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The chaotic spatio-temporal electrical activity during life-threatening cardiac arrhythmias like ventricular fibrillation is governed by the dynamics of vortex-like spiral or scroll waves. The organizing centers of these waves are called wave tips (2D) or filaments (3D) and they play a key role in understanding and controlling the complex and chaotic electrical dynamics. Therefore, in many experimental and numerical setups it is required to detect the tips of the observed spiral waves. Most of the currently used methods significantly suffer from the influence of noise and are often adjusted to a specific situation (e.g. a specific numerical cardiac cell model). In this study, we use a specific type of deep neural networks (UNet), for detecting spiral wave tips and show that this approach is robust against the influence of intermediate noise levels. Furthermore, we demonstrate that if the UNet is trained with a pool of numerical cell models, spiral wave tips in unknown cell models can also be detected reliably, suggesting that the UNet can in some sense learn the concept of spiral wave tips in a general way, and thus could also be used in experimental situations in the future (ex-vivo, cell-culture or optogenetic experiments).

The development of new approaches to control cardiac arrhythmias requires a deep understanding of spiral wave dynamics. Optogenetics offers new possibilities for this. Preliminary experiments show that sub-threshold illumination affects electrical wave propagation in the mouse heart. However, a systematic exploration of these effects is technically challenging. Here, we use state-of-the-art computer models to study the dynamic control of spiral waves in a two-dimensional model of the adult mouse ventricle, using stationary and non-stationary patterns of sub-threshold illumination. Our results indicate a light intensity-dependent increase in cellular resting membrane potentials, which together with diffusive cell-cell coupling leads to the development of spatial voltage gradients over differently illuminated areas. A spiral wave drifts along the positive gradient. These gradients can be strategically applied to ensure drift-induced termination of a spiral wave, both in optogenetics and in conventional methods of electrical defibrillation.

Background and objective: Spiral waves are phenomena observed in cardiac tissue especially during fibrillatory activities. Spiral waves are revealed through in-vivo and in-vitro studies using high density mapping that requires special experimental setup. Also, in-silico spiral wave analysis and classification is performed using membrane potentials from entire tissue. In this study, we report a characterization approach that identifies spiral wave behaviors using intracardiac electrogram (EGM) readings obtained with commonly used multipolar diagnostic catheters that perform localized but high-resolution readings. Specifically, the algorithm is designed to distinguish between stationary, meandering, and break-up rotors. Methods: The clustering and classification algorithms are tested on simulated data produced using a phenomenological 2D model of cardiac propagation. For EGM measurements, unipolar-bipolar EGM readings from various locations on tissue using two catheter types are modeled. The distance measure between spiral behaviors are assessed using normalized compression distance (NCD), an information theoretical distance. NCD is a universal metric in the sense it is solely based on compressibility of dataset and not requiring feature extraction. We also introduce normalized FFT distance (NFFTD) where compressibility is replaced with a FFT parameter. Results: Overall, outstanding clustering performance was achieved across varying EGM reading configurations. We found that effectiveness in distinguishing was superior in case of NCD than NFFTD. We demonstrated that distinct spiral activity identification on a behaviorally heterogeneous tissue is also possible. Conclusions: This report demonstrates a theoretical validation of clustering and classification approaches that provide an automated mapping from EGM signals to assessment of spiral wave behaviors and hence offers a potential mapping and analysis framework for cardiac tissue wavefront propagation patterns.

Rotating spiral waves in cardiac tissue are implicated in life threatening cardiac arrhythmias. Experimental and theoretical evidences suggest the inhomogeneities in cardiac tissue play a significant role in the dynamics of spiral waves. Based on a modified 2D cardiac tissue model, the interaction of inhomogeneity on the nearby rigidly rotating spiral wave is numerically studied. The adjacent area of the inhomogeneity is divided to two areas, when the initial rotating center of the spiral tip is located in the two areas, the spiral tip will be attracted and anchor on the inhomogeneity finally, or be repulsed away. The width of the area is significantly dependent on the intensity and size of the inhomogeneity. Our numerical study sheds some light on the mechanism of the interaction of inhomogeneity on the spiral wave in cardiac tissue.

INTRODUCTION: Successful initiation of spiral wave reentry in the neonatal rat ventricular myocyte (NRVM) monolayer implicitly assumes the presence of spatial dispersion of repolarization (DR), which is difficult to quantify. We recently introduced a NRVM monolayer that utilizes anthopleurin‐A to impart a prolonged plateau to the NRVM action potential. This was associated with a significant degree of spatial DR that lends itself to accurate quantification. METHODS AND RESULTS: We utilized the monolayer and fluorescence optical mapping of intracellular calcium transients (F(Cai)) to systematically study and compare the contribution of spatial dispersion of the duration of F(Cai) (as a surrogate of DR) to induction of spiral wave reentry around a functional core versus reentry around a fixed anatomical obstacle. We show that functional reentry could be initiated by a premature stimulus acting on a substrate of spatial DR resulting in a functional line of propagation block. Subsequent wave fronts circulated around a central core of functional obstacle created by sustained depolarization from the circulating wave front. Both initiation and termination of spiral wave reentry around an anatomical obstacle consistently required participation of a region of functional propagation block. This region was similarly based on spatial DR. Spontaneous termination of spiral wave reentry also resulted from block in the functional component of the circuit obstacle, usually preceded by beat‐to‐beat slowing of propagation. CONCLUSIONS: The study demonstrates the critical contribution of DR to spiral wave reentry around a purely functional core as well as reentry around a fixed anatomical core.

