Twisted Scroll Waves Research Articles

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.

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.

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.

Structure of wave front and organization center in excitable media

With help of establishing the moving coordinate on the wave front surface and the perturbation analysis in the boundary layer, the structure of wave front and organization center in excitable media were studied. The eikonal equation of wave front surface and general equation of organization center were obtained. These eikonal equations reveal the wave front surfaces have structures of twisted scroll wave and Mobius band, the organization centers have structures of knotted and linked ring. These theoretical results not only explain the wave pattern of BZ (Belousov-Zhabotainskii) chemical reaction but also give several possibility structure of wave front surface and organization center in general excitable media.

A tomographic study of desynchronization and complex dynamics of scroll waves in an excitable chemical reaction with a gradient

The evolution of scroll waves in an excitable chemical medium with a gradient parallel to the scroll filament is studied. Depending on the excitability and the gradient, twisted scroll waves and scroll wave attachment to the boundary of the reaction volume are observed by optical tomography, which allows the full three-dimensional reconstruction of the vortex structure in the Belousov–Zhabotinsky reaction as well as estimates on the shape and dynamics of the organizing center of the scroll (the filament). This behavior is reproduced in numerical simulations with the generic Barkley model for excitable media supplemented by a gradient along the filament. In particular, the study deals with scroll waves in a cylindrical reaction system with a free surface open to the air. Oxygen penetrating through the free surface inhibits the evolution of waves and establishes a twist of the filament. Cooling of the reaction system enhances this twist, eventually leading to a bending of the initially straight filament. Finally, the bending becomes so pronounced that the filament touches the container boundary and breaks into two pieces, each of which has its distinct rotation frequency.

Large-excitability asymptotics for scroll waves in three-dimensional excitable media.

Three-dimensional scroll waves are considered in a reaction-diffusion model of excitable media in the large excitability limit. Coordinates based on the scroll filament are defined and shown to provide a natural extension of the coordinates used for two-dimensional spiral waves. The leading-order free-boundary equations for interface motion in three dimensions are explicitly derived in these coordinates. Three specific examples are considered: straight twisted scroll waves, axisymmetric scroll waves, and helical scroll waves. The equations for the fields at leading and first order in the core region are given.

Cookbook asymptotics for spiral and scroll waves in excitable media.

Algebraic formulas predicting the frequencies and shapes of waves in a reaction-diffusion model of excitable media are presented in the form of four recipes. The formulas themselves are based on a detailed asymptotic analysis (published elsewhere) of the model equations at leading order and first order in the asymptotic parameter. The importance of the first order contribution is stressed throughout, beginning with a discussion of the Fife limit, Fife scaling, and Fife regime. Recipes are given for spiral waves and detailed comparisons are presented between the asymptotic predictions and the solutions of the full reaction-diffusion equations. Recipes for twisted scroll waves with straight filaments are given and again comparisons are shown. The connection between the asymptotic results and filament dynamics is discussed, and one of the previously unknown coefficients in the theory of filament dynamics is evaluated in terms of its asymptotic expansion. (c) 2002 American Institute of Physics.

Scroll waves in isotropic excitable media: linear instabilities, bifurcations, and restabilized states.

Scroll waves are three-dimensional analogs of spiral waves. The linear stability spectrum of untwisted and twisted scroll waves is computed for a two-variable reaction-diffusion model of an excitable medium. Different bands of modes are seen to be unstable in different regions of parameter space. The corresponding bifurcations and bifurcated states are characterized by performing direct numerical simulations. In addition, computations of the adjoint linear stability operator eigenmodes are also performed and serve to obtain a number of matrix elements characterizing the long-wavelength deformations of scroll waves.

Scroll waves in the presence of slowly varying anisotropy with application to the heart.

We consider the dynamics of scroll waves in the presence of rotating anisotropy, a model of the left ventricle of the heart in which the orientation of fibers in successive layers of tissue rotates. By choosing a coordinate system aligned with the fiber rotation and studying the phase dynamics of a straight but twisted scroll wave, we derive a Burgers' equation with forcing associated with the fiber rotation rate. We present asymptotic solutions for scroll twist, verified by numerics, using a realistic fiber distribution profile. We make connection with earlier numerical and analytical work on scroll dynamics.

Selection of twisted scroll waves in three-dimensional excitable media.

The selection of shape and rotation frequency for scroll waves in reaction-diffusion equations modeling excitable media is investigated. For scrolls with uniform twist about straight filaments, asymptotic methods are used to derive free-boundary equations at leading and first order. Both orders are validated against full solutions of the reaction-diffusion equations. Using these two orders and with no adjustable parameters, the shape and frequency of waves are correctly predicted except possibly near the point of propagation failure where the core becomes large.

