Interaction of a Pair of Parallel Scroll Waves - American Chemical

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Interaction of a Pair of Parallel Scroll Waves Dennis Kupitz and Marcus J. B. Hauser* Abteilung Biophysik, Institut für Experimentelle Physik, Otto-von-Guericke Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany ABSTRACT: Interactions of pairs of scroll waves in threedimensional excitable media were studied experimentally in the Belousov−Zhabotinsky reaction by optical tomography. The behavior of two scroll waves depended on the distance d between their filaments. When the interfilament distance was shorter than the wavelength λ of the scroll waves (but larger than the diameter of the spiral core), the filaments repelled each other. Once d ≈ λ, the two scroll waves synchronized, rotating around their filaments with both a common rotation frequency and a common pitch. The interfilament distance of synchronized scroll waves did not change. When fluctuations broke the symmetry of the rotation periods, the scroll with higher rotation frequency displaced the slower rotating one, until the latter was ousted or even annihilated. These behaviors were independent of the sense of rotation (co- or counterrotating), the filament dynamics (rigidly rotating or meandering tip motion in two-dimensional media), and the presence or absence of a gradient of excitability parallel to the filaments.

1. INTRODUCTION Excitable media play an important role in many biological and chemical systems. Typical self-organized structures in twodimensional (2D) excitable systems are rotating spiral waves. In three-dimensional (3D) systems, scroll waves are the extension of the 2D spiral waves. A scroll wave rotates around its organizing center, which is called a filament. Cardiac tissue is a prominent example of a 3D excitable medium where scroll waves are believed to be at the basis of some types of cardiac arrhythmias, for instance, ventricular tachycardia.1 In the ventricle, a scroll wave of electrical activity acts as an autonomous pacemaker that takes over the control of the dynamics of the ventricle from the natural pacemaker, the sinus node. During ventricular tachycardia the scroll wave may lose its stability and break up into two or more scroll waves. This leads to the highly dangerous ventricular fibrillation.2−5 Due to the severity of such a situation, numerous simulation studies of scroll wave dynamics in the context of cardiac arrhythmias have been performed (refs 6 and 7 and references therein). Experimental studies of scroll waves and their dynamics remain challenging, especially in the whole heart and in heart tissue. This is also due to difficulties in visualizing scroll waves of electrical activity in the opaque heart tissue. Visualization of perfused hearts or heart tissue usually relies on voltage sensitive dyes, which provide information on the surface activities of the heart muscle or tissue. The true dynamics in 3D remains hidden. However, the 3D dynamics of scroll waves and their filaments can be monitored in experimental model systems by optical tomography.8,9 This technique requires transparent experimental model systems, such as the Belousov−Zhabotin© 2013 American Chemical Society

sky (BZ) reaction, which allows for detailed experimental studies of scroll wave dynamics.9−14 It is crucial to better understand the dynamics of scroll waves and their interactions with each other to be able to manipulate and even annihilate scroll waves in the heart. Currently, the standard way to stop ventricular tachycardia is a painful defibrillation of the cardiac muscle, which may also cause tissue damage and create sites for future incidents of arrhythmia. Therefore, efforts are being made to develop reliable and more gentle ways of suppressing cardiac arrhythmias.15−18 In 2D systems, the interaction among spiral waves is wellstudied.19−24 Since spiral waves may rotate clockwise or counterclockwise, they possess topological charges. Depending on the distance between their cores, spiral waves may attract or repel each other.20,25,26 Numerical simulations showed, for instance, that counter-rotating spiral waves attract and annihilate each other if the distance between the spiral tips is smaller than the diameter of the spiral core.25,27 By contrast, if the distance between a pair of co-rotating spiral waves is smaller than the core diameter, spirals repel each other.25 When the distance between the spiral cores distinctly exceeds that of the core radius, the spiral waves repel each other and develop independently of their topological charges.20,25,26 Furthermore, spiral waves may form bound pairs with an axial symmetry or a symmetry center if the distance between the spiral cores is close to the core diameter.25,28 Received: September 17, 2013 Revised: November 5, 2013 Published: November 8, 2013 12711