The work presents a mechanism of spiral wave initiation due to the specific boundary conditions on a border of cardiac tissue defect. There are known scenarios when anatomical or functional defects in cardiac tissue may provoke the spiral wave origination, including unidirectional blockage while passing through the narrow gates, bent over critical curvature wave fronts, inhomogeneous recovery of the tissue, etc. We show a new scenario of spiral wave breakup on a small defect, which is unexcitable but permeable for ionic currents supporting the excitation wave. It was believed that such defects stabilize the rotating wave; however, as shown, instead of stabilizing it leads to the spiral breakup and subsequent multiplication of the rotating waves.

Abnormal electrical activities in neuronal system could be associated with some neuronal diseases. Indeed, external forcing can cause breakdown even collapse in nervous system under appropriate condition. The excitable media sometimes could be described by neuronal network with different topologies. The collective behaviors of neurons can show complex spatiotemporal dynamical properties and spatial distribution for electrical activities due to self-organization even from the regulating from central nervous system. Defects in the nervous system can emit continuous waves or pulses, and pacemaker-like source is generated to perturb the normal signal propagation in nervous system. How these defects are developed? In this paper, a network of neurons is designed in two-dimensional square array with nearest-neighbor connection type; the formation mechanism of defects is investigated by detecting the wave propagation induced by external forcing. It is found that defects could be induced under external periodical forcing under the boundary, and then the wave emitted from the defects can keep balance with the waves excited from external forcing.

The onset of cardiac arrhythmias depends on the electrophysiological and structural properties of cardiac tissue. Electrophysiological remodeling of myocytes due to the presence of adipocytes constitutes a possibly important pathway in the pathogenesis of atrial fibrillation. In this paper we perform an in-silico study of the effect of such myocyte remodeling on the onset of atrial arrhythmias and study the dynamics of arrhythmia sources-spiral waves. We use the Courtemanche model for atrial myocytes and modify their electrophysiological properties based on published cellular electrophysiological measurements in myocytes co-cultered with adipocytes (a 69-87 % increase in APD 90 and an increase of the RMP by 2.5-5.5 mV). In a generic 2D setup we show that adipose tissue remodeling substantially affects the spiral wave dynamics resulting in complex arrhythmia and such arrhythmia can be initiated under high frequency pacing if the size of the remodeled tissue is sufficiently large. These results are confirmed in simulations with an anatomically accurate model of the human atria. ispartof: FRONTIERS IN PHYSIOLOGY vol:9 ispartof: location:Switzerland status: published

Electrogram-guided ablation has been recently developed for allowing better detection and localization of abnormal atrial activity that may be the source of arrhythmogeneity. Nevertheless, no clear indication for the benefit of using electrograms guided ablation over empirical ablation was established thus far, and there is a clear need of improving the localization of cardiac arrhythmogenic targets for ablation. In this paper, we propose a new approach for detection and localization of irregular cardiac activity during ablation procedures that is based on dimension reduction algorithms and principal component analysis (PCA). Using an $8\times 8$ electrode array, our method produces manifolds that allow easy visualization and detection of possible arrhythmogenic ablation targets characterized by irregular conduction. We employ mathematical modeling and computer simulations to demonstrate the feasibility of the new approach for two well established arrhythmogenic sources for irregular conduction--spiral waves and patchy fibrosis. Our results show that the PCA method can differentiate between focal ectopic activity and spiral wave activity, as these two types of activity produce substantially different manifold shapes. Moreover, the technique allows the detection of spiral wave cores and their general meandering and drifting pattern. Fibrotic patches larger than 2 mm 2 could also be visualized using the PCA method, both for quiescent atrial tissue and for tissue exhibiting spiral wave activity. We envision that this method, contingent to further numerical and experimental validation studies in more complex, realistic geometrical configurations and with clinical data, can improve existing atrial ablation mapping capabilities, thus increasing success rates and optimizing arrhythmia management.

Many theoretical and experimental studies indicate that a propagation block represents an important factor in spiral wave initiation in excitable media. The analytical and numerical results we obtained for a generic two-component reaction-diffusion system demonstrate quantitative conditions for the propagation block in a one-dimensional and a two-dimensional medium due to a sharp spatial increase of the medium's excitability or the coupling strength above a certain critical value. Here, we prove that this critical value strongly depends on the medium parameters and the geometry of the inhomogeneity. For an exemplary two-dimensional medium, we show how the propagation block can be used to initiate spiral waves by a specific choice of the size and shape of the medium's inhomogeneity.

