Wave-particle dualism of spiral waves dynamics

Annihilation dynamics during spiral defect chaos revealed by particle models

Pair-annihilation events are ubiquitous in a variety of spatially extended systems and are often studied using computationally expensive simulations. Here, we develop an approach in which we simulate the pair-annihilation of spiral wave tips in cardiac models using a computationally efficient particle model. Spiral wave tips are represented as particles with dynamics governed by diffusive behavior and short-ranged attraction. The parameters for diffusion and attraction are obtained by comparing particle motion to the trajectories of spiral wave tips in cardiac models during spiral defect chaos. The particle model reproduces the annihilation rates of the cardiac models and can determine the statistics of spiral wave dynamics, including its mean termination time. We show that increasing the attraction coefficient sharply decreases the mean termination time, making it a possible target for pharmaceutical intervention.

Scroll Waves and Filaments in Excitable Media of Higher Spatial Dimension

Excitable media are ubiquitous in nature, and in such systems the local excitation tends to self-organize in traveling waves, or in rotating spiral-shaped patterns in two or three spatial dimensions. Examples include waves during a pandemic or electrical scroll waves in the heart. Here we show that such phenomena can be extended to a space of four or more dimensions and propose that connections of excitable elements in a network setting can be regarded as additional spatial dimensions. Numerical simulations are performed in four dimensions using the FitzHugh-Nagumo model, showing that the vortices rotate around a two-dimensional surface which we define as the superfilament. Evolution equations are derived for general superfilaments of codimension two in an N-dimensional space, and their equilibrium configurations are proven to be minimal surfaces. We suggest that biological excitable systems, such as the heart or brain which have nonlocal connections can be regarded, at least partially, as multidimensional excitable media and discuss further possible studies in this direction.

Neuromodulation of Chemical Synaptic Transmission Driven by THz Photons

Postsynaptic currents of chemical synapse are modulated by multitudinous neurotransmitters, such as acetylcholine, dopamine, glutamate, and γ-aminobutyric acid, many of which have been used in the treatment of neurological diseases. Here, based on molecular dynamics simulations and quantum chemical calculation, we propose that 30- to 45-THz photons can resonate with a variety of typical neurotransmitter molecules and make them absorb photon energy to activate the transition to high energy state, which is expected to be a new method of neural regulation. Furthermore, we verified the calculated results through experiments that THz irradiation could substantively change neuronal signal emission and enhance the frequency, amplitude, and dynamic properties of excitatory postsynaptic current and inhibitory postsynaptic current. In addition, we demonstrated the potential of neural information regulation by THz photons through 2-photon imaging in vivo. These findings are expected to improve the understanding of the physical mechanism of biological phenomena and facilitate the application of terahertz technology in neural regulation and the development of new functional materials.

Catheter Ablation of Ventricular Fibrillation

Ventricular fibrillation (VF) is a common cause of sudden cardiac death (SCD) and is unfortunately without a cure. Current therapies focus on prevention of SCD, such as implantable cardioverter-defibrillator (ICD) implantation and anti-arrhythmic agents. Significant progress has been made in improving our understanding and ability to target the triggers of VF, via advanced mapping and ablation techniques, as well as with autonomic modulation. However, the critical substrate for VF maintenance remains incompletely defined. In this review, we discuss the evidence behind the basic mechanisms of VF and review the current role of catheter ablation in patients with VF.

A Phase Defect Framework for the Analysis of Cardiac Arrhythmia Patterns

During cardiac arrhythmias, dynamical patterns of electrical activation form and evolve, which are of interest to understand and cure heart rhythm disorders. The analysis of these patterns is commonly performed by calculating the local activation phase and searching for phase singularities (PSs), i.e., points around which all phases are present. Here we propose an alternative framework, which focuses on phase defect lines (PDLs) and surfaces (PDSs) as more general mechanisms, which include PSs as a specific case. The proposed framework enables two conceptual unifications: between the local activation time and phase description, and between conduction block lines and the central regions of linear-core rotors. A simple PDL detection method is proposed and applied to data from simulations and optical mapping experiments. Our analysis of ventricular tachycardia in rabbit hearts (n = 6) shows that nearly all detected PSs were found on PDLs, but the PDLs had a significantly longer lifespan than the detected PSs. Since the proposed framework revisits basic building blocks of cardiac activation patterns, it can become a useful tool for further theory development and experimental analysis.