dx.doi.org/10.1021/jp409269u | J. Phys. Chem. A 2013, 117, 12711−12718

The Journal of Physical Chemistry A

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Figure 1. Initiation of pairs of scroll waves. (a) A pair of counter-rotating scroll waves were initiated by inserting one silver wire in the middle of the reactor along the partitioning sheet. The dark surface represents the semicylindrical excitation wave. (c) A pair of co-rotating scroll waves were initiated by two silver wires at opposite sides of the partitioning sheet close to the wall of the cuvette. Again, the excitation waves are shown as dark surfaces. Top view of two spiral waves with (b) opposite and (d) equal topological charges.

combination of these three factors leads to 23 = 8 cases, all of which were studied. Scroll waves were created in the excitable BZ reaction medium and observed by optical tomography. Experiments were run either in the absence or in the presence of a gradient of excitability parallel to the filaments. If such gradients are sufficiently steep, the initially straight filaments are subjected to a twist-induced instability. Then, their originally straight filaments become helical.14,39−42 Note that there is no 2D counterpart for gradients parallel to the filament, since this direction is orthogonal to the plane where the spiral exists. By contrast, when the gradient is oriented exactly perpendicular to the filament, the scroll waves remain unaffected by the gradient.40,43 However, if the filament deviates slightly from the perpendicular orientation to the gradient, a single filament may become twisted from both ends and develop twists of opposite handedness.43 The paper is organized as follows: section 2 summarizes the experimental methods and the results are presented in section 3. We first address the effects induced by a gradient of excitability oriented in parallel to the scroll wave filaments. Next, we report on the dynamics of a pair of scroll waves, and finally, we study the effects caused by fluctuations and their consequences for the dynamics of interacting filaments. The results are discussed in section 4, and conclusions presented in section 5.

In 3D, a pair of scroll waves represents the minimal system for investigating scroll wave interactions. Whereas a body of computational and theoretical work on interacting scroll waves exists, there is a lack of experimental studies on this topic. The dynamics of self-interactions of ring-shaped filaments (i.e., scroll rings) and the interplay of a scroll wave with boundaries have attracted scientific attention.29−34 The interactions between filaments become important once the interfilament distance is shorter than or comparable to the wavelengths of the scroll waves, whereas they are negligible once the filaments are several wavelengths apart from each other.35,36 The dynamics of concatenated or “knotted” filaments have often been studied because of their interesting topology.31−33 Since small scroll rings shrink due to their positive line tension, such a situation necessarily leads to filament−filament interactions, which may induce the rupture and reconnection of filaments. The necessary topological requirements for these phenomena to occur were studied theoretically.32,33,37 In addition, the filaments must come so close to each other that their interfilament distance becomes smaller than the diameter of the filament core for rupture and reconnection of filaments to take place.37 However, despite the richness of numerical and theoretical papers, the dynamics of concatenated scroll rings have not yet been studied experimentally, since the required initial conditions could not yet be realized. Scroll wave interactions also occurred in numerical studies on ventricular fibrillation. Some of the filaments interact, for instance by annihilating each other, by rupturing, or by fusing.4,38 However, these papers focus rather on the dynamics of cardiac arrhythmias than on investigating the details of the interactions between scroll waves. In the present experimental study, we report on the interactions of a pair of scroll waves with each other. We investigate different cases of initially straight scroll waves with straight and parallel filaments. The initial distance between the filaments was varied; however, this distance was always larger than the diameter of the core of the filaments. The behavior of interacting filaments was found to depend on the interfilament distance. Furthermore, the roles of three other factors on the interaction of pairs of straight scroll waves were also studied, namely, (i) the topological charge of the scroll waves (i.e., whether they have the same or opposite senses of rotation), (ii) the dynamics of the filaments (i.e., whether they describe rigid rotations or if they meander), and (iii) the absence or presence of a gradient of excitability parallel to the filaments. The

