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

Impact of Combined Modulation of Two Potassium Ion Currents on Spiral Waves and Turbulent States in the Heart

In the realm of cardiac research, the control of spiral waves and turbulent states has been a persistent focus for scholars. Among various avenues of investigation, the modulation of ion currents represents a crucial direction. It has been proved that the methods involving combined control of currents are superior to singular approaches. While previous studies have proposed some combination strategies, further reinforcement and supplementation are required, particularly in the context of controlling arrhythmias through the combined regulation of two potassium ion currents. This study employs the Luo-Rudy phase I cardiac model, modulating the maximum conductance of the time-dependent potassium current and the time-independent potassium current, to investigate the effects of this combined modulation on spiral waves and turbulent states. Numerical simulation results indicate that, compared to modulating a single current, combining reductions in the conductance of two potassium ion currents can rapidly control spiral waves and turbulent states in a short duration. This implies that employing blockers for both potassium ion currents concurrently represents a more efficient control strategy. The control outcomes of this study represent a novel and effective combination for antiarrhythmic interventions, offering potential avenues for new antiarrhythmic drug targets.

The spiral wave frequency effect in atrial fibrillation

A spiral wavefront (WF), generated by a cardiac rotor that drifts between surface electrodes during atrial fibrillation, exhibits frequency changes inconsistent with classical Doppler effect (CDE) phenomena. Recent clinical studies reveal three repeatedly observed events--1) side-dependent frequency changes across the path of the rotor, 2) one additional WF strike on the higher frequency side, and 3) a reversal of WF strike sequence--which constitute a diametrical property of spinning WF sources. A linear ray model is first used to reveal and develop the diametrical phenomena. Mathematical models of an Archimedean spiral and a spiral generated by the diffusion equation are developed and compared. Each formulation predicts the diametrical property that CDE does not capture and illuminates the occurrence of a strong side and weak side with respect to the rotor path. Whereas CDE exhibits higher and lower frequencies from approaching and receding sources of WFs, respectively, spiral rotors generate higher and lower frequencies on opposite sides of the migration path. This motivates the reconsideration of mapping and ablation strategies that have traditionally been based on identifying sites of the dominant frequency. While this research aims to characterize the path of a spiral rotor during atrial fibrillation accurately, the results are applicable in other fields of science and engineering in which rotating spiral waves occur.

T-wave inversion through inhomogeneous voltage diffusion within the FK3V cardiac model

The heart beats are due to the synchronized contraction of cardiomyocytes triggered by a periodic sequence of electrical signals called action potentials, which originate in the sinoatrial node and spread through the heart's electrical system. A large body of work is devoted to modeling the propagation of the action potential and to reproducing reliably its shape and duration. Connection of computational modeling of cells to macroscopic phenomenological curves such as the electrocardiogram has been also intense, due to its clinical importance in analyzing cardiovascular diseases. In this work, we simulate the dynamics of action potential propagation using the three-variable Fenton-Karma model that can account for both normal and damaged cells through a the spatially inhomogeneous voltage diffusion coefficient. We monitor the action potential propagation in the cardiac tissue and calculate the pseudo-electrocardiogram that reproduces the R and T waves. The R-wave amplitude varies according to a double exponential law as a function of the (spatially homogeneous, for an isotropic tissue) diffusion coefficient. The addition of spatial inhomogeneity in the diffusion coefficient by means of a defected region representing damaged cardiac cells may result in T-wave inversion in the calculated pseudo-electrocardiogram. The transition from positive to negative polarity of the T-wave is analyzed as a function of the length and the depth of the defected region.

Heart-on-a-chip systems with tissue-specific functionalities for physiological, pathological, and pharmacological studies

Recent advances in heart-on-a-chip systems hold great promise to facilitate cardiac physiological, pathological, and pharmacological studies. This review focuses on the development of heart-on-a-chip systems with tissue-specific functionalities. For one thing, the strategies for developing cardiac microtissues on heart-on-a-chip systems that closely mimic the structures and behaviors of the native heart are analyzed, including the imitation of cardiac structural and functional characteristics. For another, the development of techniques for real-time monitoring of biophysical and biochemical signals from cardiac microtissues on heart-on-a-chip systems is introduced, incorporating cardiac electrophysiological signals, contractile activity, and biomarkers. Furthermore, the applications of heart-on-a-chip systems in intelligent cardiac studies are discussed regarding physiological/pathological research and pharmacological assessment. Finally, the future development of heart-on-a-chip toward a higher level of systematization, integration, and maturation is proposed.

