Meandering Spiral Waves Research Articles

The eikonal equation and meandering of spiral waves: A geometrical approach

The eikonal equation supports the rotating spiral wave with the core rotating periodically. It is shown that this pivoting constraint can be removed by prescribing appropriate motion of the core compatible with the global motion of the spiral. A geometrical approach is presented for investigating the phenomenon not previously solvable by means of the eikonal equation: the motion of spirals whose tips are not fixed, but meandering.

Spiral-wave dynamics in an excitable medium with many excitability obstacles

Many real excitable systems can be descibed as inhomogeneous media, where the inhomogeneity is an important factor for the formation of spiral waves and the changing of their dynamics. In this paper, we investigate the effect of excitability obstacles on spiral-wave dynamics. For an excitability-reduced obstacle, the neighbor spiral tip is attracted into the obstacle. When more localized obstacles are placed, the attactive case depends on the distribution, size and excitability of the obstcales. On the basis of analyzing the small-value area of the inhibitor variable, we illustrate the mechanism of these behaviors occuring. For an excitability-enhanced obstacle, the nearby spiral tip is repelled. The tip motion after the repelsive effect depends on the type of the initial spiral wave, i.e. rigidily rotating spiral wave or meandering spiral wave. In the present of more localized obstacles, there exist different behaviors for different distributions, sizes and excitabilities of the obstcales, and different types of initial waves.

Effects of the aging and systole of heart on the dynamics of spiral wave

In this paper, the effects of the aging and systole of heart on the dynamics of spiral wave are studied by using the Greenberg-Hasting cellular automaton model. In this model the neighbor radius and the excitation threshold are increased in order to simulate the aging of heart, and the neighbor radius is changed alternately to simulate heart systole and diastole. The results show that the aging of heart can induce some influences on spiral wave, such as make spiral wave meandering, and even cause spiral wave to disappear; in addition, it can shorten the wavelength and keep period fixed, and also elongate the wavelength and increase the period of spiral wave. If the aging and the regular systole of heart take place at the same time, we observe some phenomena, such as different spiral wave patterns, the spiral wave breakup and disappearance. We also obtain the probabilities of heart failure, ventricular fibrillation and death. These results are essentially consistent with the results of relevant epidemiological survey.

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.

Dynamics of spiral waves driven by a dichotomous periodic signal

We study the effects of a dichotomous periodic force on meandering and rigidly rotating spiral waves. For a meandering state, the periodic forcing induces more modulating frequencies according to the rules of frequency-locked relations and linear combinations. It can also generate some unique closed tip orbits. On the modulating period T-axis, there exist all kinds of resonant entrainment bands. Arnold tongues exist in the period-amplitude space. The width of entrainment bands is affected by the symmetry of positive and negative parts in each signal unit. In addition, appropriate choices of T-value can be used to eliminate spiral waves. For a rigidly rotating state, the periodic forcing can induce a transition toward meandering spiral waves via generating a transitive bidirectional spiral wave. It is very interesting that, after the transition, the meandering spiral wave has the same primary rotating period as the free meandering states.

The influence of spatiotemporal modulation on spiral tip dynamics in excitable medium and its application for spiral control

In this paper, the influence of spatiotemporal modulation on tip dynamics of periodic spiral wave in excitable medium is studied first. By varying spatiotemporal modulation item, the dynamics of spiral wave changes dramatically and the system undergoes periodic spiral wave, epicycloid meandering spiral wave, traveling spiral wave and hypocycloid meandering spiral wave. An order parameter is introduced to detect the critical conditions of non-equilibrium transition between different patterns. And the variation of spiral tip radius induced by spatiotemporal modulation can also be reflected by this order parameter. When spatiotemporal modulation increases to a critical value, spiral waves break up. And spiral waves will damp to homogeneous rest state if spatiotemporal modulation increases further. The mechanisms of spiral breakup and damping are explained in the paper. Finally we apply the spatiotemporal modulation method to the meandering spiral waves and can successfully control meandering spiral waves into periodic spiral waves or homogeneous rest state.