2. MATERIALS AND METHODS Experiments were carried out using the ferroin-catalyzed BZ reaction embedded in an 1.0% (weight by volume) agarose (type VII, Sigma) gel matrix, which suppressed hydrodynamic effects. To reduce and retard the production of CO2 bubbles, the surfactant sodium dodecyl sulfate (SDS) was added at concentrations below its critical micelle concentration.13 The initial concentrations were 50 mM malonic acid (Merck), 50 mM sodium bromate (Merck), 0.2 mM SDS, and 0.5 mM ferroin. The initial concentration of sulfuric acid (Fluka) lay in the range 160−250 mM. In 2D systems the recipes with 160− 200 and 210−250 mM sulfuric acid yielded meandering and rigidly rotating spiral waves, respectively. All of the studied filaments had a positive line tension. The used recipes generated reaction fronts with sufficiently slow propagation velocities, as required for tomographic observation.44 The temperature was controlled by the air conditioning system of the laboratory, and the experiments were performed at 21 ± 1 °C. 12712



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Figure 2. Isoconcentration surfaces (a−d) of a pair of counter-rotating scroll waves and their filaments (e−h). The experiment was run in the absence of any gradient of excitability. The filaments performed a rigidly rotating tip motion in 2D. (a, e) 47 min, (b, f) 210 min, (c, g) 300 min, and (d, h) 421 min after initiation. Sample dimensions: ∼20 × 13 × 22 mm3.

of excitability the reactor was sealed with a Teflon lid immediately after initiation of the scrolls.13 In sealed reactors, vertically oriented gradients formed as the reaction progressed, due to the accumulation of CO2 at the top of the reactor. The gradients caused originally straight, vertically oriented scroll waves to twist, and once the critical twist rate was exceeded, the initially straight filaments developed a helical shape. Detailed studies on the response of scroll waves to vertical gradients of excitability were reported recently.13,14,43 The wave patterns were observed in a optical tomographic setup that acquired 100 2D projections per rotation separated by 1.8° from each other. The images were recorded by a CCD camera (Hamamatsu C3077) with a resolution of 768 × 576 pixels (px) at a rate of 25 frames s−1, digitized, and stored on a computer. The data acquisition protocol consisted of a sequence of recording 100 projections (which takes 4 s) followed by a 6 s quiescent phase during which the scroll wave propagated distances of 0.1−0.2 mm, i.e., 1−2 px. This led to a temporal resolution of 10 s. The spatial resolution was 0.1 mm px−1. The 3D structures were reconstructed slicewise from the tomographic projections by filtered back-projection of the ferriin concentration.9−14,43,45 The filaments of the scroll waves were reconstructed from the 2D slices that intersected the scrolls perpendicularly.13,14 The phase along the filaments were determined as described in ref 13. The distance d between two filaments at height z is defined as the distance between the spiral cores in the (x,y) slices at height z. The wavelengths λ of the scroll waves were determined from each (x,y) slice by measuring the distances between successive wave fronts. The helical structure of the filaments can be illustrated in more detail by smoothing the x,y coordinates of the filament using a moving average filter routine (“smooth”) from MATLAB with a span of 25 px to obtain the smoothed coordinates u and v. Together with the original z coordinates, these smoothed u,v coordinates form a “smoothed filament”. A vector plot pointing from the “smoothed filament” (u,v,z) to the original filament (x,y,z) reveals the helical structure of the

