Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity

Robustness of free and pinned spiral waves against breakup by electrical forcing in excitable chemical media

We present an investigation on the breakup of free and pinned spiral waves under an applied electrical current in the Belousov-Zhabotinsky reaction. Spiral fronts propagating towards the negative electrode are decelerated. A breakup of the spiral waves occurs when some segments of the fronts are stopped by a sufficiently strong electrical current. In the absence of obstacles (i.e., free spiral waves), the critical value of the electrical current for the wave breakup increases with the excitability of the medium. For spiral waves pinned to circular obstacles, the critical electrical current increases with the obstacle diameter. Analysis of spiral dynamics shows that the enhancement of the robustness against the breakup of both free and pinned spiral waves is originated by the increment of wave speed when either the excitability is strengthened or the obstacle size is enlarged. The experimental findings are reproduced by numerical simulations using the Oregonator model. In addition, the simulations reveal that the robustness against the forced breakup increases with the activator level in both cases of free and pinned spiral waves.

Spiral-wave dynamics in a mathematical model of human ventricular tissue with myocytes and Purkinje fibers

We present systematic numerical studies of the possible effects of the coupling of human endocardial and Purkinje cells at cellular and two-dimensional tissue levels. We find that the autorhythmic-activity frequency of the Purkinje cell in a composite decreases with an increase in the coupling strength; this can even eliminate the autorhythmicity. We observe a delay between the beginning of the action potentials of endocardial and Purkinje cells in a composite; such a delay increases as we decrease the diffusive coupling, and eventually a failure of transmission occurs. An increase in the diffusive coupling decreases the slope of the action-potential-duration-restitution curve of an endocardial cell in a composite. By using a minimal model for the Purkinje network, in which we have a two-dimensional, bilayer tissue, with a layer of Purkinje cells on top of a layer of endocardial cells, we can stabilize spiral-wave turbulence; however, for a sparse distribution of Purkinje-ventricular junctions, at which these two layers are coupled, we can also obtain additional focal activity and many complex transient regimes. We also present additional effects resulting from the coupling of Purkinje and endocardial layers and discuss the relation of our results to the studies performed in anatomically accurate models of the Purkinje network.

Fast propagation regions cause self-sustained reentry in excitable media

Self-sustained waves of electrophysiological activity can cause arrhythmia in the heart. These reentrant excitations have been associated with spiral waves circulating around either an anatomically defined weakly conducting region or a functionally determined core. Recently, an ablation procedure has been clinically introduced that stops atrial fibrillation of the heart by destroying the electrical activity at the spiral core. This is puzzling because the tissue at the anatomically defined spiral core would already be weakly conducting, and a further decrease should not improve the situation. In the case of a functionally determined core, an ablation procedure should even further stabilize the rotating wave. The efficacy of the procedure thus needs explanation. Here, we show theoretically that fundamentally in any excitable medium a region with a propagation velocity faster than its surrounding can act as a nucleation center for reentry and can anchor an induced spiral wave. Our findings demonstrate a mechanistic underpinning for the recently developed ablation procedure. Our theoretical results are based on a very general and widely used two-component model of an excitable medium. Moreover, the important control parameters used to realize conditions for the discovered phenomena are applicable to quite different multicomponent models.

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.

Model of electrical activity in cardiac tissue under electromagnetic induction

Complex electrical activities in cardiac tissue can set up time-varying electromagnetic field. Magnetic flux is introduced into the Fitzhugh-Nagumo model to describe the effect of electromagnetic induction, and then memristor is used to realize the feedback of magnetic flux on the membrane potential in cardiac tissue. It is found that a spiral wave can be triggered and developed by setting specific initials in the media, that is to say, the media still support the survival of standing spiral waves under electromagnetic induction. Furthermore, electromagnetic radiation is considered on this model as external stimuli, it is found that spiral waves encounter breakup and turbulent electrical activities are observed, and it can give guidance to understand the occurrence of sudden heart disorder subjected to heavily electromagnetic radiation.

A New Efficient Method for Detecting Phase Singularity in Cardiac Fibrillation

Background

The point of phase singularity (PS) is considered to represent a spiral wave core or a rotor in cardiac fibrillation. Computational efficiency is important for detection of PS in clinical electrophysiology. We developed a novel algorithm for highly efficient and robust detection of PS.

Methods

In contrast to the conventional method, which calculates PS based on the line integral of the phase around a PS point equal to ±2π (the Iyer-Gray method), the proposed algorithm (the location-centric method) looks for the phase discontinuity point at which PS actually occurs. We tested the efficiency and robustness of these two methods in a two-dimensional mathematical model of atrial fibrillation (AF), with and without remodeling of ionic currents.

Results

1. There was a significant association, in terms of the Hausdorff distance (3.30 ± 0.0 mm), between the PS points measured using the Iyer-Gray and location-centric methods, with almost identical PS trajectories generated by the two methods. 2. For the condition of electrical remodeling of AF (0.3 × I_CaL), the PS points calculated by the two methods were satisfactorily co-localized (with the Hausdorff distance of 1.64 ± 0.09 mm). 3. The proposed location-centric method was substantially more efficient than the Iyer-Gray method, with a 28.6-fold and 28.2-fold shorter run times for the control and remodeling scenarios, respectively.

Conclusion

We propose a new location-centric method for calculating PS, which is robust and more efficient compared with the conventionally used method.