Emergent order in hydrodynamic spin lattices

Macroscale analogues^1-3 of microscopic spin systems offer direct insights into fundamental physical principles, thereby advancing our understanding of synchronization phenomena⁴ and informing the design of novel classes of chiral metamaterials^5-7. Here we introduce hydrodynamic spin lattices (HSLs) of 'walking' droplets as a class of active spin systems with particle-wave coupling. HSLs reveal various non-equilibrium symmetry-breaking phenomena, including transitions from antiferromagnetic to ferromagnetic order that can be controlled by varying the lattice geometry and system rotation⁸. Theoretical predictions based on a generalized Kuramoto model⁴ derived from first principles rationalize our experimental observations, establishing HSLs as a versatile platform for exploring active phase oscillator dynamics. The tunability of HSLs suggests exciting directions for future research, from active spin-wave dynamics to hydrodynamic analogue computation and droplet-based topological insulators.

High-frequency pacing of scroll waves in a three-dimensional slab model of cardiac tissue

Vortices in excitable media underlie dangerous cardiac arrhythmias. One way to eliminate them is by stimulating the excitable medium with a period smaller than the period of the vortex. So far, this phenomenon has been studied mostly for two-dimensional vortices known as spiral waves. Here we present a first study of this phenomenon for three-dimensional vortices, or scroll waves, in a slab. We consider two main types of scroll waves dynamics: with positive filament tension and with negative filament tension and show that such elimination is possible for some values of the period in all cases. However, in the case of negative filament tension for relatively long stimulation periods, three-dimensional instabilities occur and make elimination impossible. We derive equations of motion for the drift of paced filaments and identify a bifurcation parameter that determines whether the filaments orient themselves perpendicular to the impeding wave train or not.

Numerical analysis of subcritical Hopf bifurcations in the two-dimensional FitzHugh-Nagumo model

It had been shown that the transition from a rigidly rotating spiral wave to a meandering spiral wave is via a Hopf bifurcation. Many studies have shown that these bifurcations are supercritical, but, by using simulations in a comoving frame of reference, we present numerical results which show that subcritical bifurcations are also present within FitzHugh-Nagumo. We show that a hysteresis region is present at the boundary of the rigidly rotating spiral waves and the meandering spiral waves for a particular set of parameters, a feature of FitzHugh-Nagumo that has previously not been reported. Furthermore, we present a evidence that this bifurcation is highly sensitive to initial conditions, and it is possible to convert one solution in the hysteresis loop to the other.

Chaotic tip trajectories of a single spiral wave in the presence of heterogeneities

Spiral waves have been observed in a variety of physical, chemical, and biological systems. They play a major role in cardiac arrhythmias, including fibrillation, where the observed irregular activation patterns are generally thought to arise from the continuous breakup of multiple unstable spiral waves. Using spatially extended simulations of different electrophysiological models of cardiac tissue, we show that a single spiral wave in the presence of heterogeneities can display chaotic tip trajectories, consistent with fibrillation. We also show that the simulated spiral tip dynamics, including chaotic trajectories, can be captured by a simple particle model which only describes the dynamics of the spiral tip. This shows that spiral wave breakup, or interactions with other waves, are not necessary to initiate chaos in spiral waves.

Response function framework for the dynamics of meandering or large-core spiral waves and modulated traveling waves

In many oscillatory or excitable systems, dynamical patterns emerge which are stationary or periodic in a moving frame of reference. Examples include traveling waves or spiral waves in chemical systems or cardiac tissue. We present a unified theoretical framework for the drift of such patterns under small external perturbations, in terms of overlap integrals between the perturbation and the adjoint critical eigenfunctions of the linearized operator (i.e., response functions). For spiral waves, the finite radius of the spiral tip trajectory and spiral wave meander are taken into account. Different coordinate systems can be chosen, depending on whether one wants to predict the motion of the spiral-wave tip, the time-averaged tip path, or the center of the meander flower. The framework is applied to analyze the drift of a meandering spiral wave in a constant external field in different regimes.

Filament Tension and Phase Locking of Meandering Scroll Waves

Meandering spiral waves are often observed in excitable media such as the Belousov-Zhabotinsky reaction and cardiac tissue. We derive a theory for drift dynamics of meandering rotors in general reaction-diffusion systems and apply it to two types of external disturbances: an external field and curvature-induced drift in three dimensions. We find two distinct regimes: with small filament curvature, meandering scroll waves exhibit filament tension, whose sign determines the stability and drift direction. In the regimes of strong external fields or meandering motion close to resonance, however, phase locking of the meander pattern is predicted and observed.

