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This article mainly studies the repulsion dynamics of spiral waves near the unexcitable zone. We propose that this repulsion behavior is caused by changes in the natural frequency of the spiral wave near the unexcitable boundary. Specifically, assuming that the drift motion of the spiral wave near the unexcitable boundary still satisfies the kinematic equation under the action of a periodic external force, the change in the natural frequency leads to the appearance of the repulsion motion. By the numerical simulation of the Barkley model with circular and flat unexcitable regions, it is found that the simulation results are consistent with the above assumptions.

Many multicellular communities propagate signals in a directed manner via excitable waves. Cell-to-cell heterogeneity is a ubiquitous feature of multicellular communities, but the effects of heterogeneity on wave propagation are still unclear. Here we use a minimal FitHugh-Nagumo-type model to investigate excitable wave propagation in a two-dimensional heterogeneous community. The model shows three dynamic regimes in which waves either propagate directionally, die out, or spiral indefinitely, and we characterize how these regimes depend on the heterogeneity parameters. We find that in some parameter regimes, spatial correlations in the heterogeneity enhance directional propagation and suppress spiraling. However, in other regimes, spatial correlations promote spiraling, a surprising feature that we explain by demonstrating that these spirals form by a second, distinct mechanism. Finally, we characterize the dependence of the spiral period on the degree of heterogeneity in the system by using techniques from percolation theory. Our results reveal that the spatial structure of cell-to-cell heterogeneity can have important consequences for signal propagation in cellular communities., 10 pages, 6 figures

In the present article, we demonstrate the emergence and existence of the spiral wave chimera-like transient pattern in coupled ecological systems, composed of prey-predator patches, where the patches are connected in a three-dimensional medium through local diffusion. We explore the transition scenarios among the several collective dynamical behaviors together with transient spiral wave chimera-like states and investigate the long time behavior of these states. The transition from the transient spiral chimera-like pattern to the long time synchronized or desynchronized pattern appears through the deformation of the incoherent region of the spiral core. We discuss the transient dynamics under the influence of the species diffusion at different time instants. By calculating the instantaneous strength of incoherence of the populations, we estimate the duration of the transient dynamics characterized by the persistence of the chimera-like spatial coexistence of coherent and incoherent patterns over the spatial domain. We generalize our observations on the transient dynamics in three-dimensional grid of diffusive ecological systems by considering two different prey-predator systems., Accepted for publication in Chaos: An Interdisciplinary Journal of Nonlinear Science

The lowest swirling wave mode arising in upright circular cylinders as a response to circular orbital excitation has been widely studied in the last decade, largely due to its high practical relevance for orbitally shaken bioreactors. Our recent theoretical study (Horstmann et al., J. Fluid Mech., vol. 891, 2020, A22) revealed a damping-induced symmetry breaking mechanism that can cause spiral wave structures manifested in the so far widely disregarded higher rotating wave modes. Building on this work, we develop a linear criterion describing the degree of spiralisation and classify different spiral regimes as a function of the most relevant dimensionless groups. The analysis suggests that high Bond numbers and shallow liquid layers favour the formation of coherent spiral waves. This result paved the way to find the predicted wave structures in our interfacial sloshing experiment. We present two sets of experiments, with different characteristic damping rates, verifying the formation of both coherent and overdamped spiral waves in conformity with the theoretical predictions.

We consider a two-dimensional array of heterogeneous nonlocally coupled phase oscillators on a flat torus and study the bound states of two counter-rotating spiral chimeras, shortly two-core spiral chimeras, observed in this system. In contrast to other known spiral chimeras with motionless incoherent cores, the two-core spiral chimeras typically show a drift motion. Due to this drift, their incoherent cores become spatially modulated and develop specific fingerprint patterns of varying synchrony levels. In the continuum limit of infinitely many oscillators, the two-core spiral chimeras can be studied using the Ott-Antonsen equation. Numerical analysis of this equation allows us to reveal the stability region of different spiral chimeras, which we group into three main classes---symmetric, asymmetric, and meandering spiral chimeras.