Reentry and Ectopic Pacemakers Emerge in a Three-Dimensional Model for a Slab of Cardiac Tissue with Diffuse Microfibrosis near the Percolation Threshold

Arrhythmias in cardiac tissue are generally associated with irregular electrical wave propagation in the heart. Cardiac tissue is formed by a discrete cell network, which is often heterogeneous. Recently, it was shown in simulations of two-dimensional (2D) discrete models of cardiac tissue that a wave crossing a fibrotic, heterogeneous region may produce reentry and transient or persistent ectopic activity provided the fraction of conducting connections is just above the percolation threshold. Here, we investigate the occurrence of these phenomena in three-dimensions by simulations of a discrete model representing a thin slab of cardiac tissue. This is motivated (i) by the necessity to study the relevance and properties of the percolation-related mechanism for the emergence of microreentries in three dimensions and (ii) by the fact that atrial tissue is quite thin in comparison with ventricular tissue. Here, we simplify the model by neglecting details of tissue anatomy, e. g. geometries of atria or ventricles and the anisotropy in the conductivity. Hence, our modeling study is confined to the investigation of the effect of the tissue thickness as well as to the comparison of the dynamics of electrical excitation in a 2D layer with the one in a 3D slab. Our results indicate a strong and non-trivial effect of the thickness even for thin tissue slabs on the probability of microreentries and ectopic beat generation. The strong correlation of the occurrence of microreentry with the percolation threshold reported earlier in 2D layers persists in 3D slabs. Finally, a qualitative agreement of 3D simulated electrograms in the fibrotic region with the experimentally observed complex fractional atrial electrograms (CFAE) as well as strong difference between simulated electrograms in 2D and 3D were found for the cases where reentry and ectopic activity were triggered by the micro-fibrotic region.

Interactions between Activation and Repolarization Restitution Properties in the Intact Human Heart: In-Vivo Whole-Heart Data and Mathematical Description

BACKGROUND

The restitution of the action potential duration (APDR) and conduction velocity (CVR) are mechanisms whereby cardiac excitation and repolarization adapt to changes in heart rate. They modulate the vulnerability to dangerous arrhythmia, but the mechanistic link between restitution and arrhythmogenesis remains only partially understood.

METHODS

This paper provides an experimental and theoretical study of repolarization and excitation restitution properties and their interactions in the intact human epicardium. The interdependence between excitation and repolarization dynamic is studied in 8 patients (14 restitution protocols, 1722 restitution curves) undergoing global epicardial mapping with multi-electrode socks before open heart surgery. A mathematical description of the contribution of both repolarization and conduction dynamics to the steepness of the APDR slope is proposed.

RESULTS

This study demonstrates that the APDR slope is a function of both activation and repolarization dynamics. At short cycle length, conduction delay significantly increases the APDR slope by interacting with the diastolic interval. As predicted by the proposed mathematical formulation, the APDR slope was more sensitive to activation time prolongation than to the simultaneous shortening of repolarization time. A steep APDR slope was frequently identified, with 61% of all cardiac sites exhibiting an APDR slope > 1, suggesting that a slope > 1 may not necessarily promote electrical instability in the human epicardium. APDR slope did not change for different activation or repolarization times, and it was not a function of local baseline APD. However, it was affected by the spatial organization of electrical excitation, suggesting that in tissue APDR is not a unique function of local electrophysiological properties. Spatial heterogeneity in both activation and repolarization restitution contributed to the increase in the modulated dispersion of repolarization, which for short cycle length was as high as 250 ms. Heterogeneity in conduction velocity restitution can translate into both activation and repolarization dispersion and increase cardiac instability. The proposed mathematical formulation shows an excellent agreement with the experimental data (correlation coefficient r = 0.94) and provides a useful tool for the understanding of the complex interactions between activation and repolarization restitution properties as well as between their measurements.

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.

Dynamics and Molecular Mechanisms of Ventricular Fibrillation in Structurally Normal Hearts

Ventricular fibrillation (VF) is the most severe cardiac rhythm disturbance and one of the most important immediate causes of sudden cardiac death. In the structurally normal heart, a small number of stable reentrant sources, perhaps 1 or 2, underlie the mechanism of VF, and the stabilization of the sources, their frequency, and the complexity of the turbulent waves they generate depend on the expression, spatial distribution, and intermolecular interactions of the 2 most important ion channels that control cardiac excitability: the inward rectifier potassium channel, Kir2.1, and the alpha subunit of the main cardiac sodium channel, NaV1.5.

Increased Late Sodium Current Contributes to the Electrophysiological Effects of Chronic, but Not Acute, Dofetilide Administration

BACKGROUND

Drugs are screened for delayed rectifier potassium current (IKr) blockade to predict long QT syndrome prolongation and arrhythmogenesis. However, single-cell studies have shown that chronic (hours) exposure to some IKr blockers (eg, dofetilide) prolongs repolarization additionally by increasing late sodium current (INa-L) via inhibition of phosphoinositide 3-kinase. We hypothesized that chronic dofetilide administration to intact dogs prolongs repolarization by blocking IKr and increasing INa-L.

METHODS AND RESULTS

We continuously infused dofetilide (6-9 μg/kg bolus+6-9 μg/kg per hour IV infusion) into anesthetized dogs for 7 hours, maintaining plasma levels within the therapeutic range. In separate experiments, myocardial biopsies were taken before and during 6-hour intravenous dofetide infusion, and the level of phospho-Akt was determined. Acute and chronic dofetilide effects on action potential duration (APD) were studied in canine left ventricular subendocardial slabs using microelectrode techniques. Dofetilide monotonically increased QTc and APD throughout 6.5-hour exposure. Dofetilide infusion during ≥210 minutes inhibited Akt phosphorylation. INa-L block with lidocaine shortened QTc and APD more at 6.5 hours than at 50 minutes (QTc) or 30 minutes (APD) dofetilide administration. In comparison, moxifloxacin, an IKr blocker with no effects on phosphoinositide 3-kinase and INa-L prolonged APD acutely but no additional prolongation occurred on chronic superfusion. Lidocaine shortened APD equally during acute and chronic moxifloxacin superfusion.