Numerical study of the drift of scroll waves by optical feedback in cardiac tissue

Nonlinear waves were found in various types of physical, chemical, and biological excitable media, e.g., in heart muscle. They can form three-dimensional (3D) vortices, called scroll waves, that are of particular significance in the heart, as they underlie lethal cardiac arrhythmias. Thus controlling the behavior of scroll waves is interesting and important. Recently, the optical feedback control procedure for two-dimensional vortices, called spiral waves, was developed. It can induce directed linear drift of spiral waves in optogenetically modified cardiac tissue. However, the extension of this methodology to 3D scroll waves is nontrivial, as optogenetic signals only penetrate close to the surface of cardiac tissue. Here we present a study of this extension in a two-variable reaction-diffusion model and in a detailed model of cardiac tissue. We show that the success of the control procedure is determined by the tension of the scroll wave filament. In tissue with positive filament tension the control procedure works in all cases. However, in the case of negative filament tension for a sufficiently large medium, instabilities occur and make drift and control of scroll waves impossible. Because in normal cardiac tissue the filament tension is assumed to be positive, we conclude that the proposed optical feedback scheme can be a robust method in inducing the linear drift of scroll waves that can control their positions in cardiac tissue.

From actin waves to mechanism and back: How theory aids biological understanding

Actin dynamics in cell motility, division, and phagocytosis is regulated by complex factors with multiple feedback loops, often leading to emergent dynamic patterns in the form of propagating waves of actin polymerization activity that are poorly understood. Many in the actin wave community have attempted to discern the underlying mechanisms using experiments and/or mathematical models and theory. Here, we survey methods and hypotheses for actin waves based on signaling networks, mechano-chemical effects, and transport characteristics, with examples drawn from Dictyostelium discoideum, human neutrophils, Caenorhabditis elegans, and Xenopus laevis oocytes. While experimentalists focus on the details of molecular components, theorists pose a central question of universality: Are there generic, model-independent, underlying principles, or just boundless cell-specific details? We argue that mathematical methods are equally important for understanding the emergence, evolution, and persistence of actin waves and conclude with a few challenges for future studies.

Front propagation and global bifurcations in a multivariable reaction-diffusion model

We study the existence and stability of propagating fronts in Meinhardt's multivariable reaction-diffusion model of branching in one spatial dimension. We identify a saddle-node-infinite-period bifurcation of fronts that leads to episodic front propagation in the parameter region below propagation failure and show that this state is stable. Stable constant speed fronts exist only above this parameter value. We use numerical continuation to show that propagation failure is a consequence of the presence of a T-point corresponding to the formation of a heteroclinic cycle in a spatial dynamics description. Additional T-points are identified that are responsible for a large multiplicity of different unstable traveling front-peak states. The results indicate that multivariable models may support new types of behavior that are absent from typical two-variable models but may nevertheless be important in developmental processes such as branching and somitogenesis.

Reconstructing in-depth activity for chaotic 3D spatiotemporal excitable media models based on surface data

Motivated by potential applications in cardiac research, we consider the task of reconstructing the dynamics within a spatiotemporal chaotic 3D excitable medium from partial observations at the surface. Three artificial neural network methods (a spatiotemporal convolutional long-short-term-memory, an autoencoder, and a diffusion model based on the U-Net architecture) are trained to predict the dynamics in deeper layers of a cube from observational data at the surface using data generated by the Barkley model on a 3D domain. The results show that despite the high-dimensional chaotic dynamics of this system, such cross-prediction is possible, but non-trivial and as expected, its quality decreases with increasing prediction depth.

