Two-dimensional Excitable Media Research Articles

Effects of dynamic change of action potential on evolution behavior of spiral wave

It is observed in cardiac patients that the steepnesses of action potential duration (APD) restitution curve of cardiomyocytes in different regions of the ventricle are significantly different from region to region. However, the steep APD restitution curve can either lead the spiral wave to break up and set up the ventricular fibrillation in certain conditions or result in no breakup of spiral wave in other conditions. The relationship between the dynamic behavior of spiral wave and steep APD restitution curve is still not completely clear. Therefore, further research is needed. In this paper, a two-dimensional excitable medium cellular automata model is used to study the influences of the APD restitution curves with different steepnesses on the dynamic behavior of spiral wave. Numerical simulation results show that the steep APD restitution curve can stabilize the meandering spiral wave, causing the stable spiral wave to roam or break, and even to disappear. When the total average slope of APD restitution curve is greater than 1, it is observed that spiral wave may be still stable or meandering. When the total average slope of APD restitution curve is much smaller than 1, the breakup of spiral waves may occur. Three types of spiral wave breakups are observed. They are the Doppler instability, Eckhaus instability, and APD alternation. The Doppler instability and Eckhaus instability are related to the total average slope of APD restitution curve greater than 1, and the spiral wave breakup caused by APD alternans may occur when the total average slope of APD restitution curve is much smaller than 1. When the total average slope of APD restitution curve is greater than 1, the phenomena that spiral waves disappear by meandering out of the system boundary and conduction barriers are observed. In addition, we also find that increasing cellular APD is beneficial to preventing spiral wave from breaking up. The physical mechanisms behind those phenomena are heuristically analyzed.

Simultaneous unpinning of multiple vortices in two-dimensional excitable media.

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.

Data-Driven Modeling and Prediction of Complex Spatio-Temporal Dynamics in Excitable Media

Spatio-temporal chaotic dynamics in a two-dimensional excitable medium is (cross-) estimated using a machine learning method based on a convolutional neural network combined with a conditional random field. The performance of this approach is demonstrated using the four variables of the Bueno-Orovio-Fenton-Cherry model describing electrical excitation waves in cardiac tissue. Using temporal sequences of two-dimensional fields representing the values of one or more of the model variables as input the network successfully cross-estimated all variables and provides excellent forecasts when applied iteratively.

Hidden structures of information transport underlying spiral wave dynamics.

A spiral wave is a macroscopic dynamics 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 the 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 the 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 the information flow over the excitable medium, hidden coherent structures emerge that effectively quantify the information transport underlying the spiral wave dynamics. Most importantly, we find that some coherent structures become more clearly defined over a longer observation period. These findings provide validity with our approach to quantitatively characterize the 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.

Lattice Symmetry-Breaking Perturbations for Spiral Waves

Spiral waves in two-dimensional excitable media have been observed experimentally and studied extensively. It is now well known that the symmetry properties of the medium of propagation drives many of the dynamics and bifurcations which are experimentally observed for these waves. Also, symmetry-breaking induced by boundaries, inhomogeneities, and anisotropy have all been shown to lead to different dynamical regimes from that which is predicted for mathematical models that assume infinite homogeneous and isotropic planar geometry. Recent mathematical analyses incorporating the concept of forced symmetry-breaking from the Euclidean group of all planar translations and rotations have given model-independent descriptions of the effects of media imperfections on spiral wave dynamics. In this paper, we continue this program by considering rotating waves in dynamical systems which are small perturbations of a Euclidean-equivariant dynamical system, but for which the perturbation preserves only the symmetry of a regular square lattice.

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.

Open-source tools for dynamical analysis of Liley's mean-field cortex model

Mean-field models of the mammalian cortex treat this part of the brain as a two-dimensional excitable medium. The electrical potentials, generated by the excitatory and inhibitory neuron populations, are described by nonlinear, coupled, partial differential equations that are known to generate complicated spatio-temporal behaviour. We focus on the model by Liley et al. (Network: Computation in Neural Systems 13 (2002) 67–113). Several reductions of this model have been studied in detail, but a direct analysis of its spatio-temporal dynamics has, to the best of our knowledge, never been attempted before. Here, we describe the implementation of implicit time-stepping of the model and the tangent linear model, and solving for equilibria and time-periodic solutions, using the open-source library PETSc. By using domain decomposition for parallelization, and iterative solving of linear problems, the code is capable of parsing some dynamics of a macroscopic slice of cortical tissue with a sub-millimetre resolution.