CONCLUSIONS

Increased INa-L contributes to chronic dofetilide effects in vivo. These data emphasize the need to include time and INa-L in evaluating the phosphoinositide 3-kinase inhibition-derived proarrhythmic potential of drugs and provide a mechanism for benefit from lidocaine administration in clinical acquired long QT syndrome.

Linked and knotted chimera filaments in oscillatory systems

While the existence of stable knotted and linked vortex lines has been established in many experimental and theoretical systems, their existence in oscillatory systems and systems with nonlocal coupling has remained elusive. Here, we present strong numerical evidence that stable knots and links such as trefoils and Hopf links do exist in simple, complex, and chaotic oscillatory systems if the coupling between the oscillators is neither too short ranged nor too long ranged. In this case, effective repulsive forces between vortex lines in knotted and linked structures stabilize curvature-driven shrinkage observed for single vortex rings. In contrast to real fluids and excitable media, the vortex lines correspond to scroll wave chimeras [synchronized scroll waves with spatially extended (tubelike) unsynchronized filaments], a prime example of spontaneous synchrony breaking in systems of identical oscillators. In the case of complex oscillatory systems, this leads to a topological superstructure combining knotted filaments and synchronization defect sheets.

Modeling Current Density Maps Using Aliev-Panfilov Electrophysiological Heart Model

Most existing studies of cardiac arrhythmia rely on surface measurements through optical or electrical mapping techniques. Current density imaging (CDI) is a method which enables us to study current pathways inside the tissue. However, this method entails implementation complexities for beating ex vivo hearts. Hence, this work presents an approach to simulate and study the current distributions in different cardiac electrophysiological states. The results are corroborated by experimental data, and they indicate that different states were distinguishable. The CDI simulations can be used for studying cardiac arrhythmias under simulation conditions which are otherwise impossible or difficult to be implemented experimentally.

Uncertainty and variability in computational and mathematical models of cardiac physiology

KEY POINTS

Mathematical and computational models of cardiac physiology have been an integral component of cardiac electrophysiology since its inception, and are collectively known as the Cardiac Physiome. We identify and classify the numerous sources of variability and uncertainty in model formulation, parameters and other inputs that arise from both natural variation in experimental data and lack of knowledge. The impact of uncertainty on the outputs of Cardiac Physiome models is not well understood, and this limits their utility as clinical tools. We argue that incorporating variability and uncertainty should be a high priority for the future of the Cardiac Physiome. We suggest investigating the adoption of approaches developed in other areas of science and engineering while recognising unique challenges for the Cardiac Physiome; it is likely that novel methods will be necessary that require engagement with the mathematics and statistics community.

ABSTRACT

The Cardiac Physiome effort is one of the most mature and successful applications of mathematical and computational modelling for describing and advancing the understanding of physiology. After five decades of development, physiological cardiac models are poised to realise the promise of translational research via clinical applications such as drug development and patient-specific approaches as well as ablation, cardiac resynchronisation and contractility modulation therapies. For models to be included as a vital component of the decision process in safety-critical applications, rigorous assessment of model credibility will be required. This White Paper describes one aspect of this process by identifying and classifying sources of variability and uncertainty in models as well as their implications for the application and development of cardiac models. We stress the need to understand and quantify the sources of variability and uncertainty in model inputs, and the impact of model structure and complexity and their consequences for predictive model outputs. We propose that the future of the Cardiac Physiome should include a probabilistic approach to quantify the relationship of variability and uncertainty of model inputs and outputs.

Atrial Conduction Velocity Correlates with Frequency Content of Bipolar Signal

BACKGROUND

Anisotropy in conduction velocity (CV) is a key substrate abnormality influencing atrial arrhythmias. In skeletal muscle fibers, CV and frequency content of the surface electromyogram signal are directly related. We hypothesized that in human atria the frequency content of the bipolar signal, recorded on the endocardial surface, is directly related to the local CV.

METHODS

In 15 patients submitted to ablation of supraventricular arrhythmias, incremental pacing was performed through an octapolar catheter inserted into the coronary sinus (CS), alternatively from both extremities in two different sequences: CS bipole 1-2 as the pacing site and CS bipole 7-8 as the detection site in the first, and vice versa in the second. The pacing cycle length (PCL) was stepwise decreased from 600 ms to 500 ms, 400 ms, 300 ms, until 250 ms. Estimation of the CV was performed as the ratio between the distance traveled by the propagating pulse and the propagation time. The frequency distribution of the signal energy was estimated using the fast Fourier transform, and the characteristic frequency (CF) was estimated as the barycenter of the frequency spectrum.

RESULTS

A total of 2,496 bipolar signals were analyzed; CV and CF were estimated and compared. The single patient and group data analysis showed a significant direct correlation between CV and CF of the local bipolar signal.

CONCLUSIONS

Comparing the degree of spectral compression among signals registered in different points of the endocardial cardiac surface in response to decreasing PCL enables to map local differences in CV, a useful arrhythmogenic substrate index.

