Suppression of spiral waves using intermittent local electric shock

In this paper, an intermittent local electric shock scheme is proposed to suppress stable spiral waves in the Barkley model by a weak electric shock （about 0.4 to 0.7） imposed on a random selected n × n grids （n =1 - 5, compared with the original 256×256 lattice） and monitored synchronically the evolutions of the activator on the grids as the sampled signal of the activator steps out a given threshold （i.e., the electric shock works on the n ~ n grids if the activator u ≤ 0.4 or u ≥ 0.8）. The numerical simulations show that a breakup of spiral is observed in the media state evolution to finally obtain homogeneous states if the electric shock with appropriate intensity is imposed.

Synchronization of spiral waves in a two-layer coupled inhomogeneous excitable system

We studied synchronization behaviours of spiral waves in a two-layer coupled inhomogeneous excitable system. It was found that phase synchronization can be observed under weak coupling strength. By increasing the coupling strength, the synchronization is broken down. With the further increase of the coupling strength, complete synchronization and phase synchronization occur again. We also found that the inhomogeneity in excitable systems is helpful to the synchronization.

Elimination of spiral waves and spatiotemporal chaos by the synchronization transmission technology of network signals

A method to eliminate spiral waves and spatiotemporal chaos by using the synchronization transmission technology of network signals is proposed in this paper. The character of the spiral waves and the spatiotemporal chaos in the Fitzhugh-Nagumo model is presented. The network error evolution equation with spatiotemporal variables and the corresponding eigenvalue equation are determined based on the stability theory, and the global synchronization condition is obtained. Simulations are made in a complex network with Fitzhugh-Nagumo models as the nodes to verify the effectiveness of the synchronization transmission principle of the network signal.

The influence of long-range links on spiral waves and their application for control

The influence of long-range links on spiral waves in an excitable medium has been investigated. Spatiotemporal dynamics in an excitable small-world network transform remarkably when we increase the long-range connection probability P. Spiral waves with few perturbations, broken spiral waves, pseudo spiral turbulence, synchronous oscillations, and homogeneous rest state are discovered under different network structures. Tip number is selected to detect non-equilibrium phase transition between different spatiotemporal patterns. The Kuramoto order parameter is used to identify these patterns and explain the emergence of the rest state. Finally, we use long-range links to successfully control spiral waves and spiral turbulence.

Size transition of spiral waves using the pulse array method

The domain size of spiral waves is an important issue in studies of two-dimensional （2D） spatiotemporal patterns. In this work, we use the 2D complex Ginzburg-Landau equation （CGLE） as our model and find that an initially big spiral can successfully transfer to several small spirals by applying a pulse array method. The impacts of several important factors, such as array density, controlling intensity and pulsing time, are investigated. This control approach may be useful for the control of 2D spatiotemporal patterns and has potential applications in the control of some realistic systems, such as meteorological and cardiac systems.

The effect of cellular aging on the dynamics of spiral waves

Cellular aging can result in deterioration of electrical coupling, the extension of the action potential duration, and lower excitability of the cell. Those factors are introduced into the Greenberg–Hastings cellular automaton model and the effects of the cellular aging on the dynamics of spiral waves are studied. The numerical results show that a 50% reduction of the coupling strength of aging cells has a little influence on spiral waves. If the coupling strength of aging cells equals zero, the ability for the medium to maintain spiral waves will be reduced by approximately 50% when the aging cell ratio increases from 0 to 0.5, where the reduction of cell excitability plays a major role in inducing disappearance of spiral waves. When the relevant parameters are properly chosen, the cellular aging can lead to the meandering of spiral waves,the emergence of the binary spiral waves, and even the disappearance of spiral waves via the stopping rotation or shrinkage of wave. Physical mechanisms of the above phenomena are analyzed briefly.

Elimination of spiral waves and spatiotemporal chaos by the pulse with a specific spatiotemporal configuration

Spiral waves and spatiotemporal chaos are sometimes harmful and should be controlled. In this paper spiral waves and spatiotemporal chaos are successfully eliminated by the pulse with a very specific spatiotemporal configuration. The excited position D of spiral waves or spatiotemporal chaos is first recorded at an arbitrary time （to）. When the system at the domain D enters a recovering state, the external pulse is injected into the domain. If the intensity and the working time of the pulse are appropriate, spiral waves and spatiotemporal chaos can finally be eliminated because counter-directional waves can be generated by the pulse. There are two advantages in the method. One is that the tip can be quickly eliminated together with the body of spiral wave, and the other is that the injected pulse may be weak and the duration can be very short so that the original system is nearly not affected, which is important for practical applications.

