Spiral Waves In Excitable Media Research Articles

Spiral waves in fractal dimensions and their elimination in λ − ω systems with less damaging intervention

The theory of spiral waves in excitable media is considered an important topic in several dynamical systems. They have been observed in various physicochemical and biological systems such as the Belousov-Zhabotinsky chemical reactions, lethal arrhythmia in the heart, amoebas Dictyostelium discoideium, disinhibited mammalian neocortex retinal spreading depression, and seizures in the brain. In this study, spiral waves in an excitable system are analytically studied in fractal dimensions based on the theory of integration and differentiation for a non-integer dimensional space. We consider the generic oscillatory model λ−ω system characterized by a rotational symmetry where spiral waves are expected to occur. It was observed that for lower fractal dimensions with respect to unity, spiral waves may be suppressed in an excitable media without destroying the propagation of normal waves. This may be considered as an alternative approach to experimental and numerical approaches found in the literature. Their eliminations are important since they forbid the transition to a chaotic state in excitable media. Our approach could contribute to an improved therapy of clinical conditions such as atrial fibrillation and suppressing arrhythmias in cardiac models.

On the Initiation of Spiral Waves in Excitable Media

On the Initiation of Spiral Waves in Excitable Media

Terminating spiral waves with a single designed stimulus: Teleportation as the mechanism for defibrillation

We identify and demonstrate a universal mechanism for terminating spiral waves in excitable media using an established topological framework. This mechanism dictates whether high- or low-energy defibrillation shocks succeed or fail. Furthermore, this mechanism allows for the design of a single minimal stimulus capable of defibrillating, at any time, turbulent states driven by multiple spiral waves. We demonstrate this method in a variety of computational models of cardiac tissue ranging from simple to detailed human models. The theory described here shows how this mechanism underlies all successful defibrillation and can be used to further develop existing and future low-energy defibrillation strategies.

Effect of straining on spiral wave dynamics in excitable media

The present article envisages to explore the dynamics of stable rotating spiral waves in excitable media under the effect of shear strain through the Oregonator model. We employ an existing unconditionally stable, implicit high order compact finite difference scheme to discretize the highly nonlinear system governing the spiral wave patterns. In the process, we also study the effect of reaction parameters on the dynamics of spiral waves in the absence of straining, which results in the transition from the stable (periodic) rotation to compound (non-periodic) rotation of spiral waves. Our study reveals the existence of three regimes with a new time range, which is different from the earlier ones reported in literature; they characterize the extremely complex behaviour of the patterns under the effect of straining. The aforesaid conclusion is arrived at by carrying out a comprehensive analysis of the power spectrum of the tip motion. Besides establishing the stability of simulated rotating waves, the accuracy of the computed solutions has been ascertained through a convergence analysis and mesh independence study.

Resonant drift of synchronized spiral waves in excitable media.

Many methods have been employed to investigate the drift behavior of spiral waves in an effort to understand and control their dynamics. Here, we propose a joint method to control the dynamics of spiral waves by applying both a rotating external field and periodic forcing. First, spirals in different regimes (including rigidly rotating, meandering, and drifting spirals) are synchronized by suitable rotating external fields. Then, under weak periodic forcing at frequencies equal to those of the external fields, the synchronized spirals undergo a directional and simple drift. Dependence of the drift velocity of the synchronized spiral on the initial rotational phase of the rotating external field and on the initial phase of the periodic forcing is studied in detail. We hope that our theoretical results can be observed in experiments, such as in the Belousov-Zhabotinsky reaction.

Control of spiral waves in excitable media under polarized electric fields

Spiral waves are ubiquitous in diverse physical, chemical, and biological systems. Periodic external fields, such as polarized electric fields, especially circularly polarized electric fields which possess rotation symmetry may have significant effects on spiral wave dynamics. In this paper, control of spiral waves in excitable media under polarized electric fields is reviewed, including resonant drift, synchronization, chiral symmetry breaking, stabilization of multiarmed spiral waves, spiral waves in subexcitable media, control of scroll wave turbulence, unpinning of spiral waves in cardiac tissues, control of spiral wave turbulence in cardiac tissues, etc.

