Spiral wave initiation in excitable media

Abstract

Spiral waves represent an important example of dissipative structures observed in many distributed systems in chemistry, biology and physics. By definition, excitable media occupy a stationary resting state in the absence of external perturbations. However, a perturbation exceeding a threshold results in the initiation of an excitation wave propagating through the medium. These waves, in contrast to acoustic and optical ones, disappear at the medium's boundary or after a mutual collision, and the medium returns to the resting state. Nevertheless, an initiation of a rotating spiral wave results in a self-sustained activity. Such activity unexpectedly appearing in cardiac or neuronal tissues usually destroys their dynamics which results in life-threatening diseases. In this context, an understanding of possible scenarios of spiral wave initiation is of great theoretical importance with many practical applications.

This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

1. Introduction

The state of many systems found in nature is far from thermodynamic equilibrium. One demonstrative example is dissipative systems which are in a state of continuous exchange of energy or matter with an environment. In the pioneering work by Turing, it was shown that a homogeneous state of a distributed dissipative system could be unstable, and this instability results in the development of spatially inhomogeneous structures [1]. Another consequence of symmetry breaking instabilities in dissipative systems is an oscillatory behaviour either homogeneous or inhomogeneous in space, predicted by Prigogine & Lefever [2].

There are also so-called excitable media, which exhibit a stationary resting state. This resting state can be excited by the application of a suprathreshold external stimulus that triggers a propagating excitation wave in a spatially distributed medium. In many cases, the transformation of the oscillatory dissipative systems into excitable ones can be achieved by variations of the medium's parameters.

Excitable dynamical behaviour occurs in many natural systems including the Belousov–Zhabotinsky (BZ) chemical reaction [3–5], autocatalytic reactions of carbon monoxide on a platinum surface [6], aggregations of Dictyostelium discoideum amoebae [7,8], Xenopus oocytes [9], chicken retina [10], disinhibited mammalian neocortex [11] and cardiac tissue [12,13].

In a uniform two-dimensional excitable medium, an excitation wave once initiated propagates through the whole medium until it reaches the medium's boundary. Such a wave represents a rapid transition from a stable resting state to an excited one followed by a slow recovery transition (refractory) back to the resting state. Under normal conditions, the wave back follows the wavefront, and they never touch each other.

However, under some special conditions, the propagating wavefront can be broken. Then the front and the back of the wave coincide at one point called a phase change point [14]. Near this point, the front and the back are moving in opposite directions and the boundary of the excited region curls around this singularity point. As a result, the broken wave is winding up into a spiral permanently rotating within the medium.

A rotating spiral wave represents a source of a self-sustained periodic activity in an excitable medium. This extraordinary activation can cause cardiac arrhythmias and even sudden death [15,16]. One obvious way to prevent such undesirable activity is to clarify and analyse possible mechanisms of spiral wave initiation in excitable media.

From the mathematical point of view, the main dynamical features of a broad class of excitable media can be simulated by a two-component reaction–diffusion system

1.1

and

1.2

Here the local kinetics of an activator u and an inhibitor v is specified by the nonlinear functions F(u, v) and G(u, v). Depending on a chosen form of these functions the concrete model obtained its own name, e.g. the FitzHugh–Nagumo [17], the Oregonator [18], the Barkley [19] model. The coefficients D and A are two important control parameters, which determine the spatio-temporal characteristics of the simulated medium. They are universal and applicable to any models.

From the experimental point of view, there is also some universality because there are many common dynamical features observed in quite different chemical and biological excitable media.

Some important results obtained in the context of spiral wave generation in experiments and computations are reviewed below. The review begins with a description of the spiral wave initiation in a chemical solution. Then the role of the vulnerable window, spatial inhomogeneity of the medium refractoriness and the obstacle geometry is illustrated. Finally, a recently discovered role of the fast propagation regions is discussed.

