Hysteresis and drift of spiral waves near heterogeneities: From chemical experiments to cardiac simulations

Dissipative patterns in excitable reaction-diffusion systems can be strongly affected by spatial heterogeneities. Using the photosensitive Belousov-Zhabotinsky reaction, we show a hysteresis effect in the transition between free and pinned spiral rotation. The latter state involves the rotation around a disk-shaped obstacle with an impermeable and inert boundary. The transition is controlled by changes in light intensity. For permeable heterogeneities of higher excitability, we observe spiral drift along both linear and circular boundaries. Our results confirm recent theoretical predictions and, in the case of spiral drift, are further reproduced by numerical simulations with a modified Oregonator model. Additional simulations with a cardiac model show that orbital motion can also exist in anisotropic and three-dimensional systems.

Figure 1

Schematic illustration of the experimental setup: (1) observation light source, (2) yellow long-pass filter, (3) condenser lens, (4) flow reactor with BZ reagent, (5) beam splitter, (6) charge-coupled device camera, (7) neutral density filter, (8) blue dichroic filter, (9) controlled light source, (10) computer with LabVIEW platform.

Figure 2

Still images of spiral waves in the photosensitive BZ system. The patterns are observed in a $0.40\pm 0.1\mathrm{mm}$ thick hydrogel matrix containing the immobilized catalyst. A circular disk of 0.65 mm radius is used as an inert and impermeable obstacle. Images show a spiral wave in (a) the pinned state and (b) shortly after unpinning. Image area: $3.3\times 2.9{\mathrm{cm}}^2$.

Figure 3

(a) Temporal tracking of the spiral tip in a typical hysteresis experiment. The solid, gray circle represents the impermeable disk. The tip trajectory traces the disk circumference in the pinned state (solid, red curve) and evolves into a more expansive curve after unpinning (solid, black circles). Thereafter, the spiral tip traces smaller paths (open, blue circles) before reestablishing the pinned state. (b) Core radius $r$ as a function of applied light intensity $P$ relative to the wavelength of the pinned spiral $\left({\lambda }_p\right)$. For a given value of $P,r$ depends on whether the spiral is in the pinned (solid, red circles) or the unpinned state (open, black and blue circles).

Figure 4

Temporal evolution of image brightness $\left(I\right)$ at a representative location in the reaction medium. The period $T$ is equivalent to the time lapse between successive peaks and has a constant value when the spiral wave is in the pinned state (solid, red curves). Upon unpinning, $T$ jumps to a large value and then decreases monotonically with decreasing light intensity (dashed, blue curve). The black arrows represent the time points when the light intensity is modified. Finally, the approximate initial period is recovered when the spiral wave returns to the pinned state.

Figure 5

(a) Image of a spiral wave in a reaction medium with nonhomogeneous illumination. The dashed, red line represents the border between the bright left region and the right region of $20\%$ lower light intensity. The wavefronts are deformed due to a difference in the propagation velocity between the two regions. (b) The tip trajectory of the spiral wave shows that the spiral tip drifts parallel to the linear heterogeneity border at a constant velocity. Note that $\mathrm{\Delta }P$ has a negative value.

Figure 6

(a) Image of a spiral wave in the vicinity of a circular heterogeneity of lower light intensity. (b) The spiral tip attempts an orbital motion but drifts away because of unintended, weak gradients in the illumination profile. Note that $\mathrm{\Delta }P$ has a negative value.

Figure 7

Tip trajectories from simulations with $\varphi =0.035$ and $\mathrm{\Delta }\varphi =-0.015$. The model parameters are provided in the text. (a) The spiral tip is initially attracted to the heterogeneity boundary which then causes a downward directed steady drift with a constant velocity. (b) The spiral tip undergoes a biperiodic orbital motion in the presence of a circular heterogeneity.

Figure 8

Two-dimensional cardiac simulations. Spiral tip trajectories (black curves) near a disk-shaped heterogeneity of increased excitability (red border). Within the disk the conductance of sodium ions is increased from 2.4 to 4.0. The three simulations correspond to (a) isotropic and (b,c) anisotropic conditions. The conductivity ratio (vertical-to-horizontal) is 5.0 and 9.0 in (b) and (c), respectively.

Figure 9

Three-dimensional cardiac simulations of scroll wave dynamics near (a, c) a cylindrical and (b, d) a cone-shaped heterogeneity of increased excitability. The red lines in (a, b) are snapshots of the scroll wave filament. The curves in (c, d) are the corresponding trajectories of the filament at the bottom boundary $\left(z=0\right)$. (e) Time required to complete a full rotation around the heterogeneity as a function of the cones' base angle. This angle equals 90 and 22 degrees in (a) and (b), respectively.