Pattern Selection in Network of Coupled Multi-Scroll Attractors

Multi-scroll chaotic attractor makes the oscillator become more complex in dynamic behaviors. The collective behaviors of coupled oscillators with multi-scroll attractors are investigated in the regular network in two-dimensional array, which the local kinetics is described by an improved Chua circuit. A feasible scheme of negative feedback with diversity is imposed on the network to stabilize the spatial patterns. Firstly, the Chua circuit is improved by replacing the nonlinear term with Sine function to generate infinite aquariums so that multi-scroll chaotic attractors could be generated under appropriate parameters, which could be detected by calculating the Lyapunov exponent in the parameter region. Furthermore, negative feedback with different gains (D1, D2) is imposed on the local square center area A2 and outer area A1 of the network, it is found that spiral wave, target wave could be developed in the network under appropriate feedback gain with diversity and size of controlled area. Particularly, homogeneous state could be reached after synchronization by selecting appropriate feedback gain and controlled size in the network. Finally, the distribution for statistical factors of synchronization is calculated in the two-parameter space to understand the transition of pattern region. It is found that developed spiral waves, target waves often are associated with smaller factor of synchronization. These results show that emergence of sustained spiral wave and continuous target wave could be effective for further suppression of spatiotemporal chaos in network by generating stable pacemaker completely.

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.

Using Skewness and the First-Digit Phenomenon to Identify Dynamical Transitions in Cardiac Models

Disruptions in the normal rhythmic functioning of the heart, termed as arrhythmia, often result from qualitative changes in the excitation dynamics of the organ. The transitions between different types of arrhythmia are accompanied by alterations in the spatiotemporal pattern of electrical activity that can be measured by observing the time-intervals between successive excitations of different regions of the cardiac tissue. Using biophysically detailed models of cardiac activity we show that the distribution of these time-intervals exhibit a systematic change in their skewness during such dynamical transitions. Further, the leading digits of the normalized intervals appear to fit Benford's law better at these transition points. This raises the possibility of using these observations to design a clinical indicator for identifying changes in the nature of arrhythmia. More importantly, our results reveal an intriguing relation between the changing skewness of a distribution and its agreement with Benford's law, both of which have been independently proposed earlier as indicators of regime shift in dynamical systems.

Analytical description of the slope of the APD-restitution curve to assess the interacting contribution of conduction and repolarization dynamics

The restitution of the action potential duration (APDR) is a mechanism whereby cardiac excitation and relaxation adapt to changes in heart rate. Several studies, mainly carried out in-vitro and in-silico, have demonstrated that a steep APDR curve is associated with increased vulnerability to fatal arrhythmias. However, the mechanisms that link the steepness of the APDR curve to arrhythmogenesis remain undetermined. Although APDR is known to interact with conduction dynamics, few studies have focused on these interactions. In this paper, an analytical expression of the slope of the APDR is derived. This expression explicitly describes the dependency of the slope of the APDR curve on the activation time and/or conduction velocity changes. The study of this expression shows that conduction dynamics are among the main determinants of the slope of the APDR curve. A small absolute increment in the steepness of the activation time restitution slope can cause the steepness of the APDR slope to dramatically increase. Theoretically, the APDR slope quickly diverges to infinity when the increase in activation time matches the decrease in the pacing interval. High density epicardial mapping performed in a patient undergoing open heart surgery, shows excellent agreement between measures of the slope of the APDR curve and its analytical prediction (linear correlation > 0.95). The in-vivo recordings suggest that activation time restitution is the main determinant of the slope of the APDR curve.

Reconstructing three-dimensional reentrant cardiac electrical wave dynamics using data assimilation

For many years, reentrant scroll waves have been predicted and studied as an underlying mechanism for cardiac arrhythmias using numerical techniques, and high-resolution mapping studies using fluorescence recordings from the surfaces of cardiac tissue preparations have confirmed the presence of visible spiral waves. However, assessing the three-dimensional dynamics of these reentrant waves using experimental techniques has been limited to verifying stable scroll-wave dynamics in relatively thin preparations. We propose a different approach to recovering the three-dimensional dynamics of reentrant waves in the heart. By applying techniques commonly used in weather forecasting, we combine dual-surface observations from a particular experiment with predictions from a numerical model to reconstruct the full three-dimensional time series of the experiment. Here, we use model-generated surrogate observations from a numerical experiment to evaluate the performance of the ensemble Kalman filter in reconstructing such time series for a discordant alternans state in one spatial dimension and for scroll waves in three dimensions. We show that our approach is able to recover time series of both observed and unobserved variables matching the truth. Where nearby observations are available, the error is reduced below the synthetic observation error, with a smaller reduction with increased distance from observations. Our findings demonstrate that state reconstruction for spatiotemporally complex cardiac electrical dynamics is possible and will lead naturally to applications using real experimental data.

Myokit: A simple interface to cardiac cellular electrophysiology

Myokit is a new powerful and versatile software tool for modeling and simulation of cardiac cellular electrophysiology. Myokit consists of an easy-to-read modeling language, a graphical user interface, single and multi-cell simulation engines and a library of advanced analysis tools accessible through a Python interface. Models can be loaded from Myokit's native file format or imported from CellML. Model export is provided to C, MATLAB, CellML, CUDA and OpenCL. Patch-clamp data can be imported and used to estimate model parameters. In this paper, we review existing tools to simulate the cardiac cellular action potential to find that current tools do not cater specifically to model development and that there is a gap between easy-to-use but limited software and powerful tools that require strong programming skills from their users. We then describe Myokit's capabilities, focusing on its model description language, simulation engines and import/export facilities in detail. Using three examples, we show how Myokit can be used for clinically relevant investigations, multi-model testing and parameter estimation in Markov models, all with minimal programming effort from the user. This way, Myokit bridges a gap between performance, versatility and user-friendliness.

