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Drift of scroll waves in thin layers caused by thickness features: asymptotic theory and numerical simulations

A scroll wave in a very thin layer of excitable medium is similar to a spiral wave, but its behaviour is affected by the layer geometry. We identify the effect of sharp variations of the layer thickness, which is separate from filament tension and curvature-induced drifts described earlier. We outline a two-step asymptotic theory describing this effect, including asymptotics in the layer thickness and calculation of the drift of so perturbed spiral waves using response functions. As specific examples, we consider drift of scrolls along thickness steps, ridges, ditches, and disk-shaped thickness variations. Asymptotic predictions agree with numerical simulations.

A scroll wave in a very thin layer of excitable medium is similar to a spiral wave, but its behaviour is affected by the layer geometry. We identify the effect of sharp variations of the layer thickness, which is separate from filament tension and curvature-induced drifts described earlier. We sketch a two-step asymptotic theory describing this effect, including averaging through the layer thickness and then drift of perturbed spiral waves using response functions. As specific examples, we consider drift of scrolls along thickness steps, ridges, ditches, and disk-shape thickness variations. Asymptotic predictions agree with numerical simulations. Spiral waves in two dimensions (2D) and scroll wave in three dimensions (3D) are regimes of self-organization observed in physical, chemical and biological dissipative systems [1]. A particularly important example is re-entrant arrhythmias in the heart [2]. In nature, 2D systems are often thin layers of 3D media, and geometry of such layers affects the dynamics of spiral/scroll waves. Known phenomena include scroll filament tension [3] and layer curvature [4], which can cause spiral waves to drift to/from thinner regions and more curved regions of the 2D excitable sheet respectively. Here we consider effects caused by sharp features of the layer thickness. There is experimental evidence that such effects play significant role in atrial fibrillation [5,6]. Here we analyse these effects theoretically by a combination of asymptotic and numerical methods, for two selected archetypical models.We start from a generic homogeneous isotropic reaction-diffusion system in 3D: where v = v( r, t), r = (x, y, z).
Let us consider first a step in thickness, i.e.
Then we have K = ln (H − )+ Θ(x−x s ), = ln(H + /H − ), and Eqs. (12,13,14) of [8] predict the drift velocity via α(r, θ; R) = e −iφh (U, r, θ, φ) dφ 2π , whereh is the perturbation h, calculated for u = U and considered in the frame corotating with the spiral, and W are (spatial) response functions of the spirals. The spirals and their response functions calculated for the two selected models using DXSpiral [7,9] are illustrated in fig. 1. Evaluation of integral (12) with account of (9) and the coordi- Eqs. (11) and (13) define the specific force produced by the thickness step which depends only on the distance between the current spiral centre and the step line and is an even function about the position of the step, The components of the function S(d) for the two selected models are shown in fig. 2(b,e). The important feature are zeros of S x for d = ±d * in both models. Assuming without loss of generality x s = 0, the drift of a spiral is then described asymptotically by Fig. 2 illustrates predictions of the theory for the case of the stepwise thickness inhomogeneity and its comparison with the direct numerical simulations of both the 2D thickness-reduced system (5) and the full 3D system (1). Numerical simulations for both selected models were done with BeatBox [7,10]. The relevant attractor for (16) is where That is, in both models the spiral attaches to the step at its thinner side and drifts along with the speed |S y (−d)|0. The speed of the drift is proportional to = ln(H + /H − ), and the direction of the drift depends on the spiral chirality: compare panels (a) and (d).
Now let us consider the following thickness profile: for some x < x r , which means a "ridge" for H i > H o and a "ditch" for H i < H o . This case is easily reduced to the previous because We use the linearity of (10)-(12) and the previous result to get the interaction force in the form  fig. 2(b)), the smaller giving w * ≈ 1.769. Fig. 3(c,d) illustrates the drift along a cuneiform ditch, which may be in the first approximation considered a negative ditch with almost constant but slowly varying width. The bifurcation width w * is designated by the dashed horizontal line. We see that below this line the spiral wave drifts in accordance with the theory prediction but slows down markedly in the vicinity of this line. It does not stop completely but proceeds further, albeit at a much slower speed. This is due to the "wedging" effect of the varying width: at w ≥ w * , the forces from the two opposite steps making the banks of the ditch, do not compensate each other exactly due to the angle between them. To estimate roughly the associated correction, let the wedge angle be ψ 1. Then the wedge-induced component of the drift speed at the bifurcation point is 2 S x (w * ) sin(ψ/2) ≈ S x (w * )ψ. For the simulation shown in fig. 3(c,d), we have ψ ≈ 0.03, and S x (w * ) ≈ 0.4142, hence the drift speed ψS x (w * ) ≈ 0.002266. This drift speed is represented by the dotted line in fig. 3 and cor-responds well with the simulations.Finally, let us consider the thickness perturbation of the form i.e. thickening (for > 0) or thinning (for < 0) in a disk- shaped area. Then we have represents the vector from the current spiral centre to the feature centre, and κ = R d 2 − 2 − r 2 /((2r ). Hence the interaction force is The radial F r ( ) and the azimuthal F a ( ) components calculated for the Oregonator model (3) for an arbitrarily chosen disk radius R d are shown in Fig. 4(a). We see there is a root of F r ( ) at = * ≈ 4.023 and the corresponding value of F * a = F a ( * ) ≈ 0.1055 which predicts longterm behaviour of a spiral starting from an appropriate initial condition as "meander" or "orbital movement" along a circle of radius * and linear speed F * a , that is with the orbit period of 2π * /( F * a ) ≈ 1314. Fig. 4(b) compares these predictions with results of 2D and 3D numerical simulations at = log(1.2). This example is similar to the case considered phenomenologically (simulations and experiment) in [6] and is analogous to "orbital motion" described in [11] for localized parametric heterogeneities.
To summarise, movement of transmural scroll waves through thin layers of excitable media of varying thickness can be approximately described by thickness-averaged twodimensional equations, and a corresponding 2D perturbation theory can be successfully applied within its limits. This theory shows the propensity of scrolls to interact with sharp features of the layer geometry, which is distinct from and not reducible to previously known geometric effects such as filament tension or curvature-induced drift, and is completely independent from other factors that may cause drift such as parametric inhomogeneities. Interaction with sharp features can manifest nontrivial attractor structures, depending on the geometric parameters. These predictions should be immediately testable in experiments with the Belousov-Zhabotinsky reaction and may have important implications for understanding of evolution of re-entrant waves excitation in the heart, particularly in atria which have abundance of geometric features.
The above derivation was done in the assumption of smoothness of the thickness profileH(x, y). Hence the applications considered in the main paper will be formally covered by this approximation and the 2D spiral perturbation theory, if the "sharp features" considered there are smooth on the scale ofH but sharper than the typical scale of the effective response functions' support. Deviation from this condition in actual 3D simulations may account for some of the discrepancies between the 3D and 2D simulations.

