Spatial Confinement Causes Lifetime Enhancement and Expansion of Vortex Rings with Positive Filament Tension

We study the impact of spatial confinement on the dynamics of three-dimensional excitation vortices with circular filaments. In a chemically active medium we observe a decreased contraction of such scroll rings and even expanding ones, despite of their positive filament tension. We propose a kinematical model which takes into account the interaction of the scroll ring with a confining Neumann boundary. The model reproduces all experimentally observed regimes of ring evolution, and correctly predicts the results obtained by numerical simulations of the underlying reaction-diffusion equations.

Confinement effects attract interest across many areas of physics as they generate a wealth of non-intuitive phenomena. One widely studied example is Brownian motion in confined geometries, which exhibits intriguing features such as a broad excess peak of the diffusion coefficient resulting in a violation of the Sutherland-Einstein fluctuation-dissipation relation [1,2] and hydrodynamically enforced entropic trapping of Brownian particles [3]. Phase separation in porous materials leads to layering, freezing, wetting and other novel phase transitions not found in the bulk system, when the pore size is on the order of the range of the forces between the confined molecules [4]. Further examples include the efficacy of insect flight [5] and vortex-related phenomena such as the onset of turbulence [6].
Vortex structures exist also in excitable systems including chemical reaction-diffusion media and information relaying biological systems [7][8][9]. In this context, confinement effects have attracted comparably little attention, although important examples, such as widely studied multicellular organisms [10] and the human heart [11], measure at most a few vortex wavelengths. Vortices in two-dimensional excitable systems are rotating spiral waves and exist only above a minimal system size [12]. Their interaction with a no-flux (Neumann) boundary induces a drift of the spiral tip along the wall in which the tip-wall distance is a system-specific value below the pattern wavelength [13]. It has been shown that the effect of a no-flux boundary is formally identical to that of a mirror image [14]. It is also known that sharp corners in the system boundary can nucleate spiral waves from nonrotating wave trains [15]. Moreover, local heterogeneities such as unexcitable discs can pin spiral tips which then orbit along the boundary of the inclusion at an altered * jantotz@itp.tu-berlin.de † h.engel@physik.tu-berlin.de ‡ steinbck@chem.fsu.edu rotation frequency [16].
In three-dimensional media, excitation vortices are called scroll waves and rotate around one-dimensional phase singularities [17]. In the limit of small curvature and twist, the local speed of these filaments is proportional to their local curvature [8,[17][18][19]. In the case of a positive filament tension α, this motion contracts circular filaments according to a simple square root law and the vortex annihilates in a finite time R 2 0 /(2α) where R 0 is the initial filament radius.
Experimental and numerical studies of the formation and evolution of free scroll rings have been reported in [20] and elsewhere. In addition, Bray and Wikswo simulated "quatrefoil reentry", which involves the interaction of two scroll rings having opposite chirality [21]. This scenario shares similarities to vortex interaction with noflux boundaries. A numerical study by Winfree noted the stabilizing case of boundary interaction, but only for negative filament tension [22]. The behavior of vortices with positive filament tension near planar no-flux boundaries, however, has never been studied experimentally.
In this letter we use the ferroin-catalzyzed Belousov-Zhabotinsky (BZ) reaction [8,19,23,24] and study the evolution of scroll rings confined to a layer (Fig. 1a). We demonstrate experimentally that the interaction of the wave fronts with the no-flux top and bottom boundaries of the layer can significantly alter the ring dynamics as compared to a scroll ring in an unbounded medium. A kinematical model leads to good agreement between measurements and numerical simulations based on a reaction diffusion model of the BZ reaction [25]. For these conditions the filament tension of an unbounded scroll ring was previously determined to be as α = 1.4 × 10 −5 cm 2 s −1 . Curvature-induced filament motion in binormal direction was shown to be vanishingly small [23]. To initiate the scroll ring, we introduce a silver wire at the interface of the two layers for about 20 s. This causes a decrease in the local concentration of inhibitory bromide ions. As soon as the developing spherical wave reaches a certain size, the system is strongly agitated to spatially homogenize the liquid top layer. After the fluid comes to rest again, the unperturbed gel-bound part of the wave extends into it and starts to curl in, thereby nucleating the scroll ring as depicted in Fig. 1(a).