Phase synchrony reveals organization in human atrial fibrillation

It remains unclear if human atrial fibrillation (AF) is spatially nonhierarchical or exhibits a hierarchy of organization sustained by sources. We utilize activation times obtained at discrete locations during AF to compute the phase synchrony between tissue regions, to examine underlying spatial dynamics throughout both atria. We construct a binary synchronization network and show that this network can accurately define regions of coherence in coarse-grained in silico data. Specifically, domains controlled by spiral waves exhibit regions of high phase synchrony. We then apply this analysis to clinical data from patients experiencing cardiac arrhythmias using multielectrode catheters to simultaneously record from a majority of both atria. We show that pharmaceutical intervention with ibutilide organizes activation by increasing the size of the synchronized domain in AF and quantify the increase in temporal organization when arrhythmia changes from fibrillation to tachycardia. Finally, in recordings from 24 patients in AF we show that the level of synchrony is spatially broad with some patients showing large spatially contiguous regions of synchronization, while in others synchrony is localized to small pockets. Using computer simulations, we show that this distribution is inconsistent with distributions obtained from simulations that mimic multiwavelet reentry but is consistent with mechanisms in which one or more spatially conserved spiral waves is surrounded by tissue in which activation is disorganized.

Mechanisms for the Termination of Atrial Fibrillation by Localized Ablation: Computational and Clinical Studies

BACKGROUND

Human atrial fibrillation (AF) can terminate after ablating localized regions, which supports the existence of localized rotors (spiral waves) or focal drivers. However, it is unclear why ablation near a spiral wave tip would terminate AF and not anchor reentry. We addressed this question by analyzing competing mechanisms for AF termination in numeric simulations, referenced to clinical observations.

METHODS AND RESULTS

Spiral wave reentry was simulated in monodomain 2-dimensional myocyte sheets using clinically realistic rate-dependent values for repolarization and conduction. Heterogeneous models were created by introduction of parameterized variations in tissue excitability. Ablation lesions were applied as nonconducting circular regions. Models confirmed that localized ablation may anchor spiral wave reentry, producing organized tachycardias. Several mechanisms referenced to clinical observations explained termination of AF to sinus rhythm. First, lesions may create an excitable gap vulnerable to invasion by fibrillatory waves. Second, ablation of rotors in regions of low-excitability (from remodeling) produced re-entry in more excitable tissue allowing collision of wavefront and back. Conversely, ablation of rotors in high-excitability regions migrated spiral waves to less excitable tissue, where they detached to collide with nonconducting boundaries. Third, ablation may connect rotors to nonconducting anatomic orifices. Fourth, reentry through slow-conducting channels may terminate if ablation closes these channels.

CONCLUSIONS

Limited ablation can terminate AF by several mechanisms. These data shed light on how clinical AF may be sustained in patients' atria, emphasizing heterogeneities in tissue excitability, slow-conducting channels, and obstacles that are increasingly detectable in patients and should be the focus of future translational studies.

Dynamical disease: Challenges for nonlinear dynamics and medicine

Dynamical disease refers to illnesses that are associated with striking changes in the dynamics of some bodily function. There is a large literature in mathematics and physics which proposes mathematical models for the physiological systems and carries out analyses of the properties of these models using nonlinear dynamics concepts involving analyses of the stability and bifurcations of attractors. This paper discusses how these concepts can be applied to medicine.

A patient-specific model of virtual ablation for atrial fibrillation

The purpose of this study was to propose a patient-specific model of atrial fibrillation (AF) and apply it to virtual radiofrequency ablation (RFA). We obtained patient-specific geometries of the left atrium (LA) from CT data and constructed three-dimensional (3D) simulation models. A bidomain Courtemanche model was used to simulate the 3D electric waves on the LA surface, and an S1-S2 protocol was applied to induce AF in the model. To identify scar areas in the models, we converted clinically measured voltage data on the LA surface to the scar maps of the simulation model. Then, after initiation of AF, we applied the virtual ablation scheme to the model and investigated whether the AF was terminated by the scheme. The computed results of AF and ablation were similar to those of clinical observation, providing a clinically important simulation method for preclinical virtual trials of AF treatment.

Models of ventricular arrhythmia mechanisms

The mechanisms that initiate and sustain ventricular arrhythmias in the human heart are clinically important, but hard to study experimentally. In this study, a monodomain model of electrical activation was used to examine how dynamics of electrophysiology at the cell scale influence the surface activation patterns of VF at the tissue scale. Cellular electrophysiology was described with two variants of a phenomenological model of the human ventricular epicardial action potential. The tissue geometry was an 8.0 × 8.0 × 1.2 cm 3D tissue slab with axially symmetric anisotropy. In both cases an initial re-entrant wave fragmented into multiple wavelets of activation. The model variant with steep action potential duration restitution produced much more complex activation, with a greater average number of filaments (13.79) than the variant with less steep restitution (3.08). More complex activation was associated with proportionally fewer transmural filaments, and so the average number of epicardial wavefronts and phase singularities per filament was lower. The average number of epicardial phase singularities and wavefronts for the model variant with less steep restitution were consistent with experimental observations in the human heart. This study shows that small changes in cell scale dynamics can have a large influence on the complexity of re-entrant activation in simulated 3D tissue, as well as on the features observed on the epicardial surface.

High-order finite element methods for cardiac monodomain simulations

Computational modeling of tissue-scale cardiac electrophysiology requires numerically converged solutions to avoid spurious artifacts. The steep gradients inherent to cardiac action potential propagation necessitate fine spatial scales and therefore a substantial computational burden. The use of high-order interpolation methods has previously been proposed for these simulations due to their theoretical convergence advantage. In this study, we compare the convergence behavior of linear Lagrange, cubic Hermite, and the newly proposed cubic Hermite-style serendipity interpolation methods for finite element simulations of the cardiac monodomain equation. The high-order methods reach converged solutions with fewer degrees of freedom and longer element edge lengths than traditional linear elements. Additionally, we propose a dimensionless number, the cell Thiele modulus, as a more useful metric for determining solution convergence than element size alone. Finally, we use the cell Thiele modulus to examine convergence criteria for obtaining clinically useful activation patterns for applications such as patient-specific modeling where the total activation time is known a priori.

