Nonlinear waves in excitable media are ubiquitous.^[1–5] The emergence of defects in cardiac tissue, such as complex anatomical structures, blood vessels, and even scars from tissue damage, may lead to serious life-threatening consequences.^[6] The defects (obstacles) give rise to the breakup of propagating wave trains initiated from the pacemaker, and subsequently results in formation of spiral waves. It is difficult to remove the pinned spirals on the defects, which represents ectopic pacing and tachyrhythmia.^[7] In some cases, the defects also lead to breakup of spiral and consequent chaotic state, which is the precursor of lethal ventricular fibrillation.^[8,9] Therefore, the studies on the interaction of traveling wave with defects had attracted much attention.

The interaction of wave trains with an anatomical defect can be quite complex especially in the spatiotemporally chaotic state associated with spiral turbulence.^[7,10,11] Indeed, experiments concerning defects in cardiac tissue have yielded various results.^[12–16] For example, some experiments reported that small defects do not affect spiral waves but, as the size of the defect is increased, such a wave can get pinned to the defect.^[17] Various other experiments have been done to discuss the role of an anatomical defect as an anchoring site for spiral waves, which showed reducing the defect size or prolong the wavelength promotes complex oscillations, conduction failure, and leads to complex spiral wave behavior.^[6,18] Simulations in excitable media showed that the rotating frequency and pulse behavior were size dependent in circular domain.^[19,20] By starting with a large defect and decreasing its radius, a continuous transition was created from periodic motion to a modulated period-2 rhythm, and then to spiral wave breakup. These results may provide a useful basis for refining cardiac ablation techniques currently in use.^[21,22]

Although great progress has been made on this topic in recent years,^{[10–13,23–25]} it is still open for its complexity and significance. When the planar wave trains (PWTs) interact with a defect, the shape and size of the defect play an important role in determining the dynamics of interaction while it has not been studied so far. In this paper, we examine the evolution and transition of the PWTs propagating through several kinds of defects in an excitable medium. We chose circle, triangle, and rectangle defects with different sizes. The PWTs breaks into two parts when it collides with a defect. Subsequently, the two broken ends of the wave front show interesting behavior, which depends on the frequency of the excitation wave as well as the shape and size of the defects. The underlying mechanism about transition of patterns will be discussed.

The simulation of an excitable medium is performed in terms of a modified FitzHugh-Nagumo model (the Bär model).^[26] This simplified mono-domain model is given by
where u is the fast variable corresponding to membrane potential, and v is a slow variable corresponding to a recovery process. In this model , and g(u,v) describes a delayed production of inhibitor with g(u,v)=−v for , for , and , for . Numerical simulations are carried out on 256 × 256 2D grid points by employing the explicit Euler method. The space and time steps are and , respectively. No-flux condition is imposed on the boundaries. ε is the ratio of their temporal scales which characterizing the excitability of the medium. In our simulation, we fix parameters ε=0.02, a=0.84, b=0.07. An unexcitable area by means of no-flux condition was defined in the middle of the system as the defect.

Three types of defects are considered, i.e. circular, triangular (up and down triangles), and rectangular defects. Periodic local pacing is utilized on the bottom boundary by applying a signal on a line containing grids where A and ω are the amplitude and the angular frequency of the periodic pacing, respectively. PWTs are stimulated from the boundary of the system to mimic the waves from a pacemaker. The frequency of the PWTs is equal to that of the local pacing. When the pacing frequency increases a critical value, it is shown that the output wave train can no longer follow each excitation pulse of the pacing, namely, the system output cannot keep 1:1 frequency relation with the input, rather it can generate only 1:n (n is larger than or equal to 2) frequency response. We record the critical frequency as ω₀.

The first examination is the propagation of PWTs through a circular defect. When we increase the value of pacing angular frequency ω, three different parameter regimes can be distinguished by the pattern of the excitation wave. When the frequency of the local pacing is not very high, such as in Fig. 1(a), the PWTs break when they encounter the front of the defect and form two broken ends. Upon circumnavigating the defect, the two broken ends of the wave front come together and fuse with small pits. Subsequently, the wave front thus recovers its previous shape and moves on. Now if we continue to increase the frequency ω, the system would experience an interesting evolution. Wave chains of “V” configuration with ω=0.90ω₀ appear in Fig. 1(b). It is obvious that the wavelength of the “V” wave trains is greatly increased. If ω is increased further, two counter-rotating spirals with ω=0.93ω₀ in Fig. 1(c) are observed in the behind of the defect, respectively.