Multi-layer networks are quite fundamental to study structural and functional properties of various biological systems. In this study, a two-layer neuronal network is considered and the interactions between the spiral pattern in one layer and a homogeneous state in the other layer, under the effect of inter-layer bidirectional connection are investigated. Spiral wave has been confirmed to play a significant role in many complex systems. In this regard, in laminar structured systems in particular, it is crucial to study the dynamics of the spiral wave affected by the inter-layer interactions. Here, for each layer (sub-network), a regular network with eight-neighbor connection is designed. For the local dynamics of each neuron, the magnetic Fitzhugh–Nagumo (FN) neuronal model is introduced. The results show that depending on the level of interactions between the two layers, four different types of collective electrical activity can occur. When the inter-layer connection is weak, the layer with spiral pattern does not change while the homogeneous state of the other layer is broken by a blurred spiral pattern. As the inter-layer connection is strengthened, the dynamics of the spiral wave changes significantly, leading to unstable spiral wave and spiral turbulence. However, by a further increase in the inter-layer coupling strength, the spiral wave does not exist at all.

The effects of spatial heterogeneity on a two-dimensional complex Ginzburg–Landau equation model are studied. In general, the interaction of a pair of spiral waves with a large degree of heterogeneity in two different media will cause three different patterns: (a) Multiple spiral waves coexist in different media; (b) the spiral wave is swept away in one medium and remains in another medium; (c) all of the spiral waves are suppressed by travelling waves having different frequencies. These travelling waves are generated from interface reported before. It is found that the interface is a wave source that can generate travelling waves with different frequencies in two submedia to compete with the original spiral waves in two different media. The competition results depend on the frequencies of the original spiral wave and the two travelling waves. Furthermore, local periodic pacing can replace the effect of the interface and reproduce the corresponding results, which gives additional evidence that the interface works as a wave source. The results give new ideas in pattern control such that we can suppress and annihilate spiral waves by generating a large degree of heterogeneity using selected parameters.

The chaotic spatio-temporal electrical activity during life-threatening cardiac arrhythmias like ventricular fibrillation is governed by the dynamics of vortex-like spiral or scroll waves. The organizing centers of these waves are called wave tips (2D) or filaments (3D) and they play a key role in understanding and controlling the complex and chaotic electrical dynamics. Therefore, in many experimental and numerical setups it is required to detect the tips of the observed spiral waves. Most of the currently used methods significantly suffer from the influence of noise and are often adjusted to a specific situation (e.g. a specific numerical cardiac cell model). In this study, we use a specific type of deep neural networks (UNet), for detecting spiral wave tips and show that this approach is robust against the influence of intermediate noise levels. Furthermore, we demonstrate that if the UNet is trained with a pool of numerical cell models, spiral wave tips in unknown cell models can also be detected reliably, suggesting that the UNet can in some sense learn the concept of spiral wave tips in a general way, and thus could also be used in experimental situations in the future (ex-vivo, cell-culture or optogenetic experiments).

The development of new approaches to control cardiac arrhythmias requires a deep understanding of spiral wave dynamics. Optogenetics offers new possibilities for this. Preliminary experiments show that sub-threshold illumination affects electrical wave propagation in the mouse heart. However, a systematic exploration of these effects is technically challenging. Here, we use state-of-the-art computer models to study the dynamic control of spiral waves in a two-dimensional model of the adult mouse ventricle, using stationary and non-stationary patterns of sub-threshold illumination. Our results indicate a light intensity-dependent increase in cellular resting membrane potentials, which together with diffusive cell-cell coupling leads to the development of spatial voltage gradients over differently illuminated areas. A spiral wave drifts along the positive gradient. These gradients can be strategically applied to ensure drift-induced termination of a spiral wave, both in optogenetics and in conventional methods of electrical defibrillation.

We report on a stripe-core mixed spiral chimera in a system of nonlocally coupled phase oscillators, located on the spherical surface, where the spiral wave consisting of phase-locked oscillators is separated by a stripe-type region of incoherent oscillators into two parts. We analyze the existence and stability of the stripe-core mixed spiral chimera state rigorously, on the basis of the Ott–Antonsen reduction theory. The stability diagram for the stationary states including the spiral chimeras as well as incoherent state is presented. Our stability analysis reveals that the stripe-core mixed spiral chimera state emerges as a unique attractor and loses its stability via the Hopf bifurcation. We verify our theoretical results using direct numerical simulations of the model system.