We propose an integrated modelling and clinical protocol for characterising local tissue properties in the left atrium and validate the resulting personalised models in four patient cases. We generate a personalised model from a set of measured local activation times (LATs) obtained by pacing the left atrium in the proximity of the coronary sinus with a programmed pacing protocol. We validate the model by evaluating the correlation between a set of measured LATS, obtained by pacing on the high right atrium and a set numerically computed LATs. We then estimate if the tissue is capable of sustaining an atrial fibrillation or a tachycardia by triggering a spiral wave on the computational model and then analysing the activation frequencies and the time elapsed until the termination of the aberrant activation.

See Article by Rodrigo et al A spiral wave is a common macroscopic behavior of excitable media observed in biological, chemical, and physical systems.1,2 In cardiac tissues, spiral wave reentry occurs when a wavefront of electric propagation encounters functionally inexcitable tissue and rotates around it in a vortex-like fashion.3 A rotor of a spiral wave is a rotation center from which a 2-dimensional spiral wave of excitation rotates outward. Phase mapping has been the de facto standard method to identify rotors and to track their trajectories in animal models of fibrillation.4 On phase maps, a rotor is defined as a phase singularity point around which the phase transitions through a complete cycle from −π to +π.3,5 Phase mapping can falsely detect phase singularities in the absence of rotors,6 and spiral waves can exist without phase singularities.7 Rotors and focal impulses have been proposed to be the underlying drivers of atrial fibrillation (AF) in human.3,8,9 This localized source hypothesis has resulted in the utilization of phase mapping in clinical practice, mainly in 2 settings: invasive focal impulse and rotor modulation (FIRM) and noninvasive electrocardiographic imaging (ECGI). FIRM uses phase mapping to locate stable rotors based on endocardial unipolar electrograms acquired by a 64-lead basket catheter.9 ECGI uses phase mapping to locate rotors based on unipolar electrograms acquired by ≈250 body surface electrocardiographic leads, which are back projected onto the epicardial surface, using a technique called inverse solution. FIRM and ECGI have …

In the past years, a number cardiac electromechanics models have been developed to better understand the excitation-contraction behavior of the heart. However, there is no agreement on whether inertial forces play a role in this system. In this study, we assess the influence of mass in electromechanical simulations, using a fully coupled finite element model. We include the effect of mechano-electrical feedback via stretch activated currents. We compare five different models: electrophysiology, electromechanics, electromechanics with mechano-electrical feedback, electromechanics with mass, and electromechanics with mass and mechano-electrical feedback. We simulate normal conduction to study conduction velocity and spiral waves to study fibrillation. During normal conduction, mass in conjunction with mechano-electrical feedback increased the conduction velocity by 8.12% in comparison to the plain electrophysiology case. During the generation of a spiral wave, mass and mechano-electrical feedback generated secondary wavefronts, which were not present in any other model. These secondary wavefronts were initiated in tensile stretch regions that induced electrical currents. We expect that this study will help the research community to better understand the importance of mechanoelectrical feedback and inertia in cardiac electromechanics.

Aims: Ventricular tachycardia (VT) and fibrillation (VF) are the most lethal cardiac arrhythmias. The degeneration of VT into VF is associated with the breakup of a spiral wave of the action potential in cardiac tissue. β-Adrenergic (βAR) signalling potentiates the L-type Ca current (ICaL) faster than the slow delayed rectifier potassium current (IKs), which transiently prolongs the action potential duration (APD) and promotes early after depolarizations. In this study, we aimed at investigating how βAR signalling affects the transition from VT to VF. Methods and results: We used a physiologically detailed computer model of the rabbit ventricular myocyte in a two-dimensional tissue to determine how spiral waves respond to βAR activation following administration of isoproterenol. A simplified mathematical model was also used to investigate the underlying dynamics. We found that the spatiotemporal behaviour of spiral waves strongly depends on the kinetics of βAR activation. When βAR activation is rapid, a stable spiral wave turns into small fragments and its electrocardiogram reveals the transition from VT to VF. This is due to the transiently steepened APD restitution induced by the faster activation of ICaLvs. IKsupon sudden βAR activation. The spiral wave may also disappear if its transient wavelength is too large to be supported by the tissue size upon sudden strong βAR activation that prolongs APD transiently. When βAR activation is gradual, a stable spiral wave remains such, because of more limited increase in both APD and slope of APD restitution due to more contemporaneous ICaLand IKsactivation. Conclusion: Changes in APD restitution during βAR activation revealed a novel transient spiral wave dynamics; this spatiotemporal characteristic strongly depends on the protocol of isoproterenol application. © The Author 2014.

Previous studies have suggested that spiral waves are responsible for the ventricular tachycardia and fibrillation. As a consequence, many contributions have been made to researching spiral waves in normal ventricular tissues. In order to work out how the spiral waves perform in the pace-making

We study a model for cell sorting based in the presence of differential chemotactic sensitivities. The chemical waves which are responsible for the cell motion propagate along some spiral waves. We prove rigorously that cells with larger chemotactic sensitivity are trapped in a region close to the center of the spiral waves if these propagate along some archimedian spirals.