Bifurcation structure of traveling pulses in type-I excitable media

We study the scenario in which traveling pulses emerge in a prototypical type-I one-dimensional excitable medium, which exhibits two different routes to excitable behavior, mediated by a homoclinic (saddle-loop) and a saddle-node on the invariant cycle bifurcations. We characterize the region in parameter space in which traveling pulses are stable together with the different bifurcations behind either their destruction or loss of stability. In particular, some of the bifurcations delimiting the stability region have been connected, using singular limits, with the two different scenarios that mediated type-I local excitability. Finally, the existence of traveling pulses has been linked to a drift pitchfork instability of localized steady structures.

Resource-Efficient Use of Modern Processor Architectures For Numerically Solving Cardiac Ionic Cell Models

A central component in simulating cardiac electrophysiology is the numerical solution of nonlinear ordinary differential equations, also called cardiac ionic cell models, that describe cross-cell-membrane ion transport. Biophysically detailed cell models often require a considerable amount of computation, including calls to special mathematical functions. This paper systematically studies how to efficiently use modern multicore CPUs for this costly computational task. We start by investigating the code restructurings needed to effectively enable compiler-supported SIMD vectorisation, which is the most important performance booster in this context. It is found that suitable OpenMP directives are sufficient for achieving both vectorisation and parallelisation. We then continue with an evaluation of the performance optimisation technique of using lookup tables. Due to increased challenges for automated vectorisation, the obtainable benefits of lookup tables are dependent on the hardware platforms chosen. Throughout the study, we report detailed time measurements obtained on Intel Xeon, Xeon Phi, AMD Epyc and two ARM processors including Fujitsu A64FX, while attention is also paid to the impact of SIMD vectorisation and lookup tables on the computational accuracy. As a realistic example, the benefits of performance enhancement are demonstrated by a 109-run ensemble on the Oakforest-PACS system, where code restructurings and SIMD vectorisation yield an 84% reduction in computing time, corresponding to 63,270 node hours.

Engineering Models of the Heart Left Ventricle

Despite capturing the imagination of scientists for decades, the goal of creating an artificial heart for transplantation proved to be significantly more challenging than initially anticipated. Toward this goal, recent ground-breaking studies demonstrate the development of functional left ventricular (LV) models. LV models are artificially constructed 3D chambers that are capable of containing liquid within the engineered cavity and exhibit the functionality of native LV including contraction, ejection of fluid, and electrical impulse propagation. Various hydrogels and polymers have been used in manufacturing of LV models, relying on techniques such as electrospinning, bioprinting, casting, and molding. Most studies scaled down the models based on the dimensions of the human or rat ventricle. Initially, neonatal rat cardiomyocytes were the cell type of choice for construction the LV models. Yet, as the stem cell biology field advanced, recent studies focused on the use of cardiomyocytes derived from human induced pluripotent stem cells. In this review, we first describe the physiological characteristics of the human heart, to establish the parameter space for modeling. We then elaborate on current advances in the field and compare recently developed LV models among themselves and with the native human left ventricle. Fabrication methods, cell types, biomaterials, functional properties, and disease modeling capability are some of the major parameters that have distinguished these models. We also highlight some of the current challenges in this field, such as vascularization, cell composition and fidelity, and discuss potential solutions to overcome them.

Pattern Propagation Driven by Surface Curvature

Pattern dynamics on curved surfaces are found everywhere in nature. The geometry of surfaces has been shown to influence dynamics and play a functional role, yet a comprehensive understanding is still elusive. Here, we report for the first time that a static Turing pattern on a flat surface can propagate on a curved surface, as opposed to previous studies, where the pattern is presupposed to be static irrespective of the surface geometry. To understand such significant changes on curved surfaces, we investigate reaction-diffusion systems on axisymmetric curved surfaces. Numerical and theoretical analyses reveal that both the symmetries of the surface and pattern participate in the initiation of pattern propagation. This study provides a novel and generic mechanism of pattern propagation that is caused by surface curvature, as well as insights into the general role of surface geometry.