Measurement and structure of spiral wave response functions

The rotating spiral waves that emerge in diverse natural and man-made systems typically exhibit a particle-like behaviour since their adjoint critical eigenmodes (response functions) are often seen to be localised around the spiral core. We present a simple method to numerically compute response functions for circular-core and meandering spirals by recording their drift response to many elementary perturbations. Although our method is computationally more expensive than solving the adjoint system, our technique is fully parallellisable, does not suffer from memory limitations and can be applied to experiments. For a cardiac tissue model with the linear spiral core, we find that the response functions are localised near the turning points of the trajectory.

BeatBox-HPC simulation environment for biophysically and anatomically realistic cardiac electrophysiology

The BeatBox simulation environment combines flexible script language user interface with the robust computational tools, in order to setup cardiac electrophysiology in-silico experiments without re-coding at low-level, so that cell excitation, tissue/anatomy models, stimulation protocols may be included into a BeatBox script, and simulation run either sequentially or in parallel (MPI) without re-compilation. BeatBox is a free software written in C language to be run on a Unix-based platform. It provides the whole spectrum of multi scale tissue modelling from 0-dimensional individual cell simulation, 1-dimensional fibre, 2-dimensional sheet and 3-dimensional slab of tissue, up to anatomically realistic whole heart simulations, with run time measurements including cardiac re-entry tip/filament tracing, ECG, local/global samples of any variables, etc. BeatBox solvers, cell, and tissue/anatomy models repositories are extended via robust and flexible interfaces, thus providing an open framework for new developments in the field. In this paper we give an overview of the BeatBox current state, together with a description of the main computational methods and MPI parallelisation approaches.

Adjoint eigenfunctions of temporally recurrent single-spiral solutions in a simple model of atrial fibrillation

This paper introduces a numerical method for computing the spectrum of adjoint (left) eigenfunctions of spiral wave solutions to reaction-diffusion systems in arbitrary geometries. The method is illustrated by computing over a hundred eigenfunctions associated with an unstable time-periodic single-spiral solution of the Karma model on a square domain. We show that all leading adjoint eigenfunctions are exponentially localized in the vicinity of the spiral tip, although the marginal modes (response functions) demonstrate the strongest localization. We also discuss the implications of the localization for the dynamics and control of unstable spiral waves. In particular, the interaction with no-flux boundaries leads to a drift of spiral waves which can be understood with the help of the response functions.

A novel technique to initiate and investigate scroll waves in thin layers of the photosensitive Belousov-Zhabotinsky reaction

While free scroll rings are non-stationary objects that either grow or contract with time, spatial confinement can have a large impact on their evolution reaching from significant lifetime extension (J.F. Totz, H. Engel, O. Steinbock, New J. Phys. 17, 093043 (2015)) up to formation of stable stationary and breathing pacemakers (A. Azhand, J.F. Totz, H. Engel, EPL 108, 10004 (2014)). Here, we explore the parameter range in which the interaction between an axis-symmetric scroll ring and a confining planar no-flux boundary can be studied experimentally in transparent gel layers supporting chemical wave propagation in the photosensitive variant of the Belousov-Zhabotinsky medium. Based on full three-dimensional simulations of the underlying modified complete Oregonator model for experimentally realistic parameters, we determine the conditions for successful initiation of scroll rings in a phase diagram spanned by the layer thickness and the applied light intensity. Furthermore, we discuss whether the illumination-induced excitability gradient due to Lambert-Beer's law as well as a possible inclination of the filament plane with respect to the no-flux boundary can destabilize the scroll ring.

Chiral selection and frequency response of spiral waves in reaction-diffusion systems under a chiral electric field

Chirality is one of the most fundamental properties of many physical, chemical, and biological systems. However, the mechanisms underlying the onset and control of chiral symmetry are largely understudied. We investigate possibility of chirality control in a chemical excitable system (the Belousov-Zhabotinsky reaction) by application of a chiral (rotating) electric field using the Oregonator model. We find that unlike previous findings, we can achieve the chirality control not only in the field rotation direction, but also opposite to it, depending on the field rotation frequency. To unravel the mechanism, we further develop a comprehensive theory of frequency synchronization based on the response function approach. We find that this problem can be described by the Adler equation and show phase-locking phenomena, known as the Arnold tongue. Our theoretical predictions are in good quantitative agreement with the numerical simulations and provide a solid basis for chirality control in excitable media.