Spiral waves are shown to undergo directional drifts in the presence of ac and polarized electric fields when their frequencies are twice of the spiral frequencies. Here, we propose a quantitative description for the spiral wave drift induced by weak electric fields, and provide the explicit equations for the spiral wave drift speed and direction. Numerical simulations are performed to demonstrate the quantitative agreement with analytical results in both weakly and highly excitable media.

To manage the development of the national innovation system used spiral waves. It is they that allows adequately reflect the focused wave interference of innovators and use spiral effect to manage their respective processes. To showcase the work of cognitive model of spiral of Archimedes spiral, used common enough in nature. This model allows you to influence the innovation processes by introducing specially designed coefficient of providing increased or decrease spiral effect in different periods of the national innovation system.

Rotating spiral waves of electrical activity in the heart can anchor to unexcitable tissue (an obstacle) and become stable pinned waves. A pinned rotating wave can be unpinned either by a local electrical stimulus applied close to the spiral core, or by an electric field pulse that excites the core of a pinned wave independently of its localization. The wave will be unpinned only when the pulse is delivered inside a narrow time interval called the unpinning window (UW) of the spiral. In experiments with cardiac monolayers, we found that other obstacles situated near the pinning centre of the spiral can facilitate unpinning. In numerical simulations, we found increasing or decreasing of the UW depending on the location, orientation and distance between the pinning centre and an obstacle. Our study indicates that multiple obstacles could contribute to unpinning in experiments with intact hearts.

Spiral tip can be considered as wave source of a spiral wave. A wave is sent out after the tip rotates one circle. Therefore, the dynamics of spiral tip is vital to understand the behavior of spiral waves. In this paper a new approach for identifying spiral tip dynamics via deterministic learning is proposed. First, a subsystem, which contains frequency and speed of a spiral tip’s movement, is used to identify spiral tip. Second, dynamics of subsystem are identified using the deterministic learning theory. It is shown that spiral tip can be accurately identified by using the proposed approach. Numerical experiments are included to demonstrate the effectiveness of the proposed method.

Spiral waves are spatiotemporal patterns, observed in complex networks of biological oscillators such as heart and brain. In this paper, at first a modified Hindmarsh–Rose model with external electromagnetic radiation and discontinuous exponential flux coupling is introduced. Then a 2-dimensional network of the proposed model is constructed. The spatiotemporal patterns of the network are investigated in two different cases of periodic and quasi-periodic external magnetic excitations, by varying the external force parameters. As a consequence, the effective factors on the formation and elimination of spiral waves are reported.

Spiral waves rigidly rotating in excitable media sometimes play a constructive role in self-organization, while in many cases they cause an undesirable and dangerous activity. An understanding of spiral wave kinematics can help to control or to prevent this self-sustained activity. A description of the spiral wave kinematics performed by use of a free-boundary approach, reveals the selection principle which determines the shape and the rotation frequency of spiral waves in an unbounded medium with a given excitability. It is shown that a rigidly rotating spiral in a medium with strongly reduced refractoriness is supported within an excitability range restricted by two universal limits. At the low excitability limit, the spiral core radius diverges, while it vanishes at the high excitability limit and the spiral wave resembles the Yin-Yang pattern.