Synchronization in renal microcirculation unveiled with high-resolution blood flow imaging

Internephron interaction is fundamental for kidney function. Earlier studies have shown that nephrons signal to each other, synchronise over short distances, and potentially form large synchronised clusters. Such clusters would play an important role in renal autoregulation, but due to the technological limitations, their presence is yet to be confirmed. In the present study, we introduce an approach for high-resolution laser speckle imaging of renal blood flow and apply it to estimate frequency and phase differences in rat kidney microcirculation under different conditions. The analysis unveiled spatial and temporal evolution of synchronised blood flow clusters of various sizes, including the formation of large (>90 vessels) long-lived clusters (>10 periods) locked at the frequency of the tubular glomerular feedback mechanism. Administration of vasoactive agents caused significant changes in the synchronisation patterns and, thus, in nephrons' co-operative dynamics. Specifically, infusion of vasoconstrictor angiotensin II promoted stronger synchronisation, while acetylcholine caused complete desynchronisation. The results confirm the presence of the local synchronisation in the renal microcirculatory blood flow and that it changes depending on the condition of the vascular network and the blood pressure, which will have further implications for the role of such synchronisation in pathologies development.

Control of spiral waves in optogenetically modified cardiac tissue by periodic optical stimulation

Resonant drift of nonlinear spiral waves occurs in various types of excitable media under periodic stimulation. Recently a novel methodology of optogenetics has emerged, which allows to affect excitability of cardiac cells and tissues by optical stimuli. In this paper we study if resonant drift of spiral waves in the heart can be induced by a homogeneous weak periodic optical stimulation of cardiac tissue. We use a two-variable and a detailed model of cardiac tissue and add description of depolarizing and hyperpolarizing optogenetic ionic currents. We show that weak periodic optical stimulation at a frequency equal to the natural rotation frequency of the spiral wave induces resonant drift for both depolarizing and hyperpolarizing optogenetic currents. We quantify these effects and study how the speed of the drift and its direction depend on the initial conditions. We also derive analytical formulas based on the response function theory which correctly predict the drift velocity and its trajectory. We conclude that optogenetic methodology can be used for control of spiral waves in cardiac tissue and discuss its possible applications.

Traveling pulses in type-I excitable media

We consider a general model exhibiting type-I excitability mediated by a homoclinic and a saddle node on the invariant circle bifurcations. We show how the distinct properties of type-I with respect to type-II excitability confer unique features to traveling pulses in excitable media. They inherit the characteristic divergence of type-I excitable trajectories at threshold exhibiting analogous scalings in the spatial thickness of the pulses. Our results pave the way to identify basic underlying mechanisms behind type-I excitable pulses based solely on the characteristics of the pulse.

Instability of the Homogeneous Distribution of Chemical Waves in the Belousov-Zhabotinsky Reaction

Chemical traveling waves play an important role in biological functions, such as the propagation of action potential and signal transduction in the nervous system. Such chemical waves are also observed in inanimate systems and are used to clarify their fundamental properties. In this study, chemical waves were generated with equivalent spacing on an excitable medium of the Belousov–Zhabotinsky reaction. The homogeneous distribution of the waves was unstable and low- and high-density regions were observed. In order to understand the fundamental mechanism of the observations, numerical calculations were performed using a mathematical model, the modified Oregonator model, including photosensitive terms. However, the homogeneous distribution of the traveling waves was stable over time in the numerical results. These results indicate that further modification of the model is required to reproduce our experimental observations and to discover the fundamental mechanism for the destabilization of the homogeneous-distributed chemical traveling waves.

Topological charge-density method of identifying phase singularities in cardiac fibrillation

Spiral waves represent the key motifs of typical self-sustained dynamical patterns in excitable systems such as cardiac tissue. The motion of phase singularities (PSs) that lies at the center of spiral waves captures many qualitative and, in some cases, quantitative features of their complex dynamics. Recent clinical studies suggested that ablating the tissue at PS locations may cure atrial fibrillation. Here, we propose a different method to determine the location of PSs. Starting from the definition of the topological charge of spiral waves, we obtain the expression of the topological charge density in a discrete case. With this expression, we can calculate the topological charge at each grid in the space directly, so as to accurately identify the position of PSs.

