Superspiral structures of meandering and drifting spiral waves

Composite spiral waves in discrete-time systems

Spiral waves are a type of typical pattern in open reaction-diffusion systems far from thermodynamic equilibrium. The study of spiral waves has attracted great interest because of its nonlinear characteristics and extensive applications. However, the study of spiral waves has been confined to continuous-time systems, while spiral waves in discrete-time systems have been rarely reported. In recent years, discrete-time models have been widely studied in ecology because of their appropriateness to systems with nonoverlapping generations and other factors. Therefore, spiral waves in discrete-time systems need to be studied. Here, we investigated a novel type of spiral wave called a composite spiral wave in a discrete-time predator-pest model, and we revealed the formation mechanism. To explain the observed phenomena, we defined and quantified a move state effect of multiperiod states caused by the coupling of adjacent stable multiperiod orbits, which is strictly consistent with the numerical results. The other move state effect is caused by an unstable focus, which is the state of the local points at the spiral center. The combined effect of these two influences can lead to rich dynamical behaviors of spiral waves, and the specific structure of the composite spiral waves is the result of the competition of the two effects in opposite directions. Our findings shed light on the dynamics of spiral waves in discrete-time systems, and they may guide the prediction and control of pests in deciduous forests.

The physics of heart rhythm disorders

The global burden caused by cardiovascular disease is substantial, with heart disease representing the most common cause of death around the world. There remains a need to develop better mechanistic models of cardiac function in order to combat this health concern. Heart rhythm disorders, or arrhythmias, are one particular type of disease which has been amenable to quantitative investigation. Here we review the application of quantitative methodologies to explore dynamical questions pertaining to arrhythmias. We begin by describing single-cell models of cardiac myocytes, from which two and three dimensional models can be constructed. Special focus is placed on results relating to pattern formation across these spatially-distributed systems, especially the formation of spiral waves of activation. Next, we discuss mechanisms which can lead to the initiation of arrhythmias, focusing on the dynamical state of spatially discordant alternans, and outline proposed mechanisms perpetuating arrhythmias such as fibrillation. We then review experimental and clinical results related to the spatio-temporal mapping of heart rhythm disorders. Finally, we describe treatment options for heart rhythm disorders and demonstrate how statistical physics tools can provide insights into the dynamics of heart rhythm disorders.

Instability of the Homogeneous Distribution of Chemical Waves in the Belousov-Zhabotinsky Reaction

Chemical traveling waves play an important role in biological functions, such as the propagation of action potential and signal transduction in the nervous system. Such chemical waves are also observed in inanimate systems and are used to clarify their fundamental properties. In this study, chemical waves were generated with equivalent spacing on an excitable medium of the Belousov–Zhabotinsky reaction. The homogeneous distribution of the waves was unstable and low- and high-density regions were observed. In order to understand the fundamental mechanism of the observations, numerical calculations were performed using a mathematical model, the modified Oregonator model, including photosensitive terms. However, the homogeneous distribution of the traveling waves was stable over time in the numerical results. These results indicate that further modification of the model is required to reproduce our experimental observations and to discover the fundamental mechanism for the destabilization of the homogeneous-distributed chemical traveling waves.

Excited state of spiral waves in oscillatory reaction-diffusion systems caused by a pulse

Previous studies claim that the dynamic behaviors of spiral waves are uniquely determined by the nature of the medium, which can be determined by control parameters. In this article, the authors break from the previous view and present an alternate stable state of spiral waves, named the excited state. The authors find that two states of the spiral wave switch to each other after a one-off pulse is applied to the medium. The dynamic behaviors of the two states are quite different, specifically, the spiral tip trajectory of the original spiral, which is named the ground-state spiral as observed in the previous studies, is a point, while the spiral tip trajectory of the excited-state spiral is a circle. Moreover, the authors study the trajectories of the spiral tip of spiral waves in both states after the pulse is applied and find two types of trajectories, a spiral trajectory and a spiral-inward-petal trajectory. The frequency of the spiral wave in the excited state is less than that in the ground state. The findings enrich the dynamics of pattern formation.

