Anchoring of drifting spiral and scroll waves to impermeable inclusions in excitable media

Spatially Conserved Spiral Wave Activity During Human Atrial Fibrillation

BACKGROUND

Atrial fibrillation is the most common cardiac arrhythmia in the world and increases the risk for stroke and morbidity. During atrial fibrillation, the electric activation fronts are no longer coherently propagating through the tissue and, instead, show rotational activity, consistent with spiral wave activation, focal activity, collision, or partial versions of these spatial patterns. An unexplained phenomenon is that although simulations of cardiac models abundantly demonstrate spiral waves, clinical recordings often show only intermittent spiral wave activity.

METHODS

In silico data were generated using simulations in which spiral waves were continuously created and annihilated and in simulations in which a spiral wave was intermittently trapped at a heterogeneity. Clinically, spatio-temporal activation maps were constructed using 60 s recordings from a 64 electrode catheter within the atrium of N=34 patients (n=24 persistent atrial fibrillation). The location of clockwise and counterclockwise rotating spiral waves was quantified and all intervals during which these spiral waves were present were determined. For each interval, the angle of rotation as a function of time was computed and used to determine whether the spiral wave returned in step or changed phase at the start of each interval.

RESULTS

In both simulations, spiral waves did not come back in phase and were out of step." In contrast, spiral waves returned in step in the majority (68%; P=0.05) of patients. Thus, the intermittently observed rotational activity in these patients is due to a temporally and spatially conserved spiral wave and not due to ones that are newly created at the onset of each interval.

CONCLUSIONS

Intermittency of spiral wave activity represents conserved spiral wave activity of long, but interrupted duration or transient spiral activity, in the majority of patients. This finding could have important ramifications for identifying clinically important forms of atrial fibrillation and in guiding treatment.

Theory and experiments of spiral unpinning in the Belousov-Zhabotinsky reaction using a circularly polarized electric field

We present the first experimental study of unpinning an excitation wave using a circularly polarized electric field. The experiments are conducted using the excitable chemical medium, the Belousov-Zhabotinsky (BZ) reaction, which is modeled with the Oregenator model. The excitation wave in the chemical medium is charged so that it can directly interact with the electric field. This is a unique feature of the chemical excitation wave. The mechanism of wave unpinning in the BZ reaction with a circularly polarized electric field is investigated by varying the pacing ratio, the initial phase of the wave, and field strength. The chemical wave in the BZ reaction unpins when the electric force opposite the direction of the spiral is equal to or above a threshold. We developed an analytical relation of the unpinning phase with the initial phase, the pacing ratio, and the field strength. This is then verified in experiments and simulations.

Intermittent trapping of spiral waves in a cardiac model

Spiral waves are found in many excitable systems and are thought to play a role in the incoherent electrical activation that underlies cardiac arrhythmias. It is well-known that spiral waves can be permanently trapped by local heterogeneities. In this paper, we demonstrate that spiral waves can also be intermittently trapped by such heterogeneities. Using simulations of a cardiac model in two dimensions, we show that a tissue heterogeneity of sufficient strength or size can result in a spiral wave that is trapped for a few rotations, after which it dislodges and meanders away from the heterogeneity. We also show that these results can be captured by a particle model in which the particle represents the spiral wave tip. For both models, we construct a phase diagram which quantifies which parameter combinations of heterogeneity size and strength result in permanent, intermittent, or no trapping. Our results are consistent with clinical observations in patients with atrial fibrillation that showed that spiral wave reentry can be intermittent.

Robust data assimilation with noise: Applications to cardiac dynamics

Reconstructions of excitation patterns in cardiac tissue must contend with uncertainties due to model error, observation error, and hidden state variables. The accuracy of these state reconstructions may be improved by efforts to account for each of these sources of uncertainty, in particular, through the incorporation of uncertainty in model specification and model dynamics. To this end, we introduce stochastic modeling methods in the context of ensemble-based data assimilation and state reconstruction for cardiac dynamics in one- and three-dimensional cardiac systems. We propose two classes of methods, one following the canonical stochastic differential equation formalism, and another perturbing the ensemble evolution in the parameter space of the model, which are further characterized according to the details of the models used in the ensemble. The stochastic methods are applied to a simple model of cardiac dynamics with fast-slow time-scale separation, which permits tuning the form of effective stochastic assimilation schemes based on a similar separation of dynamical time scales. We find that the selection of slow or fast time scales in the formulation of stochastic forcing terms can be understood analogously to existing ensemble inflation techniques for accounting for finite-size effects in ensemble Kalman filter methods; however, like existing inflation methods, care must be taken in choosing relevant parameters to avoid over-driving the data assimilation process. In particular, we find that a combination of stochastic processes-analogously to the combination of additive and multiplicative inflation methods-yields improvements to the assimilation error and ensemble spread over these classical methods.

