Refraction of Scroll-Wave Filaments at the Boundary Between Two Reaction-Diffusion Media

We explore the shape and the dynamics of scroll-wave filaments in excitable media with an abruptly changing diffusion tensor, important for cardiac applications. We show that, similar to a beam of light, the filament refracts at the boundary separating domains with different diffusion. We derive the laws of filament refraction and test their validity in computational experiments. We discovered that at small angles to the interface, the filament can become unstable and develop oscillations. The nature of the observed instabilities, as well as overall theoretical and experimental significance of the findings, is discussed.

Figure 1

Refraction of a scroll-wave filament in a medium with discontinuous diffusion tensor. (a) Snapshot of a scroll wave and its filament in the model given by Eqs. (1) and (2) (case I: $d_1=1.5$, $d_2=0.5$). The isosurfaces for $u=0.5$ are shown in transparent gray, the filament and the anchors in black, and the interface between the upper and lower halves in transparent dark gray. The filament is noticeably thinner in the upper region because $d_2<d_1$. (b) Schematic cross section through the anchor plane ($y=0$) in (a), explaining our notations. (c) Dependency of $a_x$ on $b_x$ as derived from the geodesic principle (lines) and observed in numerical experiments (symbols) for $\gamma =3$. (d) Numerical verification of Eqs. (6) and (7) for fixed anchor positions and varying $\gamma$.

Figure 2

Filament snapshots (lateral and top views) at different anchor positions in case I, with $d_1=1.5$, $d_2=0.5$ ($\gamma =3$). The values of $z_B$ indicate the distance of the anchor $B$ from the interface in units of ${\lambda }_z$, the wavelength of the scroll wave in the $z$ direction. (a) Filament refraction consistent with the geodesic principle. (b) The upper segment of the filament becomes curved. (c) Filament begins to oscillate in the vicinity of the geodesic solution. (d) The filament detaches from anchor $B$ and rotates in the clockwise direction. Numbers show consecutive filament configurations starting from the moment of detachment.

Figure 3

Refraction and “reflection” of the filament in case II ($\gamma =3$). Notations as in Fig. 2. (a) The filament consists of two rectilinear fragments consistent with geodesic predictions. (b) At $\alpha$ close ${\alpha }_c$, the upper segment of the filament runs almost parallel to the interface. (c) As the anchor $B$ moves below the interface, the filament folds and produces an additional linear segment at an angle close to ${\alpha }_c$, reminiscent of internal reflection. (d) Straightening of the filament following further reduction of $z_B$.

Figure 4

(a) Schematic illustrating the mechanism of filament deformation at small $z_B$. The portion of the filament inside the boundary layer of width $w$ experiences a force ${\mathit{F}}_b$, which pulls it off the $xz$ plane, creating a bulge as indicated by the dashed line. (b) Deviation of the upper filament segment from the straight line as a function of $w/z_B$ (case I, $\gamma =3$). Deviation was quantified by determining the distance between the predicted and actual filament coordinates in each cross section ($yz$ plane) of the medium and then calculating the root-mean-square deviation. The width of the hatched area indicates the amplitude of oscillations. The dashed vertical line indicates the detachment of the filament from anchor $B$.