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.

Spectral analysis of localized rotating waves in parabolic systems

In this paper, we study the spectra and Fredholm properties of Ornstein-Uhlenbeck operators [Formula: see text]where [Formula: see text] is the profile of a rotating wave satisfying [Formula: see text] as [Formula: see text], the map [Formula: see text] is smooth and the matrix [Formula: see text] has eigenvalues with positive real parts and commutes with the limit matrix [Formula: see text] The matrix [Formula: see text] is assumed to be skew-symmetric with eigenvalues (λ1,…,λ d )=(±iσ1,…,±iσk ,0,…,0). The spectra of these linearized operators are crucial for the nonlinear stability of rotating waves in reaction-diffusion systems. We prove under appropriate conditions that every [Formula: see text] satisfying the dispersion relation [Formula: see text]belongs to the essential spectrum [Formula: see text] in Lp For values Re λ to the right of the spectral bound for [Formula: see text], we show that the operator [Formula: see text] is Fredholm of index 0, solve the identification problem for the adjoint operator [Formula: see text] and formulate the Fredholm alternative. Moreover, we show that the set [Formula: see text]belongs to the point spectrum [Formula: see text] in Lp We determine the associated eigenfunctions and show that they decay exponentially in space. As an application, we analyse spinning soliton solutions which occur in the Ginzburg-Landau equation and compute their numerical spectra as well as associated eigenfunctions. Our results form the basis for investigating the nonlinear stability of rotating waves in higher space dimensions and truncations to bounded domains. This article is part of the themed issue 'Stability of nonlinear waves and patterns and related topics'.

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.

Formation of spiral waves with substructure in a bursting media

Formation of spiral waves in a bursting media is investigated. Due to the multiple timescale oscillation of the local dynamics, an interesting substructure of traveling wave (STW) is observed in the spiral arm. As a result of the special moving media formed by neurons in the spiral arm, STWs propagate from the spiral tip to far field with an increasing wave length and move faster along the front of the spiral arm than along the back, leading to the formation of fragments in STWs. Moreover, we find that a sharp change of stimulus current can lead to backfiring of STWs, which may break the spiral wave front and further result in the formation of a multi-spiral pattern.

Diffusion-induced periodic transition between oscillatory modes in amplitude-modulated patterns

We study amplitude-modulated waves, e.g., wave packets in one dimension, overtarget spirals and superspirals in two dimensions, under mixed-mode oscillatory conditions in a three-variable reaction-diffusion model. New transition zones, not seen in the homogeneous system, are found, in which periodic transitions occur between local 1(N-1) and 1(N) oscillations. Amplitude-modulated complex patterns result from periodic transition between (N - 1)-armed and N-armed waves. Spatial recurrence rates provide a useful guide to the stability of these modulated patterns.

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.

Induced spiral motion in cardiac tissue due to alternans

Spiral wave meander is a typical feature observed in cardiac tissue and in excitable media in general. Here, we show for a simple model of excitable cardiac tissue that a transition to alternans--a beat-to-beat temporal alternation in the duration of cardiac excitation--can also induce a transition in the spiral core motion that is related to the presence of synchronization defect lines (SDLs) or nodal lines. While this is similar to what has been predicted and indeed observed for complex-oscillatory media close to onset, we find important qualitative differences. For example, single straight SDLs rotate and induce an additional nonresonant frequency characterizing the core motion of the attached spiral. We analyze this behavior quantitatively as a function of the steepness of the restitution curve and show that the velocity and the directionality of the core motion vary monotonically with the control parameter. Our findings agree with recent observations in rat heart tissue cultures indicating that the described behavior is of rather general nature. In particular, it could play an important role in the context of potentially life-threatening cardiac arrhythmias such as fibrillation for which alternans and spiral waves are known precursors.

