Target Waves Research Articles

Emergence of target waves in neuronal networks due to diverse forcing currents

The electric activities of neurons could be changed when ion channel block occurs in the neurons. External forcing currents with diversity are imposed on the regular network of Hodgkin-Huxley (HH) neuron, and target waves are induced to occupy the network. The forcing current I1 is imposed on neurons in a local region with m0×m0 nodes in the network, neurons in other nodes are imposed with another forcing current I2. Target wave could be developed to occupy the network when the gradient forcing current (I1–I2) exceeds certain threshold, and the formation of target wave is independent of the selection of boundary condition. It is also found that the developed target wave can decrease the negative effect of ion channel block and suppress the spiral wave, and thus channel noise is also considered. The potential mechanism of formation of target wave could be that the gradient forcing current (I1–I2) generates quasi-periodical signal in local area, and the propagation of quasi-periodical signal induces target-like wave due to mutual coupling among neurons in the network.

Measurement of Large Spiral and Target Waves in Chemical Reaction-Diffusion-Advection Systems: Turbulent Diffusion Enhances Pattern Formation

In the absence of advection, reaction-diffusion systems are able to organize into spatiotemporal patterns, in particular spiral and target waves. Whenever advection is present that can be parametrized in terms of effective or turbulent diffusion D(*), these patterns should be attainable on a much greater, boosted length scale. However, so far, experimental evidence of these boosted patterns in a turbulent flow was lacking. Here, we report the first experimental observation of boosted target and spiral patterns in an excitable chemical reaction in a quasi-two-dimensional turbulent flow. The wave patterns observed are ~50 times larger than in the case of molecular diffusion only. We vary the turbulent diffusion coefficient D(*) of the flow and find that the fundamental Fisher-Kolmogorov-Petrovsky-Piskunov equation, v(f) proportional sqrt[D(*)], for the asymptotic speed of a reactive wave remains valid. However, not all measures of the boosted wave scale with D(*) as expected from molecular diffusion, since the wave fronts turn out to be highly filamentous.

Spiral–pacemaker interactions in a mathematical model of excitable medium

Interactions of a spiral wave with a pacemaker is studied in a mathematical model of two dimensional excitable medium. Faster pacemakers emitting target waves can abolish spirals by driving them to the border of the medium. Our study shows that a slower pacemaker can modify spiral wave behavior by changing the motion of the spiral core. We analyze the dynamics of the spiral wave near the spiral core and away from the core as a function of size and period of the pacemaker. The pacemaker can cause the spiral wave to drift towards it, and either speed up or slow down the reentrant activity. Furthermore, the drift induced by the pacemaker can result in irregular or quasiperiodic dynamics even at sites away from the pacemaker. These results highlight the influence of pacemakers on complex spiral wave dynamics.

Evolution of spiral waves in indirectly coupled excitable medium with time-delayed coupling

The interaction between two spiral waves in two-layer excitable medium coupled indirectly through a passive medium and with time-delayed coupling is investigated by using the Bär model. The numerical results show that time-delayed coupling can either facilitate the synchronization of two spiral waves or lead to the transition from spiral wave to collective oscillation, different target waves, spatiotemporal chaos or rest state. Period-2 and period-3 spiral waves and the meander or drift of spiral wave are observed in the coupled excitable medium. The physical mechanism underlying these phenomena is discussed.

3D Imaging of Dead Sea Area Using Weighted Multipath Summation: A Case Study

The formation of sinkholes along the Dead Sea is caused by the rapid decline of the Dead Sea level, as a possible result of human extensive activity. According to one of the geological models, the sinkholes in several sites are clustered along a narrow coastal strip developing along lineaments representing faults in NNW direction. In order to understand the relationship between a developing sinkhole and its tectonic environment, a high-resolution (HR) three-dimensional (3D) seismic reflection survey was carried out at the western shoreline of the Dead Sea. A recently developed 3D imaging approach was applied to this 3D dataset. Imaging of subsurface is performed by a spatial summation of seismic waves along time surfaces using recently proposed multipath summation with proper weights. The multipath summation is performed by stacking the target waves along all possible time surfaces having a common apex at the given point. This approach does not require any explicit information on parameters since the involved multipath summation is performed for all possible parameters values within a wide specified range. The results from processed 3D time volume show subhorizontal coherent reflectors at approximate depth of 50–80 m which incline on closer location to the exposed sinkhole and suggest a possible linkage between revealed fault and the sinkholes.