A 8.7 mL volume of the reaction solution was added to a cylindrical reaction cuvette of 21 mm inner diameter and 25 mm height. A pair of scroll waves with initially straight filaments was initiated by a modified partition method (Figure 1). The reaction volume was divided into two compartments by a poly(ethylene terephthalate) (PET) sheet. Counter-rotating scroll waves (i.e., with opposite topological charges) were initiated by insertion of a silver wire (0.5 mm diameter) exactly in the middle of the reactor along the PET sheet (Figure 1a). This created a semicylindrical excitation wave that formed two contact lines with the sheet. The contact lines moved away from their ignition site toward the wall of the cuvette. Before the contact lines touched the reactor wall, the partitioning sheet and the silver wire were carefully removed. The reaction fronts were left as open wave edges in the bulk of the reaction volume and curled in to form two scroll waves with opposite senses of rotation (Figure 1b). Co-rotating scroll waves (with the same sense of rotation, i.e., equal topological charges) were initiated by insertion of two silver wires on opposite sides of the partitioning sheet but both in the immediate vicinity of the PET sheet, and close to the wall of the cuvette (Figure 1c). This generated two segments of semicylindrical excitation waves, one at each side of the partitioning sheet. These segments formed four contact lines with the sheet, which then moved away from their ignition sites: two toward the center and the other two toward the wall of the cuvette. When the latter anchored at the cuvette wall, the partitioning sheet and the silver wires were carefully removed. The inwardly moving reaction fronts were left as open wave edges and curled in to form two scroll waves with the same sense of rotation (Figure 1d). Although the tomographic experiments were run in batch reactors, the reaction medium aged slowly. In fact, the used recipes allow for the observation of the dynamics of a pair of interacting scroll waves up to 7−8 h before the reaction system started to drift considerably.10,44 For investigations of a pair of scroll waves without excitation gradient the reactor was left open to the atmosphere, whereas for studies of a pair of scroll waves in the presence of a gradient 12713



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Figure 3. Isoconcentration surfaces (a−f) of a pair of co-rotating scroll waves and their filaments (g−m). The experiment was run in the presence of an excitation gradient parallel to the filaments. The filaments performed a rigidly rotating tip motion in 2D. (a, g) 31 min, (b, h) 120 min, (c, i) 200 min, (d, j) 265 min, (e, l) 300 min, and (f, m) 360 min after initiation. The horizontal red line in (g)−(m) represents z = 5 px ≙ 0.5 mm below the lid of the reactor. Sample dimensions: ∼22 × 22 × 22 mm3.

Figure 4. Pair of counter-rotating scroll waves with a gradient of excitability parallel to the filaments. (x,y) slice of the scroll waves 0.5 mm below the lid of the reactor (a) 100 and (b) 320 min after initiation. (c) Helical structure of both filaments (red and blue vectors) as obtained from the vector pointing from the “smoothed filaments” to the original filaments. (d) Phase angle Ψ at different heights z of the filaments (red and blue lines). The average distance between the filaments in (a) is d̅ = 7.5 mm, and the average wavelength is λ̅ = 4.5 mm. The average distance between the filaments in (b) is d̅ = 7.5 mm, and the average wavelength is λ̅ = 4.8 mm.

filament. Furthermore, the vectors yield the phase along the helix.

In presence of a gradient of excitability oriented parallel to a single straight scroll wave, the reaction medium at the top of the reactor was more excitable. Therefore, the scroll wave rotated with a higher frequency at the top of the reactor than in its bulk. Hence, the scroll wave twisted with time.13,14 Once the twist exceeded the critical value, the filament became helical.14 Pairs of scroll waves were found to behave similarly. This is shown in Figure 3 for a pair of co-rotating scroll waves in a BZ medium with a gradient of excitability parallel to the filaments. The filaments performed a rigidly rotating tip motion in 2D. Initially, the isoconcentration surfaces were straight (Figure 3a), but after ≈100 min the scroll waves began to twist starting at the top of the reactor, and with time the twist extended downward along the filament (Figure 3b−d). Initially, the filaments of the two scroll waves were straight (Figure 3g−i), but after ≈200 min the twist of the scroll waves exceeded the critical twist and the filaments started to undulate (Figure 3j). These helical deformations of the filaments set in at the top of the scroll waves once the twist exceeded the critical twist ωc = 41−44 deg mm−1,14 and the undulation spread to