2. An instructive example of spiral wave initiation

Winfree has demonstrated initiation of spiral waves in a slightly modified Belousov–Zhabotinsky solution containing a little less acid and a little more bromide [20]. Owing to this modification, the solution's bulk oscillations have been eliminated without affecting its ability to support waves of chemical activity. This modified red-dish-orange solution, called as Z reagent, remains in a stable resting state when it is left alone. However, when it is stimulated by an ‘infectious’ droplet of a blue wave from another dish or by a touch of a heated needle, a single sharp blue expanding ring propagates through it at a steady propagation velocity of a few millimetres per minute.

In the procedure illustrated in figure 1, a blue circular wave was induced by touching the surface of the red solution with a hot filament, and then the dish was rocked gently to break the expanding wavefront. The free ends of the fragmented circular wave curl around a pivot near each open end of the broken wave, where the wavefront coincides with the wave back. As a result, two counter-rotating spiral waves have been created. Wherever two waves collide head-on, they do not interpenetrate like ripples in a pound, but vanish like colliding grass fires. The reason for this is the presence of the refractoriness, which restricts the possibility of initiating a secondary excitation immediately after the first one.

Figure 1.

Spirals of chemical activity form in a shallow dish of red Z reagent. Adapted from Winfree [20].

3. Vulnerable window

As already mentioned, each propagating excitation wave is followed by a region of recovering medium. If an external supra-threshold stimulus is applied near the wave back within a certain vulnerable region, a triggered front will propagate only in the retrograde direction. In a two-dimensional medium, the triggered wave can evolve into two counter-rotating spirals.

This mechanism of spiral wave initiation is illustrated in figure 2. Here, a concentric wave has been initially triggered at the top left corner of a square-shaped medium. An additional stimulus has been applied at a slightly different place in the vicinity of the wave back. The triggered wave starts to propagate in the retrograde direction with respect to the initially induced one. Its front has a semicircular shape and coincides with the wave back at a phase change point q. The wavefront starts to curl around this point which is simultaneously moving through the media. Finally, it results in the creation of a well-developed spiral steadily rotating within the medium.

Figure 2.

Spiral wave initiation by means of additional stimulation within a vulnerable region near the back of a primary wave. Thick (thin) solid represents the wavefront (back). Adapted from Gulko & Petrov [21].

It is important to note that the place for application of the secondary stimulus was chosen not at a diagonal of the square-shaped medium. Thus, the spatial symmetry has been broken, which results in the creation of only one spiral wave, in contrast to symmetrical counter-rotating waves shown in figure 1.

It is also important to note that the width of the vulnerable window is rather thin. The stimulus cannot be applied too early since no new wave will be generated in this case. On the other hand, if the stimulus is applied too late, an almost perfect circular wave will be generated. In this case, no phase change points and, consequently, no spiral waves will be created. The width of the vulnerable window, of course, depends on the stimulus amplitude and duration, as well as on the application area.

4. Spatial inhomogeneity of the medium's refractoriness

As mentioned earlier, the medium's refractoriness plays a very important role in spiral wave initiation. In particular, a spatial inhomogeneity of the medium's refractoriness is quite often a cause of wave breakage and spiral wave creation. One simple example of this scenario is illustrated in figure 3.

Figure 3.

Spiral wave initiation in a medium with inhomogeneity of the pulse duration and the refractory period. Thick (thin) solid represents the wavefront (back). Dotted line represents the trajectory of the phase change point. Adapted from Zykov & Petrov [22].

In this model case, a planar wave is initiated at the left medium's boundary and propagates to the right. The medium consists of two intrinsically homogeneous parts connected along a dashed line. The excitation pulse duration and the refractoriness within the upper part of the medium are shorter than within the lower part. The moment of a repeated stimulation is chosen so that the upper half of the medium had already completely recovered its excitability, while the lower half was still in the relative refractory period. For this reason, a secondary stimulation also applied at the left boundary can initiate a secondary wave only within the upper part. As a result, a formation of a wave segment is observed which contains a phase change point, where the wavefront coincides with the wave back. The wavefront curls up around this singularity point. After a short transient process, a steadily rotating spiral wave is created.