Spiral wave is an important dynamic behavior in several physiological signaling processes, such as the cardiac fibrillation. Since antispiral wave recently been discovered in microemulsion system, it has been intensively studied and discussed. In this work, we find that the spatial noise induced heterogeneity in homogenous system may also induce antispiral formation. Quantitative characterization prove spatiotemporal coherence resonance phenomenon appears, and spatiotemporal noise is more favorable to optimally sustain the antispirals.

Therapy for atrial fibrillation (AF) remains suboptimal, in large part because its mechanisms are unclear. While pulmonary vein ectopy may trigger AF, it remains uncertain how AF, once triggered, is actually sustained. Recent discoveries show that human AF is maintained by a small number of rotors or focal sources. AF sources are widely distributed in patient-specific locations, often remote from pulmonary veins and in the right atrium and stable for prolonged periods of time. In a multicentre experience, brief targeted ablation at sources (focal impulse and rotor modulation [FIRM]) terminated AF predominantly to sinus rhythm prior to pulmonary vein isolation and eliminated AF on rigorous followup. This review summarises the evidence for stable rotors and focal sources of human AF and their clinical role as ablation targets to eliminate paroxysmal, persistent and long-standing persistent AF.

We present results from a suite of three-dimensional global hydrodynamic simulations which show that spiral density waves propagating in circumstellar disks are unstable to the growth of a parametric instability that leads to break-down of the flow into turbulence. This spiral wave instability (SWI) arises from a resonant interaction between pairs of inertial waves, or inertial-gravity waves, and the background spiral wave. The development of the instability in the linear regime involves the growth of a broad spectrum of inertial modes, with growth rates on the order of the orbital time, and results in a nonlinear saturated state in which turbulent velocity perturbations are of a similar magnitude to those induced by the spiral wave. The turbulence induces angular momentum transport, and vertical mixing, at a rate that depends locally on the amplitude of the spiral wave (we obtain a stress parameter $\alpha \sim 5 \times 10^{-4}$ in our reference model). The instability is found to operate in a wide-range of disk models, including those with isothermal or adiabatic equations of state, and in viscous disks where the dimensionless kinematic viscosity $\nu\le 10^{-5}$. This robustness suggests that the instability will have applications to a broad range of astrophysical disk-related phenomena, including those in close binary systems, planets embedded in protoplanetary disks (including Jupiter in our own Solar System) and FU Orionis outburst models. Further work is required to determine the nature of the instability, and to evaluate its observational consequences, in physically more complete disk models than we have considered in this paper., Comment: Accepted for publication in ApJ, 22 pages, 22 figures, 2 tables, some figures are reduced in size to meet the arxiv requirement

Diversity in the neurons and noise are inevitable in the real neuronal network. In this paper, parameter diversity induced spiral waves and multiple spatial coherence resonances in a two-dimensional neuronal network without or with noise are simulated. The relationship between the multiple resonances and the multiple transitions between patterns of spiral waves are identified. The coherence degrees induced by the diversity are suppressed when noise is introduced and noise density is increased. The results suggest that natural nervous system might profit from both parameter diversity and noise, provided a possible approach to control formation and transition of spiral wave by the cooperation between the diversity and noise.

As part of a 3-wk intersession workshop funded by a National Science Foundation Expeditions in Computing award, 15 undergraduate students from the City University of New York1collaborated on a study aimed at characterizing the voltage dynamics and arrhythmogenic behavior of cardiac cells for a broad range of physiologically relevant conditions using an in silico model. The primary goal of the workshop was to cultivate student interest in computational modeling and analysis of complex systems by introducing them through lectures and laboratory activities to current research in cardiac modeling and by engaging them in a hands-on research experience. The success of the workshop lay in the exposure of the students to active researchers and experts in their fields, the use of hands-on activities to communicate important concepts, active engagement of the students in research, and explanations of the significance of results as the students generated them. The workshop content addressed how spiral waves of electrical activity are initiated in the heart and how different parameter values affect the dynamics of these reentrant waves. Spiral waves are clinically associated with tachycardia, when the waves remain stable, and with fibrillation, when the waves exhibit breakup. All in silico experiments were conducted by simulating a mathematical model of cardiac cells on graphics processing units instead of the standard central processing units of desktop computers. This approach decreased the run time for each simulation to almost real time, thereby allowing the students to quickly analyze and characterize the simulated arrhythmias. Results from these simulations, as well as some of the background and methodology taught during the workshop, is presented in this article along with the programming code and the explanations of simulation results in an effort to allow other teachers and students to perform their own demonstrations, simulations, and studies.

Recent development of tissue engineering techniques allows creating and maintaining almost indefinitely networks of excitable cells with desired architecture. We coupled the network of live excitable cardiac cells with a common computer by sensitizing them to light, projecting a light pattern on the layer of cells, and monitoring excitation with the aid of fluorescent probes (optical mapping). As a sensitizing substance we used azobenzene trimethylammonium bromide (AzoTAB). This substance undergoes cis-trans-photoisomerization and trans-isomer of AzoTAB inhibits excitation in the cardiac cells, while cis-isomer does not. AzoTAB-mediated sensitization allows, thus, reversible and dynamic control of the excitation waves through the entire cardiomyocyte network either uniformly, or in a preferred spatial pattern. Technically, it was achieved by coupling a common digital projector with a macroview microscope and using computer graphic software for creating the projected pattern of conducting pathways. This approach allows real time interactive photocontrol of the heart tissue.

