Meander pattern of spiral wave and the spatial distribution of its cycle length

One of the most interesting dynamics of rotating spiral waves in an excitable medium is meandering. The tip of a meandering spiral wave moves along a complex trajectory, which often takes the shape of an epitrochoid or hypotrochoid with inward or outward petals. The cycle lengths (CLs) of a meandering spiral wave are not constant; rather, they depend on the meandering dynamics. In this paper, we show that the CLs take two mean values, outside $T^{\mathrm{out}}$ and inside $T^{\mathrm{in}}$ the meandering trajectory with a ratio of $T^{\mathrm{in}}/T^{\mathrm{out}}=(n+1)/n$ for the inward and $(n-1)/n$ for the outward petals, where $n$ is the number of petals in the tip trajectory. We illustrate this using four models of excitable media and prove this result. These formulas are shown to be suitable for a meandering spiral wave in an anatomical model of the heart. We also show that the effective periods of overdrive pacing of meandering spiral waves depend on the electrode location relative to the tip trajectory. Overall, our approach can be used to study the meandering pattern from the CL data; it should work for any type of dynamics that produces complex tip trajectories of the spiral wave, for example, for a drift due to heterogeneity.

Figure 1

Left ventricle model. Section by plane $y=0$. Axes in mm. Red line, epicardium; blue line, endocardium; vertical black line, apical segment with $x=y=0, \psi =\pi /2, \phi$ undefined, $h$ is the apical thickness of the myocardial wall. Electrode 1 had $\psi >1.4$; its borders are shown as dashed black lines. Two horizontal black lines, the base with $z=60$ mm, $\psi =0$.

Figure 1

Left ventricle model. Section by plane $y=0$. Axes in mm. Red line, epicardium; blue line, endocardium; vertical black line, apical segment with $x=y=0, \psi =\pi /2, \phi$ undefined, $h$ is the apical thickness of the myocardial wall. Electrode 1 had $\psi >1.4$; its borders are shown as dashed black lines. Two horizontal black lines, the base with $z=60$ mm, $\psi =0$.

Figure 2

Results for Model 1 with inward petals. (a) Spiral wave (state variable $u$), its tip (black dot), and tip trajectory (red curve). (b) Tip trajectory, points $A, B$, four detectors (circle, square, dot, and plus), and the subdomains. The disk is inside the black dashed circle; the ring is between the black and blue dashed circles; the outside is the remaining part of the domain. (c) Potential $100u$ for two points with coordinates $A(24,20)$ (red, inside the curve) and $B(7,10)$ (blue, outside the curve). Abscissa axis, time in model units. Vertical lines mark ends of a segment of time with 9 APs in point $A$ and 10 APs in point $B.$ The tip follows one full flower trajectory during that time segment. (d) Cycle lengths at the detectors outside the flower [see (b)]. (e) Average CLs in the entire domain (axes show indices of the nodes, not model units). (f) Tip trajectory with pacing (details in Sec. 3d). (a), (b), (f) Axes in MULs.

Figure 3

Results for Model 2 with outward petals. (a) Spiral wave (state variable $u$), its tip (black dot), and tip trajectory (red curve). (b) Tip trajectory, points $C, D$, four detectors, and the subdomains. The disk is inside the black dashed circle; the ring is between the black and blue dashed circles; and the outside is the remaining part of the domain. (c) Potential $100u$ for two points with coordinates $C(10,15)$ (red, outside the curve) and $D(15,15)$ (blue, inside the curve). Abscissa axis, time in model units. Vertical lines mark ends of a segment of time with 4 APs in point $C$ and 5 APs in point $D.$ The tip drew one full flower during that time segment. (d) Cycle lengths at the detectors outside the flower [see (b)]. (e) Average CLs in the entire domain (axes show indices of the nodes, not model units). (a), (b) Axes in MULs.

Figure 4

Schematic of spiral wave forefront (blue solid curve), backfront (blue dashed curve), and tip trajectory fragment (red solid curve). $G$, a tip at an arbitrary time; $B$, a point at the disk or outside; $s$ (blue straight line), the ray tangent to the forefront; $w$ (black straight line), the ray tangent to the tip trajectory. The arrow shows the direction of tip motion.

Figure 5

Schematic of discrete full angle $\varphi$. $C$, arbitrary disk point; blue dashed line, arbitrary axis; red curve, tip trajectory; red circles, discrete tip positions (for clarity, every fifth tip is drawn); black arrow displays the direction of tip motion near the $(i-1)\mathrm{th}$ tip. Green arrow and line show the lifted angle ${\varphi }_{i-1}$ for the $(i-1)\mathrm{th}$ tip, and gray arrow and line show ${\varphi }_i$ for the $i\mathrm{th}$ tip. Note that $k_i=1$ here to neutralize the discontinuity of atan2.

Figure 6

Results for a hypermeandering spiral wave, FHN model 3. (a) Tip curve for 180 $\mathsf{T}$. (b) Tip points (not connected) for ${10}^4 \mathsf{T}$. The disk, ring, and outside are clearly observable. (c) Tip coordinates $x, y$ vs. time. Axes in all plots in model units.

Figure 7

Simulation results with Model 4 in the LV model and in two dimensions. (a) The LV model, epicardium (semitransparent colored surface, the color denotes $z$ coordinate), endocardium (meshed surface, opaque in the left panel, semitransparent in the right panel), tip trajectory on the epicardium (for time 700–7000 ms, black curve), electrode 1 border on the epicardium (red curve), and electrode 2 border on the epicardium (yellow curve). Side and up view (left panel); bottom view (right panel). (b) Scroll wave shown on the model surface, up view. $X$ axis (red); $Y$ axis (yellow); color denotes potential $u$. The circular wave is emerged by electrode 1. Pacing period of 189 ms. (c) Tip trajectory in two dimensions. Axes in mm.

Figure 8

A schematic comparison of CLs and pacing periods, ms, for Model 4. CLs for the 2D spiral wave (empty bars on the left), for the 3D scroll wave in the LV model (hatched bars in the middle), and pacing periods for the 3D LV scroll wave (filled bars on the right). Red, a node inside the tip curve and electrode 1 [red in Fig. 7], which is also located inside the curve. Yellow, a node outside the tip curve and electrode 2 [yellow in Fig. 7] also located outside the curve.