Determining individual phase response curves from aggregate population data

Phase reduction is an invaluable technique for investigating the dynamics of nonlinear limit cycle oscillators. Central to the implementation of phase reduction is the ability to calculate phase response curves (PRCs), which describe an oscillator's response to an external perturbation. Current experimental techniques for inferring PRCs require data from individual oscillators, which can be impractical to obtain when the oscillator is part of a much larger population. Here we present a simple methodology to calculate PRCs of individual oscillators using an aggregate signal from a large homogeneous population. This methodology is shown to be accurate in the presence of interoscillator coupling and noise and can also provide a good estimate of an average PRC of a heterogeneous population. We also find that standard experimental techniques for PRC measurement can produce misleading results when applied to aggregate population data.

Unstable spiral waves and local Euclidean symmetry in a model of cardiac tissue

This paper investigates the properties of unstable single-spiral wave solutions arising in the Karma model of two-dimensional cardiac tissue. In particular, we discuss how such solutions can be computed numerically on domains of arbitrary shape and study how their stability, rotational frequency, and spatial drift depend on the size of the domain as well as the position of the spiral core with respect to the boundaries. We also discuss how the breaking of local Euclidean symmetry due to finite size effects as well as the spatial discretization of the model is reflected in the structure and dynamics of spiral waves. This analysis allows identification of a self-sustaining process responsible for maintaining the state of spiral chaos featuring multiple interacting spirals.

Entoptic perceptions of spiral waves and rare inward spirals

This report concerns Entoptic Rotating Spiral Waves as observed and documented by the author over a period of 46 years (1962-2008). The manifestations of these state-dependent, elusive rotating spiral entities were brief, emerging only during sleep-to-waking arousal epochs (in limbo). The images were seen only with closed lids in favorable ambient lighting-here, termed the umbral view. The clusters of rotating spiral entities emerge briefly to conscious view; their angular subtenses are estimated to be between 1° and 4°, and the rotations at ten-turns per second. Epochs of these activities commonly continued for about 20 s, with longevity of each visible entity up to 4 s. 90% of all observed entities were circular and outwardly levorotary; 5% were elliptical, appearing only as horizontal (prolate) entities. Overlapping units were rare, and were chiefly elliptical. Observations of twin spirals were also rare, seen in counter rotations, each twin inwardly rotating.

Unusual spiral wave dynamics in the Kessler-Levine model of an excitable medium

The Kessler-Levine model is a two-component reaction-diffusion system that describes spatiotemporal dynamics of the messenger molecules in a cell-to-cell signaling process during the aggregation of social amoeba cells. An excitation wave arising in the model has a phase wave at the wave back, which simply follows the wave front after a fixed time interval with the same propagation velocity. Generally speaking, the medium excitability and the refractoriness are two important factors which determine the spiral wave dynamics in any excitable media. The model allows us to separate these two factors relatively easily since the medium refractoriness can be changed independently of the medium excitability. For rigidly rotating waves, the universal relationship has been established by using a modified free-boundary approach, which assumes that the front and the back of a propagating wave are thin in comparison to the wave plateau. By taking a finite thickness of the domain boundary into consideration, the validity of the proposed excitability measure has been essentially improved. A novel method of numerical simulation to suppress the spiral wave instabilities is introduced. The trajectories of the spiral tip observed for a long refractory period have been investigated under a systematic variation of the medium refractoriness.

Propagation of spiral waves pinned to circular and rectangular obstacles

We present an investigation of spiral waves pinned to circular and rectangular obstacles with different circumferences in both thin layers of the Belousov-Zhabotinsky reaction and numerical simulations with the Oregonator model. For circular objects, the area always increases with the circumference. In contrast, we varied the circumference of rectangles with equal areas by adjusting their width w and height h. For both obstacle forms, the propagating parameters (i.e., wavelength, wave period, and velocity of pinned spiral waves) increase with the circumference, regardless of the obstacle area. Despite these common features of the parameters, the forms of pinned spiral waves depend on the obstacle shapes. The structures of spiral waves pinned to circles as well as rectangles with the ratio w/h∼1 are similar to Archimedean spirals. When w/h increases, deformations of the spiral shapes are observed. For extremely thin rectangles with w/h≫1, these shapes can be constructed by employing semicircles with different radii which relate to the obstacle width and the core diameter of free spirals.

Influence of the medium's dimensionality on defect-mediated turbulence

Spatiotemporal chaos in oscillatory and excitable media is often characterized by the presence of phase singularities called defects. Understanding such defect-mediated turbulence and its dependence on the dimensionality of a given system is an important challenge in nonlinear dynamics. This is especially true in the context of ventricular fibrillation in the heart, where the importance of the thickness of the ventricular wall is contentious. Here, we study defect-mediated turbulence arising in two different regimes in a conceptual model of excitable media and investigate how the statistical character of the turbulence changes if the thickness of the medium is changed from (quasi-) two- dimensional to three dimensional. We find that the thickness of the medium does not have a significant influence in, far from onset, fully developed turbulence while there is a clear transition if the system is close to a spiral instability. We provide clear evidence that the observed transition and change in the mechanism that drives the turbulent behavior is purely a consequence of the dimensionality of the medium. Using filament tracking, we further show that the statistical properties in the three-dimensional medium are different from those in turbulent regimes arising from filament instabilities like the negative line tension instability. Simulations also show that the presence of this unique three-dimensional turbulent dynamics is not model specific.

Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization

Impulse propagation in biological tissues is known to be modulated by structural heterogeneity. In cardiac muscle, improved understanding on how this heterogeneity influences electrical spread is key to advancing our interpretation of dispersion of repolarization. We propose fractional diffusion models as a novel mathematical description of structurally heterogeneous excitable media, as a means of representing the modulation of the total electric field by the secondary electrical sources associated with tissue inhomogeneities. Our results, analysed against in vivo human recordings and experimental data of different animal species, indicate that structural heterogeneity underlies relevant characteristics of cardiac electrical propagation at tissue level. These include conduction effects on action potential (AP) morphology, the shortening of AP duration along the activation pathway and the progressive modulation by premature beats of spatial patterns of dispersion of repolarization. The proposed approach may also have important implications in other research fields involving excitable complex media.

Optical Mapping of Ventricular Fibrillation Dynamics

There is very limited information regarding the dynamic patterns of the electrical activity during ventricular fibrillation (VF) in humans. Most of the data used to generate and test hypotheses regarding the mechanisms of VF come from animal models and computer simulations and the quantification of VF patterns is non-trivial. Many of the experimental recordings of the dynamic spatial patterns of VF have been obtained from mammals using "optical mapping" or "video imaging" technology in which "phase maps" are derived from high-resolution transmembrane recordings from the heart surface. The surface manifestation of the unstable reentrant waves sustaining VF can be identified as "phase singularities" and their number and location provide one measure of VF complexity. After providing a brief history of optical mapping of VF, we compare and contrast a quantitative analysis of VF patterns from the heart surface for four different animal models, hence providing physiological insight into the variety of VF dynamics among species. We found that in all four animal models the action potential duration restitution slope was actually negative during VF and that the spatial dispersion of electrophysiological parameters were not different during the first second of VF compared to pacing immediately before VF initiation. Surprisingly, our results suggest that APD restitution and spatial dispersion may not be essential causes of VF dynamics. Analyses of electrophysiological quantities in the four animal models are consistent with the idea that VF is essentially a two-dimensional phenomenon in small rabbit hearts whose size are near the boundary of the "critical mass" required to sustain VF, while VF in large pig hearts is three-dimensional and exhibits the maximal theoretical phase singularity density, and thus will not terminate spontaneously.

Computational cardiac electrophysiology: implementing mathematical models of cardiomyocytes to simulate action potentials of the heart

Mathematical models are now an important tool for studying cardiac electrophysiology and arrhythmias. Utilizing these models to quantify behavior and make predictions requires solving the models computationally using numerical schemes. We discuss different solution methods and other computational considerations for simulating cardiac action potentials in single cells and multicellular preparations.

Macro-micro imaging of cardiac-neural circuits in co-cultures from normal and diseased hearts

The autonomic nervous system plays an important role in the modulation of normal cardiac rhythm, but is also implicated in modulating the heart's susceptibility to re-entrant ventricular and atrial arrhythmias. The mechanisms by which the autonomic nervous system is pro-arrhythmic or anti-arrhythmic is multifaceted and varies for different types of arrhythmia and their cardiac substrates. Despite decades of research in this area, fundamental questions related to how neuron density and spatial organization modulate cardiac wave dynamics remain unanswered. These questions may be ill-posed in intact tissues where the activity of individual cells is often experimentally inaccessible. Development of simplified biological models that would allow us to better understand the influence of neural activation on cardiac activity can be beneficial. This Symposium Review summarizes the development of in vitro cardiomyocyte cell culture models of re-entrant activity, as well as challenges associated with extending these models to include the effects of neural activation.

The role of short term memory and conduction velocity restitution in alternans formation

Alternans is the periodic beat-to-beat short-long alternation in action potential duration (APD), which is considered to be a precursor of ventricular arrhythmias and sudden cardiac death. In extended cardiac tissue, electrical alternans can be either spatially concordant (SCA, all cells oscillate in phase) or spatially discordant (SDA, cells in different regions oscillate out of phase). SDA gives rise to an increase in the spatial dispersion of repolarization, which is thought to be proarrhythmic. In this paper, we investigated the effect of two aspects of short term memory (STM) (α, τ) and their interplay with conduction velocity (CV) restitution on alternans formation using numerical simulations of a mapping model with two beats of memory. Here, α quantifies the dependence of APD restitution on pacing history and τ characterizes APD accommodation, which is an exponential change of APD over time once basic cycle length (BCL) changes. Our main findings are as follows: In both single cell and spatially coupled homogeneous cable, the interplay between α and τ affects the dynamical behaviors of the system. For the case of large APD accommodation (τ ≥ 290 ms), increase in α leads to suppression of alternans. However, if APD accommodation is small (τ ≤ 250 ms), increase in α leads to appearance of additional alternans region. On the other hand, the slope of CV restitution does not change the regions of alternans in the cable. However, steep CV restitution leads to more complicated dynamical behaviors of the system. Specifically, SDA instead of SCA are observed. In addition, for steep CV restitution and sufficiently large τ, we observed formations of type II conduction block (CB2), transition from type I conduction block (CB1) to CB2, and unstable nodes.