Influence of Control Parameters on the Competition Between Spiral Waves and Target Waves

In this paper we systematically investigate the influence of control parameters on the competition results between spiral waves and target waves. Driving frequency f, amplitude A and injection area n of the input signals are three important parameters and the competition results between spiral waves and target waves are influenced by these three parameters remarkably. Based on these understandings we can control spiral waves effectively by suitable combination these parameters to generate faster target waves. And the effective controllable parameter regions are also studied.

Design and application of feedback-sustained target waves in excitable medium

In this paper, we present a formulas of rational ratios between the frequency of external stimulus and its paced target waves, considering the existence of the absolute refractory period and the relative refractory period. Based on this, we design a self-sustained high-frequency target waves by feeding back few periphery field points to the center to responsibly play the role of source for pacemaker. Such constructed target waves, to all appearances, are shown as a repetition of their initial condition. We explore further in application that those feedback-sustained targets can eliminate both a freely rotating spiral waves and a pinned ones only if frequencies, feedback lengths and radius of obstacles are appropriately chosen. After feedback stimuli are removed, the system eventually returns to the rest state.

Revising the Role of Species Mobility in Maintaining Biodiversity in Communities with Cyclic Competition

One of the most crucial tasks faced by biologists today is revealing the mechanisms which account for biodiversity, yet we are still far from a full understanding of these mechanisms, and in particular the role of spatially heterogeneous population distributions. Recently, the spatially heterogeneous coexistence seen in cyclic competition models--in which species compete as in the game rock-paper-scissors--has brought them to the fore as a paradigm for biodiversity. Research into cyclic competition has so far been focused almost exclusively on stochastic lattice models with discrete space, which ignore several key dynamical aspects. In particular, such models usually assume that species disperse at the same speed. This paper aims to extend our understanding of cyclic competition by applying a reaction-diffusion Lotka-Volterra scheme to the problem, which allows us to vary the mobility of each species, and lets us take into account cyclic competition with more complex underlying mechanisms. In this paper we reveal an entirely new kind of cyclic competition-'conditional' cyclic competition, with a different underlying mechanism to 'classic' cyclic competition-and we show that biodiversity in communities with cyclic competition in fact depends heavily on the ratios between the species mobilities. Furthermore, we show that this dependence can be completely different for conditional and classic cyclic competition. We also present a wide range of spatiotemporal patterns which are formed in the system, including spiral and target waves, spiralling patches, and irregular chaotic patches. We show that the previously unknown case of conditional cyclic competition is host to a scenario of patchy co-invasion, where the spread of the population front takes place via the formation, splitting and propagation of patches of high species density. This is also an example of invasional meltdown because one competitor facilitates the invasion of the other, but unlike well-known cases of invasional meltdown the co-invaders in this system are not mutualists but antagonistic competitors, and the overall result mitigates, rather than amplifies, the damage done to the native ecosystem.

Effects of sodium and potassium ion channel fluctuation on the spatiotemporal patterns of neuronal network

Taking into account sodium and potassium ion channel noises, the evolution of the patterns of neuronal networks is investigated. No matter what kind of ion channel noise is working, with coupling coefficient increasing, the spatiotemporal patterns of the neuronal network can be evolved into spiral waves when temperature and noise strength are given, and there is a coupling coefficient threshold for forming a spiral wave. The analysis shows that sodium ion channel noise contributes to the formation of spiral waves in neurons network, while the potassium ion channel noise is not conducive to the formation of spiral waves. In addition, it is found that lower temperature can make the neurons network more sensitive to noise. Finally, the transformation of spiral waves into target waves, in the case of specific parameters is discussed.

Self-sustained target waves in excitable media with only a long-range link

In this paper we investigate spatiotemporal pattern formation in excitable media with only a long-range link. Besides the trivial solutions of spiral patterns, we find the asymptotic self-sustained target waves in the autonomous tissues. The wave source supporting this kind of new pattern is the oscillatory one-dimensional Winfree-loop self-organized under the presence of a long-range link, which is explored by the dominant phase-advanced driving method. Based on this understanding we can effectively regulate the oscillations of excitable media by suitably arranging the long-range link, including construction of self-sustained target waves with controllable period and wave length, or manipulation of system states between different patterns.