Cardiac arrhythmias can be caused by the occurrence of electrical spiral waves and spatiotemporal chaos in the cardiac tissues, as well as by the changes of cardiac tissues resulting from the electrical coupling of cardiomyocytes to fibroblasts (M-F coupling). How to control the arrhythmia induced by spiral wave or spatiotemporal chaos is the problem which has attracted much attention of scientists. In this paper, a two-layer composite medium is constructed by using cardiomyocytes and fibroblasts. Luo-Rudy phase I cardiac model and passive model of fibroblast are used to study the effects of the M-F coupling on the formation of spiral wave and the control of spiral wave and spatiotemporal chaos in a two-layer composite medium. A control scheme that the spiral waves and spatiotemporal chaos are controlled by increasing the coupling strength between cells is proposed. The numerical results show that the M-F coupling has an important influence on the dynamics of spiral wave. With the increase of the density of fibroblasts, the M-F coupling may result in spiral wave meandering and spiral wave breaking into spatiotemporal chaos, and even induce the transition from spatiotemporal chaos (or spiral wave) to no wave. The eliminating spiral wave and spatiotemporal chaos in the composite medium by increasing the coupling strength between cells is only effective in most of cases, depending on the role played by fibroblasts. When fibroblasts act as current sinks for the cardiomyocyte, the spiral wave and spatiotemporal chaos are effectively eliminated only in most of cases by increasing the coupling strength between cells, and the controlled area is small. When fibroblasts act as a current source for the cardiomyocyte, increasing the coupling strength between cells to a value higher than a critical value can effectively terminate spiral wave and spatiotemporal chaos, and the controllable area is greatly increased compared with the former. Increasing the coupling strength between cardiomyocytes is a key factor in controlling the spiral waves and spatiotemporal chaos.

Spiral waves have been found to appear alternatively with plane waves in the brain cerebral cortex, which has a significant effect on neuron firing behaviors. In this paper, we propose a functional firing network based on the correlated firing behaviors among neuronal populations and use the complex network method to investigate the effects of spiral waves and plane waves on the structure and function of the network. We first analyze the correlation coefficient and the largest eigenvalue of the functional firing network. We find a larger range distribution of correlation coefficients and greater largest eigenvalue of the functional firing network for spiral waves than those for plane waves, which indicates that spiral waves induce higher network synchronization. In addition, we explore the topological structure of the functional firing network using the complex network method. We find that the functional firing network for spiral waves has a larger degree and global efficiency and a lower modularity and characteristic path length than that for plane waves, revealing that spiral waves contribute to neural information transmission and strengthen the functional integration. Our work not only provides new insights for studying spatiotemporal patterns, but is also helpful for explaining the modulation of spiral waves on brain function.

The work presents a mechanism of spiral wave initiation due to the specific boundary conditions on a border of cardiac tissue defect. There are known scenarios when anatomical or functional defects in cardiac tissue may provoke the spiral wave origination, including unidirectional blockage while passing through the narrow gates, bent over critical curvature wave fronts, inhomogeneous recovery of the tissue, etc. We show a new scenario of spiral wave breakup on a small defect, which is unexcitable but permeable for ionic currents supporting the excitation wave. It was believed that such defects stabilize the rotating wave; however, as shown, instead of stabilizing it leads to the spiral breakup and subsequent multiplication of the rotating waves.

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.

The spiral tip is vital to the understanding of the spiral wave behaviors. Most studies of spiral tip dynamics focused on the prevention, control, and elimination of spiral wave, while few studies focused on the recognition of spiral wave. In real systems with the spiral wave, the recognition of the spiral wave should be before control or elimination. In the paper, we study the recognition of the spiral tip via deterministic learning. It mainly consists of two phases: the identification phase and the recognition phase. In the identification phase, the dynamics of spiral tips of the training set is accurately identified by using deterministic learning. In the recognition phase, a set of errors is obtained for a test spiral tip by employing an estimator model. Finally, the recognition of test spiral tip is achieved according to the smallest error principle. Numerical simulations based on the spiral tip generated by the Barkley model are performed to demonstrate the effectiveness and feasibility of the proposed method.

We propose a model for describing oscillatory chemical reactions involving a variety of immiscible solvents. This model describes the interaction between chemical oscillations on both sides of the interface of different solvents and the effect of the interface on chemical oscillations. Based on this model, we study the spiral chemical wave in a chemical oscillation with interfaces of different solvents, and report a new type of spiral wave. The interface does not hinder the propagation of chemical waves, which can continue to propagate through the interface. The frequency of the chemical wave is increased after the chemical wave passed through the interface. The work of this paper enriched the categories of the dynamics of pattern formation. We believe that the results of this study can provide direction and guidance for chemical experimental research about interfacial oscillatory chemical reactions.