A charged coupled device camera mounted over the system records transmission image sequences. Grayscale values in the recorded images are proportional to the light transmission integrated across the total height of the active medium. The faint contrast between the blue wave and the red refractory wake (corresponding to the oxidized and reduced state of the catalyst, respectively) is enhanced using an additive dichroic filter.
Our experiments show that the close vicinity of a Neumann boundary can strongly delay the contraction of the filament loop (Fig. 1f). For slightly different experimental conditions we even observe quasi-stationary, almost immortal scroll rings (Fig. 1g). Moreover, ring contraction can be superseded by expansion (Fig. 1h) despite of positive filament tension. Conversely, the life time can be drastically diminished because of sudden annihilation of the filament with a nearby boundary (Fig. 1d). Quan-titative details of these different dynamics are shown in Fig. 2. Notice the nearly linear increase of R 2 (open squares) for the experimental conditions also shown in Fig. 1h. The corresponding increase in R equals 39% over 3.5 hours.
The experimentally observed delayed contraction and even expansion of the confined scroll ring cannot be explained within earlier proposed models that describe filament motion kinematically in terms of curvature flow [17]. Therefore, we propose an extended kinematical model for the ring radius, R(t), and the distance of the filament plane to the boundary, z(t), (see Fig. 1a) The first term on the right-hand side of Eq. (1) describes the contraction of the unconfined scroll ring. Functions c 2d (z) and c 2d ⊥ (z) denote the normal and tangential velocity components of the boundary-induced drift of a twodimensional spiral wave interacting with a planar no-flux boundary. Velocity components c 2d ⊥ (R) and c 2d (R) account for the self-interaction inside very small scroll rings, which for symmetry reasons is equivalent to boundary induced drift with the distance z replaced with the filament radius R.
The components of the drift velocity field are difficult to measure experimentally, especially within the interaction range of the scroll ring and the no-flux boundary. Therefore, we have determined the dependencies c 2d (z) and c 2d ⊥ (z) in two-dimensional numerical simulations (see Fig. 3a) with the Rovinsky model underlying the ferroincatalyzed BZ reaction which is given as Here u and v are proportional to the concentrations of bromous acid and the ferriin, respectively. The values for the parameters a, b, , q and µ have been calculated from the employed concentrations of the BZ medium using the rate constants given in [25]. Diffusion coefficients D u and D v are the same as in [21,25]. From simulations of the free spiral wave, we obtain the wavelength λ sim = 0.56 cm and the rotation period T sim = 372 s. The corresponding values measured experimentally are λ exp = 0.58 cm and T exp = 390 s. For the filament tension we find α sim = 0.024 λ 2 sim T −1 sim and α exp = 0.026 λ 2 exp T −1 exp . This agreement is exceedingly good. Figure 3 displays the numerically calculated dependencies c 2d ⊥ (z) as well as c 2d (z) (a) and the drift velocity field (b), respectively.
The normal velocity component c 2d ⊥ (z) vanishes at z = z rep = 0.28 λ sim (dashed line in Fig. 3b) and z = z att = 0.82 λ sim . z rep defines a critical distance separating spiral waves that finally annihilate with the boundary from those attracted into a stable regime of constant drift parallel to the boundary at distance z att . Note that for the chosen parameters, the drift velocity at distances comparable to z att is already very small. This is due to the exponential decay of the interaction strength between a spiral wave and a Neumann boundary [26].
In Fig. 4 we compare the predictions for the evolution of the confined scroll ring based on the extended kinematical model to numerical simulations of the Rovinsky equations using the parameter values obtained from the concentrations in the chemical experiment. In the simulation, cylindrical coordinates have been employed assuming that the scroll rings remain axis-symmetric during their evolution. Several full three-dimensional calculations confirm this assumption (see Supplemental Material).
All five regimes for the evolution of the scroll ring observed in the experiments are reproduced in the extended kinematical model, Eqs. (1)-(2), as shown in Fig.  4. To facilitate comparisons between numerical and experimental results, space and time are scaled in units of wavelength λ and rotation period T of an unperturbed two-dimensional spiral wave, respectively.
