ (the rhombuses), $\epsilon =0.001$ (the triangles), and $\epsilon =0.0015$ (the squares).