Fig. 1.

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	Fig. 1. (Color online) Dynamical patterns of the planar wave trains propagating through a circular defect (radius with three different pacing frequencies ω, i.e., ω=0.81ω0 in (a), ω=0.9ω0 in (b), and ω=0.93ω0 in (c), at t = 300. The arrow shows the direction of the propagating wave trains.

To understand the propagation of PWTs and observed patterns in Fig. 1, we present the dynamics evolution with ω = 0.93ω₀ in Fig. 2. After initial process of fusion before t = 80 in Fig. 2(a), it is shown that many broken ends cannot touch each other again after they propagate through the circular defect as that in Fig. 2(b) at t=86. That is because the wave trains with high velocity (since ω=0.93ω₀ approaches ω₀) are drag down by the defect and the two ends collide with considerable overlap (one can find this point by comparing Fig. 1(a) with Fig. 2(b)), which gives rise to a refractory region resulting from the dispersion relation at the back of the defect.^[27] The two broken ends depart each other when they propagate around the refractory region and then fuse again when they pass through it, as shown in Fig. 2(b) at t=86 and Fig. 2(c) at t=100. In the behind of the defect, a region with sparse waves appears, which can be seen in Figs. 2(c) and 2(d). This region grows up and two counter-rotating tips are formed in Figs. 2(d) and 2(e). Ultimately, two counter-rotating spiral waves are observed in Fig. 2(e). Note that the region that initial wave trains cannot enter into has fan-shaped configuration, which is consistent with the simulation in Ref. [28] where the train waves cannot propagate into the the fan-shaped region in the behind of the defect to suppress the turbulence.

Fig. 2.

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	Fig. 2. (Color online) The time evolution shows the wave breakup and formation of a pair of counter-clock rotating spirals on a circular defect at t = 80 (a), t = 86 (b), t = 100 (c), t = 120 (d), t = 140 (e), and t = 300 (f). Other parameters: and ω=0.93ω0.

Now, we consider the propagation of PWTs through a rectangle defect. The phenomenon is similar with that through the circular defect: when the frequency of pacing is increased, three different dynamical regimes are observed, as shown in Figs. 3(a)–3(c). The area of the fan-shaped region can be characterized by the central angle θ which is illustrated in Fig. 3(b). Interestingly, it is shown that the increase of ω makes the increase of fan-shape region. Comparing Fig. 3(b) with Fig. 3(c), one can see this point. We calculate the dependence of θ on ω/ω₀. The curves in Fig. 3(d) shows that the values of θ increase quickly as ω/ω₀. Simulation finds that the transition from “V” waves to spiral waves occurs when θ is about π/2. As ω/ω₀ gradually reaches 1.0, the θ approaches an asymptotic value 2π/3.

Fig. 3.

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	Fig. 3. (Color online) Patterns of PWTs interacting with a rectangle defect with height ( ) and width ( ) at t = 300. The pacing frequencies are ω=0.81ω0 in (a), ω=0.83ω0 in (b), and ω=0.88ω0 in (c), respectively. (d) The dependence of the central angle θ of the fan region on the ω/ω0. The definition of the θ is shown in (b).

On an up-triangle defect, the wave trains also experience three dynamical regimes when the pacing frequency is increased, which is illustrated in Figs. 4(a)–4(c). Once a planar wave encounters the defects, it is split into two parts. Subsequently, the two ends move along the edge of the triangle with decreased width. On the vertex of the triangle, the two ends touch each other and then leave the triangle, as that in Fig. 4(a). In Fig. 4(b), one can see the “V” trains initiate from the vertex of the triangle. If we fix the height of the triangle (H), while increase the width of the base of the triangle (L), the two ends will experience more pronounced change when they propagate on the defect, from splitting to fusion, which gives rise to the breakup of wave trains. This point is confirmed in Fig. 4(d) where the minimal frequency for the formation of “V” wave trains decreases when the L is increased. Thus, the increase of the width of a triangle defect may result in breakup of wave trains easier. In Fig. 4(e), we present the phase diagram in the H-L plane with fixed ω/ω₀. With small width, i.e. L = 5, the wave trains always fuse in the behind the defect, even the height of the defect is increased much. With the increase of width, i.e. L = 10, the PWTs can form “V” wave in the behind of the defect. In this case, increase of the height may quickly lead to the fusion of the wave train on the defect. When L is large, three different regimes appear in turn in the phase diagram if the values of H are increased. For fixed H, increase of L will give rise to the formation of “V” wave and counter rotating spiral waves. From the phase diagram, one can get a conclusion that increase of H may contribute to the propagation of PWTs by preventing the formation of “V” wave and spiral waves, while increase of L leads to the breakup of PWTs and formation of “V” wave and spiral waves.