3. RESULTS 3.1. Effects of Imposed Gradient of Excitability. In the absence of any gradient of excitability, pairs of scroll waves as well as their filaments remained nearly straight over the duration of the experiment. Figure 2 shows the isoconcentration surfaces of a pair of counter-rotating scroll waves, whose filaments performed a rigidly rotating tip motion in a 2D BZ medium. Initially, the isoconcentration surfaces of the scroll waves were straight (Figure 2a), and they remained straight for ≈6 h (Figure 2b,c). After 360 min the scroll waves twisted slightly starting at the top of the medium. The filaments of the counter-rotating scroll waves were initially slightly curved (Figure 2e). They straightened due to their positive line tension and became linear (Figure 2f). The filaments remained straight over the entire duration of the experiment (Figure 2g,h). 12714



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the bottom of the filaments. If ω < ωc, the filaments of the twisted scroll waves remained straight. These behaviors were found in all examined pairs of scroll waves, independently of their senses of rotation (co- or counter-rotating) and their filament dynamics (rigidly rotating or meandering tip motion in 2D). 3.2. Dynamics of a Pair of Scroll Waves. In the absence of fluctuations, pairs of scroll waves rotated at an almost common frequency ν (i.e., ν ± 3%) without being noticeably disturbed by each other. This is illustrated by a pair of counterrotating scroll waves (Figure 4), where the scroll waves remained symmetric. The two scroll waves synchronized their frequencies and pitches, and rotated around their filaments with a common frequency ν (ν ± 3%, Figure 4a,b) and pitch δ (δ = 2.9 ± 0.3 mm, Figure 4d). Gradients of excitability caused the scroll waves to twist and the filaments to undulate. The structure of the helical filaments in Cartesian space is depicted in Figure 4c. The phase angle Ψ of the two helical filaments developed symmetrically along the height of the filaments, except for their signs, since the two scroll waves had opposite topological charges (Figure 4d). The twists of both helical filaments were (dΨ/dz)1 = 122.8 ± 0.5 deg mm−1 and (dΨ/ dz)2 = −122.3 ± 0.4 deg mm−1. Both twists (dΨ/dz)1 and (dΨ/dz)2 were highly correlated, showing that the two scroll waves rotated in a symmetric and synchronous manner. The behavior of a pair of scroll waves was sensitive to the distance between its filaments. If the distance d between the filaments was smaller than the wavelength λ of the scroll waves but still larger than the core diameter (R) of the filaments (i.e., R < d < λ), d increased until it corresponded to the wavelength of the scroll waves. At d ≈ λ the two scroll waves synchronized their rotation frequencies. This was also the case for the filaments shown in Figure 3: In the upper region of the reactor the filaments pointed to each other at height z = 0.5 mm (Figure 3g) and the distance between the filaments was smaller than the wavelength (d ≈ 0.75λ). In the first 2 h of the experiment the distance between the filaments increased (Figure 5, i.e., for 0 ≤ t ≤ 120 min) until it reached the wavelength of the scroll waves. Once d ≈ λ, the two scroll waves rotated synchronously, i.e., without being noticeably disturbed by each other (Figure 3h,i), and the distance d remained constant. For d ≥ λ, synchronous rotation of scroll waves was observed, independently from their senses of rotation (co- or counter-rotating), their filament dynamics (rigidly rotating or meandering tip motion in 2D), and the presence or absence of a gradient of excitability. 3.3. Dynamics of a Pair of Scroll Waves in the Presence of Fluctuations. Fluctuations in the BZ medium may locally lead to symmetry breaking of the rotation periods of the scroll waves, such that one scroll wave rotates faster than the other. If the rotation periods of the scroll waves were sufficiently different (i.e., when the rotation frequencies differed by more than 6% from each other), the scroll wave rotating with the lower frequency was displaced by that with the higher rotation frequency. This type of displacement contingently occurred toward the end of the experiments when the distance between the filaments already exceeded the wavelength. The increase of the interfilament distance observed at t > 265 min (Figure 5) is due to such a fluctuation-induced symmetry breaking. The evolution of the distances between the filaments was not necessarily homogeneous along the filament; it rather depended