Self-organization of multiarmed spiral waves in excitable media.

We present an investigation of self-organized multiarmed spiral waves pinned to unexcitable circular obstacles in a thin layer of the excitable Belousov-Zhabotinsky reaction and in simulations using the Oregonator model. The multiarmed waves are initiated by a series of wave stimuli. In the proximity of the obstacle boundary, the wave rotation around the obstacle causes damped oscillations of the wave periods of all spiral arms. The dynamics of wave periods cause the wave velocities as well as the angular displacements between the adjacent arms to oscillate with decaying amplitudes. Eventually, all displacements approach approximately the same stable value so that all arms are distributed evenly around the obstacle. A further theoretical analysis reveals that the temporal dynamics of the angular displacements can be interpreted as underdamped harmonic oscillations. Far from the obstacles, the wave dynamics are less pronounced. The wave period becomes stable very soon after the initiation. When the number of spiral arms increases, the rotation of individual arms slows down but the wave period of the multiarmed spiral waves decreases. Due to their short period, multiarmed spiral waves emerging in the heart potentially result in severe pathological conditions.

Novel high-order compact approach for dynamics of spiral waves in excitable media

• Comprehensive spectral density analysis of the spiral tip to investigate the dynamics of the spiral wave. • Time and space grid independent computation. • Modified differential equation analysis of the explicit scheme for reaction-diffusion equations. • Numerical study on the stability of rotating spiral wave solutions. • Effect of an obstacle to the periodic rotating spiral waves. The current study envisages to investigate the dynamics of spiral wave in and without the presence of obstacles in excitable media through a comprehensive spectral density analysis. While most of the previous studies on spiral wave dynamics are seen to have been obtained by exploiting the state variables using lower order explicit schemes to discretize the governing equations, we accomplish the same by exploiting the spiral tip path with data obtained from a recently developed higher order compact finite difference scheme. This scheme which is implicit in nature and unconditionally stable is seen to efficiently resolve the spiral wave patterns. On the other hand, owing to implicit dispersion, diffusion and reaction terms inherently present in the explicit data, the spiral tip resulting from it was seen to behave differently for different time steps. These issues are illustrated through a modified differential equation analysis of the explicit scheme. For the range of the parameters considered, we further observed that the role of the obstacle was only to change the trajectory paths, as they were seen to settle into a periodic motion eventually. In the process, we also establish the grid independence and the rate of convergence of the simulated data, apart from exploring the stability of the computed rotating spiral wave solutions.

Elimination of spiral waves in excitable media by magnetic induction

The formation of spiral waves in excitable media is a fascinating example of the beauty of nonlinear dynamics in spatiotemporal systems. Apart from the beauty of the patterns, the subject also has many practical application. For example, the emergence of spiral waves in cardiac tissue can lead to arrhythmias. Cortical spiral waves are also involved in epileptic seizures. Motivated by this, we here study the effects of magnetic induction on the formation of spiral waves in excitable media. An external sinusoidal magnetic induction with different amplitudes and angular frequencies is applied in order to study whether spiral waves could be eliminated. We use a network of coupled neurons as a model for the excitable medium. The four-variable magnetic Hindmarsh–Rose model is used for the local dynamics of each isolated neuron. The distribution of the cell membrane potential over time, affected by magnetic induction, is determined and the results are depicted as snapshots of the 2D network. Our research reveals that the continuance of rotating spiral seeds is impaired by high-amplitude magnetic induction. Moreover, we show that low-frequency induction is not capable of breaking the reorganizing rhythm of the spiral seeds, while much higher frequencies can be too fast to overcome this special rhythm.