Density wave propagation of a wave train in a closed excitable medium

A wave train in an excitable reaction-diffusion medium shows a variety of spatiotemporal patterns as a result of interactions between the individual waves. In this paper, we report a novel spatiotemporal pattern in a wave train in a closed excitable medium. We carried out experiments using a photosensitive Belousov-Zhabotinsky reaction with Ru(bpy)(3)(2+) as a catalyst and a numerical calculation using the FitzHugh-Nagumo equation. A wave train was locally distributed as an initial condition and the number of waves was systematically varied. In both the experiment and numerical calculation, density wave propagation was formed in a wave train during relaxation with a large number of waves. Our results suggest that density wave propagation originates from inhibitory interaction between the waves.

Projective synchronization of two coupled excitable spiral waves

Interaction of two identical excitable spiral waves in a bilayer system is studied. We find that the two spiral waves can be completely synchronized if the coupling strength is sufficiently large. Prior to the complete synchronization, we find a new type of weak synchronization between the two coupled systems, i.e., the spiral wave of the driven system has the same geometric shape as the spiral wave of the driving system but with a much lower amplitude. This general behavior, called projective synchronization of two spiral waves, is similar to projective synchronization of two coupled nonlinear oscillators, which has been extensively studied before. The underlying mechanism is uncovered by the study of pulse collision in one-dimensional systems.

Spiral instabilities in media supporting complex oscillations under periodic forcing

The periodically forced Brusselator model displays temporal mixed-mode and quasiperiodic oscillations, period doubling, and chaos. We explore the behavior of such media as reaction-diffusion systems for investigating spiral instabilities. Besides near-core breakup and far-field breakup resulting from unstable modes in the radial direction or Doppler-induced instability (destabilization of the core's location), the observed complex phenomena include backfiring, spiral regeneration, and amplitude modulation from line defects. Amplitude modulation of spirals can evolve to chambered spirals resembling those found in nature, such as pine cones and sunflowers. When the forcing amplitude is increased, the spiral-tip meander evolves from simple rotation to complex petals, corresponding to transformation of the local dynamics from simple oscillations to mixed-mode, period-2, and quasiperiodic oscillations. The number of petals is related to the complexity of the mixed-mode oscillations. Spiral turbulence, standing waves, and homogeneous synchronization permeate the entire system when the forcing amplitude is further increased.

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.

Arm splitting and backfiring of spiral waves in media displaying local mixed-mode oscillations

The behavior of spiral waves is investigated in a model of reaction-diffusion media supporting local mixed-mode oscillations for a range of values of a control parameter. This local behavior is accompanied by the formation of nodes, at which the arms of the simple spiral waves begin to split. With further parameter changes, this nodal structure loses stability, becoming quite irregular, eventually evolving into turbulence, while the local dynamics increases in complexity. The breakup of the spiral waves arises from a backfiring instability of the nodes induced by the arm splitting. This process of spiral breakup in the presence of mixed-mode oscillations represents an alternative to previously described scenarios of instability of line defects and superspirals in media with period-doubling and quasiperiodic oscillations, respectively.

Three-dimensional coherent optical waves localized on trochoidal parametric surfaces

We theoretically demonstrate that the three-dimensional (3D) coherent laser waves formed by the degenerate Laguerre-Gaussian modes with different longitudinal indices are well localized on rotating trochoidal parametric surfaces. We further use a large-Fresnel-number laser system to realize the existence of the laser modes related to trochoidal coherent states. Experimental results reveal that the exotic laser modes generally originate from a superposition of two degenerate standing-wave trochoidal coherent states.

Breakup of propagating waves through the development of a transient unexcitable regime

Spontaneous breakup of propagating waves was investigated in this paper with a ferroin-catalyzed Belousov-Zhabotinsky reaction modified with the inclusion of a second substrate, 1,4-cyclohexanedione (CHD). The presence of CHD, which forms a separate chemical oscillator with acidic bromate, led to the development of a scattered unexcitable regime in the studied medium, where as waves passed through these unexcitable media, the propagation was disrupted, causing the creation of free ends. It thus presents a new avenue through which a chemical wave breaks up to form spirals. By manipulating the concentrations of CHD, sulfuric acid, and bromate, unexcitable regimes of different sizes with different survival times were obtained. Kinetic data on wave speed prior to and after the unexcitable window illustrates that the occurrence of an unexcitable regime is not due to depletion of components needed for wave propagation.