Spiral waves represent an important example of dissipative structures observed in many distributed systems in chemistry, biology and physics. By definition, excitable media occupy a stationary resting state in the absence of external perturbations. However, a perturbation exceeding a threshold results in the initiation of an excitation wave propagating through the medium. These waves, in contrast to acoustic and optical ones, disappear at the medium's boundary or after a mutual collision, and the medium returns to the resting state. Nevertheless, an initiation of a rotating spiral wave results in a self-sustained activity. Such activity unexpectedly appearing in cardiac or neuronal tissues usually destroys their dynamics which results in life-threatening diseases. In this context, an understanding of possible scenarios of spiral wave initiation is of great theoretical importance with many practical applications. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

There are many examples of excitable media, such as the heart, that can show complex dynamics and where control is a challenging task. Heavy means like a strong electric shock are nowadays still necessary to control and terminate ventricular fibrillation (VF). It is known that heterogeneities in an excitable medium can stabilize the activity, e.g., spiral waves can pin to such obstacles. This might also be a reason for the persistence of VF and the difficulty to control it. Previous studies investigated systems with a single pinned spiral wave and demonstrated how the spiral can be unpinned. In this article, we extend this case and investigate a generic excitable system with multiple pinned spiral waves. We describe a control technique that allows the simultaneous unpinning of pinned spiral waves. Apart from theoretical considerations, we provide numerical evidence that the proposed technique is superior to the underdrive pacing method that has reportedly high success rates when applied to a single pinned spiral.

Propagation of non-linear waves is key to the functioning of diverse biological systems. Such waves can organize into spirals, rotating around a core, whose properties determine the overall wave dynamics. Theoretically, manipulation of a spiral wave core should lead to full spatiotemporal control over its dynamics. However, this theory lacks supportive evidence (even at a conceptual level), making it thus a long-standing hypothesis. Here, we propose a new phenomenological concept that involves artificially dragging spiral waves by their cores, to prove the aforementioned hypothesis in silico, with subsequent in vitro validation in optogenetically modified monolayers of rat atrial cardiomyocytes. We thereby connect previously established, but unrelated concepts of spiral wave attraction, anchoring and unpinning to demonstrate that core manipulation, through controlled displacement of heterogeneities in excitable media, allows forced movement of spiral waves along pre-defined trajectories. Consequently, we impose real-time spatiotemporal control over spiral wave dynamics in a biological system., eLife digest From a spinning galaxy to a swarm of honeybees, rotating spirals are widespread in nature. Even within the muscles of the heart, waves of electrical activity sometimes rotate spirally, leading to irregular heart rhythms or arrhythmia – a condition that can be fatal. Irrespective of where they occur, spiral waves organize around a center or core with different biophysical properties compared to the rest of the medium. The properties of the core determine the overall dynamics of the spiral. This means that, theoretically, it should be possibly to completely control a spiral wave just by manipulating its core. Now, Majumder, Feola et al. have tested this long-standing hypothesis using a combination of computer modeling and experiments with single layers of rat heart cells grown in a laboratory. First, the heart cells were genetically modified so that their electrical properties could be altered with light; in other words, the cells were put under optical control. Next, by using of a narrow beam of light, Majumder, Feola et al. precisely controlled the electrical properties of a small number of cells, which then attracted and supported a rotating spiral wave by acting as its new core. Moving the light beam allowed the core of the spiral wave to be shifted too, meaning the spiral wave could now be steered along any desired path in the cell layer. Majumder, Feola et al. hope that these underlying principles may one day provide the basis of new treatments for irregular heartbeats that are more effective and less damaging to the heart than existing options. Yet first, more work is needed to translate these findings from single layers of cells to actual hearts.

Excitable media have been extensively used to study the pattern formations, most prominently the spiral wave (2D) and the scroll wave (3D) patterns in chemical reaction, cardiac tissue, etc. through different forms of the Barkley model, governed by a system of nonlinear reaction-diffusion (R-D) equations. To the best of our knowledge, most of the previous numerical results have been obtained by explicit schemes which are conditionally stable and lower order accurate. In the current study, we reconstruct an existing higher order compact (HOC) finite difference scheme to discretize the highly nonlinear equations governing the patterns. The scheme which is implicit in nature and unconditionally stable is seen to efficiently capture the patterns. Furthermore, we also determine the spiral tip path by post processing our HOC data where we specifically deal with petal formations arising out of the tip trajectories.