Parametric wave induces straight drift of spiral waves in excitable medium

Suppressing or eliminating spiral waves in an excitable medium is of practical significance. In this paper, the parametric wave is used to eliminate the spiral wave in an excitable medium modeled by Barkley equations. We found that the parametric wave could push the spiral tip out of the medium, thereby eliminating the spiral wave. Interestingly, parametric waves with particular amplitude and wave velocity could make the spiral tip drift along straight lines, which may correspond to efficient suppression of the spiral wave. Numerical results show that the drift velocity increases with the amplitude of the parametric wave prominently, while the frequency does not influence V significantly. The drift direction is determined by both the amplitude and frequency of the parametric wave. The parametric wave can push the spiral to drift along a totally different direction, even the opposite one, depending on the amplitude and frequency.

Dynamics of pulses and spiral waves in excitable media with an anomalous diffusion

Spiral waves and pulses in the excitable medium with an anomalous diffusion are studied. In the medium with an one-sided fractional diffusion in the [Formula: see text]-direction and a normal diffusion in the [Formula: see text]-direction, a pulse, traveling along the positive [Formula: see text]-direction, has a smaller velocity, which is different from the diffusion of a source in the other media. Its propagating velocity is a linear and increasing function of the square root of diffusion parameter, whose increasing rate depends on the fractional order. A minimal value of the diffusion parameter is needed for successfully propagating pulses, and the threshold becomes large with a decrease of the fractional order. For pulse trains, the frequency-locked bands are shifted along the increasing direction of the perturbation period when the fractional order is decreased. In the propagating process of a spiral wave, the tip drift is induced by the one-sided fractional diffusion, which may be explained by analyzing the SV area in front of the tip.

Identification of the FitzHugh–Nagumo Model Dynamics via Deterministic Learning

In this paper, a new method is proposed for the identification of the FitzHugh–Nagumo (FHN) model dynamics via deterministic learning. The FHN model is a classic and simple model for studying spiral waves in excitable media, such as the cardiac tissue, biological neural networks. Firstly, the FHN model described by partial differential equations (PDEs) is transformed into a set of ordinary differential equations (ODEs) by using finite difference method. Secondly, the dynamics of the ODEs is identified using the deterministic learning theory. It is shown that, for the spiral waves generated by the FHN model, the dynamics underlying the recurrent trajectory corresponding to any spatial point can be accurately identified by using the proposed approach. Numerical experiments are included to demonstrate the effectiveness of the proposed method.

Analytical approximations for spiral waves

We propose a non-perturbative attempt to solve the kinematic equations for spiral waves in excitable media. From the eikonal equation for the wave front we derive an implicit analytical relation between rotation frequency Ω and core radius R(0). For free, rigidly rotating spiral waves our analytical prediction is in good agreement with numerical solutions of the linear eikonal equation not only for very large but also for intermediate and small values of the core radius. An equivalent Ω(R(+)) dependence improves the result by Keener and Tyson for spiral waves pinned to a circular defect of radius R(+) with Neumann boundaries at the periphery. Simultaneously, analytical approximations for the shape of free and pinned spirals are given. We discuss the reasons why the ansatz fails to correctly describe the dependence of the rotation frequency on the excitability of the medium.

Attractive and repulsive contributions of localized excitability inhomogeneities and elimination of spiral waves in excitable media

The attracting and repelling of spiral waves in a two-dimensional excitable medium in the presence of localized excitability inhomogeneities are studied. The choice of two effects depends on the comparison of excitabilities inside and outside the localized obstacle. We inspect the changes in attracting and repelling behaviors with respect to the size of the obstacle and the initial distance between the center of the spiral core and the obstacle. To understand the occurrence of these phenomena, we investigated the small v-value areas near the tip and the function of the wave velocity as the excitability parameter ε. Considering the attributes of the attractive obstacle, an eliminating scheme of spiral waves is proposed in which the attractive obstacle is rapidly moved at several fixed times. This method can avoid the high-amplitude and high-frequency stimulus in most conventional methods.

Non-specular reflections in a macroscopic system with wave-particle duality: Spiral waves in bounded media

Spiral waves in excitable media possess both wave-like and particle-like properties. When resonantly forced (forced at the spiral rotation frequency) spiral cores travel along straight trajectories, but may reflect from medium boundaries. Here, numerical simulations are used to study reflections from two types of boundaries. The first is a no-flux boundary which waves cannot cross, while the second is a step change in the medium excitability which waves do cross. Both small-core and large-core spirals are investigated. The predominant feature in all cases is that the reflected angle varies very little with incident angle for large ranges of incident angles. Comparisons are made to the theory of Biktashev and Holden. Large-core spirals exhibit other phenomena such as binding to boundaries. The dynamics of multiple reflections is briefly considered.

