Effect of excitability on partially pinned scroll waves in excitable chemical media

We present an investigation of excitability effects on the dynamics of scroll waves partially pinned to inert cylindrical obstacles in three-dimensional Belousov-Zhabotinsky excitable media. We also report on corresponding numerical simulations with the Oregonator model. The excitability varies according to the concentration of sulfuric acid [H_{2}SO_{4}] in the Belousov-Zhabotinsky (BZ) reaction and the parameter ɛ^{-1} in the Oregonator model. Initially, the freely rotating scroll segment rotates faster than the pinned one. The difference in the frequency of the two parts results in a transition from a straight pinned scroll wave to a twisted one, which helically wraps around the entire obstacle. The wave frequency in the whole volume is equal to that of the freely rotating scroll wave. When the excitability is increased, the time for the transition to the twisted wave structure decreases while the average speed s of the development increases. After the transition, the twisted wave remains stable. In media with higher excitability, the helical pitch is shorter but the twist rate ω is higher. Analysis presented in this study together with our previous findings of the effect of the cylindrical obstacle diameter on the wave dynamics results in common features: The average speed s and the twist rate ω of both studies fit well to functions of the difference in the initial frequency Δf of the freely rotating and untwisted pinned waves. We also demonstrate the robustness of the partially pinned scroll waves against perturbations from spontaneous waves emerging during the wave generation in the BZ medium with high [H_{2}SO_{4}]. Even though the scroll wave is partly disturbed at the beginning of the experiment, the spontaneous waves are gradually suppressed and the typical wave structure is finally developed.

Twisted scroll wave dynamics: partially pinned waves in excitable chemical media

We present an investigation of the dynamics of scroll waves that are partially pinned to inert cylindrical obstacles of varying lengths and diameters in three-dimensional Belousov-Zhabotinsky excitable media. Experiments are carried out in which a scroll wave is initiated with a special orientation to be partially pinned to the obstacle. Numerical simulations with the Oregonator model are also carried out, where the obstacle is placed in the region of the core of a preexisting freely rotating scroll wave. In both cases, the effect of the obstacle on the wave dynamics is almost immediately observable, such that after the first revolution of the wave, the pinned region of the scroll wave has a longer period than that of the freely rotating scroll wave. The dependence of the scroll wave period on the obstacle position gives rise to a transition from a straight scroll wave to a twisted scroll wave in the pinned region, while the form of the freely rotating wave remains unchanged. The twisted scroll wave arises from the interaction of the freely rotating scroll wave with the obstacle, giving rise to a pinned twisted wave with the same period. The twisted scroll wave gradually advances, displacing the slower untwisted scroll wave until the scroll wave helically wraps around the entire obstacle. At this point, the period of the entire wave has a single value equal to that of the freely rotating scroll wave. The time for the transition to the twisted wave structure increases when either the obstacle length is increased or the obstacle diameter is decreased, while the average speed of the development increases with both the obstacle length and diameter. After the transition, the twisted wave remains stable, with its structure depending on the obstacle diameter - the larger the diameter, the shorter the helical pitch but the higher the twist rate.

Dynamics of scroll waves in a cylinder jacket geometry

The dynamics of scroll waves in a narrow cylinder jacket-shaped reactor is investigated experimentally by optical tomography. The fate of the scroll waves of excitation in the Belousov-Zhabotinsky reaction depends on the thickness of the cylinder jacket. While at sufficiently wide cylinder jackets vertically oriented scroll waves remain stable, the probability that the filament of a scroll hits a lateral wall increases as the cylinder jacket narrows. This may lead to the rupture of the initial filament and pinning of the filament ends at the lateral walls. Filaments that pin to opposite lateral walls shrink and reorient to a horizontal orientation; such a reorientation corresponds to a transition from an intramural to a transmural scroll wave. The kinetics of the reorientation and shrinkage of the scrolls were studied. Furthermore, we find that no new filaments were generated upon collision of excitation waves at the side of the cylinder jacket opposite to the scroll wave. Thus, under the studied conditions, we do not observe any new generation of filaments due to a phenomenon like reentry.

Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that is triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media with applications to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact for cardiac arrhythmias.

From chemical systems to systems chemistry: Patterns in space and time

We present a brief, idiosyncratic overview of the past quarter century of progress in nonlinear chemical dynamics and discuss what we view as the most exciting recent developments and some challenges and likely areas of progress in the next 25 years.

Three-dimensional convection-driven fronts of the exothermic chlorite-tetrathionate reaction

Horizontally propagating autocatalytic reaction fronts in fluids are often accompanied by convective motion in the presence of gravity. We experimentally and numerically investigate the stable complex three-dimensional pattern arising in the exothermic chlorite-tetrathionate reaction as a result of the antagonistic thermal and solutal contribution to the density change. By particle image velocimetry measurements, we construct the flow field that stabilizes the front structure. The calculations applied for incompressible fluids using the empirical rate-law model reproduce the experimental observations with good agreement.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Dependence of scroll-wave dynamics on the orientation of a gradient of excitability

The dynamics of scroll waves with a variable orientation to a vertically oriented gradient of excitability is studied by optical tomography in the ferroin-catalyzed Belousov-Zhabotinsky reaction. An almost perpendicular orientation between the scroll wave and gradient induces a pair of twists of opposite handedness on the scroll wave. The position of the nodal plane formed between the twists is governed by the time delay of the twist formation and therefore leads to a symmetric or asymmetric twisted scroll wave. Larger inclinations between scroll wave and gradient cause a drift of the filament along the reactor wall until it reaches the bottom of the reaction container. In this case, the scroll wave does not twist, suggesting that a drift acts as an alternative mechanism of responding to the gradient.

Interaction of a pair of parallel scroll waves

Interactions of pairs of scroll waves in three-dimensional 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 counter-rotating), 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.

Twists of opposite handedness on a scroll wave

The dynamic interaction of scroll waves in the Belousov-Zhabotinsky reaction with a vertically orientated gradient of excitability is studied by optical tomography. This study focuses on scroll waves, whose filaments were oriented almost perpendicular to the gradient. Whereas scroll waves with filaments exactly perpendicular to the gradient remain unaffected, filaments with a component parallel to the gradient develop a twist. Scroll waves with U-shaped filaments exhibit twists starting from both of its ends, resulting in scroll waves whose filaments display a pair of twists of opposite handedness. These twists are separated by a nodal plane where the filament remains straight and untwisted. The experimental findings were reproduced by numerical simulations using the Oregonator model and a linear gradient of excitability almost perpendicular to the orientation of the filament.

Helical deformation of the filament of a scroll wave

In three-dimensional excitable systems scroll waves may lose their originally straight shape through different instabilities. In experiments, the formation of zigzag-shaped or helical filaments is often observed. Such a deformation may be due to either a twist-induced instability or a 3D meandering instability. We performed a systematic study using a Belousov-Zhabotinsky (BZ) system with a vertical gradient of excitability and determined the necessary twist for the onset of undulation in the filament. Thus, we demonstrate that in the case of a BZ system with a gradient parallel to the filament, the deformation of the filament is induced by a twist-induced instability.

Scroll wave meandering induced by phase difference in a three-dimensional excitable medium

We investigated scroll waves in an inhomogeneous excitable 3D system with gradient of excitability. The gradient promotes twisting of the scroll waves. Sufficiently large excitability gradient enhances the twisting and causes simple scroll waves to transition to meandering scroll waves. For the twist-induced instability of scroll waves, we analyzed the stability of 2D spiral waves sliced from the twisted scroll in the vertical direction. The 3D problem is simplified by taking into account the diffusive coupling in the third direction as a time-delayed perturbation to the 2D spiral wave. An additional "negative mass" term measuring the twist thus arises in the 2D system and induces the transition from simple rotation to meandering. A further increase in the gradient ruins partially the unity of the meandering scrolls and generates semiturbulence, the analogs of which were observed in the Belousov-Zhabotinski reaction. We also generated the phase diagram in the parameter space by adjusting the threshold for excitation of the media.