Synchronization of spiral waves in an excitable bilayer with multiple coupled patches is studied. The bilayer, with a larger coupling strength, can show a variety of behaviors, and in order of decreasing the distance between two adjacent patches, they include single spiral synchronization, multispiral synchronization, turbulent asynchronization and spiral-target asynchronization. For the latter two cases, the function SYN(t), introduced by us, has a periodical asymptotic behavior, which means the existence of phase-locked phenomena. The similar dynamics can also be observed by increasing the size of the coupling patches with a fixed distance. A larger coupling strength is helpful for synchronization, which can be understood by checking the dynamics of the coupled two-point model.

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 novel result shows that spiral wave breakup, or interactions with other waves, are not necessary to initiate chaos in spiral waves.

Pinning of vortices by obstacles plays an important role in various systems. In the heart, anatomical reentry is created when a vortex, also known as the spiral wave, is pinned to an anatomical obstacle, leading to a class of physiologically very important arrhythmias. Previous analyses of its dynamics and instability provide fine estimates in some special circumstances, such as large obstacles or weak excitabilities. Here, to expand theoretical analyses to all circumstances, we propose a general theory whose results quantitatively agree with direct numerical simulations. In particular, when obstacles are small and pinned spiral waves are destabilized, an accurate explanation of the instability in two-dimensional media is provided by the usage of a mapping rule and dimension reduction. The implications of our results are to better understand the mechanism of arrhythmia and thus improve its early prevention.

Self-sustained waves of electrophysiological activity can cause arrhythmia in the heart. These reentrant excitations have been associated with spiral waves circulating around either an anatomically defined weakly conducting region or a functionally determined core. Recently, an ablation procedure has been clinically introduced that stops atrial fibrillation of the heart by destroying the electrical activity at the spiral core. This is puzzling because the tissue at the anatomically defined spiral core would already be weakly conducting, and a further decrease should not improve the situation. In the case of a functionally determined core, an ablation procedure should even further stabilize the rotating wave. The efficacy of the procedure thus needs explanation. Here, we show theoretically that fundamentally in any excitable medium a region with a propagation velocity faster than its surrounding can act as a nucleation center for reentry and can anchor an induced spiral wave. Our findings demonstrate a mechanistic underpinning for the recently developed ablation procedure. Our theoretical results are based on a very general and widely used two-component model of an excitable medium. Moreover, the important control parameters used to realize conditions for the discovered phenomena are applicable to quite different multicomponent models.

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.

Autapse is a type of synapse that connects axon and dendrites of the same neuron, and the effect is often detected by close-loop feedback in axonal action potentials to the owned dendritic tree. An artificial autapse was introduced into the Hindmarsh-Rose neuron model, and a regular network was designed to detect the regular pattern formation induced by autapse. It was found that target wave emerged in the network even when only a single autapse was considered. By increasing the (autapse density) number of neurons with autapse, for example, a regular area (2×2, 3×3, 4×4, 5×5 neurons) under autapse induced target wave by selecting the feedback gain and time-delay in autapse. Spiral waves were also observed under optimized feedback gain and time delay in autapses because of coherence-like resonance in the network induced by some electric autapses connected to some neurons. This confirmed that the electric autapse has a critical role in exciting and regulating the collective behaviors of neurons by generating stable regular waves (target waves, spiral waves) in the network. The wave length of the induced travelling wave (target wave, spiral wave), because of local effect of autapse, was also calculated to understand the waveprofile in the network of neurons.

This paper mainly researched the effect of dichotomous noise on the dynamics of spiral wave, provided a method to eliminate spiral waves by dichotomous noise. Spiral wave dynamics show regular changes under the action of dichotomous noise with noise strength, time correlation parameter, this rule change can be explained by spectral analysis.

