Multiarmed Spiral Waves Research Articles

Pattern-Selective Feedback Stabilization of Ginzburg–Landau Spiral Waves

The complex Ginzburg–Landau equation serves as a paradigm of pattern formation and the existence and stability properties of Ginzburg–Landau m-armed spiral waves have been investigated extensively. However, many multi-armed spiral waves are unstable and thereby rarely visible in experiments and numerical simulations. In this article we selectively stabilize certain significant classes of unstable spiral waves within circular and spherical geometries. As a result, stable spiral waves with an arbitrary number of arms are obtained for the first time. Our tool for stabilization is the symmetry-breaking control triple method, which is an equivariant generalization of the widely applied Pyragas control to the setting of PDEs.

Multi-armed spiral waves under parametric excitation in a cell with an inhomogeneous boundary

Multi-armed spiral waves under parametric excitation in a cell with an inhomogeneous boundary

Multiarmed spiral waves generated by periodic stimuli in excitable systems

Rotating spiral waves occur in different excitable systems. They also play an important role in the human health since the spiral waves of electrical signals correspond to some cardiac arrhythmia like tachycardia and fibrillations. Pinning of a spiral wave is found to stabilize extreme conditions such as the superconductivity and the superfluidity and it also causes longer cardiac arrhythmia. Multiarmed spiral waves are multiple spiral arms rotating in the same direction around a common center. In heart, they are expected to be seriously dangerous since the wave frequency increases with the number of spiral arms. In this article, we present a partition method with periodic stimuli to generate multiarmed spiral waves pinned to a circular obstacle. Periodic stimuli applied on one side of a thin line partition ignite multiple waves with large distances between them so only a few wave fronts can be distributed around the obstacle and subsequently result in multiarmed spiral waves with a few arms after the removal of the partition. In contrast, Periodic stimuli on both two sides of the partition create waves with smaller distances between them and more wave fronts can fit around the obstacle boundary. Therefore, multiarmed spiral waves with more arms can be generated using the latter method.

Control of spiral waves in excitable media under polarized electric fields

Spiral waves are ubiquitous in diverse physical, chemical, and biological systems. Periodic external fields, such as polarized electric fields, especially circularly polarized electric fields which possess rotation symmetry may have significant effects on spiral wave dynamics. In this paper, control of spiral waves in excitable media under polarized electric fields is reviewed, including resonant drift, synchronization, chiral symmetry breaking, stabilization of multiarmed spiral waves, spiral waves in subexcitable media, control of scroll wave turbulence, unpinning of spiral waves in cardiac tissues, control of spiral wave turbulence in cardiac tissues, etc.

Self-organization of multiarmed spiral waves in excitable media.

We present an investigation of self-organized multiarmed spiral waves pinned to unexcitable circular obstacles in a thin layer of the excitable Belousov-Zhabotinsky reaction and in simulations using the Oregonator model. The multiarmed waves are initiated by a series of wave stimuli. In the proximity of the obstacle boundary, the wave rotation around the obstacle causes damped oscillations of the wave periods of all spiral arms. The dynamics of wave periods cause the wave velocities as well as the angular displacements between the adjacent arms to oscillate with decaying amplitudes. Eventually, all displacements approach approximately the same stable value so that all arms are distributed evenly around the obstacle. A further theoretical analysis reveals that the temporal dynamics of the angular displacements can be interpreted as underdamped harmonic oscillations. Far from the obstacles, the wave dynamics are less pronounced. The wave period becomes stable very soon after the initiation. When the number of spiral arms increases, the rotation of individual arms slows down but the wave period of the multiarmed spiral waves decreases. Due to their short period, multiarmed spiral waves emerging in the heart potentially result in severe pathological conditions.

