Selection of Spiral Waves in Excitable Media with a Phase Wave at the Wave Back

Universal relationships between the medium excitability and the angular velocity and the core radius of rigidly rotating spiral waves in excitable media are derived for situations where the wave front is a trigger wave and the wave back is a phase wave. Two universal limits restricting the region of existence of spiral waves in the parameter space are demonstrated. The predictions of the free-boundary approach are in good quantitative agreement with results from numerical reaction-diffusion simulations performed on the Kessler-Levine model.

Figure 1

The selected shape of a spiral wave rigidly rotating at $\omega =0.552$ obtained as a solution of the free-boundary problem (1, 7, 8, 9, 10, 11) for $c_p=2.0$, $c_t=0.8$, $D=1$, and $\tau =0.943$.

Figure 2

The selected values of the dimensionless angular velocity $\Omega$ (a) and the tangential velocity $C_t$ (b) obtained from numerical solutions of the free-boundary problem (1, 7, 8, 9, 10, 11) as functions of the dimensionless parameter $B$ (solid lines). The results of direct integration of the KL model obtained for varying $\tau$ with different threshold values are depicted as dotted lines for $a^{\prime }=0.37$, dashed lines for $a^{\prime }=0.27$ and dashed-dotted lines for $a^{\prime }=0.107$. Thick dotted lines show similar dependences found for TT waves in 9.

Figure 3

Selection of the dimensionless angular velocity in the KL model with $a_{\min }=0.085$ and ${\tau }_{\mathrm{rr}}=5$. Solid line represents the relation $\Omega (B)$ in the medium without refractoriness. Dashed lines depict the generalized dispersion relations ${\Omega }_T(B)$ obtained for $\tau =0.75$, 0.43, 0.29, and 0.21 from left to right.

Figure 4

The angular velocity ${\Omega }_r$ (a) and spiral tip velocity $C_{\mathrm{tr}}$ (b) predicted in the medium with refractoriness ${\tau }_{\mathrm{rr}}=5$ and ${\tau }_{\mathrm{ar}}=0.4$ are shown by (*). Functions $\Omega (B)$ and $C_t(B)$ obtained above with neglected refractoriness are depicted by solid lines. The data from numerical simulations with the KL model with $a_{\min }=0.085$ and $a_{\max }=0.1$ are shown by (+) for ${\tau }_{\mathrm{rr}}=5$ and by (⋄) without refractoriness.