Surfactant-induced gradients in the three-dimensional Belousov-Zhabotinsky reaction

Scroll waves are prominent patterns formed in three-dimensional excitable media, and they are frequently considered highly relevant for some types of cardiac arrhythmias. Experimentally, scroll wave dynamics is often studied by optical tomography in the Belousov-Zhabotinsky reaction, which produces CO(2) as an undesired product. Addition of small concentrations of a surfactant to the reaction medium is a popular method to suppress or retard CO(2) bubble formation. We show that in closed reactors even these low concentrations of surfactants are sufficient to generate vertical gradients of excitability which are due to gradients in CO(2) concentration. In reactors open to the atmosphere such gradients can be avoided. The gradients induce a twist on vertically oriented scroll waves, while a twist is absent in scroll waves in a gradient-free medium. The effects of the CO(2) gradients are reproduced by a numerical study, where we extend the Oregonator model to account for the production of CO(2) and for its advection against the direction of gravity. The numerical simulations confirm the role of solubilized CO(2) as the source of the vertical gradient of excitability in reactors closed to the atmosphere.

Tomography of reaction-diffusion microemulsions reveals three-dimensional Turing patterns

Spatially periodic, temporally stationary patterns that emerge from instability of a homogeneous steady state were proposed by Alan Turing in 1952 as a mechanism for morphogenesis in living systems and have attracted increasing attention in biology, chemistry, and physics. Patterns found to date have been confined to one or two spatial dimensions. We used tomography to study the Belousov-Zhabotinsky reaction in a microemulsion in which the polar reactants are confined to aqueous nanodroplets much smaller than the scale of the stationary patterns. We demonstrate the existence of Turing patterns that can exist only in three dimensions, including curved surfaces, hexagonally packed cylinders, spots, and labyrinthine and lamellar patterns.

Spatial desynchronization of glycolytic waves as revealed by Karhunen-Loeve analysis

The dynamics of glycolytic waves in a yeast extract have been investigated in an open spatial reactor. At low protein contents in the extract, we find a transition from inwardly moving target patterns at the beginning of the experiment to outwardly moving spiral- or circular-shaped waves at later stages. These two phases are separated by a transition phase of more complex spatiotemporal dynamics. We have analyzed the pattern dynamics in these three intervals at different spatial scales by means of a Karhunen-Loeve (KL) decomposition. During the initial phase of the experiment, the observed patterns are sufficiently described by the two dominant KL modes independently of the spatial scale. However, during the last stage of the experiment, at least 6 KL modes are needed to account for the observed patterns at spatial scales larger than 3 mm, while for smaller scales, 2 KL modes are still sufficient. This indicates that in the course of the experiment, the local glycolytic oscillators become desynchronized at spatial scales larger than 3 mm. Possible reasons for the desynchronization of the glycolytic waves are discussed.

Three-dimensional spiral waves in an excitable reaction system: initiation and dynamics of scroll rings and scroll ring pairs

We report experimental results on spiral and scroll waves in the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction. The propagating concentration waves are detected by two-dimensional photometry and optical tomography. Wave pulses can disappear in front-to-front and front-to-back collisions. This anomaly causes the nucleation of vortices from collisions of three nonrotating waves. In three-dimensional systems, these vortices are scroll rings that rotate around initially circular filaments. Depending on reactant concentrations, the filaments shrink or expand indicating positive and negative filament tensions, respectively. Shrinkage results in vortex annihilation. Expansion is accompanied by filament buckling and bending, which is interpreted as developing Winfree turbulence. We also describe the initiation of scroll ring pairs in four-wave collisions. The two filaments are stacked on top of each other and their motion suggests filament repulsion.