Formation of multi-armed spiral waves in neuronal network induced by adjusting ion channel conductance

The Hodgkin–Huxley neuron model is used to describe the local dynamics of nodes in a two-dimensional regular network with nearest-neighbor connections. Multi-armed spiral waves emerge when a group of spiral waves rotate the same core synchronously. Here we have numerically investigated how multi-armed spiral waves are formed in such a system. Under the appropriate conditions, multi-armed spiral waves were able to develop as a result of adjusting the conductance of ion channels of particular neurons in the network. In a realistic neuron model, it can be practiced by blocking potassium of ion channels embedded in the membrane of neurons. For example, decreasing the potassium channel conductance in some neurons with a certain transient period can lead to the development of a group of double spirals in a localized area of the network. Furthermore, decreasing the excitability and the external forcing current to zero led to the growth of these double spirals and the formation of a stable multi-armed spiral wave that occupied the network under inhomogeneity.

Selection of Multiarmed Spiral Waves in a Regular Network of Neurons

Formation and selection of multiarmed spiral wave due to spontaneous symmetry breaking are investigated in a regular network of Hodgkin-Huxley neuron by changing the excitability and imposing spatial forcing currents on the neurons in the network. The arm number of the multiarmed spiral wave is dependent on the distribution of spatial forcing currents and excitability diversity in the network, and the selection criterion for supporting multiarmed spiral waves is discussed. A broken spiral segment is measured by a short polygonal line connected by three adjacent points (controlled nodes), and a double-spiral wave can be developed from the spiral segment. Multiarmed spiral wave is formed when a group of double-spiral waves rotate in the same direction in the network. In the numerical studies, a group of controlled nodes are selected and spatial forcing currents are imposed on these nodes, and our results show that l-arm stable spiral wave (l = 2, 3, 4,...8) can be induced to occupy the network completely. It is also confirmed that low excitability is critical to induce multiarmed spiral waves while high excitability is important to propagate the multiarmed spiral wave outside so that distinct multiarmed spiral wave can occupy the network completely. Our results confirm that symmetry breaking of target wave in the media accounts for emergence of multiarmed spiral wave, which can be developed from a group of spiral waves with single arm under appropriate condition, thus the potential formation mechanism of multiarmed spiral wave in the media is explained.

On nonlinear multiarmed spiral waves in slowly rotating fluid systems

Stable nonlinear equilibria of convection in the form of quasistationary, multiarmed spiral waves, up to a maximum of six spiral arms, are found in a slowly rotating fluid confined within a thin spherical shell governed by the three-dimensional Navier–Stokes equation, driven by a radial unstable temperature gradient and affected by a weak Coriolis force. It is shown that three essential ingredients are generally required for the formation of the multiarmed spirals: the influence of slow rotation, large-aspect-ratio geometry and the effect of weak nonlinearity.

Crossflow instability of finite Bödewadt flows: Transients and spiral waves

The flow in an enclosed rotating cylinder with a stationary lower end wall is investigated numerically. For fast rotation rates, the flow in the interior is primarily in the azimuthal direction, with an angular momentum distribution very close to that corresponding to solid-body rotation for about the inner-half radius. The differential rotation sets up a large-scale circulation that is primarily present in the boundary layers on the rotating top and sidewalls and the stationary bottom wall, with a very weak effusive component throughout the bulk interior providing a matching between the boundary layer flows on the top and bottom. The top end wall boundary layer has a profile that very closely matches the von Kármán solution for a rotating disk boundary layer; it is stable and very robust to finite disturbances for all rotation rates considered. The boundary layer on the stationary bottom end wall has a profile that agrees with the Bödewadt solution for a stationary disk with an ambient flow in solid-body rotation. This boundary layer is not robust, suffering crossflow instability to multiarmed spiral waves via a supercritical Hopf bifurcation, as well as being susceptible to axisymmetric circular waves that travel radially inward where the boundary layer profile is most inflectional. In the absence of any external forcing, the circular waves are transitory, but low amplitude forcing can sustain them indefinitely, whereas the spiral waves are essentially unaffected by the external forcing.

Stabilization of multiarmed spiral waves by circularly polarized electric fields

The influence of circularly polarized electric fields (CPEFs) on the stability of multiarmed spiral waves is investigated. It is shown that CPEFs can change the period of the multiarmed spirals. The average period is an important quantity of multiarmed spiral and it must be larger than a threshold for stable multiarmed spiral. After a counter-rotating CPEF with suitable amplitude and period is applied, the average period of the multiarmed spiral may increase, which stabilizes the multiarmed spiral.