Buffering and total calcium levels determine the presence of oscillatory regimes in cardiac cells

Calcium oscillations and waves induce depolarization in cardiac cells which are believed to cause life-threathening arrhythimas. In this work, we study the conditions for the appearance of calcium oscillations in both a detailed subcellular model of calcium dynamics and a minimal model that takes into account just the minimal ingredients of the calcium toolkit. To avoid the effects of homeostatic changes and the interaction with the action potential we consider the somewhat artificial condition of a cell without pacing and with no calcium exchange with the extracellular medium. Both the full subcellular model and the minimal model present the same scenarios depending on the calcium load: two stationary states, one with closed ryanodine receptors (RyR) and most calcium in the cell stored in the sarcoplasmic reticulum (SR), and another, with open RyRs and a depleted SR. In between, calcium oscillations may appear. The robustness of these oscillations is determined by the amount of calsequestrin (CSQ). The lack of this buffer in the SR enhances the appearance of oscillations. The minimal model allows us to relate the stability of the oscillating state to the nullcline structure of the system, and find that its range of existence is bounded by a homoclinic and a Hopf bifurcation, resulting in a sudden transition to the oscillatory regime as the cell calcium load is increased. Adding a small amount of noise to the RyR behavior increases the parameter region where oscillations appear and provides a gradual transition from the resting state to the oscillatory regime, as observed in the subcellular model and experimentally.

In cardiac cells, calcium plays a very important role. An increase in calcium levels is the trigger used by the cell to initiate contraction. Besides, calcium modulates several transmembrane currents, affecting the cell transmembrane potential. Thus, dysregulations in calcium handling have been associated with the appearance of arrhythmias. Often, this dysregulation results in the appearance of periodic calcium waves or global oscillations, providing a pro-arrhythmic substrate. In this paper, we study the onset of calcium oscillations in cardiac cells using both a detailed subcellular model of calcium dynamics and a minimal model that takes into account the essential ingredients of the calcium toolkit. Both reproduce the main experimental results and link this behavior with the presence of different steady-state solutions and bifurcations that depend on the total amount of calcium in the cell and in the level of buffering present. We expect that this work will help to clarify the conditions under which calcium oscillations appear in cardiac myocytes and, therefore, will represent a step further in the understanding of the origin of cardiac arrhythmias.

Why a Large-Scale Mode Can Be Essential for Understanding Intracellular Actin Waves

During the last decade, intracellular actin waves have attracted much attention due to their essential role in various cellular functions, ranging from motility to cytokinesis. Experimental methods have advanced significantly and can capture the dynamics of actin waves over a large range of spatio-temporal scales. However, the corresponding coarse-grained theory mostly avoids the full complexity of this multi-scale phenomenon. In this perspective, we focus on a minimal continuum model of activator-inhibitor type and highlight the qualitative role of mass conservation, which is typically overlooked. Specifically, our interest is to connect between the mathematical mechanisms of pattern formation in the presence of a large-scale mode, due to mass conservation, and distinct behaviors of actin waves.

Excitable solitons: Annihilation, crossover, and nucleation of pulses in mass-conserving activator-inhibitor media

Excitable pulses are among the most widespread dynamical patterns that occur in many different systems, ranging from biological cells to chemical reactions and ecological populations. Traditionally, the mutual annihilation of two colliding pulses is regarded as their prototypical signature. Here we show that colliding excitable pulses may exhibit solitonlike crossover and pulse nucleation if the system obeys a mass conservation constraint. In contrast to previous observations in systems without mass conservation, these alternative collision scenarios are robustly observed over a wide range of parameters. We demonstrate our findings using a model of intracellular actin waves since, on time scales of wave propagations over the cell scale, cells obey conservation of actin monomers. The results provide a key concept to understand the ubiquitous occurrence of actin waves in cells, suggesting why they are so common, and why their dynamics is robust and long-lived.