Front propagation in channels with spatially modulated cross section

Propagation of traveling fronts in a three-dimensional channel with spatially varying cross section is reduced to an equivalent one-dimensional reaction-diffusion-advection equation with boundary-induced advection term. Treating the advection term as a weak perturbation, an equation of motion for the front position is derived. We analyze channels whose cross sections vary periodically with L along the propagation direction of the front. Taking the Schlögl model as a representative example, we calculate analytically the nonlinear dependence of the front velocity on the ratio L/l where l denotes the intrinsic front width. In agreement with finite-element simulations of the three-dimensional reaction-diffusion dynamics, our theoretical results predicts boundary-induced propagation failure for a finite range of L/l values. In particular, the existence of the upper bound of L/l can be completely understood based on the linear eikonal equation. Last, we demonstrate that the front velocity is determined by the suppressed diffusivity of the reactants for L≪l.

Asymptotic dynamics of reflecting spiral waves

Resonantly forced spiral waves in excitable media drift in straight-line paths, their rotation centers behaving as pointlike objects moving along trajectories with a constant velocity. Interaction with medium boundaries alters this velocity and may often result in a reflection of the drift trajectory. Such reflections have diverse characteristics and are known to be highly nonspecular in general. In this context we apply the theory of response functions, which via numerically computable integrals, reduces the reaction-diffusion equations governing the whole excitable medium to the dynamics of just the rotation center and rotation phase of a spiral wave. Spiral reflection trajectories are computed by this method for both small- and large-core spiral waves in the Barkley model. Such calculations provide insight into the process of reflection as well as explanations for differences in trajectories across parameters, including the effects of incidence angle and forcing amplitude. Qualitative aspects of these results are preserved far beyond the asymptotic limit of weak boundary effects and slow resonant drift.

Capture of fixation by rotational flow; a deterministic hypothesis regarding scaling and stochasticity in fixational eye movements

Visual scan paths exhibit complex, stochastic dynamics. Even during visual fixation, the eye is in constant motion. Fixational drift and tremor are thought to reflect fluctuations in the persistent neural activity of neural integrators in the oculomotor brainstem, which integrate sequences of transient saccadic velocity signals into a short term memory of eye position. Despite intensive research and much progress, the precise mechanisms by which oculomotor posture is maintained remain elusive. Drift exhibits a stochastic statistical profile which has been modeled using random walk formalisms. Tremor is widely dismissed as noise. Here we focus on the dynamical profile of fixational tremor, and argue that tremor may be a signal which usefully reflects the workings of oculomotor postural control. We identify signatures reminiscent of a certain flavor of transient neurodynamics; toric traveling waves which rotate around a central phase singularity. Spiral waves play an organizational role in dynamical systems at many scales throughout nature, though their potential functional role in brain activity remains a matter of educated speculation. Spiral waves have a repertoire of functionally interesting dynamical properties, including persistence, which suggest that they could in theory contribute to persistent neural activity in the oculomotor postural control system. Whilst speculative, the singularity hypothesis of oculomotor postural control implies testable predictions, and could provide the beginnings of an integrated dynamical framework for eye movements across scales.

Effective dynamics of twisted and curved scroll waves using virtual filaments

Scroll waves are three-dimensional excitation patterns that rotate around a central filament curve; they occur in many physical, biological, and chemical systems. We explicitly derive the equations of motion for scroll wave filaments in reaction-diffusion systems with isotropic diffusion up to third order in the filament's twist and curvature. The net drift components define at every instance of time a virtual filament which lies close to the instantaneous filament. Importantly, virtual filaments obey simpler, time-independent laws of motion which we analytically derive here and illustrate with numerical examples. Stability analysis of scroll waves is performed using virtual filaments, showing that filament curvature and twist add as quadratic terms to the nominal filament tension. Applications to oscillating chemical reactions and cardiac tissue are discussed.

Reversal of spiral waves in an oscillatory system caused by an inhomogeneity

Spatial heterogeneities are commonly found in realistic systems and play significant roles in dynamics of spiral waves. We here demonstrate a novel phenomenon that a localized inhomogeneity put around the spiral core could lead to the reversal of spiral waves in an oscillatory system, e.g., the complex Ginzburg-Landau equation. With the amplitude-phase representation, we analyze underling mechanism and conditions of the wave reversal in detail, which is found to agree with the numerical evidence.