A ring-like heterogeneous medium-induced self-sustained target waves in excitable media

The formation of self-sustained target wave in a two-dimensional excitable medium into which a ring-like heterogeneous excitable medium is introduced is investigated. The numerical results show that the initial perturbation can produce a self-sustained target wave in an excitable medium when the excitabilities of two media and the size of the ring are appropriately chosen. The physical mechanism responsible for the formation of self-sustained target wave is analyzed.

The inverse excitation-induced abnormal spiral wave

The effects of a local constant forcing on spiral waves in two-dimensional excitable media described by Bär model are investigated. A constant external forcing is imposed on the core of spiral wave, leading to parameter variability of a medium. It is found that the forcing can significantly alter the shape and rotation period of spiral wave when the values of related parameters are properly chosen. The change of wave structure is attributed to the transition from normal excitation to inverse excitation in the forced medium. An abnormal spiral wave with a very thick spiral arm has been observed. The physical mechanism underlying these phenomena is theoretically analyzed.

Hysteresis phenomenon in the dynamics of spiral waves rotating around a hole

Hysteresis in the pinning–depinning transitions of spiral waves rotating around a hole in a circular shaped two-dimensional excitable medium is studied both by use of the continuation software AUTO and by direct numerical integration of the reaction–diffusion equations for the FitzHugh–Nagumo model. In order to clarify the role of different factors in this phenomenon, a kinematical description is applied. It is found that the hysteresis phenomenon computed for the reaction–diffusion model can be reproduced qualitatively only when a nonlinear eikonal equation (i.e. velocity–curvature relationship) is assumed. However, to obtain quantitative agreement, the dispersion relation has to be taken into account.

The cellular automaton model for the nonlinear waves in the two-dimensional excitable media

In this paper，we have studied the nonlinear waves using the Greenberg-Hasting Cellular Automaton model. The dependence of the propagation speed of plane wave on the neighbor radius and the excitation threshold is analyzed with the no-flux boundary condition by computer simulation, and then the excitation condition is obtained. The orbit of the tip of spiral wave is affected by the number of refractory states and the excitation threshold, and then the mechanism is analyzed.

Active and passive control of spiral turbulence in excitable media

The influence of a spatially localized heterogeneity defect, defined by failure of the diffusion effect, on spiral turbulence suppression in two-dimensional excitable media is studied numerically, based on the Bär model. It is shown that in certain parameter regions spiral turbulence without the defect can be suppressed by a boundary periodic forcing (called active control) if the forcing frequency is properly chosen. However, with a sufficiently large defect this active control method no longer works due to the wake turbulence following the defect. We suggest an auxiliary method of enclosing the defect with a thin layer of material of high excitability (called passive control) to screen the interaction between the defect and the turbulence and to restore the global control effect of the periodic forcing. The possible application of the method in cardiac defibrillation is discussed.

Pattern deformation and annihilation in two-dimensional excitable media in oscillatory domains

The effects of oscillatory domains on the dynamics of the FitzHugh–Nagumo equation in two dimensions is investigated as a function of the amplitude and frequency of the boundary motion. It is shown that the moving-boundary problem introduces anisotropy through the diffusion terms and an advection-like term in the direction of the boundary motion. If the advection-like term is neglected, it is shown that spiral wave solutions of the FitzHugh–Nagumo equation are robust and do not lose their integrity under the anisotropic effects induced by the moving domain, albeit undergo stretching and compression in the direction of the boundary motion. However, when the advection-like terms are accounted for, the anisotropy and stretching/compression of the initial spiral wave result in a homogeneous state at high frequencies, and the time required to achieve such a uniformity is mainly a function of the amplitude of the boundary motion. For frequencies comparable to that of the spiral wave in a fixed domain, it is shown that the spiral wave preserves its integrity for low amplitudes of the boundary motion and is annihilated at high amplitudes.