Abstract 1349: Overexpression And Dominant-negative Suppression Of Inwardly Rectifying Potassium Channel Protein (Kir2.1) Alters Electrophysiology And Spiral Wave Dynamics Of An In Vitro Model Of Cardiac Myocytes

Introduction : An important role for the inwardly rectifying potassium current (I K1 ) has been postulated in controlling the stability and frequency of rotors responsible for ventricular tachycardia and fibrillation. We investigated the effects of Kir2.1 overexpression and Kir2.1AAA dominant-negative mutant suppression on the electrophysiology and inducibility, stability and frequency of spiral waves in an in vitro cardiac tissue model. Methods/Results : Neonatal rat ventricular myocytes (NRVMs) were transduced by lentiviral vectors encoding Kir2.1 or Kir2.1AAA. Immunostaining revealed Kir2.1 or mutant Kir2.1 protein overexpression and whole cell-clamp confirmed the predicted effects on I K1 , resting potential, and action potential duration (APD 80 ). Optical mapping was performed on confluent NRVM monolayers containing a 5 mm diameter central island of gene-modified NRVMs created by a stenciling technique. APs propagated with increased CV (25.1±2.7 cm/sec, n=7) and shortened APD 80 (73±11 msec, n=7) in islands of Kir2.1 overexpression, or decreased CV (13.1±1.1 cm/sec, n=7) and prolonged APD 80 (263±14 msec, n=7) in islands of Kir2.1AAA suppression, compared with normal CV and APD 80 of 19.2±0.4 cm/sec and 169±14 msec (n=7) in non-transduced islands. Reentry was initiated by rapid pacing. With Kir2.1 overexpression, reentrant waves anchored to the island and remained stable (89±15 minutes, n=3) with a frequency of 8±2 Hz. Superfusion with 0.5 mM BaCl 2 to block I K1 slowed reentry to 1 Hz and terminated it shortly after initiation. NRVM monolayers with islands of Kir2.1AAA suppression (n=3) displayed rapid spontaneous activity. Rapid pacing of these monolayers initiated an unstable figure-of-eight reentry (n=3) that degraded into single and multi-armed spiral waves, anchored to varying parts of the island with a maximum frequency of 2±1 Hz. Importantly, no reentry could be initiated in monolayers with non-transduced islands (n=3). Conclusion : Functional reentrant waves induced by rapid pacing are anchored to islands of localized Kir2.1 overexpression whereas they drop in frequency and meander in islands of dominant-negative suppression of Kir2.1, confirming the importance of I K1 for the stability of these waves in cardiac tissue.

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.

Dynamics of bound states of same-chirality spiral waves

We describe the dynamics of bound states of same-chirality spirals in a generic numerical model of an excitable medium. For each bound state, we analyze its tip trajectory patterns and determine its characteristic frequencies. We report two previously unidentified bound states: for spiral pairs, a state that exhibits alternating cycles of small and large distances between collisions (A2); for triplets, the first example of a meandering bound state (M3). In parameter space, A2 lies in between the previously described oscillating pairs (O2) and master-slave pairs (MS). We present numerical evidence that the transition O2-->A2 occurs via a supercritical period-doubling bifurcation, while the transition A2-->MS occurs via a symmetry breaking secondary Hopf bifurcation. A classification of all regimes according to dynamical systems theory exposes the wealth of phenomena exhibited by multiarmed spiral waves.

Front aggregation in multiarmed excitation vortices

Using the Belousov-Zhabotinsky reaction, we study the pinning of multiarmed spiral waves to nonexcitable obstacles. With increasing obstacle size, the individual arms switch from a repulsive to an attractive state. This transition yields densely aggregated spiral arms and is caused by anomalous dispersion. A kinematic model reproduces the measurements quantitatively and identifies the transition as a supercritical pitchfork bifurcation.

CAMP receptor affinity controls wave dynamics, geometry and morphogenesis in Dictyostelium.