Pattern formation in reaction-diffusion systems in the presence of non-Markovian diffusion

We study reaction-diffusion systems beyond the Markovian approximation to take into account the effect of memory on the formation of spatiotemporal patterns. Using a non-Markovian Brusselator model as a paradigmatic example, we show how to use reductive perturbation to investigate the formation and stability of patterns. Focusing in detail on the Hopf instability and short-term memory, we derive the corresponding complex Ginzburg-Landau equation that governs the amplitude of the critical mode and we establish the explicit dependence of its parameters on the memory properties. Numerical solution of this memory-dependent complex Ginzburg-Landau equation as well as direct numerical simulation of the non-Markovian Brusselator model illustrates that memory changes the properties of the spatiotemporal patterns. Our results indicate that going beyond the Markovian approximation might be necessary to study the formation of spatiotemporal patterns even in systems with short-term memory. At the same time, our work opens up a new window into the control of these patterns using memory.

Curved surface geometry-induced topological change of an excitable planar wavefront

On the curved surfaces of living and nonliving materials, planar excitable wavefronts frequently exhibit a directional change and subsequently undergo a discontinuous (topological) change; i.e., a series of wavefront dynamics from collision, annihilation to splitting. Theoretical studies have shown that excitable planar stable waves change their topology significantly depending on the initial conditions on flat surfaces, whereas the directional change of the waves occurs based on the geometry of curved surfaces. However, it is not clear if the geometry of curved surfaces induces this topological change. In this study, we first demonstrated that the curved surface geometry induces bending, collision, and splitting of a planar stable wavefront by numerically solving an excitable reaction-diffusion equation on a bell-shaped surface. We determined two necessary conditions for inducing the topological change: the characteristic length of the curved surface (i.e., the height of the bell-shaped structure) should be greater than the width of the wave, and the ratio of the height to the width of the bell shape should be greater than a threshold. As for the geometrical mechanism of the latter, we found that a bifurcation of the geodesics on the curved surface provides the alternative minimal paths of the wavefront, which circumvent the surface region with a high local curvature, thereby resulting in the topological change. These conditions imply that the topological change of the wavefront can be predicted on the basis of the curved surfaces, whose structures are larger than the wave width.

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

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

Resonance of scroll rings with periodic external fields in excitable media

By direct numerical simulations of a chemical reaction-diffusion system coupled to a periodic external AC electric field with frequency equal to double frequency of the scroll wave rotation, we find that scroll rings resonate with the electric field and exhibit various dynamical behaviors, for example, their reversals, collapses, or growths, depending both on the initial phase of AC electric fields and on the initial phase of scroll rings. A kinematical model characterizing the drift velocity of the scroll rings along their radial directions as well as that of the scroll rings along their symmetry axes is proposed, which can effectively account for the numerical observations and predict the behaviors of the scroll rings. Besides, the existence of the equilibrium state of a scroll ring under the AC electric fields is predicted by the kinematical model and the predictions agree well with the simulations.

Discretization-dependent model for weakly connected excitable media

Pattern formation has been widely observed in extended chemical and biological processes. Although the biochemical systems are highly heterogeneous, homogenized continuum approaches formed by partial differential equations have been employed frequently. Such approaches are usually justified by the difference of scales between the heterogeneities and the characteristic spatial size of the patterns. Under different conditions, for example, under weak coupling, discrete models are more adequate. However, discrete models may be less manageable, for instance, in terms of numerical implementation and mesh generation, than the associated continuum models. Here we study a model to approach discreteness which permits the computer implementation on general unstructured meshes. The model is cast as a partial differential equation but with a parameter that depends not only on heterogeneities sizes, as in the case of quasicontinuum models, but also on the discretization mesh. Therefore, we refer to it as a discretization-dependent model. We validate the approach in a generic excitable media that simulates three different phenomena: the propagation of action membrane potential in cardiac tissue, in myelinated axons of neurons, and concentration waves in chemical microemulsions.