Note that while the medium's parameters simply jump at the middle dashed line, the pulse duration and the refractoriness are changing smoothly due to diffusion. Of course, this diffusion influence reduces the parameter range where a wave breakage can be observed.

Note that the trajectory of the phase change point is located within the lower part of the medium, where the pulse duration is larger. Within a homogeneous medium with such parameters, a spiral wave is rotating faster than within the upper part.

5. Sharp corner of an obstacle boundary

Another example of spiral wave generation is shown in figure 4. Here a homogeneous medium includes an obstacle with a no-flux boundary. The obstacle has a rectangular shape, and the corners play a crucial role in spiral wave creation [23].

Figure 4.

Initiation of two counterrotating spiral waves in an excitable media containing a rectangular obstacle with a no-flux boundary. (a) Low-frequency stimulation. (b,c) High frequency stimulation. Solid lines indicate the location of a wavefront at successive times. Adapted from Panfilov & Keener [23].

As can be seen in figure 4a, an initially planar wave initiated at the left medium's boundary begins to break after a collision with the obstacle. Two-phase change points appear as a result of this collision. They continue to propagate along the obstacle boundary. The medium's excitability is so high that they are even able to rotate around the sharp corners and do not detach from the obstacle. Then two counterrotating waves meet each other at the symmetry axis of the obstacle. After this collision, both phase change points disappear, and a smooth wavefront is reconstructed. This restored front is, of course, not planar, but strongly curved. However, in the course of time, it achieves a planar shape.

The dynamics shown in figure 4b,c are strongly different since here a periodic sequence of external stimuli is generated at a high frequency. Owing to the refractoriness, the medium's excitability is effectively decreased. For this decreased excitability, the created phase change points are not able to rotate around the corners. They are detached from the obstacle and start to describe rather complicated trajectories. If the external stimulation is stopped now, two counter-rotating waves remain in the medium.

Thus, this scenario predicts a possibility to initiate self-sustained rotors due to a suitable choice of the obstacle shape and size, as well as an excitation period. Moreover, the location and orientation of an obstacle within a medium of a final size are of importance [24,25].

These model predictions have been verified in experiments with the BZ reaction [26], in cardiac muscle [27] and on composite catalytic surfaces [28].

6. Fast propagation regions

Another important mechanism of spiral wave generation has been proposed recently [29]. Generally speaking, this mechanism is based on a well-known observation that an excitation wavefront can be stopped due to a phenomenon termed source–sink mismatch in the cardiology literature [30]. It was found that an excitation wave can be broken at a boundary of a region, where the diffusivity D or the parameter A are significantly larger than in the surrounding tissue [29,31]. Increased values of these parameters result in a faster propagation velocity of an excitation wave, which is proportional to .

As an example let us consider a two-dimensional excitable medium with a circular shaped fast propagation region (FPR). Figure 5 shows numerical results of the wave propagation simulated for the excitable media model described by equations (1.1) and (1.2). Everywhere, except the circular FPR, D = 1/2, while within the left half of the FPR D = 1 and in the right half D = 2/3. A planar left-travelling excitation wave initiated at the right propagates unbroken towards the left boundary. It even speeds up slightly within the FPR (figure 5b). Hence the FPR cannot be considered as an obstacle. Finally, the wave approaches the left medium's boundary and disappears due to the no-flux boundary conditions.

Figure 5.

Initiation of two counterrotating spiral waves in an excitable media containing a circular shaped fast propagation region. The trajectories of two created phase change points are shown by white solid lines. Adapted from Zykov et al. [29].