Drift laws for spiral waves on curved anisotropic surfaces

Rotating spiral waves organize spatial patterns in chemical, physical, and biological excitable systems. Factors affecting their dynamics, such as spatiotemporal drift, are of great interest for particular applications. Here, we propose a quantitative description for spiral wave dynamics on curved surfaces which shows that for a wide class of systems, including the Belousov-Zhabotinsky reaction and anisotropic cardiac tissue, the Ricci curvature scalar of the surface is the main determinant of spiral wave drift. The theory provides explicit equations for spiral wave drift direction, drift velocity, and the period of rotation. Depending on the parameters, the drift can be directed to the regions of either maximal or minimal Ricci scalar curvature, which was verified by direct numerical simulations.

Chiralities of spiral waves and their transitions

The chiralities of spiral waves usually refer to their rotation directions (the turning orientations of the spiral temporal movements as time elapses) and their curl directions (the winding orientations of the spiral spatial geometrical structures themselves). Traditionally, they are the same as each other. Namely, they are both clockwise or both counterclockwise. Moreover, the chiralities are determined by the topological charges of spiral waves, and thus they are conserved quantities. After the inwardly propagating spirals were experimentally observed, the relationship between the chiralities and the one between the chiralities and the topological charges are no longer preserved. The chiralities thus become more complex than ever before. As a result, there is now a desire to further study them. In this paper, the chiralities and their transition properties for all kinds of spiral waves are systemically studied in the framework of the complex Ginzburg-Landau equation, and the general relationships both between the chiralities and between the chiralities and the topological charges are obtained. The investigation of some other models, such as the FitzHugh-Nagumo model, the nonuniform Oregonator model, the modified standard model, etc., is also discussed for comparison.

Non-specular reflections in a macroscopic system with wave-particle duality: spiral waves in bounded media

Spiral waves in excitable media possess both wave-like and particle-like properties. When resonantly forced (forced at the spiral rotation frequency) spiral cores travel along straight trajectories, but may reflect from medium boundaries. Here, numerical simulations are used to study reflections from two types of boundaries. The first is a no-flux boundary which waves cannot cross, while the second is a step change in the medium excitability which waves do cross. Both small-core and large-core spirals are investigated. The predominant feature in all cases is that the reflected angle varies very little with incident angle for large ranges of incident angles. Comparisons are made to the theory of Biktashev and Holden. Large-core spirals exhibit other phenomena such as binding to boundaries. The dynamics of multiple reflections is briefly considered.

Evolution of spiral and scroll waves of excitation in a mathematical model of ischaemic border zone

Abnormal electrical activity from the boundaries of ischemic cardiac tissue is recognized as one of the major causes in generation of ischemia-reperfusion arrhythmias. Here we present theoretical analysis of the waves of electrical activity that can rise on the boundary of cardiac cell network upon its recovery from ischaemia-like conditions. The main factors included in our analysis are macroscopic gradients of the cell-to-cell coupling and cell excitability and microscopic heterogeneity of individual cells. The interplay between these factors allows one to explain how spirals form, drift together with the moving boundary, get transiently pinned to local inhomogeneities, and finally penetrate into the bulk of the well-coupled tissue where they reach macroscopic scale. The asymptotic theory of the drift of spiral and scroll waves based on response functions provides explanation of the drifts involved in this mechanism, with the exception of effects due to the discreteness of cardiac tissue. In particular, this asymptotic theory allows an extrapolation of 2D events into 3D, which has shown that cells within the border zone can give rise to 3D analogues of spirals, the scroll waves. When and if such scroll waves escape into a better coupled tissue, they are likely to collapse due to the positive filament tension. However, our simulations have shown that such collapse of newly generated scrolls is not inevitable and that under certain conditions filament tension becomes negative, leading to scroll filaments to expand and multiply leading to a fibrillation-like state within small areas of cardiac tissue.