The dynamics of spiral waves in three-layer excitable medium with circular feedback coupling

The dynamics of spiral wave in a three-layer excitable medium with circular feedback coupling is studied, based on the Bär model. The numerical results show that the drifting or meandering of spiral waves in the subsystems can be observed when the coupling strength is small. When the coupling strength is slightly big, the interaction between subsystems may cause spiral waves in some subsystems to move out of the boundaries of the subsystems. The subsystems return to rest state. In addition, the interaction may generate the transition from spiral wave state to target wave or turbulence states in some subsystems. The phenomenon that the asymptotic state of a subsystem depends on the initiation condition is observed. With the further increase of the coupling strength, the approximate generalized synchronization of the spiral waves in three subsystems is established. When the coupling strength is bigger, the spiral waves evolve into turbulence.

Eliminate spiral wave and spatiotemporal chaos by using short-term cardiac memory

Cardiac memory effect, which exhibits the time-delayed coupling of membrane voltage, is introduced into Luo-Rudy phase I cardiac model. The effect of the cardiac memory on spiral waves is investigated. The numerical results show that the cardiac memory can lead to irregular meandering of spiral waves. When the delay-time is properly chosen, the increase of memory strength causes the decrease of spiral wave frequency. If memory strength exceeds the critical value, the cardiac memory results in the disappearances of both spiral waves and spatiotemporal chaos in system because the time-dependent potassium ion current is over suppressed by cardiac memory.

Dynamics of Meandering Spiral Waves Driven by Two-Point Feedback

We numerically study the dynamics of meandering spiral waves in the excitable system subjected to a feedback signal coming from two measuring points located on a straight line together with the initial spiral core. The core location and size radius of the final attractors are computed, and they change with the position of the moving measuring point in a unique way. By the Fourier Spectral analysis, we find the frequency-locked behaviors different from the driving scheme of the external periodic force. It is also found that the meandering spiral wave can be eliminated when the moving measuring point approaches closely the boundary and its feedback gain is large enough. This offers an effective and convenient method for eliminating meandering spiral waves.

Chemical wave studies in the bromate-pyrocatechol beads system

We studied the chemical wave activity of the pyrocatechol-acidic bromate system in the presence of ferroin-loaded beads. The wave activity lasted for more than 24 h while meandering spirals continued for up to 10 h. Rigid and meandering spiral waves were investigated. We have analyzed the wave propagation speed and spiral tip trajectory versus the initial concentrations of all reagents as well as the age of the solution. Wave velocity depends on [H+] and [BrO] concentrations by the relationship v=k[H+]1/2[BrO]1/2, which is in agreement with other studies. This system is ideal to study wave activity and spiral waves as it does not produce precipitates under the studied conditions. © 2011 Wiley Periodicals, Inc. Int J Chem Kinet 43: 198–203, 2011

The numerical study on the effect of the earlyafterdepolarization on spiral wavesin discrete excitable media

The early afterdepolarization (EAP) behavior is introduced into the Greeberg-Hasting model of discrete excitable medium through considering that some of the refractory states can be excited. The effect of the EAP on spiral wave is studied. The numerical results show that the EAP has significant influences on spiral waves when the related parameters are suitably chosen. These influences include that the EAP causes spiral wave drifting and meandering,and spiral breakup. The pattern of spiral wave is distorted. The spiral wave varies alternately between those with thick and thin arms. The period of spiral wave alters alternately between two values. The EAP induces the transition from stable spiral wave to breathing spiral wave or antispiral wave. When the excitation threshold of refractory states is very high,the EAP has no affect on the spiral wave. The EAP induced phenomena are briefly discussed.

Origin choice and petal loss in the flower garden of spiral wave tip trajectories.