The often cited life time R 2 0 /(2α) and dynamics of free scroll rings result if the contribution due to filament curvature is dominating over the boundary effects. This is the case when the filament radius is small and/or the distance between filament plane and no-flux boundary leads to comparably small c 2d (z) values, Fig. 1,4(e) and Fig.  2 (filled circles).
The regimes of boundary-affected filament dynamics are located in a relatively narrow range of distances around z rep . Here, collapsing scroll rings evolve with a considerably increased life time, Fig. 1,4(f) and Fig.  2 (filled triangles), including "immortal" ones contracting at an exceptional, nearly vanishingly small rate, Fig.  1,4(g) and Fig. 2 (filled squares). In this case, a large enough scroll ring starts to grow and simultaneously departs from the no-flux boundary. This movement in turn weakens the interaction with the boundary and finally causes the ring to contract.
For sufficiently large filament radii even expanding scroll rings are predicted in agreement with experimental observation, Fig. 1,4(h) and Fig. 2 (open squares).
Below initial distances z 0 ≤ z rep the scroll ring is pushed by the repellor (dashed line in Fig. 3b) into the no-flux boundary where it is annihilated before complete collapse. This is the scenario behind the experimentally observed collision case, Fig. 1,4(d), Fig. 2 (open circles).
For small initial ring radii within R 0 0.6 λ, we found that the interaction between the spiral wave fronts forming the scroll ring has to be taken into account to improve the agreement between kinematical and numerical results. On the phenomenological level, this effect is de- scribed by the terms c 2d (R) and c 2d ⊥ (R) in Eqs. (1)-(2), as for self-interaction inside the ring the filament radius R plays the role of the distance z. These two terms are responsible for the drift of the ring along its symmetry axis; it becomes negligible small for larger core radii.
We emphasize that the transient dynamics in the drift velocity field is crucial to understand the experi- To summarize, we have demonstrated experimentally that scroll rings confined within a layer of excitable medium with positive filament tension can exhibit entirely new phenomena not observed in the unbounded system. In particular, due to the interaction with a noflux boundary we found a substantial increase in the life time of the rings and even an expansion of scroll rings previously thought to be impossible in systems with negative filament tension. Although the observed quasi-stationary rings are a transient phenomenon, the radius changes less than 3.3% during 37 periods of rotation (Fig. 1g), thereby closely resembling an autonomous pacemaker. The scenarios of ring evolution in (Fig. 1d,e,f) confirm previous numerical simulations by Bray et al. [21] for the first time experimentally. The evolution of the confined scroll ring can be understood as the superposition of the well-known intrinsic dynamics of the free scroll ring and boundary effects. The latter are calculated approximately from the interaction of a two-dimensional spiral wave with a no-flux boundary. We confirmed our hypothesis in full three-dimensional numerical simulations of the underlying Rovinsky equations.
Since a general theory for strong perturbations, such as a no-flux boundary, does not exist, our extended kinematical approach is by necessity phenomenological. Only weak perturbations arising from small parametric inhomogeneities, for example, can be treated in the context of the so-called response functions which are the eigenmodes of the adjoint linear stability operator [27,28].
Related results that emphasize the importance of con-finement effects on the behavior of scroll waves have been recently reported by Dierckx et al. [29]. The authors studied the buckling instability of a straight scroll wave confined within a layer of excitable medium with negative line tension in dependence of the layer widths. The expected scroll wave turbulence develops only in sufficiently thick layers. Slightly above a critical width, the scroll wave undergoes a buckling instability where its filament assumes a constant S-like shape precessing at fixed angular velocity. A behavior corresponding to this intermediate regime was observed by Alonso and Panfilov in the Luo-Rudy model of heart tissue [30]. The understanding of possible confinement effects on scroll waves is very important for the correct interpretation of experimental data obtained in layers of the BZ reaction or in cell cultures of cardiac tissue if the layer thickness is comparable to the wavelength of the spiral wave. Regarding cardiac arrhythmias it can be expected that the dynamics of transmural scroll waves in the comparatively thin atrial tissue will be affected by the interaction with the tissue boundaries.
J.F.T. and H.E. thank the German Science Foundation (DFG) for financial support through the Research Training Group 1558 (GRK 1558). O.St. acknowledges support by the National Science Foundation under Grant No. 1213259.
Supplementary information can be found at URL and includes videos from the BZ experiments and details about the chemical preparation and numerical simulations.