Chaotic tip trajectories of a single spiral wave in the presence of heterogeneities

Spiral waves have been observed in a variety of physical, chemical, and biological systems. They play a major role in cardiac arrhythmias, including fibrillation, where the observed irregular activation patterns are generally thought to arise from the continuous breakup of multiple unstable spiral waves. Using spatially extended simulations of different electrophysiological models of cardiac tissue, we show that a single spiral wave in the presence of heterogeneities can display chaotic tip trajectories, consistent with fibrillation. We also show that the simulated spiral tip dynamics, including chaotic trajectories, can be captured by a simple particle model which only describes the dynamics of the spiral tip. This shows that spiral wave breakup, or interactions with other waves, are not necessary to initiate chaos in spiral waves.

Defects formation and spiral waves in a network of neurons in presence of electromagnetic induction

Complex anatomical and physiological structure of an excitable tissue (e.g., cardiac tissue) in the body can represent different electrical activities through normal or abnormal behavior. Abnormalities of the excitable tissue coming from different biological reasons can lead to formation of some defects. Such defects can cause some successive waves that may end up to some additional reorganizing beating behaviors like spiral waves or target waves. In this study, formation of defects and the resulting emitted waves in an excitable tissue are investigated. We have considered a square array network of neurons with nearest-neighbor connections to describe the excitable tissue. Fundamentally, electrophysiological properties of ion currents in the body are responsible for exhibition of electrical spatiotemporal patterns. More precisely, fluctuation of accumulated ions inside and outside of cell causes variable electrical and magnetic field. Considering undeniable mutual effects of electrical field and magnetic field, we have proposed the new Hindmarsh-Rose (HR) neuronal model for the local dynamics of each individual neuron in the network. In this new neuronal model, the influence of magnetic flow on membrane potential is defined. This improved model holds more bifurcation parameters. Moreover, the dynamical behavior of the tissue is investigated in different states of quiescent, spiking, bursting and even chaotic state. The resulting spatiotemporal patterns are represented and the time series of some sampled neurons are displayed, as well.

Emergent dynamics of spatio-temporal chaos in a heterogeneous excitable medium

Self-organized activation patterns in excitable media such as spiral waves and spatio-temporal chaos underlie dangerous cardiac arrhythmias. While the interaction of single spiral waves with different types of heterogeneity has been studied extensively, the effect of heterogeneity on fully developed spatio-temporal chaos remains poorly understood. We investigate how the complexity and stability properties of spatio-temporal chaos in the Bär-Eiswirth model of excitable media depend on the heterogeneity of the underlying medium. We employ different measures characterizing the chaoticity of the system and find that the spatial arrangement of multiple discrete lower excitability regions has a strong impact on the complexity of the dynamics. Varying the number, shape, and spatial arrangement of the heterogeneities, we observe strong emergent effects ranging from increases in chaoticity to the complete cessation of chaos, contrasting the expectation from the homogeneous behavior. The implications of our findings for the development and treatment of arrhythmias in the heterogeneous cardiac muscle are discussed.

Suppression of turbulence by heterogeneities in a cardiac model with fiber rotation

Electrical scroll wave turbulence in human ventricles is associated with ventricular fibrillation and sudden cardiac death. We perform three-dimensional simulations on the basis of the anisotropic Fenton-Karma model and show that macroscopic, insulating heterogeneities (e.g., blood vessels) can cause the spontaneous formation of pinned scroll waves. The wave field of these vortices is periodic, and their frequencies are sufficiently high to push the free, turbulent vortices into the system boundaries where they annihilate. Our study considers cylindrical heterogeneities with radii in the range of 0.1 to 2 cm that extend either in the transmural or a perpendicular direction. Thick cylinders cause the spontaneous formation of multi-armed rotors according to a radius-dependence that is explained in terms of two-dimensional dynamics. For long cylinders, local pinning contacts spread along the heterogeneity by fast and complex self-wrapping.

