Characterization and Control of Spiral Wave Instabilities

Nature of Spiral Wave Instabilities

The eigenmode method can also be applied to the case of rotating action potential waves (spiral waves) in two spatial dimensions by employing the version of eigenmode theory that assumes the viewing time interval is infinitesimal (as described above). Technical details have been published previously2 and also appear in abbreviated form in the Appendix. To find the steady state and eigenmodes, the theoretical approach, as described by Barkley,3 for a spiral wave system was applied. The three-variable Fenton-Karma model11 was used as the underlying ion channel model. It was modified so that it exhibited both unstable alternans eigenmodes and also eigenmodes that characterize the tendency of the spiral wave to meander. The model was then used to study the control of these unstable modes.

The steady state for this case was simply a rigidly rotating spiral wave (i.e., a wave that rotates with constant angular velocity without changing form). If viewed in the frame

Figure 9: Eigenvalues for the two-dimensional, modified three-variable Fenton-Karma model. The solid dots, the asterisk, the open square, and the open circle denote the alternans eigenvalues, the meandering mode eigenvalue, the eigenvalue associated with the trans-lational symmetry, and the eigenvalue associated with rotational symmetry, respectively (modified from Allexandre and Otani, Fig. 2, copyright American Physical Society)2

of reference rotating with the wave, such a wave appears to be completely stationary and unchanging at all times, thus satisfying the requirement for a steady state previously discussed when the viewing time interval is infinitesimal.

The role of perturbations in this case was then to modify this rigid waveform. Inclusion of perturbations thus allowed the wave to widen and/or narrow, and speed up and/or slow down, as it rotated. Furthermore, these alterations did not necessarily apply to the wave as a whole; it was possible for some portions of the wave to widen while others narrowed, for example. Addition of perturbations also made it possible for the spiral wave tip to simultaneously move toward or away from the center of rotation, creating a meandering pattern commonly seen in spiral wave computer simulations. The profile of the wave could also be changed by the presence of perturbations.

When the eigenmode theory was applied, it was found that the total number of eigenmodes was tremendous - equal to the total number of points in the system times the number of variables. In fact, the former is technically infinite; but remains finite although large in the computer calculation due to the spatial discretization scheme used in the analysis. Fortunately, only a small number of these modes are unstable. The eigenvalues of these unstable modes (along with a few of the most prominent stable modes) are shown in Fig. 9. Of all the alternans eigenmodes (represented by the solid dots), four were found to be unstable, those lying to the right of the dashed vertical line. There was also one unstable meandering eigenmode (represented by the asterisk). All the alternans modes had one property in common: at every point in the tissue through which the spiral wave rotated, the APD of the wave alternated between being slightly longer and slightly shorter than the steady state APD, each time the wave came around, consistent with the definition of alternans. The alternans modes differed from one another through their spatial structure -that is, the size, shape, and locations of the regions that differed in phase relative to one another were different for each of the different modes.

Figure 9: Eigenvalues for the two-dimensional, modified three-variable Fenton-Karma model. The solid dots, the asterisk, the open square, and the open circle denote the alternans eigenvalues, the meandering mode eigenvalue, the eigenvalue associated with the trans-lational symmetry, and the eigenvalue associated with rotational symmetry, respectively (modified from Allexandre and Otani, Fig. 2, copyright American Physical Society)2

Elimination of Alternans in a Rotating Spiral Wave

As with the single cell case, eigenmode theory also provided us with information about when and where control stimuli might best be applied. Most of the eigenmodes were found to be most sensitive to modification when the stimulus was situated close to the center of rotation, in the recovery phase of the spiral wave. The stimulus location and timing were chosen accordingly. To simplify the study, the initial perturbation to the steadily rotating wave was chosen in a special way - it was calculated so that a single stimulus would eliminate all the unstable alternans modes. The development of a plan to kill all the alternans modes when an arbitrary perturbation is present is somewhat more complicated, and is a topic currently under investigation. The results for the case of the specially chosen initial conditions are shown in Fig. 10. The upper panels show that the alternans modes are indeed unstable, leading to increasingly large perturbations as shown in Fig. 10c, d. In contrast, when a single stimulus is applied at the location outlined by the small circle in Fig. 10e with the timing and amplitude provided by the theory to a simulation that is otherwise identical, the result is the near-complete suppression of alternans for four rotational periods of the spiral wave, as shown in the four lower panels of Fig. 10.

Figure 10: Snapshots of the membrane potential perturbation at several selected times, for the case where no control stimulus is applied (top panels) and for the case in which a control stimulus is applied at time t = 10 ms (bottom panels). The two cases were initialized with specially chosen, identical perturbations. Viewed in color, the color shading gives amplitude of the perturbation. Viewed in black and white, the darker shades correspond to the larger perturbations. Level contours of the total membrane potential (steady state plus linearized perturbation) are shown as black lines. A small black circle was drawn in each of the lower subplots to highlight the region in which the stimulus was applied (from Allexandre and Otani, Fig. 8, copyright American Physical Society)2

Figure 10: Snapshots of the membrane potential perturbation at several selected times, for the case where no control stimulus is applied (top panels) and for the case in which a control stimulus is applied at time t = 10 ms (bottom panels). The two cases were initialized with specially chosen, identical perturbations. Viewed in color, the color shading gives amplitude of the perturbation. Viewed in black and white, the darker shades correspond to the larger perturbations. Level contours of the total membrane potential (steady state plus linearized perturbation) are shown as black lines. A small black circle was drawn in each of the lower subplots to highlight the region in which the stimulus was applied (from Allexandre and Otani, Fig. 8, copyright American Physical Society)2

Perhaps the most surprising aspect of these studies is that a single stimulus at a single location can affect the dynamics over such a large region. This assertion stands despite the fact that the initial conditions are specially chosen - the simulation still represents a clear example of how a point stimulus can influence an area of linear dimension much larger than the space constant, which is the usual decay length associated with perturbations in the membrane potential in excitable tissue. The preliminary explanation for this phenomenon is that the effect of the stimulus must use the dynamics of the spiral wave itself to spread its influence, effectively "surfing" on the wave to propagate across the system. Further study of this effect is ongoing.

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