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.

Dynamical chaos, an irregular behavior of deterministic systems, has been widely shown in nature. It also has been demonstrated in cardiac myocytes in many studies, including rapid pacing induced irregular beat-to-beat action potential alterations and slow pacing induced irregular early afterdepolarizations, etc. Here we review the roles of chaos in the genesis of cardiac arrhythmias, the transition to ventricular fibrillation, and the spontaneous termination of fibrillation, based on evidence from computer simulation of mathematical models and experiments of animal models.

The control of spiral wave breakup due to Doppler instability has been investigated. It is found that by applying the spatiotemporal modulation method with suitable control parameters, spiral wave breakup can be prevented. Further numerical simulations show that preventing spiral wave breakup is a result of the decrease of meandering motion of the spiral tip.

Torsades de pointes (TdP. is characterized not only by its electrocardiographic morphology but also by a tendency to spontaneously terminate. Although clinical and experimental studies suggested that TdP is triggered exclusively by early afterdepolarization, the reentrant mechanism seems to play a certain role in its maintenance. In this article, I review the studies that investigated the origin and activation sequences of the twisting QRS complexes of TdP, and discuss whether it is fortunate or unfortunate for us if TdP has something to do with reentry.

Pattern estimation and selection in media can give important clues to understand the collective response to external stimulus by detecting the observable variables. Both reaction–diffusion systems (RDs) and neuronal networks can be treated as multi-agent systems from molecular level, intrinsic cooperation, competition. An external stimulus or attack can cause collapse of spatial order and distribution, while appropriate noise can enhance the consensus in the spatiotemporal systems. Pattern formation and synchronization stability can bridge isolated oscillators and the network by coupling these nodes with appropriate connection types. As a result, the dynamical behaviors can be detected and discussed by developing different spatial patterns and realizing network synchronization. Indeed, the collective response of network and multi-agent system depends on the local kinetics of nodes and cells. It is better to know the standard bifurcation analysis and stability control schemes before dealing with network problems. In this review, dynamics discussion and synchronization control on low-dimensional systems, pattern formation and synchronization stability on network, wave stability in RDs and neuronal network are summarized. Finally, possible guidance is presented when some physical effects such as polarization field and electromagnetic induction are considered.

Spiral waves have been reported to be existent in the neocortex, during pharmacologically induced oscillations and sleep-like states. In the last decades, theoretical studies have demonstrated an underlying mechanism of the generation of spiral waves in a heart system. Nevertheless, how can a neural system produce spontaneous spiral wave and whether this behavior is sensitive to the dynamics of isolated neurons have not been systematically studied yet. In this paper we propose a modified Hindmarsh-Rose (HR) neuron model to study whether spiral wave can occur spontaneously in a two-dimensional array of HR neurons, which evolves from the initial state with a random phase distribution. The simulation results show that whether spiral wave can occur spontaneously in the system depends on the state of the single HR neuron, initial state of system and coupling strength. Especially, the state of the single HR neuron plays a central role. When the single HR neuron is in the state of period 1 spike, multiple spiral waves and spiral pairs can be generated spontaneously in the system for a certain range of coupling strength. In this case, the formations of spiral waves are completely independent of the initial state of the system, and as long as choosing an appropriate coupling strength, a single spiral wave can be found in the system. Furthermore, when the coupling strength exceeds a certain threshold value, the system will exhibit three kinds of dynamical behaviors, and correspond to three kinds of the different initial states, respectively. When system evolves from the first kind of initial state, the single spiral wave can be found occasionally in the system. When the system evolves from the second or third kind of initial state, the oscillation with intermittently global synchronization and oscillation death can be observed in the system, respectively. When a single HR neuron is in the state of period 2 spike, the spiral wave can appear spontaneously in the system only when the phase distribution of the initial state approaches to a uniform distribution. Moreover, the range of coupling strength on the generation of spiral wave is smaller than that of period 1 spike. When the single HR neuron is in a higher periodic state, it is difficult to generate spontaneously spiral wave in the system. These results are useful in understanding the spontaneous generation of spiral waves in the neocortex.

Standing waves of depolarisation produced by periodic low-voltage driving eliminate propagation activity in the heart, thus providing a defibrillating effect. The phenomenon cannot be reproduced by mono- or bidomain models of cardiac tissue, where voltage perturbations decay exponentially with a space constant λ 1 ≈1 mm. Extension of the bidomain model taking into account effects of the external bathing medium allows simulation of the standing waves which eliminate spiral wave activity in the tissue. Mechanisms of such a defibrillating effect can be explained by the existence of an additional, unusually long space constant, λ 2 ≈20 mm, in the bidomain model with a bath, which emerges due to redistribution of the applied voltage by the external conductive medium.

We study pattern formation induced by a spiral wave developing from heterogeneities in an excitable medium. Turbulence can be suppressed by a spiral wave from the heterogeneity, forming multiple coexistent systems of regular geometrical patterns. We find that the types of these patterns depend critically on the degree of heterogeneity. The underlying mechanism is due to dispersion relation which is characterized by excitability.