Applying a global pulse disturbance to eliminate spiral waves in models of cardiac muscle

Removal of spiral waves in cardiac muscle is necessary because of their threat to life.Common methods for this removal are to apply a local disturbance to the media,such as a periodic forcing.However,most of these methods accelerate the beating of the cardiac muscle,resulting in the aggravation of the ventricular tachycardia,which directly threatens life.In the present study,in order to clear off spiral waves,a global pulse-disturbance is applied to the media based on three models of cardiac muscle.It is found that the spiral waves are eliminated and the frequency of the cardiac muscle is decreased in a short time,and finally,the state of the medium reaches the normal oscillation,which supports a target waves.Our method sheds light on the removal of spiral waves in cardiac muscle and can prevent the ventricular tachycardia as well as the ventricular fibrillation.

Effects of abnormal excitation on the dynamics of spiral waves

The effect of physiological and pathological abnormal excitation of a myocyte on the spiral waves is investigated based on the cellular automaton model. When the excitability of the medium is high enough, the physiological abnormal excitation causes the spiral wave to meander irregularly and slowly. When the excitability of the medium is low enough, the physiological abnormal excitation leads to a new stable spiral wave. On the other hand, the pathological abnormal excitation destroys the spiral wave and resultsin the spatiotemporal chaos, which agrees with the clinical conclusion that the early after depolarization is the pro-arrhythmic mechanism of some anti-arrhythmic drugs. The mechanisms underlying these phenomena are analyzed.

Controlling intracellular Ca^(2+) spiral waves by the local agonist in the cell membrane

A modified spatially extended Tang-Othmer Ca2＋ model is used to study intracellular Ca2＋ spiral waves numerically. It is found that, as a local stimulation, the local agonist-binding on the cell membrane, which enhances the local concentration of the messenger molecule inositol 1,4,5-trisphosphate（IP3）, can influence the dynamics of the spiral waves. 1） Strong enough stimuli can change the spiral wave from a meandering to a rigidly rotating one. 2） On the other hand, strong enough stimuli can suppress the spiral wave from the system. It provides the theoretical clue for controlling the spiral waves by stimulating the cell membrane.

Synchronizing spiral waves in a coupled Rssler system

The synchronisation of spiral patterns in a drive-response R6ssler system is studied. The existence of three types of synchronisation is revealed by inspecting the coupling parameter space. Two transient stages of phase synchronisation and partial synchronisation are observed in a comparatively weak feedback coupling parameter regime, whilst complete synchronisation of spirals is found with strong negative couplings. Detailed observations of the synchronous process, such as oscillatory frequencies, parameters mismatches and amplitude variations, etc, are investigated via numerical simulations.

Unpinning the spiral waves by using parameter waves

The spiral waves anchored to heterogeneous areas are more difficult to control and eliminate than freely rotating ones in homogenous mediums.To eliminate pinned spiral waves,the resistant force should be provided to resist the pinning force.Other than advection field,we introduce parametric wave to play the role of providing resistant force.It is found that the parametric wave with large enough amplitude and proper frequency can unpin and eliminate the spiral wave successfully.The capability of parametric wave in providing resistant force is dependent on its amplitude and frequency sensitively.On the basis of parametric wave,the dependence of pinning force on the size and level of heterogeneity is further confirmed.

TARGET PATTERNS AND SPIRAL WAVES IN REACTION-DIFFUSION SYSTEMS WITH LIMIT CYCLE KINETICSAND WEAK DIFFUSION

In this paper, we study periodic waves in reaction-diffusion systems with limit cycle kinetics and weak diffusion. The results are based on the analysis of a nonlinear partial differential equation which is derived from singular perturbation techniques. We obtain the first order approximation expression of the dispersion curve for periodic wave trains and then show that in the generic case of limit cycle kinetics there is a one parameter family of target patterns and the rotating spiral waves of Archimedian type exist. Finally, we apply these results to the general λ-ω systems.