Figure 5. Ratio of the interfilament distance d and the wavelength λ of the scroll waves of Figure 3. The ratio (d/λ) was monitored at height z = 0.5 mm below the lid of the reactor. Three distinct types of behavior were found: At the beginning of the experiment, the distance between the filaments was smaller than the wavelength (d < λ). Therefore, the distance between the filaments increased until d ≈ λ (for 0 ≤ t ≤ 120 min). Once the distance between the filaments attained the wavelength of the scroll waves, the scroll waves synchronized. Their interfilament distance d did not change (120 ≤ t ≤ 265 min). While the distance d corresponded to λ, fluctuations occurred, breaking the symmetry in the rotation frequencies of the scroll waves. At t > 265 min this led to the displacement of one scroll wave by the other, and hence to a further increase in the distance between the filaments, so that d > λ.

on the vertical position z along the scroll wave. This is illustrated in Figure 6, where z = 0.5 mm and z = 22.0 mm correspond to 0.5 and 22.0 mm below the lid of the reactor, respectively. At the beginning of the experiment, the distance between the filaments varied in the range d = (2.2−5.8) ± 0.1 mm, and the wavelength was λ = 3.1 ± 0.3 mm. Only at the top of the reactor (i.e., in the slices z = 0.5 mm and z = 5.0 mm)

Figure 6. Temporal evolution of the distance d between the filaments of the scroll waves of Figure 3 in five different horizontal slices z. The dashed line represents the averaged wavelength λ̅. Note that λ̅ changes with time due to aging of the reaction medium. z = 0.5 mm is located 0.5 mm below the lid of the reactor, and z = 22.0 mm corresponds to the bottom of the scroll waves. 12715



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Figure 7. Displacement of the slower scroll wave by the faster scroll wave. (a−d) Top views of the scroll waves of Figure 3, (a) 200, (b) 265, (c) 300, and (d) 360 min after initiation. (a) Both scroll waves rotate undisturbedly with the same frequency. (b) First phase of displacement: the faster scroll wave (right) unwinds the slower scroll wave (left). (c), (d) Second phase of displacement: the slower scroll wave drifts away from the faster scroll wave.

both scroll waves were equally twisted (ω ≈ 60 deg mm−1, Figure 3d). During displacement the twist of the slower scroll wave decreased in the top part of the filament. After 360 min, the twist shrunk to ωtop ≈ 24 deg mm−1 at the top layers. The twist of the displaced scroll wave in the lower regions, where the displacement process had not yet set in, remained at ωbottom ≈ 55 deg mm−1. That twist is similar to the twist of the faster scroll wave, ω ≈ 61 deg mm−1. Where the twist drops below the critical value, the helical filaments return to their originally straight shape.