Rotating spiral waves have been observed in numerous biological and physical systems. These spiral waves can be stationary, meander, or even degenerate into multiple unstable rotating waves. The spatiotemporal behavior of spiral waves has been extensively quantified by tracking spiral wave tip trajectories. However, the precise methodology of identifying the spiral wave tip and its influence on the specific patterns of behavior remains a largely unexplored topic of research. Here we use a two-state variable FitzHugh-Nagumo model to simulate stationary and meandering spiral waves and examine the spatiotemporal representation of the system's state variables in both the real (i.e., physical) and state spaces. We show that mapping between these two spaces provides a method to demarcate the spiral wave tip as the center of rotation of the solution to the underlying nonlinear partial differential equations. This approach leads to the simplest tip trajectories by eliminating portions resulting from the rotational component of the spiral wave.

Some origins of the low excitability of a discrete excitable medium

The simple model of discrete excitable medium is applied. In order to make spiral wave meander，we introduce some death cells and fatigue cells into a homogeneous medium，respectively，or periodically change the excitation threshold of cells. By monitoring the fraction of excitation cells，we find that there is a remarkable common feature for spiral wave meandering induced by different factors，i.e.，the fraction of excitation cells under spiral wave meandering is smaller than that under stable spiral wave，which indicates that the low excitability of a medium is related to the number of death cells and fatigue cells in the medium，and the periodical change of the excitation threshold of cells，respectively. The numerical results show that the decrease of the excitability of media always accompanies the decrease of the frequency of spiral wave. We hope the present work can provide help for medical workers on the prevention and treatment of heart disease.

Kinematics of rigidly rotating spiral waves

Spiral waves rigidly rotating in excitable media are studied by use of a free-boundary approach. This study 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 a strongly reduced refractoriness is supported within a range of the medium excitability restricted by two universal limits. At the low excitability limit the spiral core radius diverges, while at the high excitability limit it vanishes. The simulations performed for the medium excitability higher than the high excitability limit reveal nonstationary rotating waves, which considerably differ from well-studied meandering spiral waves. It is shown how the proposed free-boundary approach can be extended to the case of an arbitrary refractoriness. The predictions of the free-boundary approach are in good agreement with the results from numerical simulations of the underlying reaction-diffusion model and with asymptotics derived earlier for highly and weakly excitable media.

Negative Filament Tension at High Excitability in a Model of Cardiac Tissue

One of the fundamental mechanisms for the onset of turbulence in 3D excitable media is negative filament tension. Thus far, negative tension has always been obtained in media under low excitability. For this reason, its application to normal (nonischemic) cardiac tissue has been questionable, as such cardiac turbulence typically occurs at high excitability. Here, we report expansion of scroll rings (low curvature negative filament tension) in a medium with high excitability by numerical integration of the Luo-Rudy model of cardiac tissue. We discuss the relation between negative tension and the meandering of 2D spiral waves and the possible applications to cardiac modeling.

Annihilation and reflection of spiral waves at a boundary for the Beeler-Reuter model

We employ a reaction diffusion equation with local dynamics specified by the Beeler-Reuter model to study the meandering of spiral waves. With the appropriate choice for the conductances of sodium and calcium channels, the trajectory of the tip of a spiral wave lies on a straight line. The phenomenon of annihilation or reflection of a spiral at the boundaries of the domain is studied. This phenomenon is analyzed in terms of the variable j , which controls the reactivation of the sodium channel in the Beeler-Reuter model. The results presented can have potential applications in the study of cardiac arrhythmias by providing insight on the interaction between spiral waves and obstacles in the heart.

Suppression of Spiral Waves by Generating Self-exciting Target Wave

A new scheme is proposed to suppress stable and meandering spiral waves in excitable media by generating a self-exciting target wave in a local area. An arbitrary selected grid in the media is sampled, and the sampled signal is fed back into a local area in the media. Numerical simulation results confirm its effectiveness when the scheme is introduced into anisotropic (the diffusion coefficient is perturbed vs. time and/or space) and isotropic media. Results also show the scheme's robustness to spatiotemporal noise.