Control of electrical turbulence by periodic excitation of cardiac tissue

Electrical turbulence in cardiac tissue is associated with arrhythmias such as life-threatening ventricular fibrillation. Recent experimental studies have shown that a sequence of low-energy electrical far-field pulses is able to terminate fibrillation more gently than a single high-energy pulse which causes severe side effects. During this low-energy antifibrillation pacing (LEAP), only tissue near sufficiently large conduction heterogeneities, such as large coronary arteries, is activated. In order to optimize LEAP, we performed extensive simulations of cardiac tissue perforated by blood vessels, employing two alternative cellular models that exhibit electrical turbulence at a similar length scale. Moreover, the scale of blood vessels in our two-dimensional simulations was chosen such that the threshold for single pulse defibrillation matches experimental values. For each of the 100 initial conditions, we tested different electrical field strengths, pulse shapes, numbers of pulses, and periods between the pulses. LEAP is successful for both models, albeit with substantial differences. One model exhibits a spectrum of chaotic activity featuring a narrow peak around a dominant frequency. In this case, the optimal period between low-energy pulses matches this frequency and LEAP greatly reduces the required energy for successful defibrillation. For pulses with larger energies, the system is perturbed such that underdrive pacing becomes advantageous. The spectrum of the second model features a broader peak, resulting in a less pronounced optimal pacing period and a decreased energy reduction. In both cases, pacing with five or six pulses which are separated by the dominant period maximizes the energy reduction.

Complex restitution behavior and reentry in a cardiac tissue model for neonatal mice

Spatiotemporal dynamics in cardiac tissue emerging from the coupling of individual cardiomyocytes underlie the heart's normal rhythm as well as undesired and possibly life-threatening arrhythmias. While single cells and their transmembrane currents have been studied extensively, systematically investigating spatiotemporal dynamics is complicated by the nontrivial relationship between single-cell and emergent tissue properties. Mathematical models have been employed to bridge this gap and contribute to a deepened understanding of the onset, development, and termination of arrhythmias. However, no such tissue-level model currently exists for neonatal mice. Here, we build on a recent single-cell model of neonatal mouse cardiomyocytes by Wang and Sobie (Am. J. Physiol. Heart Circ. Physiol 294:H2565) to predict properties that are commonly used to gauge arrhythmogenicity of cardiac substrates. We modify the model to yield well-defined behavior for common experimental protocols and construct a spatially extended version to study emergent tissue dynamics. We find a complex action potential duration (APD) restitution behavior characterized by a nonmonotonic dependence on pacing frequency. Electrotonic coupling in tissue leads not only to changes in action potential morphology but can also induce spatially concordant and discordant alternans not observed in the single-cell model. In two-dimensional tissue, our results show that the model supports stable functional reentry, whose frequency is in good agreement with that observed in adult mice. Our results can be used to further constrain and validate the mathematical model of neonatal mouse cardiomyocytes with future experiments.

Temperature, geometry, and bifurcations in the numerical modeling of the cardiac mechano-electric feedback

This article characterizes the cardiac autonomous electrical activity induced by the mechanical deformations in the cardiac tissue through the mechano-electric feedback. A simplified and qualitative model is used to describe the system and we also account for temperature effects. The analysis emphasizes a very rich dynamics for the system, with periodic solutions, alternans, chaotic behaviors, etc. The possibility of self-sustained oscillations is analyzed in detail, particularly in terms of the values of important parameters such as the dimension of the system and the importance of the stretch-activated currents. It is also shown that high temperatures notably increase the parameter ranges for which self-sustained oscillations are observed and that several attractors can appear, depending on the location of the initial excitation of the system. Finally, the instability mechanisms by which the periodic solutions are destabilized have been studied by a Floquet analysis, which has revealed period-doubling phenomena and transient intermittencies.

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

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

Memory effects, transient growth, and wave breakup in a model of paced atrium

The mechanisms underlying cardiac fibrillation have been investigated for over a century, but we are still finding surprising results that change our view of this phenomenon. The present study focuses on the transition from normal rhythm to spiral wave chaos associated with a gradual increase in the pacing rate. While some of our findings are consistent with existing experimental, numerical, and theoretical studies of this problem, one result appears to contradict the accepted picture. Specifically we show that, in a two-dimensional model of paced homogeneous atrial tissue, transition from discordant alternans to conduction block, wave breakup, reentry, and spiral wave chaos is associated with the transient growth of finite amplitude disturbances rather than a conventional instability. It is mathematically very similar to subcritical, or bypass, transition from laminar fluid flow to turbulence, which allows many of the tools developed in the context of fluid turbulence to be used for improving our understanding of cardiac arrhythmias.

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.

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.