Control of scroll wave turbulence in a three-dimensional reaction-diffusion system with gradient

In this paper, we summarize our recent experimental and theoretical works on observation and control of scroll wave (SW) turbulence. The experiments were conducted in a three-dimensional Belousov-Zhabotinsky reaction-diffusion system with chemical concentration gradients in one dimension. A spatially homogeneous external forcing was used in the experiments as a control; it was realized by illuminating white light on the light sensitive reaction medium. We observed that, in the oscillatory regime of the system, SW can appear automatically in the gradient system, which will be led to spatiotemporal chaos under certain conditions. A suitable periodic forcing may stabilize inherent turbulence of SW. The mechanism of the transition to SW turbulence is due to the phase twist of SW in the presence of chemical gradients, while modulating the phase twist with a proper periodic forcing can delay this transition. Using the FitzHugh-Nagumo model with an external periodic forcing, we confirmed the control mechanism with numerical simulation. Moreover, we also show in the simulation that adding temporal external noise to the system may have the same control effect. During this process, we observed a new state called "intermittent turbulence," which may undergo a transition into a new type of SW collapse when the noise intensity is further increased. The intermittent state and the collapse could be explained by a random process.

Twisted vortex filaments in the three-dimensional complex Ginzburg-Landau equation

The structure and dynamics of vortex filaments that form the cores of scroll waves in three-dimensional oscillatory media described by the complex Ginzburg-Landau equation are investigated. The study focuses on the role that twist plays in determining the bifurcation structure in various regions of the (alpha,beta) parameter space of this equation. As the degree of twist increases, initially straight filaments first undergo a Hopf bifurcation to helical filaments; further increase in the twist leads to a secondary Hopf bifurcation that results in supercoiled helices. In addition, localized states composed of superhelical segments interspersed with helical segments are found. If the twist is zero, zigzag filaments are found in certain regions of the parameter space. In very large systems disordered states comprising zigzag and helical segments with positive and negative senses exist. The behavior of vortex filaments in different regions of the parameter space is explored in some detail. In particular, an instability for nonzero twist near the alpha=beta line suggests the existence of a nonsaturating state that reduces the stability domain of straight filaments. The results are obtained through extensive simulations of the complex Ginzburg-Landau equation on large domains for long times, in conjunction with simulations on equivalent two-dimensional reductions of this equation and analytical considerations based on topological concepts.

Stability of scroll ring orientation in an advective field

The stability of the orientation of scroll rings in the excitable Belousov-Zhabotinsky reaction under an applied electrical current was investigated in experiments and simulations. The parallel and antiparallel orientations of the scroll ring unit vector with respect to the current are two stationary states, the first one unstable, the latter stable. For any other orientation, the scroll rings were forced to rotate by the current. At the stable stationary orientation, the scroll rings may contract or expand under the same applied current depending on the radius of the scroll rings. In simulations, delicate adjustments caused a scroll ring to propagate with a constant radius in an advective field.

Scroll wave instabilities in an excitable chemical medium

Two kinds of scroll wave instabilities were studied experimentally in the excitable Belousov-Zhabotinsky reaction: three-dimensional meandering and negative line tension of the scroll wave filament. The filament displays a flat zigzag shape in the initial stages of the experiment. As the chemical medium ages, the filament assumes a wiggly shape while its length increases substantially. Numerical simulations underpin the experimental findings and their interpretation.

Wave-pinned filaments of scroll waves

Scroll waves are three-dimensional excitation patterns that rotate around one-dimensional space curves. Typically these filaments are closed loops or end at the system boundary. However, in excitable media with anomalous dispersion, filaments can be pinned to the wake of traveling wave pulses. This pinning is studied in experiments with the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction and a three-variable reaction-diffusion model. We show that wave-pinned filaments are related to the coexistence of rotating and translating wave defects in two dimensions. Filament pinning causes a continuous expansion of the total filament length. It can be ended by annihilating the pinning pulse in a frontal wave collision. Following such an annihilation, the filament connects itself to the system boundary. Its postannihilation shape that is initially the exposed rim of the scroll wave unwinds continuously over numerous rotation periods.

Control of spiral turbulence by periodic forcing in a reaction-diffusion system with gradients

We report an experimental result on successfully controlling spiral turbulence in a reaction-diffusion system. The control is realized by periodic forcing in a three-dimensional Belousov-Zhabotinsky reaction-diffusion system, which has chemical concentration gradients in the third dimension. We observe that, in the oscillatory regime of the system, a suitable periodic forcing may stabilize scroll waves (SWs), which otherwise undergo a transition to spiral turbulence. Relating the spiral phase shift due to gradients and the forcing frequency, the mechanism of the control can be well understood by modulating the phase twist of SWs. We use the FitzHugh-Nagumo model to demonstrate this mechanism.