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.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Spiral-wave dynamics in ionically realistic mathematical models for human ventricular tissue: the effects of periodic deformation

We carry out an extensive numerical study of the dynamics of spiral waves of electrical activation, in the presence of periodic deformation (PD) in two-dimensional simulation domains, in the biophysically realistic mathematical models of human ventricular tissue due to (a) ten-Tusscher and Panfilov (the TP06 model) and (b) ten-Tusscher, Noble, Noble, and Panfilov (the TNNP04 model). We first consider simulations in cable-type domains, in which we calculate the conduction velocity θ and the wavelength λ of a plane wave; we show that PD leads to a periodic, spatial modulation of θ and a temporally periodic modulation of λ; both these modulations depend on the amplitude and frequency of the PD. We then examine three types of initial conditions for both TP06 and TNNP04 models and show that the imposition of PD leads to a rich variety of spatiotemporal patterns in the transmembrane potential including states with a single rotating spiral (RS) wave, a spiral-turbulence (ST) state with a single meandering spiral, an ST state with multiple broken spirals, and a state SA in which all spirals are absorbed at the boundaries of our simulation domain. We find, for both TP06 and TNNP04 models, that spiral-wave dynamics depends sensitively on the amplitude and frequency of PD and the initial condition. We examine how these different types of spiral-wave states can be eliminated in the presence of PD by the application of low-amplitude pulses by square- and rectangular-mesh suppression techniques. We suggest specific experiments that can test the results of our simulations.

Spiral defect chaos in an advection-reaction-diffusion system

This paper comprises numerical and theoretical studies of spatiotemporal patterns in advection-reaction-diffusion systems in which the chemical species interact with the hydrodynamic fluid. Due to the interplay between the two, we obtained the spiral defect chaos in the activator-inhibitor-type model. We formulated the generalized Swift-Hohenberg-type model for this system. Then the evolution of fractal boundaries due to the effect of the strong nonlinearity at the interface of the two chemical species is studied numerically. The purpose of the present paper is to point out that spiral defect chaos, observed in model equations of the extended Swift-Hohenberg equation for low Prandtl number convection, may actually be obtained also in certain advection-reaction-diffusion systems.

Cardiac potassium channel subtypes: new roles in repolarization and arrhythmia

About 10 distinct potassium channels in the heart are involved in shaping the action potential. Some of the K+ channels are primarily responsible for early repolarization, whereas others drive late repolarization and still others are open throughout the cardiac cycle. Three main K+ channels drive the late repolarization of the ventricle with some redundancy, and in atria this repolarization reserve is supplemented by the fairly atrial-specific KV1.5, Kir3, KCa, and K2P channels. The role of the latter two subtypes in atria is currently being clarified, and several findings indicate that they could constitute targets for new pharmacological treatment of atrial fibrillation. The interplay between the different K+ channel subtypes in both atria and ventricle is dynamic, and a significant up- and downregulation occurs in disease states such as atrial fibrillation or heart failure. The underlying posttranscriptional and posttranslational remodeling of the individual K+ channels changes their activity and significance relative to each other, and they must be viewed together to understand their role in keeping a stable heart rhythm, also under menacing conditions like attacks of reentry arrhythmia.

Mechanistic insights into hypothermic ventricular fibrillation: the role of temperature and tissue size

AIMS

Hypothermia is well known to be pro-arrhythmic, yet it has beneficial effects as a resuscitation therapy and valuable during intracardiac surgeries. Therefore, we aim to study the mechanisms that induce fibrillation during hypothermia. A better understanding of the complex spatiotemporal dynamics of heart tissue as a function of temperature will be useful in managing the benefits and risks of hypothermia.

METHODS AND RESULTS

We perform two-dimensional numerical simulations by using a minimal model of cardiac action potential propagation fine-tuned on experimental measurements. The model includes thermal factors acting on the ionic currents and the gating variables to correctly reproduce experimentally recorded restitution curves at different temperatures. Simulations are implemented using WebGL, which allows long simulations to be performed as they run close to real time. We describe (i) why fibrillation is easier to induce at low temperatures, (ii) that there is a minimum size required for fibrillation that depends on temperature, (iii) why the frequency of fibrillation decreases with decreasing temperature, and (iv) that regional cooling may be an anti-arrhythmic therapy for small tissue sizes however it may be pro-arrhythmic for large tissue sizes.

CONCLUSION

Using a mathematical cardiac cell model, we are able to reproduce experimental observations, quantitative experimental results, and discuss possible mechanisms and implications of electrophysiological changes during hypothermia.

Capture of fixation by rotational flow; a deterministic hypothesis regarding scaling and stochasticity in fixational eye movements

Visual scan paths exhibit complex, stochastic dynamics. Even during visual fixation, the eye is in constant motion. Fixational drift and tremor are thought to reflect fluctuations in the persistent neural activity of neural integrators in the oculomotor brainstem, which integrate sequences of transient saccadic velocity signals into a short term memory of eye position. Despite intensive research and much progress, the precise mechanisms by which oculomotor posture is maintained remain elusive. Drift exhibits a stochastic statistical profile which has been modeled using random walk formalisms. Tremor is widely dismissed as noise. Here we focus on the dynamical profile of fixational tremor, and argue that tremor may be a signal which usefully reflects the workings of oculomotor postural control. We identify signatures reminiscent of a certain flavor of transient neurodynamics; toric traveling waves which rotate around a central phase singularity. Spiral waves play an organizational role in dynamical systems at many scales throughout nature, though their potential functional role in brain activity remains a matter of educated speculation. Spiral waves have a repertoire of functionally interesting dynamical properties, including persistence, which suggest that they could in theory contribute to persistent neural activity in the oculomotor postural control system. Whilst speculative, the singularity hypothesis of oculomotor postural control implies testable predictions, and could provide the beginnings of an integrated dynamical framework for eye movements across scales.