Propagating waves control Dictyostelium discoideum morphogenesis

The morphogenesis of Dictyostelium results from the coordinated movement of starving cells to form a multicellular aggregate (mound) which transforms into a motile slug and finally a fruiting body. Cells differentiate in the mound and sort out to form an organised pattern in the slug and fruiting body. During aggregation, cell movement is controlled by propagating waves of the chemo-attractant cAMP. We show that mounds are also organised by propagating waves. Their geometry changes from target or single armed spirals during aggregation to multi-armed spiral waves in the mound. Some mounds develop transiently into rings in which multiple propagating wave fronts can still be seen. We model cell sorting in the mound stage assuming cell type specific differences in cell movement speed and excitability. This sorting feeds back on the wave geometry to generate twisted scroll waves in the slug. Slime mould morphogenesis can be understood in terms of wave propagation directing chemotactic cell movement.

Twisted scroll waves organize Dictyostelium mucoroides slugs.

Cellular slime moulds (Dictyosteloids) are characterised by at least two different modes of slug migration. Most species, e.g. Dictyostelium mucoroides, produce a stalk continuously during slug migration, while a few species, e.g. Dictyostelium discoideum are characterised by stalk-less slug migration and only produce a stalk upon culmination. Experiments on D. discoideum and theoretical model calculations have shown that D. discoideum slugs are organized by a cAMP scroll wave in the tip which produces planar waves in the back. These waves guide cell movement in slugs: spiralling in the tip and forward movement parallel to the slug axis in the back. Simple changes in model parameters can lead to the formation of a twisted scroll wave which extends throughout the slug. In order to investigate whether such twisted scroll waves occur naturally we have analysed the movement of fluorescently labelled single cells in migrating D. mucoroides slugs. The results show that cells in the prespore zone of D. mucoroides slugs move in a spiral path. Although the velocity of single cells in D. mucoroides is faster than in D. discoideum, the net forward component of their movement is less due to their spiral trajectories. As a result D. mucoroides slugs move more slowly than D. discoideum slugs. The entire D. mucoroides slug also describes a spiralling path leaving corkscrew shaped stalks behind. Based on these observations we propose that cell movement in D. mucoroides slugs is controlled by a propagating twisted scroll wave of cAMP which extends throughout the length of the slug.

Twisted scroll waves organize Dictyostelium mucoroides slugs.

Cellular slime moulds (Dictyosteloids) are characterised by at least two different modes of slug migration. Most species, e.g. Dictyostelium mucoroides, produce a stalk continuously during slug migration, while a few species, e.g. Dictyostelium discoideum are characterised by stalk-less slug migration and only produce a stalk upon culmination. Experiments on D. discoideum and theoretical model calculations have shown that D. discoideum slugs are organized by a cAMP scroll wave in the tip which produces planar waves in the back. These waves guide cell movement in slugs: spiralling in the tip and forward movement parallel to the slug axis in the back. Simple changes in model parameters can lead to the formation of a twisted scroll wave which extends throughout the slug. In order to investigate whether such twisted scroll waves occur naturally we have analysed the movement of fluorescently labelled single cells in migrating D. mucoroides slugs. The results show that cells in the prespore zone of D. mucoroides slugs move in a spiral path. Although the velocity of single cells in D. mucoroides is faster than in D. discoideum, the net forward component of their movement is less due to their spiral trajectories. As a result D. mucoroides slugs move more slowly than D. discoideum slugs. The entire D. mucoroides slug also describes a spiralling path leaving corkscrew shaped stalks behind. Based on these observations we propose that cell movement in D. mucoroides slugs is controlled by a propagating twisted scroll wave of cAMP which extends throughout the length of the slug.

Analysis of Cell Movement and Signalling during Ring Formation in an Activated Gα1 Mutant ofDictyostelium discoideumThat Is Defective in Prestalk Zone Formation

Mound formation in the cellular slime mouldDictyosteliumresults from the chemotactic aggregation of competent cells. Periodic cAMP signals propagate as multiarmed spiral waves and coordinate the movement of the cells. In the late aggregate stage the cells differentiate into prespore and several prestalk cell types. Prestalk cells sort out chemotactically to form the tip, which then controls all further development. The tip organises cell movement via a scroll wave that converts to planar waves in the prespore zone leading to rotational cell movement in the tip and periodic forward movement in the prespore zone. Expression of an activated Gα1 protein under its own promoter leads to a severely altered morphogenesis from the mound stage onwards. Instead of forming a tipped mound, the cells form a ring-shaped structure without tip. Wave propagation pattern and dynamics during aggregation and mound formation in the mutant are indistinguishable from the parental strain AX3. However, at the time of tip formation the spiral waves that organise the late aggregate do not evolve in a scroll-organising centre in the tip but transform into a circularly closed (twisted) scroll ring wave. This leads to the formation of a doughnut-shaped aggregate. During further development, the doughnut increases in diameter and the twisted scroll wave converts into a train of planar waves, resulting in periodic rotational cell movement. Although biochemical consequences resulting from this mutation are still unclear, it must affect prestalk cell differentiation. The mutant produces the normal proportion of prespore cells but is unable to form functional prestalk cells, i.e., prestalk cells with an ability to sort out from the prespore cells and form a prestalk zone. Failure of sorting leads to an altered signal geometry, ring-shaped scroll waves, that then directs ring formation. This mutant demonstrates the importance of prestalk cell sorting for the stabilisation of the scroll wave that organises the tip.