Negative filament tension of scroll rings in an excitable system

Scroll rings are three-dimensional spiral waves of excitation that rotate around circular filaments. In a modified Belousov-Zhabotinsky reaction, these filaments expand, buckle, and build up gradients in rotation phase. The instability is caused by negative filament tension (-4.3x10;{-4}cm;{2}s) . Initial deformations are strongest in the direction normal to the filament's osculating plane, and their growth rates decrease rapidly with increasing wave number.

Scroll Wave Filaments Terminate in the Back of Traveling Fronts

Experiments with the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction demonstrate that three-dimensional scroll waves can rotate around filaments that end in the wake of a traveling excitation pulse. The vortex structures nucleate during the collision of three nonrotating excitation pulses. The nucleation process and the wave-termination of filaments are direct consequences of the system's anomalous dispersion relation. Vortex filaments are found to expand with about twice the speed of their anchoring wave fronts. Filament expansion is accompanied by the build-up of phase differences in spiral rotation creating strongly twisted wave structures. Experiments employ optical tomography for the reconstruction of the three-dimensional wave patterns.

Nucleation and collapse of scroll rings in excitable media

We describe a novel nucleation mechanism of scroll rings in three-dimensional reaction-diffusion systems with anomalous dispersion. The vortices form after the collision of two spherical wave fronts from a third, trailing wave that only partially annihilates in the wake of its predecessor. Depending on the relative positions of the three relevant wave sources, one obtains untwisted or twisted scroll rings. The formation of both vortex structures is demonstrated for a modified Belousov-Zhabotinsky reaction.

Negative-Tension Instability of Scroll Waves and Winfree Turbulence in the Oregonator Model

Excitable media support self-organized scroll waves which can be unstable and give rise to three-dimensional wave chaos. Winfree turbulence of scroll waves results from the negative-tension instability of scroll waves; it plays an important role in the cardiac tissue where it may lead to ventricular fibrillation. By numerical simulations of the Oregonator model, we show that this instability and, thus, the Winfree turbulence may also be observed in the Belousov-Zhabotinsky reaction. The region in the parameter space, where the instability takes place, is determined, and a relationship between the negative-tension instability and the meandering behavior of spiral waves is found. The application of global periodic forcing to control such turbulence in the Oregonator model is discussed.

Spiral wave dynamics in excitable media with spherical geometries

We describe the spatial and temporal organization of spiral and scroll waves in spherical shells of different sizes and solid spheres. We present simulation results for the evolution of the dynamics and clustering of spiral waves as a function of the excitability of the medium. The excitability, topology, and size of the domain places restrictions on how single and multiarmed spiral waves are organized in space. The results in spherical geometries are compared with those in planar two-dimensional media. These studies are relevant to the dynamics of spiral waves in a variety of media including the heart, and chemical reactions on spherical surfaces.

Emergent reaction-diffusion phenomena in capillary tubes

Pattern formation in the Belousov-Zhabotinsky reaction experiments carried out by filling capillary glass tubes with catalyst-immobilized gel for the reaction is reported. Under unperturbed and oscillatory conditions, helicoidal waves appear spontaneously. Quantitative structural data of those helices are obtained by devising an optical tomography technique for extracting rotationally symmetric structures from time-lapse data. Space-time representation of the catalyst oxidation reveals wave transmission phenomenon that is studied further by numerical simulations of a reduced spatial model.

Nonequilibrium ribbon model of twisted scroll waves

We formulate a reduced model to analyze the motion of the core of a twisted scroll wave. The model is first shown to provide a simple description of the onset and nonlinear evolution of the helical state appearing in the sproing bifurcation of scroll waves. It then serves to examine the experimentally studied case of a medium with spatially varying excitability. The model shows the role of sproing in this more complex setting and highlights the differences between the convective and absolute sproing instabilities.