In this paper, we investigate the possibility to control the spiral waves and spatiotemporal chaos in an excitable media by subthreshold ordered waves, which is by definition in this paper the spatially ordered periodical subthreshold fluctuations of system variables. It is found that both the spiral wave and the spatiotemporal chaos could be driven out of the control domain by perturbation of periodical subthreshold ordered waves. The effective control time declines as the amplitude of subthreshold ordered waves are increased, or their frequencies are decreased. Furthermore, we show that the effectiveness of this method is also dependent on the spatial arrangement of the subthreshold ordered waves. We discuss the possible applications of this method, especially in the control of heart fibrillation.

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.

The mechanisms underpinning human cardiac fibrillation remain elusive. In his 1913 paper ‘On dynamic equilibrium in the heart’, Mines proposed that an activation wave front could propagate repeatedly in a circle, initiated by a stimulus in the vulnerable period. While the dynamics of activation and recovery are central to cardiac fibrillation, these physiological data are rarely used in clinical mapping. Fibrillation is a rapid irregular rhythm with spatiotemporal disorder resulting from two fundamental mechanisms – sources in preferred cardiac regions or spatially diffuse self‐sustaining activity, i.e. with no preferred source. On close inspection, however, this debate may also reflect mapping technique. Fibrillation is initiated from triggers by regional dispersion in repolarization, slow conduction and wavebreak, then sustained by non‐uniform interactions of these mechanisms. Notably, optical mapping of action potentials in atrial fibrillation (AF) show spiral wave sources (rotors) in nearly all studies including humans, while most traditional electrogram analyses of AF do not. Techniques may diverge in fibrillation because electrograms summate non‐coherent waves within an undefined field whereas optical maps define waves with a visually defined field. Also fibrillation operates at the limits of activation and recovery, which are well represented by action potentials while fibrillatory electrograms poorly represent repolarization. We conclude by suggesting areas for study that may be used, until such time as optical mapping is clinically feasible, to improve mechanistic understanding and therapy of human cardiac fibrillation.

Computational verb cellular networks (CVCNs) are cellular networks, of which the local rules are computational verb rules. Many patterns had been found in CVCN previously. In this paper, newly-discovered one-, two-, three- and four-armed spiral waves in CVCNs are presented. Eight-armed spiral waves appear briefly co-existing with four-armed patterns at a low frequency. When initial states are uniformly randomly chosen from interval between 0 and 1, more than one spiral wave may be formed in a CVCN. The rotors of four-armed rotating spiral waves move slowly. All these spiral waves co-exist with checkerboard patterns, which are more visible when the array size is small.

1 Cardiac Arrhythmias: Mechanistic Knowledge and Innovation from Computer Models. Natalia A. Trayanova and Patrick M. Boyle.- 2 Discrete Mechanical Modeling of Mechanoelectrical Feedback in Cardiac Tissue: Novel Mechanisms of Spiral Wave Initiation. Louis D. Weise and Alexander V. Panfilov.- 3 Multiscale Modelling of Cardiac Perfusion. Jack Lee, Andrew Cookson, Radomir Chabiniok, Simone Rivolo, Eoin Hyde, Matthew Sinclair, Christian Michler, Taha Sochi and Nicolas Smith.- 4 Using image-based CFD to investigate the intracardiac turbulence. C. Chnafa, S. Mendez, R. Moreno and F. Nicoud.- 5 Multiscale Modeling of Sickle Cell Anemia. Huan Lei and George Em Karniadakis.- 6 A three-dimensional continuum model of active contraction in single cardiomyocytes. Alessio Gizzi, Ricardo Ruiz-Baier, Simone Rossi, Aymen Laadhari, Christian Cherubini, and Simonetta Filippi.- 7 On the Assumption of Laminar ow in Physiological Flows: Cerebral Aneurysms as an Illustrative Example. Oyvind Evju and Kent-Andre Mardal.- 8 Effects of poroelasticity on fluid-structure interaction in arteries: a computational sensitivity study. M. Bukac, I. Yotov, R. Zakerzadeh, P. Zunino.- 9 Personalized Anatomical Meshing of Human Body with Applications. Yuri Vassilevski, Alexander Danilov, Yuri Ivanov, Sergey Simakov and Timur Gamilov.