Спонтанная остановка дрейфа спиральной волны в однородной возбудимой среде

In computer simulations, we found a new type of spiral wave drift in а homogeneous two-dimensional excitable medium, namely, a circular drift of the spiral wave with decrease of the drift velocity right up to its total cessation. We have investigated certain quantitative characteristics of the new spiral wave behavior. As a result, we have demonstrated that the new spiral wave behavior essentially differs from the types of its behavior that was known before. This discovery can improve comprehension of mechanisms of some potentially life- threatening cardiac arrhythmias.

Dynamics of spiral pairs induced by unidirectional propagating pulses

The dynamics of unidirectionally propagating pulses in a two-dimensional uniform excitable reaction-diffusion medium is investigated. It is shown that under weak diffusion coupling between medium points such a pulse can evolve into a pair of counter-rotating spirals (spiral pair). We analyze the drift of such a pair and examine the collisions between several drifting pairs. It is demonstrated that collisions can result in a special type of reflection or, alternatively, in new types of complex stationary spiral structures. A possible application of these findings for the diagnosis of cardiac arrhythmias is suggested.

Successive splitting of autowaves in a nonlinear chemical reaction medium

The phenomenon of wave splitting is investigated in a two-dimensional excitable light-sensitive Belousov-Zhabotinsky reaction medium after extremely changing the intensity of illuminated light for a short time. It is found that successive wave splitting and nonannihilation collision between two waves of different amplitudes occur spontaneously under narrow experimental conditions. Experimental observations are approximately reproduced in the specific parameter range by a numerical simulation with a Bär-Eiswirth model.

Noise-induced spatial dynamics in the presence of memory loss

We study the spatial dynamics of noise-induced waves in two-dimensional excitable media in dependence on the duration of the artificially imposed refractory time that is introduced to each constitutive system unit after an excitation. Due to the introduction of refractory times, a randomly induced spatial wave is temporarily unable to transmit information to the opposite site of its propagation direction. Thus, once the wave leaves the absorbing boundaries of the spatial grid the system has little or no recollection, depending on the duration of the refractory time, of its existence. We show that even in the presence of such memory loss, self-organization of excitatory events leads to noise-induced spatial periodicity in the media. We present a simple analytical treatment of a two-unit system to capture and explain the essence of the observed phenomenon. Since refractory times are widespread in biological systems, our results provide interesting insights into functioning of real-life organisms at the cellular as well as tissue level.

SPONTANEOUS FORMATION OF MULTIARMED SPIRAL WAVES IN VARIOUS SIMPLE MODELS OF WEAKLY EXCITABLE MEDIA

We investigate a spontaneous formation of stable multiarmed spirals in two-dimensional excitable media, an effect observed in various biological and chemical systems. A previous study based on FitzHugh–Nagumo-type Pushchino model reported a robust effect of stable two- and three-armed spiral formation from nearby vortices, when the spirals rotate around unexcited cores, i.e. when excitability of the medium is low. In this study, we used a powerful parallel computer cluster to perform an extensive parameter search in two other widely used FitzHugh–Nagumo-type models, as well as in the two-component Oregonator model. We observed formation of stable n-armed spirals, with 2 ≤ n ≤ 10, whenever the excitability of the medium was sufficiently low. Thus, we conclude that the formation and persistence of stable multiarmed spirals (MAS) is not an artifact of one particular model, but, rather, it is an amazing higher-level self-organization property of a generic weakly excitable medium. We also establish quantitatively that such multiarmed spirals serve as high-frequency wave sources — a finding that has a direct relevance to cardiac defibrillation research.

Removal of a pinned spiral by generating target waves with a localized stimulus

Pinning of spiral waves by defects in cardiac muscle may cause permanent tachycardia. We numerically study the removal of a pinned spiral by a localized stimulus at the boundary of a two-dimensional excitable medium. It is shown that target waves may be generated by an external local force, and then the target waves will interact with the pinned spiral. When the external force is appropriately chosen, the generated target waves may suppress the pinned spiral, and the system is finally dominated by the target waves.

On propagation failure in one- and two-dimensional excitable media.

We present a nonperturbative technique to study pulse dynamics in excitable media. The method is used to study propagation failure in one-dimensional and two-dimensional excitable media. In one-dimensional media we describe the behavior of pulses and wave trains near the saddle node bifurcation, where propagation fails. The generalization of our method to two dimensions captures the point where a broken front (or finger) starts to retract. We obtain approximate expressions for the pulse shape, pulse velocity, and scaling behavior. The results are compared with numerical simulations and show good agreement.