Suppression of meandering spiral waves in the excitable media due to a perturbation with stochastic phase

A new scheme is proposed to suppress the meandering spiral waves in the excitable media, which is described with two-dimensional variables. An external electronic signal with stochastic phase is imposed on a local area; few grids of the system are perturbed, for example, 4×4 and/or 5×5 grid nodes among all the 256×256 nodes, so that a new target wave may occur and competition between the newly generated target wave and the intrinsic meandering spiral waves develops dynamically. The numerical simulation results confirm that it is effective to suppress the meandering spiral wave in the Barkley model. Compared with a simple periodical driving in a local area, it shows some advantages such as shorter transient period to kill the spiral wave, and it is robust to spatiotempaoral noise. A new target wave could come into being and will kill the meandering spiral wave even in the anisotropic media,of which the diffusion coefficient is a function of space, when the intensity of the spatiotemporal noise is not too strong.

Substrate size as a determinant of fibrillatory activity maintenance in a mathematical model of canine atrium

Tissue size has been considered an important determinant of atrial fibrillation (AF), but recent work has questioned the critical size hypothesis. Here, we use a previously developed mathematical model of the two-dimensional canine atrium with realistic action potential, ionic, and conduction properties to address substrate size effects on the maintenance of fibrillatory activity. Cholinergic AF was simulated at different acetylcholine (ACh) concentrations ([ACh]) and distributions, with substrate area varied 11.1-fold. Automated phase singularity detection was used to facilitate the analysis of arrhythmic activity. The duration of activity induced by a single extrastimulus increased with increasing substrate dimensions. Two general mechanisms underlying activity were observed and were differentially affected by substrate size. For large mean [ACh], single primary rotors anchored in low-[ACh] zones maintained activity and substrate dimensions were not critical. At lower mean [ACh], extensive spiral wave meander prevented the emergence of single stable rotors. Prolonged activity was favored when substrate size permitted a sufficiently large number of simultaneous longer-lasting rotors that extinction of all was unlikely. Thus either single dominant rotor or multiple reentrant spiral generator mechanisms could maintain fibrillatory activity in this model and were differentially dependent on substrate size. These results speak to recent debates about the role in AF of single driver rotors versus multiple reentrant circuit mechanisms by suggesting that either may maintain fibrillatory atrial activity depending on atrial size and electrophysiological properties.

Defibrillation Depends on Conductivity Fluctuations and the Degree of Disorganization in Reentry Patterns

Defibrillation depends on conductivity and disorganization. Cardiac fibrillation is the deterioration of the heart's normally well-organized activity into one or more meandering spiral waves, which subsequently break up into many meandering wave fronts. Delivery of an electric shock (defibrillation) is the only effective way of restoring the normal rhythm. This study focuses on examining whether higher degrees of disorganization requires higher shock strengths to defibrillate and whether microscopic conductivity fluctuations favor shock success. We developed a three-dimensional computer bidomain model of a block of cardiac tissue with straight fibers immersed in a conductive bath. The membrane behavior was described by the Courtemanche human atrial action potential model incorporating electroporation and an acetylcholine- (ACh) dependent potassium current. Intracellular conductivities were varied stochastically around nominal values with variations of up to 50%. A single rotor reentry was initiated and, by adjusting the spatial ACh variation, the level of organization could be controlled. The single rotor could be stabilized or spiral wave breakup could be provoked leading to fibrillatory-like activity. For each level of organization, multiple shock timings and strengths were applied to compute the probability of shock success as a function of shock strength. Our results suggest that the level of the small-scale conductivity fluctuations is a very important factor in defibrillation. A higher variation significantly lowers the required shock strength. Further, we demonstrated that success also heavily depends on the level of organization of the fibrillatory episode. In general, higher levels of disorganization require higher shock strengths to defibrillate.