Pattern formation in oscillatory complex networks consisting of excitable nodes

Oscillatory dynamics of complex networks has recently attracted great attention. In this paper we study pattern formation in oscillatory complex networks consisting of excitable nodes. We find that there exist a few center nodes and small skeletons for most oscillations. Complicated and seemingly random oscillatory patterns can be viewed as well-organized target waves propagating from center nodes along the shortest paths, and the shortest loops passing through both the center nodes and their driver nodes play the role of oscillation sources. Analyzing simple skeletons we are able to understand and predict various essential properties of the oscillations and effectively modulate the oscillations. These methods and results will give insights into pattern formation in complex networks and provide suggestive ideas for studying and controlling oscillations in neural networks.

Traveling Bumps and Their Collisions in a Two-Dimensional Neural Field

A neural field is a continuous version of a neural network model accounting for dynamical pattern forming from populational firing activities in neural tissues. These patterns include standing bumps, moving bumps, traveling waves, target waves, breathers, and spiral waves, many of them observed in various brain areas. They can be categorized into two types: a wave-like activity spreading over the field and a particle-like localized activity. We show through numerical experiments that localized traveling excitation patterns (traveling bumps), which behave like particles, exist in a two-dimensional neural field with excitation and inhibition mechanisms. The traveling bumps do not require any geometric restriction (boundary) to prevent them from propagating away, a fact that might shed light on how neurons in the brain are functionally organized. Collisions of traveling bumps exhibit rich phenomena; they might reveal the manner of information processing in the cortex and be useful in various applications. The trajectories of traveling bumps can be controlled by external inputs.

Pattern formation, synchronization, and outbreak of biodiversity in cyclically competing games

Species in nature are typically mobile over diverse distance scales, examples of which range from bacteria run to long-distance animal migrations. These behaviors can have a significant impact on biodiversity. Addressing the role of migration in biodiversity microscopically is fundamental but remains a challenging problem in interdisciplinary science. We incorporate both intra- and inter-patch migrations in stochastic games of cyclic competitions and find that the interplay between the migrations at the local and global scales can lead to robust species coexistence characterized dynamically by the occurrence of remarkable target-wave patterns in the absence of any external control. The waves can emerge from either mixed populations or isolated species in different patches, regardless of the size and the location of the migration target. We also find that, even in a single-species system, target waves can arise from rare mutations, leading to an outbreak of biodiversity. A surprising phenomenon is that target waves in different patches can exhibit synchronization and time-delayed synchronization, where the latter potentially enables the prediction of future evolutionary dynamics. We provide a physical theory based on the spatiotemporal organization of the target waves to explain the synchronization phenomena. We also investigate the basins of coexistence and extinction to establish the robustness of biodiversity through migrations. Our results are relevant to issues of general and broader interest such as pattern formation, control in excitable systems, and the origin of order arising from self-organization in social and natural systems.

Towards Arithmetic Circuits in Sub‐Excitable Chemical Media

AbstractA sub‐excitable Belousov–Zhabotinsky medium exhibits localized travelling excitations (in contrast to an excitable medium exhibiting target or spiral waves). Initially assymetric perturbations give birth to excitation wave‐fragments. The shape and velocity vectors of the wave‐fragments are conserved, meaning they can travel for substantial distances in the reaction media. When the wave‐fragments collide they may reflect, merge, or annihilate. We interpret wave‐fragments as values of logical variables, and the post‐collision states of the fragments as outputs of logical gates. We show how to cascade logical gates in primitive arithmetical circuits.

Reinfection induced disease in a spatial SIRI model

Spatial models are widely used in epidemiology to investigate persistence and extinction of disease as well as their spatial patterns. One of the most important issues in studying epidemic models is the role of infection on the persistence and extinction of the disease. In this paper, we investigate a spatial susceptible-infected-recovered-infected model using cellular automata. We show that, in the regime where disease disappears in the susceptible-infected-recovered-susceptible model, spiral and target waves will emerge in the two-dimensional space due to the reinfection. The obtained results may point out that reinfection has great influence on the epidemic spreading, which enriches the findings of spatiotemporal dynamics in epidemic models.