This evolution is in strong contrast to the dynamics of a right-travelling wave. There a plane wave created near the left side of the medium is blocked at the FPR due to the sharp increase of the parameter D exceeding the critical value for a wave blockage [29]. This results in a wave break and the creation of two phase change points where the wavefront coincides with the wave back, as shown in figure 5d. Owing to the propagation block, the phase change points move along the boundary of the FPR until they reach its vertical symmetry axis. There the increase of the parameter D is not sufficient to block the wave anymore, and the spiral tips penetrate into the FPR (solid white lines in figure 5e). Moreover, the waves can penetrate into the left-hand part of the FPR like the left propagating wave does in figure 5b. They start to rotate around the phase change points, and two counter-rotating spirals separate from the FPR, as shown in figure 5f . The further dynamics of these counter-rotating spirals are determined by the parameters of the medium surrounding the FPR and the boundary conditions. It is remarkable that they were created immediately with the first incoming wave and no additional stimulations have been applied.

Note that a similar scenario works perfectly if an FPR is created by variations of the parameter A instead of D [29]. If the symmetry of the FPR is broken, a single rotating spiral wave can be created [29,32].

It is important to stress that the shape and size of the FPR initiated spiral waves can be varied [32,33]. One example which clarifies the importance of the FPR geometry is shown in figure 6. In this case, the parameter D is fixed to D = 2 within the whole FPR of an asymmetric shape, while D = 1 is in the surrounding part of the medium. The left boundary of the FPR has a circular shape, while the right boundary has a sinusoidal one. Such uneven boundary shape creates conditions for a unidirectional block.

Figure 6.

Initiation of two counter-rotating spiral waves in an excitable media containing a fast propagation region of a specific shape. The trajectories of two created phase change points are shown by white solid lines. Adapted from Zykov et al. [33].

Indeed, the wave propagating from the right medium's boundary penetrates into and passes through the FPR as shown in figure 6a–c. This penetration is due to the strongly curved parts of the right FPR boundary, where the wave is not blocked. As a result, the wavefront is not broken at the right FPR boundary but only slightly deformed since its propagation velocity is faster within the inhomogeneity region with larger coupling strength D.

By contrast, the wave propagating from left to right breaks up when it reaches the inhomogeneity region, as shown in figure 6d. Two generated phase change points first move along the inhomogeneity boundary. However, they cannot follow a very strongly curved concave part of the right FPR boundary. On the other hand, they can penetrate into the inhomogeneity near a convex part of the right boundary. As a result, two counter-rotating spiral waves are created, as shown in figure 6e. The further movement of these waves shown in figure 6f is determined by the chosen medium's parameters.

7. Conclusion

Spiral waves represent an important example of spatio-temporal structures developed in dissipative systems far from thermodynamic equilibrium. Several generic mechanisms for spiral waves generation have been presented. The general idea of all mechanisms is to break up the propagating excitation wave and to create phase change points. In the first case of the BZ reaction, it was achieved by a pure mechanical influence on the liquid reagents. Of course, it is not applicable to such biological objects as cardiac or neuronal tissue. The following three examples are well known and are many times confirmed in direct experiments with chemical and biological excitable media.

In addition to this, the last two examples relating to FPR have been proposed quite recently. While they have been reproduced through the widely used detailed model of cardiac activity [29], until now they have had no experimental verification. A specific feature of these mechanisms is that they do not need an application of a secondary excitation. Hence a probability of spiral wave creation in a medium with the FPR is considerably larger. This, probably, explains the high efficiency of the ablation procedure in terminating atrial fibrillation [34]. In this context, many questions are open for future theoretical and experimental investigations.

It is worth mentioning another important aspect of spiral wave dynamics. In the reviewed examples, all created spiral waves were stable. However, a transition from spirals to defect turbulence has been reported due to quite different kinds of instabilities [35–39]. Spiral wave turbulence can also be considered as an important spatio-temporal structure in dissipative systems.

Data accessibility

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Competing interests

I declare I have no competing interests.

Funding

This work was supported by the Max Planck Society and the German Center for Cardiovascular Research (DZHK).