Fig. 4.

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Fig. 4. (Color online) Patterns of PTWs interacting with an up-triangle defect with height ( ) and width ( ) at t = 300. The pacing frequencies are ω=0.81ω0 in (a), ω=0.93ω0 in (b), and ω=0.964ω0 in (c), respectively. (d) The dependence of ωmin/ω0 on the defect width L. The PWTs fuse after colliding with the defect when while they break as . The height of the triangle H is fixed to be . (e) The phase diagram in the H-L plane describes three dynamical regimes with ω=0.94ω0. The square, triangle, and circle mean the states of fusion, breakup to “V” pattern, and two counter-rotating spirals, respectively.

To study the influences of defect shape on the propagation of PWTs further, we present four types of defects in Fig. 5. For comparison, we set their widths and heights same. It is shown that different shapes of defect result in different minimal frequencies. The value of is smallest in the down triangle, that means this type of defect is the most dangerous shape for breakup of wave trains when the facing frequency is increased. The ω_min of a rectangle is slight bigger. For the circular defect, the value of ω_min is greatly increased to 0.93ω₀. The up triangle defect has the biggest ω_min (=0.96ω₀), which indicates this shape is much safer than other shapes. From the comparison, it seems that the narrowing of a defect may contribute to the persistence of waves train with high frequency. The phenomena can be qualitatively discussed in terms of the linear Eikonal relations
where C_p and C are the velocities of a propagating wave ( ) and a planar wave (K=0), respectively, D is the diffusion coefficient, and K is the local curvature of the propagating wave.^[29] When the wave trains reach the position on obstacles with maximal length, the situations are different in four cases. In Figs. 5(a) and 5(b), the two broken waves depart from the obstacle suddenly while they gradually leave the narrowing defects in Figs. 5(c) and 5(d). In Figs. 5(a) and 5(b), besides the initial propagating velocity, the broken ends show rapid growing tangential velocities that decrease their propagating velocities. Subsequently, the ends become bend with increased curvature K. Then, C_P is decreased, which leads to increased refractory time in the behind of the obstacle.^[30] Consequently, the waves trains break with lower frequency ω_min. As to the case in Fig. 5(c), especially the case in Fig. 5(d), the narrowing of defect is slow. The effect is not obvious, which results in bigger ω_min.

Fig. 5.

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	Fig. 5. (Color online) Patterns of wave trains interacting on defects with different shapes. The pacing frequencies are ω/ωmin=0.82 in (a), ω/ωmin=0.83 in (b), ω/ωmin=0.93 in (c), and ω/ωmin=0.96 in (d), respectively. The size is .

On the other hand, the area of the fan-shaped region in the behind of the defect is increased from Fig. 5(a) to 5(d). The fan-shaped region in Fig. 5(a) is similar with that in Fig. 5(b). It is interesting that the area is increased dramatically when the defects become narrower along the propagation of the wave train. Note that if turbulence exists in this fan-shaped region, local pacing is difficult to suppress they.

In conclusion, we have studied the evolution and transition of planar wave trains propagating through four kinds of defects in an excitable medium. Based on the frequency of a local pacing, three dynamical regimes are distinguished in terms of the pattern formation in the behind of the defects: fusion, “V” pattern, and two counter-rotating spirals. The dynamical process is discussed. For a rectangle defect, the area of fan-shaped region decreases with the increase of local pacing. For a triangle defect, the increase of L makes the wave train easier to breakup at lower ω_min. Also, we present a phase diagram to illustrate the influences of height (H) and width (L) of the triangle defect. The increase of width gives rise to the breakup of wave trains to form “V” pattern and spirals, while the increase of width is beneficial for the fusion of the wave trains. The narrowed defect along the propagation of the wave results in two effects: the minimal frequency for breakup of wave trains and the area of the fan-shaped region are both increased. Although the shapes of defects in cardiac tissue are complex, we hope the results studied here may contribute to the understanding of interaction between the wave trains from the pacemaker and defects.