Stabilization of collapsing scroll waves in systems with random heterogeneities

In three-dimensional reaction-diffusion systems, excitation waves may form and rotate around a one-dimensional phase singularity called the filament. If the filament forms a closed curve, it will shrink over time and eventually collapse. However, filaments may pin to non-reactive objects present in the medium, reducing their rate of collapse or even allowing them to persist indefinitely. We use numerical simulations to study how different arrangements of non-reactive spheres affect the dynamics of circular filaments. As the filament contracts, it gets closer to and eventually touches and pins to objects in its path. This causes two possible behaviors. The filament can detach from the spheres in its path, slowing down the rate of contraction, or it can remain pinned to a collection of spheres. In general, more or larger spheres increase the chance that the filament remains pinned, but there are exceptions. It is possible for a small number of small spheres to support the filament and possible for the filament to pass through a large number of large spheres. Our work yields insights into the pinning of scroll waves in excitable tissue such as cardiac muscle, where scar tissue acts in a way similar to the non-reactive domains.

Determining conduction patterns on a sparse electrode grid: Implications for the analysis of clinical arrhythmias

We present a general method of utilizing bioelectric recordings from a spatially sparse electrode grid to compute a dynamic vector field describing the underlying propagation of electrical activity. This vector field, termed the wave-front flow field, permits quantitative analysis of the magnitude of rotational activity (vorticity) and focal activity (divergence) at each spatial point. We apply this method to signals recorded during arrhythmias in human atria and ventricles using a multipolar contact catheter and show that the flow fields correlate with corresponding activation maps. Further, regions of elevated vorticity and divergence correspond to sites identified as clinically significant rotors and focal sources where therapeutic intervention can be effective. These flow fields can provide quantitative insights into the dynamics of normal and abnormal conduction in humans and could potentially be used to enhance therapies for cardiac arrhythmias.

Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that is triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media with applications to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact for cardiac arrhythmias.

Removal of pinned scroll waves in cardiac tissues by electric fields in a generic model of three-dimensional excitable media

Spirals or scroll waves pinned to heterogeneities in cardiac tissues may cause lethal arrhythmias. To unpin these life-threatening spiral waves, methods of wave emission from heterogeneities (WEH) induced by low-voltage pulsed DC electric fields (PDCEFs) and circularly polarized electric fields (CPEFs) have been used in two-dimensional (2D) cardiac tissues. Nevertheless, the unpinning of scroll waves in three-dimensional (3D) cardiac systems is much more difficult than that of spiral waves in 2D cardiac systems, and there are few reports on the removal of pinned scroll waves in 3D cardiac tissues by electric fields. In this article, we investigate in detail the removal of pinned scroll waves in a generic model of 3D excitable media using PDCEF, AC electric field (ACEF) and CPEF, respectively. We find that spherical waves can be induced from the heterogeneities by these electric fields in initially quiescent excitable media. However, only CPEF can induce spherical waves with frequencies higher than that of the pinned scroll wave. Such higher-frequency spherical waves induced by CPEF can be used to drive the pinned scroll wave out of the cardiac systems. We hope this remarkable ability of CPEF can provide a better alternative to terminate arrhythmias caused by pinned scroll waves.

Hysteresis and drift of spiral waves near heterogeneities: From chemical experiments to cardiac simulations

Dissipative patterns in excitable reaction-diffusion systems can be strongly affected by spatial heterogeneities. Using the photosensitive Belousov-Zhabotinsky reaction, we show a hysteresis effect in the transition between free and pinned spiral rotation. The latter state involves the rotation around a disk-shaped obstacle with an impermeable and inert boundary. The transition is controlled by changes in light intensity. For permeable heterogeneities of higher excitability, we observe spiral drift along both linear and circular boundaries. Our results confirm recent theoretical predictions and, in the case of spiral drift, are further reproduced by numerical simulations with a modified Oregonator model. Additional simulations with a cardiac model show that orbital motion can also exist in anisotropic and three-dimensional systems.