Local wave grouping in a parameter-gradient system and its formation mechanism

In a ferroin-catalyzed Belousov-Zhabotinsky (BZ) reaction-diffusion system with reagent concentration gradients, we observed in the experiment a type of spirals with local waves forming groups. Here, we propose an interpretation of the wave grouping phenomenon. The wave grouping mechanism can be well explained in terms of the cooperation of the excitability gradient and the Doppler effect induced by spiral tip's meandering. In the simulation based on three-dimensional reaction-diffusion system using Oregonator model, spiral patterns analogous to the experiment observation are well reproduced when the parameter gradient in the z direction is introduced.

Spiral waves in linearly coupled reaction-diffusion systems

The dynamics of spiral waves in a pair of linearly coupled reaction-diffusion systems is investigated. We find that the spiral dynamics depends on the coupling strength between the two subsystems. When the coupling strength is weak, the frequency and wavelength of the spiral wave in each subsystem remain unchanged. The interaction between the two subsystems induces the drift of spiral waves. When the coupling strength is strong, synchronization between the two subsystems is established. The two subsystems play different roles in the collective dynamics: one subsystem is always dominant and enslaves the other.

Spiral wave drift in an electric field and scroll wave instabilities

Here, I present the numerical computation of speed and direction of the drift of a spiral wave in an excitable medium in the presence of an electric field. The drift speed presents a strong variation close to the parameter value where the drift-speed component along the field direction from parallel becomes antiparallel. Using a simple phenomenological model and results from a numerical linear stability analysis of scroll waves, I show that this behavior can be attributed to a resonance of the meander modes with the translation modes of the spiral wave. Extending this phenomenological model to scroll waves also clarifies the link between the drift and long wavelength instabilities of scroll waves.

Doppler effect of nonlinear waves and superspirals in oscillatory media

Nonlinear waves emitted from a moving source are studied. A meandering spiral in a reaction-diffusion medium provides an example in which waves originate from a source exhibiting a back-and-forth movement in a radial direction. The periodic motion of the source induces a Doppler effect that causes a modulation in wavelength and amplitude of the waves ("superspiral"). Using direct simulations as well as numerical nonlinear analysis within the complex Ginzburg-Landau equation, we show that waves subject to a convective Eckhaus instability can exhibit monotonic growth or decay as well as saturation of these modulations depending on the perturbation frequency. Our findings elucidate recent experimental observations concerning superspirals and their decay to spatiotemporal chaos.

Analytical approach to the drift of the tips of spiral waves in the complex Ginzburg-Landau equation

In this paper, we investigate the motion of spiral waves in the complex Ginzburg-Landau equation (CGLE) analytically and numerically. We find that the tip of the spiral wave drifts primarily in the direction of the electric field and there is a smaller component of the drift that is perpendicular to the field when a uniform field is applied to the system. The velocity of the tip is uniform and its component along the electric field is equal to the strength of the field. When the CGLE system is driven by white noise, a diffusion law for the vortex core of the spiral wave is derived at long time explicitly. The diffusion constant is found to be D=T/C(2), in which T is the noise strength and C is the core asymptotic factor of the spiral wave. When the external force is a simple oscillation we find that the tip of the spiral wave drifts if the frequency of the external force is the same as that of the system. Our analytical results are verified using numerical simulations.

Three-state cyclic voter model extended with Potts energy

The cyclically dominated voter model on a square is extended by taking into consideration the variation of Potts energy during the nearest neighbor invasions. We have investigated the effect of surface tension on the self-organizing patterns maintained by the cyclic invasions. A geometrical analysis is also developed to study the three-color patterns. These investigations clearly indicate that in the "voter model" limit the pattern evolution is governed by the loop creation due to the overhanging during the interfacial roughening. Conversely, in the presence of surface tension the evolution is governed by spiral formation whose geometrical parameters depend on the strength of cyclic dominance.