Impact of tissue geometry on simulated cholinergic atrial fibrillation: a modeling study

Atrial fibrillation (AF), arising in the cardiac atria, is a common cardiac rhythm disorder that is incompletely understood. Numerous characteristics of the atrial tissue are thought to play a role in the maintenance of AF. Most traditional theoretical models of AF have considered the atrium to be a flat two-dimensional sheet. Here, we analyzed the relationship between atrial geometry, substrate size, and AF persistence, in a mathematical model involving heterogeneity. Spatially periodic properties were created by variations in times required for reactivation due to periodic acetylcholine concentration [ACh] distribution. The differences in AF maintenance between the sheet and the cylinder geometry are found for intermediate gradients of inexcitable time (intermediate [ACh]). The maximum difference in AF maintenance between geometry decreases with increasing tissue size, down to zero for a substrate of dimensions 20 × 10 cm. Generators have the tendency to be anchored to the regions of longer inexcitable period (low [ACh]). The differences in AF maintenance between geometries correlate with situations of moderate anchoring for which rotor-core drifts between low-[ACh] regions occur, favoring generator disappearance. The drift of generators increases their probability of disappearance at the tissue borders, resulting in a decreased maintenance rate in the sheet due to the higher number of no-flux boundaries. These interactions between biological variables and the role of geometry must be considered when selecting an appropriate model for AF in intact hearts.

Alternative stable scroll waves and conversion of autowave turbulence

Rotating spiral and scroll waves (vortices) are investigated in the FitzHugh-Nagumo model of excitable media. The focus is on a parameter region in which there exists bistability between alternative stable vortices with distinct periods. Response functions are used to predict the filament tension of the alternative scrolls and it is shown that the slow-period scroll has negative filament tension, while the filament tension of the fast-period scroll changes sign within a hysteresis loop. The predictions are confirmed by direct simulations. Further investigations show that the slow-period scrolls display features similar to delayed after-depolarization and tend to develop into turbulence similar to ventricular fibrillation (VF). Scrolls with positive filament tension collapse or stabilize, similar to monomorphic ventricular tachycardia (VT). Perturbations, such as boundary interaction or shock stimulus, can convert the vortex with negative filament tension into the vortex with positive filament tension. This may correspond to transition from VF to VT unrelated to pinning.

Computation of the drift velocity of spiral waves using response functions

Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological nature. In the presence of a small perturbation, the spiral wave's center of rotation and fiducial phase may change over time, i.e., the spiral wave drifts. In linear approximation, the velocity of the drift is proportional to the convolution of the perturbation with the spiral's response functions, which are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues λ=0,±iω . Here, we demonstrate that the response functions give quantitatively accurate prediction of the drift velocities due to a variety of perturbations: a time dependent, periodic perturbation (inducing resonant drift); a rotational symmetry-breaking perturbation (inducing electrophoretic drift); and a translational symmetry-breaking perturbation (inhomogeneity induced drift) including drift due to a gradient, stepwise, and localized inhomogeneity. We predict the drift velocities using the response functions in FitzHugh-Nagumo and Barkley models, and compare them with the velocities obtained in direct numerical simulations. In all cases good quantitative agreement is demonstrated.

Riding a spiral wave: numerical simulation of spiral waves in a comoving frame of reference

We describe an approach to numerical simulation of spiral waves dynamics of large spatial extent, using small computational grids.

Orbital motion of spiral waves in excitable media

Spiral waves in active media react to small perturbations as particlelike objects. Here we apply asymptotic theory to the interaction of spiral waves with a localized inhomogeneity, which leads to a novel prediction: drift of the spiral rotation center along circular orbits around the inhomogeneity. The stationary orbits have fixed radii and alternating stability, determined by the properties of the bulk medium and the type of inhomogeneity, while the drift speed along an orbit depends on the strength of the inhomogeneity. Direct numerical simulations confirm the validity and robustness of the theoretical predictions and show that these unexpected effects should be observable in experiment.

Resonant drift of two-armed spirals by a periodic advective field and periodic modulation of excitability

The drift behavior of two-armed spirals induced by periodic advective field and periodic modulation of excitability is investigated. It is shown that the two-armed spirals controlled by periodic advective field and periodic modulation of excitability drift in completely different ways. For periodic advective field, the two tips of the two-armed spiral drift in the same direction and the two-armed spiral is stable. While for periodic modulation of excitability, the two tips drift in the opposite direction and the two-armed spiral splits into two single-armed spirals. Analytical results based on a kinematic theory of rotating spirals in weakly excitable media are consistent with the numerical results.