Serpentine G-protein-coupled cAMP receptors are key components in the detection and relay of the extracellular cAMP waves that control chemotactic cell movement during Dictyostelium development. During development the cells sequentially express four closely related cAMP receptors of decreasing affinity. In this study, we investigated the effect of cAMP receptor type and affinity on the dynamics of cell-cell signalling in vivo, by measuring the dynamics of wave initiation and propagation in a variety of cAMP receptor mutants. We found that receptor affinity controls the frequency of wave initiation, but it does not determine wave propagation velocity, thus resulting in dramatic changes in wave geometry. In the limiting case, the affinity of the receptor is so low that waves can still be initiated but no stable centres form - thus, the cells cannot aggregate. In mounds, expression of low affinity receptors results in slow concentric waves instead of the normally observed multi-armed spiral waves. Under these conditions there is no rotational cell movement and the hemispherical mounds cannot transform into slugs. These results highlight the importance of receptor number and affinity in the proper control of cell-cell signalling dynamics required for the successful completion of development.

Theory of spiral wave dynamics in weakly excitable media: asymptotic reduction to a kinematic model and applications.

In a weakly excitable medium, characterized by a large threshold stimulus, the free end of an isolated broken plane wave (wave tip) can either rotate (steadily or unsteadily) around a large excitable core, thereby producing a spiral pattern, or retract, causing the wave to vanish at boundaries. An asymptotic analysis of spiral motion and retraction is carried out in this weakly excitable large core regime starting from the free-boundary limit of the reaction-diffusion models, valid when the excited region is delimited by a thin interface. The wave description is shown to naturally split between the tip region and a far region that are smoothly matched on an intermediate scale. This separation allows us to rigorously derive an equation of motion for the wave tip, with the large scale motion of the spiral wave front slaved to the tip. This kinematic description provides both a physical picture and exact predictions for a wide range of wave behavior, including (i) steady rotation (frequency and core radius), (ii) exact treatment of the meandering instability in the free-boundary limit with the prediction that the frequency of unstable motion is half the primary steady frequency, (iii) drift under external actions (external field with application to axisymmetric scroll ring motion in three dimensions, and spatial- or/and time-dependent variation of excitability), and (iv) the dynamics of multiarmed spiral waves with the prediction that steadily rotating waves with two or more arms are linearly unstable. Numerical simulations of FitzHugh-Nagumo kinetics are used to test several aspects of our results. In addition, we discuss the semiquantitative extension of this theory to finite cores and pinpoint mathematical subtleties related to the thin interface limit of singly diffusive reaction-diffusion models.

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.

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.

Rotating spiral waves in excitable media: The analytical results

In this paper we calculate the rotation frequency of spiral waves as a function of the parameters of the excitable medium. We give the complete analytical solution in the special case of the Rinzel-Keller model and suggest an analytical procedure for general two-component relaxational reaction-diffusion models. This procedure is based on the Greenberg equation and stability analysis; it is applicable when the core of a spiral wave is large as compared with the characteristic diffusion length. Construction of spiral wave solutions for the waves inside ring channels, circular regions, or around holes is discussed. Multi-armed spiral waves are investigated.

Spiral Waves in Reaction-Diffusion Equations

We consider the reaction-diffusion system \[\begin{gathered} R_T = \nabla ^2 R + R\left( {1 - R^2 - \vec \nabla \theta \cdot \vec \nabla \theta } \right), \hfill R\theta _T = R\nabla ^2 \theta + 2\vec \nabla R \cdot \vec \nabla \theta + qR^3 \hfill \end{gathered} \]This system governs the solutions of reaction-diffusion systems near a Hopf bifurcation point. In two spatial dimensions we use formal asymptotic expansions to contruct one-armed and multi-armed Archimedean spiral waves for small values of the parameter q. We then show that the one-armed spiral waves are probably stable and the multi-armed ones are unstable for $| q |$ small. Next, by numerical continuation methods, we construct spiral waves for all q. These calculations show that the one-armed spiral waves are unstable when $| q | > 1.397 \cdots $. We also find the explicit one-dimensional analogues of spiral waves for all q, and show that they are unstable for $| q |$ small.