were the distances between the filaments shorter than the wavelength. Therefore, the distance d between the filaments only increased in these slices (as described in section 3.2). The distance increased to d = 4.6 ± 0.1 mm, which corresponded to the wavelength of the scroll waves. From t = 120 min to t = 265 min the two scroll waves rotated synchronously. Fluctuations (occurring at t ≈ 250−260 min) caused one of the scroll waves to rotate faster than the other. Consequently, the faster scroll wave began to displace the slower one, and hence the distance between the filaments increased (for t > 265 min). It is worth noting that the increase in d sets in at the top of the filament, i.e., at slice z = 0.5 mm. After a delay, the filaments also began to recede from each other further down the filament. In the two uppermost sections (z = 0.5 mm and z = 5.0 mm) the rate of displacement of the filaments was 1.5 ± 0.3 mm h−1. At deeper layers of the scroll wave (z = 10.0, 15.0, and 22.0 mm) the distance between the filaments remained essentially constant. The fact that the displacement of the scroll waves started at one point, rather than starting simultaneously along the whole length of the scroll waves, underlined the local character of the fluctuations. In contrast to the increase in interfilament distance described in section 3.2, the distance between the filaments was always larger than the wavelength of the scroll wave (i.e., d > λ) during displacement and ousting. The dynamics accompanying the symmetry breaking of the rotation frequencies of the two scroll waves and the repulsion of one scroll wave by the other are illustrated in Figure 7. Initially, the distance between the filaments in Figure 7a slightly exceeded the wavelength of the scroll waves and the two scrolls rotated in synchrony. The fluctuation-induced displacement process can be divided into two phases. First, the faster scroll wave unwound the one with the slower frequency (Figure 7b). With each revolution the wave front of the faster scroll wave invaded deeper into the domain of the slower one. During the unwinding process the distance between the filaments did not change (Figure 5, t = 265 min, d ≈ λ). Once the slow scroll was unwound, the filament of the slower scroll wave began to drift away from the other filament. The wave fronts of the faster scroll wave interacted directly with the filament of the slower scroll, repelling the latter (Figure 7c,d). Consequently, the distance between the filaments increased (Figure 5, t > 265 min, d > λ). If a pair of twisted scroll waves (i.e., in the presence of a gradient of excitability oriented parallel to the filaments) was subjected to a fluctuation-induced displacement, the twist of the displaced slower scroll wave decreased whereas the twist of the faster scroll wave remained unchanged. This behavior is depicted in Figure 3d−f: at the beginning of the experiment,

4. DISCUSSION The interaction of a pair of straight, parallel scroll waves showed three major dynamical behaviors: First, the scroll waves repelled each other when the distance d between the filaments was smaller than their wavelength λ. The filaments displaced each other, until the interfilament distance d became comparable to the wavelength λ. Second, when the interfilament distance slightly exceeded the wavelength, both scroll waves could accommodate each other by rotating synchronously, thus adopting both a common rotation frequency and a common pitch. Third, fluctuations were found to break the symmetry between the two scroll waves. When the rotation frequencies of the scroll waves differed by more than 6%, the faster rotating scroll wave unwound and ousted its slower rotating counterpart, leading to interfilament distances which were considerably larger than the wavelength of the scrolls, i.e., d > λ. As shown in numerical studies using the Oregonator model, two interacting, parallel scroll waves repel each other, provided the distance exceeds the core radius R of the scroll waves (i.e., d > R). The repulsion is predicted to decay with the interfilament distance d, and to become negligible once d reaches a couple of wavelengths.36 This situation is also observed in our experiments: the repulsion is pronounced, as long as the interfilament distance d is shorter than the wavelength λ of the scroll. However, for interfilament distances d > λ, there are slight discrepancies between the numerical predictions and our experimental observations concerning the range of the repulsive domain. These differences stem from the fact that the Oregonator is not a quantitative model for the BZ system. In experiments, the range of the repulsive interaction between the scroll waves also exceeds the wavelength λ. This is seen from experiments where fluctuations led to the displacement and ousting of the scroll wave with lower rotation frequency. Interestingly, once the interfilament distance d becomes comparable to the wavelength λ, a pair of scroll waves can accommodate to this situation by synchronizing to both a 12716



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than the core radius of the filaments (R < d). However, spiral pairs are known to show interesting dynamics, when d < R, such as the attraction or annihilation of pairs of spiral waves as well as the formation of bound pairs.20,25,28 Under such conditions, numerical studies predict that scroll waves may rupture and reconnect.31−33,37,46 Unfortunately, scroll waves with such short interfilament distances were not accessible with the present protocol; however, experiments along these lines are in progress.