Computation of the response functions of spiral waves in active media

Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological natures. A small perturbation causes gradual change in spatial location of spiral's rotation center and frequency, i.e., drift. The response functions (RFs) of a spiral wave are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues lambda=0,+/-iomega. The RFs describe the spiral's sensitivity to small perturbations in the way that a spiral is insensitive to small perturbations where its RFs are close to zero. The velocity of a spiral's drift is proportional to the convolution of RFs with the perturbation. Here we develop a regular and generic method of computing the RFs of stationary rotating spirals in reaction-diffusion equations. We demonstrate the method on the FitzHugh-Nagumo system and also show convergence of the method with respect to the computational parameters, i.e., discretization steps and size of the medium. The obtained RFs are localized at the spiral's core.

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.

Drift of spiral waves controlled by a polarized electric field

The drift behavior of spiral waves under the influence of a polarized electric field is investigated in the light that both the polarized electric field and the spiral waves possess rotation symmetry. Numerical simulations of a reaction-diffusion model show that the drift velocity of the spiral tip can be controlled by changing the polarization mode of the polarized electric field and some interesting drift phenomena are observed. When the electric field is circularly polarized and its rotation follows that of the spiral, the drift speed of the spiral tip reaches its maximal value. On the contrary, opposite rotation between the spiral and electric field locks the drift of the spiral tip. Analytical results based on the weak deformation approximation are consistent with the numerical results. We hope that our theoretical results will be observed in experiments, such as the Belousov-Zhabotinsky reaction.

Multistability of reentrant rhythms in an ionic model of a two-dimensional annulus of cardiac tissue

The dynamics of reentry in a model of a two-dimensional annulus of homogeneous cardiac tissue, with a Beeler-Reuter type formulation of the membrane ionic currents, is examined. The bifurcation structure of the sustained propagated solutions is described as a function of Rin and Rout, the inner and outer radii of the annulus. The transition from periodic to quasiperiodic reentry occurs at a critical Rin, which first diminishes and then saturates as Rout is increased. The reduction of the critical Rin is a consequence of the increase of the wave-front curvature. There is a range of Rin below the critical radius in which two distinct quasiperiodic solutions coexist. Each of these solutions disappears at its own specific value of Rin, and their annihilation is preceded by a new type of bifurcation leading to a regime of propagation with transient successive detachments of the wave front from the inner border of the annulus.

Causodynamics of autowave patterns

Perturbative dynamics of spiral and scroll waves involves the "response functions," i.e., critical eigenvectors of the adjoint linearized operator, dual to the Goldstone modes. A well known method of calculating the Goldstone modes is time integration of the linearized equation. We suggest that backward-time integration of the adjoint linearized equation, which we call causodynamics, can be used to calculate the response functions. This new method is more robust and easier to implement than existing methods. We illustrate how it works for propagating and rotating autowaves in reaction-diffusion systems. The method reveals unexpected qualitative difference between similarly looking regimes.

Of circles and spirals: Bridging the gap between the leading circle and spiral wave concepts of cardiac reentry

The “leading circle model” was the first detailed attempt at understanding the mechanisms of functional reentry, and remains a widely-used notion in cardiac electrophysiology. The “spiral wave” concept was developed more recently as a result of modern theoretical analysis and is the basis for consideration of reentry mechanisms in present biophysical theory. The goal of this paper is to present these models in a way that is comprehensible to both the biophysical and electrophysiology communities, with the idea of helping clinical and experimental electrophysiologists to understand better the spiral wave concept and of helping biophysicists to understand why the leading circle concept is so attractive and widely used by electrophysiologists. To this end, the main properties of the leading circle and spiral wave models of reentry are presented. Their basic assumptions and determinants are discussed and the predictions of the two concepts with respect to pharmacological responses of arrhythmias are reviewed. A major difference between them lies in the predicted responses to Na+-channel blockade, for which the spiral wave paradigm appears more closely to correspond to the results of clinical and experimental observations. The basis of this difference is explored in the context of the fundamental properties of the models.

Controlled vortex-sound interactions in atomic Bose-Einstein condensates

The low temperature dynamics of a vortex in a trapped quasi-two-dimensional Bose-Einstein condensate are studied quantitatively. Precession of an off-centered vortex in a dimple trap, embedded in a weaker harmonic trap, leads to the emission of sound in a dipolar radiation pattern. Sound emission and reabsorption can be controlled by varying the depth of the dimple. In a shallow dimple, the power emitted is proportional to the vortex acceleration-squared over the precession frequency, whereas for a deep dimple, periodic sound reabsorption stabilizes the vortex against radiation-induced decay.