Target-wave to spiral-wave pattern transition in a discrete Belousov-Zhabotinsky reaction driven by inactive resin beads

Wave pattern formation and transition in chemical and biochemical reaction systems can reveal the system properties. We investigate the pattern transition from target waves to spiral waves upon the increment of inactive beads in a discrete system model, where ion-exchange resin loaded with Belousov-Zhabotinsky catalyst corresponds to the active beads. The results show that inactive beads slow the propagation speed of target waves and increase the wave frequency. As the population of inactive beads increases, clusters are formed, which then break waves into segments where bounded spiral pairs are generated and separated into individual spirals. From this observation, we conclude that the population of inactive resin beads acts as the bifurcation parameter controlling the wave pattern transition from targets to spirals.

PARAMETER FLUCTUATION-INDUCED PATTERN TRANSITION IN THE COMPLEX GINZBURG–LANDAU EQUATION

Parameter fluctuation, which is often induced by the noise, temperature, deformation of the media etc., plays an important role in changing the dynamics of the system. In this paper, the problem of parameter fluctuation-induced pattern transition in the Complex Ginzburg–Landau equation (CGLE) is investigated. At first, the perpendicular-gradient initial values are used to generate spiral wave and spiral turbulence under appropriate parameters. At second, the parameter is perturbed with the periodical and/or random signal to simulate the parameter fluctuation, respectively. Then a class of linear error feedback is used to induce transition of the spiral wave and spiral turbulence. It is found that target waves can be induced by the complete feedback forcing, while the local feedback forcing seldom induce a target wave. In the case of spiral turbulence, spiral wave is generated and the spiral turbulence is removed by the new appeared spiral wave as the linear error feedback began to work on the whole media. Finally, the common negative feedback is also used to control the parameter-fluctuated CGLE, and the results are compared with the linear error feedback control, it is found that the whole system become homogeneous when the negative feedback is imposed on the whole media, and the local negative feedback can induce new target wave to remove the spiral wave while it is in vain to generate new target or spiral wave to overcome and eliminate the spiral turbulence.

Diverse self-sustained oscillatory patterns and their mechanisms in excitable small-world networks

Diverse self-sustained oscillatory patterns and their mechanisms in small-world networks (SWNs) of excitable nodes are studied. Spatiotemporal patterns of SWNs are sensitive to long-range connection probability P and coupling intensity D . By varying P in wide range with fixed D , we observe totally six types of asymptotic states: pure spiral waves, pure self-sustained target waves, patterns of mixtured spirals and target waves, pseudospiral turbulence, synchronizing oscillations, and rest state. The parameter conditions for all these states are specified, and the mechanisms of these states are heuristically explained. In particular, the mechanism of emergence and annihilation of synchronizing oscillations is explained by using the shortest path length analysis.

Pattern formation in spatial games

Pattern formation is crucial for maintaining biodiversity and cooperation in nature and human society. We review recent works on the formation of spatial patterns in the framework of evolutionary games, involving rock-paper-scissors game and social dilemmas. The rock-paper-scissors game, as a basic paradigm for characterizing nonhierarchical cyclical competition in ecosystems, has been widely used for studying biodiversity sustained by species coexistence. Spiral waves and target waves have been observed in spatially dispersed population with mobility, which has been deemed play significant roles in species coexistence. For social dilemmas, the formation of cooperator clusters is a key factor for the survival of cooperators who can resist the invasion of defectors in the pattern of clusters. Several underlying mechanisms contributing to the formation of various spatial patterns will be reviewed in spatial games.

On excitable β-skeletons

A β-skeleton, β ≥ 1, is a planar proximity undirected graph of an Euclidean point set where nodes are connected by an edge if their lune-based neighborhood contains no other points of the given set. Parameter β determines size and shape of the nodes’ neighborhoods. In an excitable β-skeleton every node takes three states—resting, excited and refractory, and updates its state in discrete time depending on states of its neighbors. We design families of β-skeletons with absolute and relative thresholds of excitability and demonstrate that several distinct classes of space–time excitation dynamics can be selected using β. The classes include spiral and target waves of excitation, branching domains of excitation and oscillating localizations.