Refraction of scroll-wave filaments at the boundary between two reaction-diffusion media

We explore the shape and the dynamics of scroll-wave filaments in excitable media with an abruptly changing diffusion tensor, important for cardiac applications. We show that, similar to a beam of light, the filament refracts at the boundary separating domains with different diffusion. We derive the laws of filament refraction and test their validity in computational experiments. We discovered that at small angles to the interface, the filament can become unstable and develop oscillations. The nature of the observed instabilities, as well as overall theoretical and experimental significance of the findings, is discussed.

Influence of excitability on unpinning and termination of spiral waves

Application of electrical forcing to release pinned spiral waves from unexcitable obstacles and to terminate the rotation of free spiral waves at the boundary of excitable media has been investigated in thin layers of the Belousov-Zhabotinsky (BZ) reaction, prepared with different initial concentrations of H_{2}SO_{4}. Increasing [H_{2}SO_{4}] raises the excitability of the reaction and reduces the core diameter of free spiral waves as well as the wave period. An electric current with density stronger than a critical value Junpin causes a pinned spiral wave to drift away from the obstacle. For a given obstacle size, Junpin increases with [H_{2}SO_{4}]. Under an applied electrical current, the rotation center of a free spiral wave drifts along a straight path to the boundary. When the current density is stronger than a critical value Jterm, the spiral tip is forced to hit the boundary, where the spiral wave is terminated. Similar to Junpin for releasing a pinned spiral wave, Jterm also increases with [H_{2}SO_{4}]. These experimental findings were confirmed by numerical simulations using the Oregonator model, in which the excitability was adjusted via the ratio of the excitation rate to the recovery rate of the BZ reaction. Therefore, our investigation shows that decreasing the excitability can facilitate elimination of spiral waves by electrical forcing, either in the presence of obstacles or not.

Spatial symmetry breaking determines spiral wave chirality

Chirality represents a fundamental property of spiral waves. Introducing obstacles into cardiac monolayers leads to the initiation of clockwise-rotating, counterclockwise-rotating, and pairs of spiral waves. Simulations show that the precise location of the obstacle and the pacing frequency determine spiral wave chirality. Instabilities predicted by curves relating the action potential duration and the pacing frequency at different spatial locations predict sites of wave break initiation and, hence, spiral wave chirality.

Mechanism of unpinning spirals by a series of stimuli

Antitachycardia pacing (ATP) is widely used to terminate tachycardia before it proceeds to lethal fibrillation. The important prerequisite for successful ATP is unpinning of the spirals anchored to the obstacle by a series of stimuli. Here, to understand the mechanism of unpinning spirals by ATP, we propose a theoretical explanation based on a nonlinear eikonal relation and a kinematical model. The theoretical results are quantitatively consistent with the numerical simulations in both weak and high excitabilities.

Unpinning of spiral waves by electrical forcing in excitable chemical media

We present experimental observations on the electrically forced release of spiral waves pinned to unexcitable circular obstacles in the Belosov-Zhabotinsky reaction. When the applied electric current density reaches the necessary current density J(unpin), the spiral tip is detached and subsequently drifts away from the obstacle. J(unpin) is found to increase with the obstacle diameter d. The growth rate ΔJ(unpin)/Δd is much higher for obstacles larger than the free spiral core compared to that for smaller obstacles. The experimental findings are confirmed by numerical simulations using the Oregonator model. The results imply that it is more difficult to release spiral waves pinned to larger obstacles, especially when the obstacle size exceeds that of the free spiral core.

Initiation and dynamics of a spiral wave around an ionic heterogeneity in a model for human cardiac tissue

In relation to cardiac arrhythmias, heterogeneity of cardiac tissue is one of the most important factors underlying the onset of spiral waves and determining their type. In this paper, we numerically model heterogeneity of realistic size and value and study formation and dynamics of spiral waves around such heterogeneity. We find that the only sustained pattern obtained is a single spiral wave anchored around the heterogeneity. Dynamics of an anchored spiral wave depend on the extent of heterogeneity, and for certain heterogeneity size, we find abrupt regional increase in the period of excitation occurring as a bifurcation. We study factors determining spatial distribution of excitation periods of anchored spiral waves and discuss consequences of such dynamics for cardiac arrhythmias and possibilities for experimental testings of our predictions.