Circular step
Similarly, the quadrature for interaction with a disk involves α described by (19), and so is also singular, as it contains denominator √ 1 − κ 2 which becomes zero at both ends of the integration interval: where r min = |R d − |, r max = |R d + |. Doing as before, we get r mid = 1 2 (r min + r max ) , ∆ρ = (r max − r min )/N, r j = r min + j∆ρ, j = 0, . . . , N,

Two-dimensional simulations
We use explicit Euler timestepping with time step ∆t and central differencing for the diffusion term in (5), with the following discretization scheme where (i, j) are 2D indices of the regular space grid of the size N x × N y with step ∆x and I = {(−1, 0), (1, 0), (0, −1), (0, 1)}.
We employ standard non-flux boundary conditions. The discretization parameters used for different results are described in Table I.  The discretization in 3D is a natural extension of the 2D scheme, except instead of a fancy diffusion operator of (5) we now have the plain diffusion of (1). The complication now comes from the more complicated geometry of the domain, which requires special attention to the boundary conditions. We have employed the following discretization: where χ i,j,k = 1 if the grid point (i, j, k) is within the domain and 0 otherwise, and the neighbourhood template is I = {(−1, 0, 0), (1, 0, 0), (0, −1, 0), (0, 1, 0), (0, 0, −1), (0, 0, 1)}. The space grid is regular with step ∆x, rectangular N x × N y in the horizontal direction, and with k ∈ {1, . . . , N z (i, j)} where N z (i, j) represents the thickness profile. In all our examples, N z (i, j) takes only two values, denoted as N z,1 and N z,2 . The discretization parameters used for different results are described in Table II.

Response function computations
For DXSpiral computations of the FitzHugh-Nagumo model, we use disk radius ρ max = 25, number of radial intervals N ρ = 1280 and number of azimuthal intervals N θ = 64. For the Oregonator model, we have correspondingly ρ max = 15, N ρ = 128 and N θ = 64.