The influence of fractal heterogeneity on a spiral wave in an excitable system is numerically studied based on the Barkley model. The heterogeneity is implemented by letting the diffusive coefficient in the heterogeneous area be different from the other area. The results show that fruitful transitions of the spiral tip trajectories are induced by the fractal heterogeneity. In particular, when the heterogeneity increases to a sufficiently high level the spiral tip trajectory always changes to a stable rotating trajectory (closed-circle tip trajectory), whatever transitions have been induced by a lower level of heterogeneity. We qualitatively ascribe the transitions to the attraction on the spiral tip exerted by the heterogeneous area. (C) 2013 Elsevier B.V. All rights reserved.

We study a model for cell sorting based in the presence of differential chemotactic sensitivities. The chemical waves which are responsible for the cell motion propagate along some spiral waves. We prove rigorously that cells with larger chemotactic sensitivity are trapped in a region close to the center of the spiral waves if these propagate along some archimedian spirals.

We investigate experimentally and numerically the synchronization of two-dimensional spiral wave patterns in the Belousov-Zhabotinsky reaction due to point-to-point coupling of two separate domains. Different synchronization modalities appear depending on the coupling strength and the initial patterns in each domain. The behavior as a function of the coupling strength falls into two qualitatively different regimes. The weakly coupled regime is characterized by inter-domain interactions that distorted but do not break wave fronts. Under weak coupling, spiral cores are pushed around by wave fronts in the other domain, resulting in an effective interaction between cores in opposite domains. In the case where each domain initially contains a single spiral, the cores form a bound pair and orbit each other at quantized distances. When the starting patterns consist of multiple randomly positioned spiral cores, the number of cores decreases with time until all that remains are a few cores that are synchronized with a partner in the other domain. The strongly coupled regime is characterized by interdomain interactions that break wave fronts. As a result, the wave patterns in both domains become identical.

A spiral wave is a macroscopic dynamic of excitable media that plays an important role in several distinct systems, including the Belousov-Zhabotinsky reaction, seizures in the brain, and lethal arrhythmia in the heart. Because spiral wave dynamics can exhibit a wide spectrum of behaviors, its precise quantification can be challenging. Here we present a hybrid geometric and information-theoretic approach to quantifying spiral wave dynamics. We demonstrate the effectiveness of our approach by applying it to numerical simulations of a two-dimensional excitable medium with different numbers and spatial patterns of spiral waves. We show that, by defining information flow over the excitable medium, hidden coherent structures emerge that effectively quantify the information transport underlying spiral wave dynamics. Most importantly, we find that some coherent structures become more clearly defined over a longer observation period. These findings validate our approach to quantitatively characterize spiral wave dynamics by focusing on information transport. Our approach is computationally efficient and is applicable to many excitable media of interest in distinct physical, chemical and biological systems. Our approach could ultimately contribute to an improved therapy of clinical conditions such as seizures and cardiac arrhythmia by identifying potential targets of interventional therapies., 27 pages, 6 figures

The interaction of waves and obstacles is simulated by adding heterogeneities to a FitzHugh-Nagumo model and a cellular automata model. The cellular automata model is formulated to account for heterogeneities by modelling the interaction between current sources and current sinks. In both models, wave fronts propagate if the size of the heterogeneities is small, and block if the size of the heterogeneities is large. For intermediate values, wave fronts break up into numerous spiral waves. The theoretical models give insights concerning spiral wave formation in heterogeneous excitable media. (c) 2002 American Institute of Physics.

The control of spiral wave breakup due to Doppler instability has been investigated. It is found that by applying the spatiotemporal modulation method with suitable control parameters, spiral wave breakup can be prevented. Further numerical simulations show that preventing spiral wave breakup is a result of the decrease of meandering motion of the spiral tip.

International audience; During cardiac arrhythmia, functional reentries may take the form of spiral waves. The purpose of this study was to induce spiral waves by an electrical stimulation of cultured neonatal rat cardiomyocytes using a microelectrode arrays technology. In basal conditions, cardiac muscle cells in monolayer culture displayed a planar wavefront propagation. External electrical impulse trains induced severe arrhythmia and spiral waves appeared. This in vitro generation of spiral wave opens a new way to test the anti-arrhythmic drugs and for strategies at microscopically scale.