Development and transition of spiral wave in the coupled Hindmarsh-Rose neurons in two-dimensional space

The dynamics and the transition of spiral waves in the coupled Hindmarsh-Rose （H-R） neurons in two-dimensional space are investigated in the paper. It is found that the spiral wave can be induced and developed in the coupled HR neurons in two-dimensional space, with appropriate initial values and a parameter region given. However, the spiral wave could encounter instability when the intensity of the external current reaches a threshold value of 1.945. The transition of spiral wave is found to be affected by coupling intensity D and bifurcation parameter r. The spiral wave becomes sparse as the coupling intensity increases, while the spiral wave is eliminated and the whole neuronal system becomes homogeneous as the bifurcation parameter increases to a certain threshold value. Then the coupling action of the four sub-adjacent neurons, which is described by coupling coefficient D＇, is also considered, and it is found that the spiral wave begins to breakup due to the introduced coupling action from the sub-adjacent neurons （or sites） and together with the coupling action of the nearest-neighbour neurons, which is described by the coupling intensity D.

Stabilization of spiral wave and turbulence in the excitable media using parameter perturbation scheme

This paper proposes a scheme of parameter perturbation to suppress the stable rotating spiral wave, meandering spiral wave and turbulence in the excitable media, which is described by the modified Fitzhug-Nagumo （MFHN） model. The controllable parameter in the MFHN model is perturbed with a weak pulse and the pulse period is decided by the rotating period of the spiral wave approximatively. It is confirmed that the spiral wave and spiral turbulence can be suppressed greatly. Drift and instability of spiral wave can be observed in the numerical simulation tests before the whole media become homogeneous finally.

Spiral-wave dynamics in excitable medium with excitability modulated by rectangle wave

We numerically study the dynamics of spiral waves in the excitable system with the excitability modulated by a rectangle wave. The tip trajectories and their variations with the modulation period T are explained by the corre- sponding spectrum analysis. For a large T, the external modulation leads to the occurrence of more frequency peaks and these frequencies change with the modulation period according to their specific rules, respectively. Some of the frequencies and a primary frequency fl determine the corresponding curvature periods, which are locked into rational multiplies of the modulation period. These frequency-locking behaviours and the limited life-span of the frequencies in their variations with the modulation period constitute many resonant entrainment bands in the T axis. In the main bands, which follow the relation T/T12 = m/n, the size variable Rx of the tip trajectory is a monotonic increasing function of T. The rest of the frequencies are linear combinations of the two ones. Due to the complex dynamics, many unique tip trajectories appear at some certain T. We find also that spiral waves are eliminated when T is chosen from the end of the main resonant bands. This offers a useful method of controling the spiral wave.

Controlling spiral wave with target wave in oscillatory systems

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.

Motion of spiral tip driven by local forcing in excitable media

We study the motion of a spiral wave controlled by a local periodic forcing imposed on a region around the spiral tip in an excitable medium. Three types of trajectories of spiral tip are observed： the epicycloid-like meandering, the resonant drift, and the hypocycloid-like meandering. The frequency of the spiral is sensitive to the local periodic forcing. The dependency of spiral frequency on the amplitude and size of local periodic forcing are presented. In addition, we show how the drift speed and direction are adjusted by the amplitude and phase of local periodic forcing, which is consistent with a theoretical analysis based on the weak deformation approximation.

Formation and instability of spiral wave induced by Gaussian coloured noise

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.

Numerical study of IP_3-induced Ca^(2+) spiral pattern evolution

The effect of change in concentration of messenger molecule inositol 1,4,5-trisphosphate （IP3） on intracellular Ca^2＋spiral pattern evolution is studied numerically. The results indicate that when the IP3 concentration decreases from 0.27 μM, a physiologically reasonable value, to different values, the spiral centre drifts to the edge of the medium and disappears for a small enough IP3 concentration. The instability of spiral pattern can be understood in terms of excitability-change controlled by the IP3 concentration. On the other hand, when the IP3 concentration increases from 0.27 μM, a homogeneous area with a high Ca^2＋ concentration emerges and competes with the spiral pattern. A high enough IP3 concentration can lead the homogeneous area to occupy the whole medium. The instability of spiral pattern is ascribed to the change in stability of a stationary state with a high Ca^2＋ concentration.