common rotation frequency and a common pitch, thus avoiding any further displacement. Whether the synchronization led to a slight increase in the rotation frequency of the interacting scroll waves as compared to a single, undisturbed scroll wave could not be determined from the experiments, given the experimental time resolution of 10 s. The displacement of two filaments originally located at a distance below the wavelength of the scroll waves (d < λ) has also been observed in 2D. However, the extension into the third dimension shows a new feature: The repulsion of the scroll waves is limited to those domains where the filaments approach each other closer than the wavelength (i.e., d < λ), while the parts of the scrolls where d ≥ λ remained synchronized and maintained their distance. Thus, the displacement of closely positioned filaments is a local phenomenon, and not a global one affecting the entire scroll waves. Since the interactions depend on the interfilament distance d, it is desirable to prepare pairs of scroll waves whose filaments may have any distance from each other. In experiments, pairs of scroll waves were initiated by a partition method, which unfortunately did not allow creation of scroll waves with arbitrarily short initial distances d0 between their filaments. Independently from the sense of rotation of the scroll waves, the shortest initial interfilament distance that could be achieved experimentally was d0 = 2.2 mm. This distance corresponds to ≈(1/2)λ for a meandering and ≈(2/3)λ for a rigidly rotating tip motion in a 2D medium. However, this initial interfilament distance d0 is always larger than the diameter R of the filament core, which amounts to d0 ≈ 2.5R for meandering scroll waves and d0 ≈ 5.0R for rigidly rotating scrolls, respectively. This means that any interaction where the filaments approached each other as close as their core radii were not accessible experimentally. Fluctuations are always present in real, experimental systems. Fluctuations in any of the system parameters, such as the homogeneity of the gel matrix, the formation of CO 2 microbubbles, the local temperature in the 3D medium, etc., may lead to a spontaneous symmetry breaking, such that the rotation frequency of one scroll became larger than that of the other. Once filaments caused the frequencies of the scrolls to differ by more than 6% from each other, the faster scroll repelled and eventually ousted and annihilated the slower rotating scroll wave. This behavior is analogous to the fluctuation-induced breaking of the symmetry of the rotation frequencies in a pair of (2D) spirals.19,20,22−24,28 As expected, also in 3D the fluctuations arose locally (e.g., close to the top of the scroll wave in Figure 6) and spread slowly along the filaments. Thus, the displacement of one scroll wave did not take place simultaneously in the 3D medium. For twisted pairs of scroll waves (i.e., in the presence of a gradient of excitability) the ousting of the slower scroll wave by the faster one shows an additional 3D effect, which does not have a 2D analogue: As the wave fronts of the faster scroll wave invaded the domain of the slower scroll and approached the filament of the latter, the slower scroll began to unwind (e.g., the left scroll wave in Figure 3d−f). Such filament interaction dependent losses of twist are not encountered in twodimensional settings. Neither the topological charge nor the filament dynamics (i.e., whether they rotated rigidly or meandered in 2D) of the scroll wave had any effect on the interactions of a pair of scroll waves as long as the distance between the filaments was larger

5. CONCLUSIONS Straight and parallel scroll waves repel each other, provided that the distance d between the two filaments is smaller than the wavelength λ of the scroll waves. The repulsive interaction decays with the interfilament distance d, but it lasts over several wavelengths λ of the scroll wave. However, once d ≈ λ, two scroll waves may accommodate to each other by synchronizing and rotating with a common frequency of rotation and a common pitch. When this occurred, the scroll waves kept their distance. This behavior was found to be independent of the sense of rotation (co- or counter-rotating scrolls) and of the filament dynamics (rigidly rotating or meandering filament motion) of the scrolls as well as of the presence or absence of any gradient of excitability. Fluctuations may lead one scroll wave to rotate faster than the other. Since the scroll waves can no longer accommodate to each other, the slower scroll is repelled by the faster one, ending in the ousting or annihilation of the slower rotating scroll wave. When the interacting scroll waves were twisted, the vanishing, slower rotating scroll wave was subjected to a gradual lost of twist, whereas the twist of the dominant (faster rotating) scroll wave remained constant.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Graduiertenförderung des Landes Sachsen-Anhalt for financial support for D.K. REFERENCES

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Interaction of a Pair of Parallel Scroll Waves - American Chemical

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