Using a Barkley model as an example, we study spiral waves and spiral tips in a gradient excitable medium. The gradient distribution of parameters is introduced to depict the inhomogeneous medium. It is found that the parameter fluctuations play an important role in the morphology of spiral pattern and the movements of spiral tips. For varied gradient parameters, it is observed that there exist three kinds of spiral behaviors, stable rotation, rebound of spiral tip from the boundary, and spiral breakup.

The interactions of several spiral waves with different independent rotation frequencies are studied in a model of two-dimensional complex Ginzburg-Laudau equation. We find a general coexistence phenomenon, non-local non-phase-locking-invasion coexistence, that is, the non-slowest spiral wave call survive and not be killed by the fastest spiral wave as it is insulated from the fastest one with the sacrifice of the slowest one, which stays in the spatial position between the fastest spiral and the non-slowest one. Both the parameter non-monotonicity and the non-phase-locking invasion between the fastest and the slowest spiral waves play key roles in this phenomenon. Importantly, the results could give a general idea for extensively observed coexistence of spiral waves in various inhomogeneous circumstances. (C) 2007 Elsevier Ltd. All rights reserved.

In this paper, we studied the effect of Gaussian coloured noise on the formation and instability of spiral waves described by one class of modified FitzHugh–Nagumo equation. It was found that Gaussian coloured noise plays a constructive role in the formation, transition and instability of spiral wave. Too weak or too strong noise may act against the formation of spiral waves. At a certain noise level, spiral wave is maintained in a medium, in which spiral wave cannot be observed in the absence of the noise. It is difficult to make a stable spiral wave into unstable state by Gaussian coloured noise, unless the noise level is very high. The parameter regions of Gaussian coloured noise for spiral forming and spiral instability were given and discussed with numerical simulations.

The phenomenon of wave grouping, in which the dense waves and the sparse waves can form groups in front of the spiral tip when the spiral wave is meandering, has been reported in a chemical reaction system recently. We present a method to realize the phenomenon of wave grouping by applying an external field to the system. The numerical simulations are carried out on the basis of the FitzHugh–Nagumo equations.

Spiral waves have been controlled by generating target waves with a localized inhomogeneity in the oscillatory medium. The competition between the spiral waves and target waves is discussed. The effect of the localized inhomogeneity size has also been studied.

Spiral waves and spatiotemporal chaos are sometimes harmful and should be controlled. In this letter we present a feedback scheme to eliminate them. We first collect feedback signals at a certain time t0. Then wait for the system at the excitable position to enter the recovering state. When the time comes, the feedback signals are added. This scheme has two advantages. Firstly, the tip can be eliminated together with the body of spiral wave. Secondly, the injected feedback signals can be very weak and the duration can be very short so that the original system is nearly not to be affected, which is important for practical applications.

Spontaneous spiral breakup scenarios are investigated in a reaction-diffusion glycolytic medium exhibiting birhythmic dynamics, in which, depending on the direction the system is driven toward the birhythmic window, near-core breakup and far-field breakup have been observed. Our analysis suggests that the breakup is because the emergence of a second attractor splits the concentration domain that spans the spiral wave activity into two attraction basins, causing that different parts of the spiral evolve, respectively, toward the two limit cycles and then break up. Affected by the attraction strength of the new limit cycle, intermittent breakup behavior is uncovered during the near-core breakup.

We numerically investigate the boundary-induced spiral wave drift in the complex Ginzburg–Landau equation. We find some novel phenomena for the spiral drifting dynamics such as the chaotic behaviors, the transient chaos and asymmetrical attractors.

Computational verb cellular networks (CVCNs) are cellular networks, of which the local rules are computational verb rules. Many patterns had been found in CVCN previously. In this paper, newly-discovered one-, two-, three- and four-armed spiral waves in CVCNs are presented. Eight-armed spiral waves appear briefly co-existing with four-armed patterns at a low frequency. When initial states are uniformly randomly chosen from interval between 0 and 1, more than one spiral wave may be formed in a CVCN. The rotors of four-armed rotating spiral waves move slowly. All these spiral waves co-exist with checkerboard patterns, which are more visible when the array size is small.

The spontaneous drift of the spiral wave in a finite domain in the complex Ginzburg–Landau equation is investigated numerically. By using the interactions between the spiral wave and its images, we propose a phenomenological theory to explain the observations.