Background— Functional reentry in the heart takes the form of spiral waves. Drifting spiral waves can become pinned to anatomic obstacles and thus attain stability and persistence. Lidocaine is an antiarrhythmic agent commonly used to treat ventricular tachycardia clinically. We examined the ability of small obstacles to anchor spiral waves and the effect of lidocaine on their attachment. Methods and Results— Spiral waves were electrically induced in confluent monolayers of cultured, neonatal rat cardiomyocytes. Small, circular anatomic obstacles (0.6 to 2.6 mm in diameter) were situated in the center of the monolayers to provide an anchoring site. Eighty reentry episodes consisting of at least 4 revolutions were studied. In 36 episodes, the spiral wave attached to the obstacle and became stationary and sustained, with a shorter reentry cycle length and higher rate. Spiral waves could attach to obstacles as small as 0.6 mm, with a likelihood for attachment that increased with obstacle size. After attachment, both conduction velocity of the wave-front tip and wavelength near the obstacle adapted from their pre-reentry values and increased linearly with obstacle size. In contrast, reentry cycle length did not correlate significantly with obstacle size. Addition of lidocaine 90 μmol/L depressed conduction velocity, increased reentry cycle length, and caused attached spiral waves to become quasi- attached to the obstacle or terminate. Conclusions— Anchored spiral waves exhibit properties of both unattached spiral waves and anatomic reentry. Their behavior may be representative of functional reentry dynamics in cardiac tissue, particularly in the setting of monomorphic tachyarrhythmias.

In this article, three methods for the analysis of the relationships among atrial electrical activities in different sites recorded during sinus rhythms (SR) and atrial fibrillation (AF) will be presented and discussed. The properties of pairs of atrial signals (ASs) will be characterized through a linear parameter obtained from the estimation of the cross-correlation (CC) function, using a nonlinear association (NLA) estimator and through a synchronization index based on the corrected cross-conditional entropy (CCE). These parameters will be employed to assess possible changes in the determinism and dynamics of AS induced by adrenergic stimulation, which is supposed, but not yet demonstrated, to promote both spiral wave maintenance and wave break-up thus fragmenting atrial conduction. The sympathetic stimulation is mimicked by isoproterenol infusion, and the effects of this drug are evaluated in a group of patients in four experimental conditions: 1) SR, 2) SR during isoproterenol (SRISO) administration, 3) AF, and 4) AF during isoproterenol administration (AFISO)

We study dynamics of spiral waves in excitable media under spatio-temporal forcing in the form of a traveling-wave modulation of a control parameter. Within a certain range of the modulation frequency a synchronized drift of a spiral wave along a straight line is obtained. A complicated cycloidal-like motion occurs outside this synchronization band. Although circulation of the spiral wave tip is in-phase with external modulation, its rotation frequency is not locked by the applied perturbation. These numerical results obtained for the Oreganator model are confirmed and complemented by analysis of a simplified kinematical model. It is shown that the observed paradoxical phase synchronization is caused by Doppler shift of the modulation frequency in a coordinate system comoving with the spiral wave core.

The spiral waves in a system of two-dimensional coupled oscillators with spatial period-2 structure are investigated. We find a sandwiched spiral wave where any adjacent oscillators are in anti-phase. We show that the propagation rate of the sandwiched spiral wave is insensitive to the change of the coupling constant. The influences of the local kinetics on the sandwiched spiral wave are also investigated.

The dynamic behaviour of spiral tip in the light-sensitive Belousov–Zhabotinsky reaction under the influence of an externally applied light gradient was experimentally studied. The gradient causes different drifts for different spiral patterns. The centre of the spiral wave moved toward the region of lower light intensity. The direction of an additional perpendicular drift depended on the chirality of the spiral wave. The dependences of the drifting angle and the drifting velocity on light gradient have been measured.

Background— Two types of ventricular fibrillation (VF) have been demonstrated in isolated rabbit hearts during D600 infusion. Type 1 VF is characterized by the presence of multiple, wandering wavelets, whereas type 2 VF shows local spatiotemporal periodicity. We hypothesized that a single mother rotor underlies type 2 VF. Methods and Results— One (protocol I) or 2 (protocol II) cameras were used to map the epicardial ventricular activations in Langendorff-perfused rabbit hearts. Multiple episodes of type 2 VF were induced in 22 hearts by high-concentration (≥2.5 mg/L) D600 (protocol I). During type 2 VF, a single spiral wave (n=19) and/or an epicardial breakthrough pattern (n=11) was present in 14 hearts. These spiral waves either slowly drifted or intermittently anchored on the papillary muscle (PM) of the left ventricle. Dominant-frequency (DF) analyses showed that the highest local DF was near the PM (12.5±1.1 Hz). There was an excellent correlation between the highest local DF of these spiral waves and breakthroughs (11.8±1.7 Hz) and the DF of simultaneously obtained global pseudo-ECG (11.2±1.8 Hz, r =0.97, P Conclusions— Either a stationary or a slowly drifting mother rotor can result in type 2 VF. Colocalization of the stationary mother rotors with the PM suggests the importance of underlying anatomic structures in mother rotor formation.

It has become widely accepted that ventricular fibrillation, the most dangerous cardiac arrhythmias, is a major cause of death in the industrialized world. Alternans and conduction block have recently been related to the progression from ventricular tachycardia to ventricular fibrillation. From the point of view in cellular electrophysiology, ventricular tachycardia is the formation of reentrant wave in cardiac tissue. And ventricular fibrillation arises from subsequent breakdown of reentrant wave into multiple drifting and meandering spiral waves. In this paper, we numerically study pulse and vortex dynamics in cardiac tissue. Our numerical results include 1:1 normal sinus rhythm, 2:1 conduction block, complete conduction block, spiral wave, and spiral breakup. All of our numerical findings can be corresponding to clinical measurements in electrocardiogram. Various electrical activities in cardiac tissue will be discussed in detail in the present manuscript.