An elegant method to study an isolated spiral wave in a thin layer of a batch Belousov–Zhabotinsky reaction under oxygen-free conditions

A method to prepare a uniform thin layer of a batch Belousov-Zhabotinsky (BZ) reaction under oxygen-free conditions for the study of an isolated spiral wave is presented. After a first layer of gel soaked with the BZ solution has been delivered into the reactor, a single spiral wave was initiated, and finally the remaining reactor volume was filled with gel and BZ medium. The completely filled reactor is sealed gas-tightly, yielding oxygen-free, and thus more controlled, reaction conditions. A systematic study of the behaviour of an isolated spiral wave in a ferroin-catalyzed BZ reaction under batch conditions has been performed. Recipes for BZ media that support a slowly rotating meandering spiral were developed. In cases of extremely low excitability (i.e., relative large stimuli are required to induce a propagating wave), the number of petals in the trajectory of a spiral tip decreased due to aging of the reaction system. Since oxygen-free conditions are necessary for the study of the dynamics in three-dimensional excitable media, and the wave velocities of a spiral are sufficiently low, the developed chemical recipes are suitable for studies of the behaviour of scroll waves in three-dimensional systems by optical tomography.

Spontaneous scroll ring creation and scroll instability in oscillatory medium with gradients

Scroll waves in a quasi-three-dimensional reaction-diffusion medium with a gradient in the third dimension are studied by numerical simulations using the Fitzhugh-Nagumo model. Under a simple initial condition with only one straight filament, we show a spontaneous creation of twisted scroll rings surrounding the initial filament when the control parameter is increased across a threshold. We find that due to the presence of the gradient, the difference of oscillation frequencies in the third dimension is the underlying cause of the phenomenon. Further increase of the control parameter will lead to the spiral breakups, as a result of the interaction of filaments. Observations in the experiments conducted in the same type of medium with the Belousov-Zhabotinsiky reaction qualitatively agree with our simulation results.

Magnetic Resonance Imaging of Flow-Distributed Oscillations

The formation of stationary concentration patterns in a packed-bed reactor (PBR), using a manganese-catalyzed Belousov-Zhabotinsky (BZ) reaction in a mixed sulfuric-phosphoric acid medium, was studied using magnetic resonance imaging (MRI). The PBR was composed of a column filled with glass beads, which was fed by a continuous stirred tank reactor (CSTR). As the reactor is optically opaque, investigation of the three-dimensional (3D) structure of these reaction-diffusion-advection waves is not possible using conventional image capture techniques. MRI has been used to probe this system and the formation, 3D structure, and development of these waves has been studied. At reactor startup, traveling waves were observed. After this initial period the waves stabilized and became stationary. Once fixed, they were found to be remarkably stable. There was significant heterogeneity of the reaction fronts, which were not flat, as would be expected from a plug-flow reactor. Instead, the reaction wave fronts were observed to be conical in shape due to the local hydrodynamics of the bed and specifically the higher velocities and therefore lower residence times close to the wall of the reactor.

Expanding scroll rings and negative tension turbulence in a model of excitable media

Scroll waves in excitable media, described by the Barkley model, are studied. In the parameter region of weak excitability, negative tension of wave filaments is found. It leads to expansion of scroll rings and instability of wave filaments. A circular filament tends to stretch, bend, loop, and produce an expanding tangle that fills up the volume. The filament does not undergo fragmentation before it touches the boundaries. Statistical properties of such Winfree turbulence of scroll waves are numerically investigated.

Intermittent self-organization of scroll wave turbulence in three-dimensional excitable media

We study the asymptotic behavior of scroll wave turbulence in large three-dimensional excitable media modeled by FitzHugh-Nagumo equations. The focus is on the type of turbulence caused by negative tension of scroll wave filaments, which is considered to be one of the mechanisms of cardiac fibrillation. We discovered that the initial increase in turbulence complexity can be followed by intermittent self-organization, when complex filament tangles are replaced by a small number of relatively stable triple filament strands. The intermittency is the result of a competition between the destabilizing effect of negative tension and mutual attraction of filaments with similar orientation.