Selection of Spiral Waves in Excitable Media with a Phase Wave at the Wave Back

Universal relationships between the medium excitability and the angular velocity and the core radius of rigidly rotating spiral waves in excitable media are derived for situations where the wave front is a trigger wave and the wave back is a phase wave. Two universal limits restricting the region of existence of spiral waves in the parameter space are demonstrated. The predictions of the free-boundary approach are in good quantitative agreement with results from numerical reaction-diffusion simulations performed on the Kessler-Levine model.

Spiral waves in excitable media due to noise and periodic forcing

We investigate the jointly driven effects of external periodic forcing and Gaussian white noise on meandering spiral waves in excitable media with FitzHugh–Nagumo local dynamics. Interesting phenomena resulted from various forcing periods are found, for example, piece-wise line drift, intermittent straight-line drift and so on. We also observe new type of breakup of spiral wave between entrainment bands with 1:1 and 2:1. It is believed that the occurrence of the new type is relevant to the appearance of local bidirectional propagation window. There exist optimized noise intensities which can induce the broadest entrainments and Arnold tongues. Such a phenomenon is referred to as stochastic resonance. It is also observed that the noise makes significant effects on the spiral wave with straight-line drift. Via the tip Fourier spectrum, the varying of tip motion with external periods on the resonance band is interpreted.

Orbital Motion of Spiral Waves in Excitable Media

Spiral waves in active media react to small perturbations as particlelike objects. Here we apply asymptotic theory to the interaction of spiral waves with a localized inhomogeneity, which leads to a novel prediction: drift of the spiral rotation center along circular orbits around the inhomogeneity. The stationary orbits have fixed radii and alternating stability, determined by the properties of the bulk medium and the type of inhomogeneity, while the drift speed along an orbit depends on the strength of the inhomogeneity. Direct numerical simulations confirm the validity and robustness of the theoretical predictions and show that these unexpected effects should be observable in experiment.

The Large Core Limit of Spiral Waves in Excitable Media: A Numerical Approach

We modify the freezing method introduced by Beyn and Thümmler in 2004 for analyzing rigidly rotating spiral waves in excitable media. The proposed method is designed to stably determine the rotation frequency and the core radius of rotating spirals, as well as the approximate shape of spiral waves in unbounded domains. In particular, we introduce spiral wave boundary conditions based on geometric approximations of spiral wave solutions by Archimedean spirals and by involutes of circles. We further propose a simple implementation of boundary conditions for the case when the inhibitor is nondiffusive, a case which had previously caused spurious oscillations. We then utilize our method to numerically analyze the large core limit. The proposed method allows us to investigate the case close to criticality where spiral waves acquire infinite core radius $r_c$ and zero rotation frequency $\omega$, before they begin to develop into retracting fingers. We confirm the linear scaling regime of a drift bifurcation for the rotation frequency and the core radius of spiral wave solutions close to criticality. This regime is unattainable with conventional numerical methods.

Frozen state of spiral waves in excitable media

It has long been observed in experiments that frozen state of spiral waves can spontaneously appear in both excitable and oscillatory media, in which several spiral waves coexist and are well separated by thin walls (shocks). In this work, we study the global structure of frozen state of spiral waves in excitable media and find that different from stationary shocks in oscillatory media, in excitable media the shock dynamics actually depends on the status of single spiral, which is also determined by the system parameter, i.e., the shock line can be stationary if the single spiral is rigidly rotating, whereas it can wiggle if the single spiral is meandering, and the wiggling amplitude increases with the degree of meandering. We also find the stationary structure of shock line can still be well predicted by a previous theory, which has been developed for the frozen state of spiral waves in oscillatory media. All these findings are generic and expected to be verified in experiments.