We investigate the spiral wave control in the two-dimensional complex Ginzburg–Landau equation. External drivings which are not resonant with spiral waves but with intrinsic system modes are used to successfully annihilate spiral waves and direct the system to various target states. The novel control mechanism is intuitively explained and the richness and flexibility the control results are emphasized.

Sudden cardiac death, secondary to ventricular fibrillation (VF), remains the leading cause of death in many developed countries. Substantial experimental and theoretical support exists for the idea that VF is caused by spiral wave re-entry. The initiation and subsequent break-up of spiral waves have been linked to electrical alternans, a phenomenon typically associated with a steeply sloped restitution relationship. Interventions that reduce the slope of the restitution relationship have been shown to prevent the induction of VF and to terminate existing VF in experimental models. These results suggest that electrical restitution may be a promising new target for antiarrhythmic therapies.

The vulnerable window in a heterogeneous virtual LRl cardiac tissue, with a linear gradient in GK, is wider when following propagation down the gradient, towards tissue with longer action potential duration, than when following propagation up the gradient. Spiral wave solutions in a uniform linear gradient in GK drift, with a velocity component along the gradient of the order of mm/s, towards tissue with a longer APD.

In numerical experiments with the Fitzhugh-Nagumo set of reaction-diffusion equations describing two-dimensional excitable media, unusual solutions are found that correspond to a concave spiral wave steadily rotating round a circular obstacle in a finite-size medium. Such a wave arises in the region of parameters corresponding to the solitonlike regime (see text); it appears due to the interaction between the peripheral areas of a “seed” spiral wave with a convex front and the echo waves incoming from the outer boundaries of a medium. The solutions obtained are in contradiction with intuition and represent a numerical counterexample to the known theories that forbid steadily moving excitation waves with concave fronts. Nevertheless, a concave spiral wave is a stable object; being transformed to the usual spiral wave with a convex front by suppressing echo at the outer boundaries of the medium, it is again recovered upon restoring the echo conditions. In addition to the single-arm spiral concave wave, solutions are obtained that describe multiarm waves of this type; for this reason, the concave fronts of these waves are a coarse property.

Introduction: The ventricular apex has a helical arrangement of myocardial fibers called the “vortex cordis.” Experimental studies have demonstrated that the first postshock activation originates from the ventricular apex, regardless of the electrical shock outcome; however, the related underlying mechanism is unclear. We hypothesized that the vortex cordis contributes to the initiation of postshock activation. To clarify this issue, we numerically studied the transmembrane potential distribution produced by various electrical shocks. Methods and Results: Using an active membrane model, we simulated a two-dimensional bidomain myocardial tissue incorporating a typical fiber orientation of the vortex cordis. Monophasic or biphasic shock was delivered via two line electrodes located at opposite tissue borders. Transmembrane potential distribution during the monophasic shock at the center of the vortex cordis showed a gradient high enough to initiate postshock activation. The postshock activation from the center of the vortex cordis was not suppressed, regardless of the initiation of spiral wave reentry. Spiral wave reentry was induced by the monophasic shock when the center area of the vortex cordis was partially excited by the nonuniform virtual electrode polarization. Postshock activation following the biphasic shock also originated from the center of the vortex cordis, but it tended to be suppressed due to the narrower excitable gap around the center of the vortex cordis. The electroporation effect, which was maximal at the center of the vortex cordis, is another possible mechanism of postshock activation. Conclusion: Our simulations suggest that the vortex cordis may cause postshock activation.(J Cardiovasc Electrophysiol, Vol. 14, pp. 295-302, March 2003)

Phase Singularities and Termination of Reentry.Introduction: Recent defibrillation studies show that electric fields interact with reentrant activity in myocardial tissue through virtual electrode polarization (VEP). This study focuses on determining how VEP relates to the creation and survival of postshock phase singularities in cardiac tissue and demonstrating that interactions between VEP and preshock tissue state engender the probabilistic nature of defibrillation. Methods and Results: We constructed a two-dimensional tissue model representing a ventricular cross-section with fiber architecture and surrounding bath. We initiated spiral wave reentry and subjected the tissue to a monophasic shock. We estimated the shock success probability for a given strength by testing 16 coupling intervals throughout a single rotation of the wavefront. Over a range of shock strengths, our model exhibits dose-response behavior similar to experimental defibrillation efficacy curves. At the 50% effective strength (ED50), successful termination of reentry depends upon the interaction between preshock excitable gap and postshock phase singularities. We also found that increasing the stimulus strength toward ED50 increases the number of postshock singularities, whereas further strength increases above ED50 decrease the number of singularities. Conclusion: Our results show for the first time that a computational model can account for the probabilistic nature of defibrillation as VEP interacts with the dynamics of an ongoing reentrant wavefront. Further, we demonstrate that success of a shock depends on the annihilation of the phase singularities that arise after any strong stimulus. Our findings imply that VEP completely overrides the preshock tissue state in shocks that are highly likely to defibrillate (ED95).