Three dimensional imaging of the Belousov-Zhabotinsky reaction using magnetic resonance

The combination of 3D magnetic resonance imaging data with polygon based computer graphic display software is ideally suited to the study of the Belousov-Zhabotinsky reaction in extended volumes. In this paper we present the first true three dimensional visualization of experimental data from the Belousov-Zhabotinsky reaction. The time evolution of a twisted scroll wave like isoconcentration surface and its organizing filament are demonstrated for a manganese-catalyzed B-Z mixture. These techniques extend the experimental study of the B-Z reaction as a class of pattern-forming systems to the third dimension. The limitations of the technique are discussed.

ON THE GENERATION OF SCROLL WAVES IN A THREE-DIMENSIONAL DISCRETE ACTIVE MEDIUM

This paper reports on the simulation, in three-dimensional cellular-neural-network (CNN) arrays of Chua’s circuits, of basic three-dimensional scroll wave patterns observed previously from other media. Among the simulated patterns are the straight scroll wave, twisted scroll wave in both homogeneous and inhomogeneous media, as well as the scroll ring. These types of waves have been obtained for only one set of circuit parameters by varying the initial conditions.

SPATIOTEMPORAL STRUCTURES IN DISCRETELY-COUPLED ARRAYS OF NONLINEAR CIRCUITS: A REVIEW

Spatiotemporal pattern formation occurring in discretely-coupled nonlinear dynamical systems has been studied numerically. Reaction-diffusion systems can be viewed as an assembly of a large number of identical local subsystems which are coupled to each other by diffusion. Here, the local subsystems are defined by a system of nonlinear ordinary differential equations. While for continuous systems, the characteristic time scale corresponding to the diffusion is slower than that corresponding to the local subsystems, in discretely-coupled systems, both time scales can be of the same order of magnitude. Discrete systems can exhibit behaviors different from those exhibited by their equivalent continuous model: the wave propagation failure phenomenon occurring in nerve-pulse propagation due to transmission blockage is a case in point. In this case, it is found that the wave fails to propagate at or below some critical value of the coupling coefficient. Systems of coupled cells can be found to occur in the transformation and transport processes in living cells, tissues, neuron networks, physiological systems and ecosystems, as well as in all forms of chemical, biochemical reactors and combustion systems. In this paper, we review the possibilities of using arrays of discretely-coupled nonlinear electronic circuits to study these systems. The possibility of building large arrays of these circuits via VLSI technology makes this approach a unique tool for real time applications. Classical examples occurring in other continuous media, such as spiral wave initiation and propagation, and Turing pattern formation, are depicted here. Because of the discrete nature of our system, the influence of inhomogeneities arising from damaged cells, or from an anisotropic media, is analyzed for spiral wave propagation. Spiral wave initiation and vulnerability effects are considered and compared with their corresponding effects in continuous media. More complex spatiotemporal structures are also studied via three-dimensional arrays of discretely-coupled circuits. Straight and twisted scroll waves, as well as scroll ring waves, are shown to exist in these arrays, where their properties can be easily measured. Sidewall forcing of Turing patterns is shown to be capable of driving the system into a perfect spatial organization, namely, a rhombic pattern, where no defects occur. The dynamics of the two layers supporting Turing and Hopf modes, respectively, is analyzed as a function of the coupling strength between them. The competition between these two modes is shown to increase with the diffusion between layers.

Three-dimensional waves of excitation during Dictyostelium morphogenesis.

Cells in Dictyostelium slugs follow well-defined patterns of motion. We found that the chemotactic cell response is controlled by a scroll wave of messenger concentration in the highly excitable prestalk zone of the slug that decays in the less-excitable prespore region into planar wave fronts. This phenomenon is investigated by numerical solutions of partial differential equations that couple local nonlinear kinetics and diffusive transport of the chemotactic signal. In the interface of both regions a complex twisted scroll wave is formed that reduces the wave frequency in the prespore zone. The spatio-temporal dynamics of waves and filaments are followed over 33 periods of rotation. These results yield an explanation of collective self-organized cell motion in a multicellular organism.

Front geometries of chemical waves under anisotropic conditions

An experimental study of chemical waves in the Belousov-Zhabotinsky reaction under anisotropic conditions of substrate concentrations extending over the system was carried out by use of ferroin-immobilized silica gels as reacting matrices. Unidirectionally moving wave trains (similar to the so-called “chemical pinwheels”) followed by the spontaneous fragmentation of wave fronts were observed in rectangular two-dimensional gels. A helicoidal chemical wave (or twisted scroll wave), which was predicted by numerical calculation, was detected in a micro-cylindrical gel system. A space-time representation of the chemical helix was constructed and its structural features are discussed qualitatively.

Evolution of twist of an autowave vortex

An equation is derived for the evolution of an inhomogeneously twisted scroll wave in a general reaction-diffusion system, with twist regarded as an order parameter and not assumed to be small. The equation describes the interaction of dispersion and diffusion of the twist. A criterion is given of twist alignment in the absence of twisting forces.