Introduction to Focus Issue: Complex Cardiac Dynamics

A number of papers in this issue address aspects of mathematical and computational modeling. Some authors address developing or extending models to reproduce particular behavior. Although the basic mechanisms of propagation through discrete gap junctions or, within a continuum context, following the cable equation have been understood for many years, questions persist concerning propagation when cells are weakly coupled and the precise physiological mechanisms governing gap junction behavior. Weinberg (2017) studies the effects of voltage-gated gap junctions and ephaptic coupling when cells are coupled weakly. He shows that although the voltage-gating of the reduced number of gap junctions tends to promote failure of conduction through a positive feedback loop, coupling through extracellular electric fields can allow propagation when it otherwise would fail, especially for fast heart rates. The standard cable model exhibits some unphysiological properties, which Rossi and Griffith (2017) address. Specifically, they aim to prevent propagation of perturbations at infinite speed by extending cardiac conduction models to include axial inductances. They show that conduction velocity is faster in their revised model, and in some cases, initiation of reentry may be facilitated. They also find that their approach does not introduce numerical challenges but rather can increase the accuracy of solutions. Other approaches to propagation have been proposed recently, including models that account for the porous structure of cardiac tissue. Along these lines, Gizzi et al. (2017) extend the standard cable equation by including a nonlinear voltage dependence in the diffusion term; they also include thermo-electric effects in the reaction terms. They find that nonlinear diffusion affects repolarization and facilitates spiral-wave breakup at temperatures higher and lower than standard physiological temperatures. Along with updated models for propagation, understanding the development of emergent behavior in which spontaneous activity can arise from heterogeneous groups of excitable and oscillatory cells is an important modeling challenge with implications for the development of biological pacemakers. Aghighi and Comtois (2017) study how a stochastic pattern formation algorithm interacts with pacemaker cell characteristics. In particular, they find that weaker pacemaker cells require clusters with greater size and density to produce spontaneous activity.

Several papers are focused on simplified models or reductions of complex models. Although many simplifications of complex models have been proposed, the reductions have tended to be made in a somewhat ad hoc manner. Here, Lombardo and Rappel (2017) apply a geometric and information theoretic technique to perform a reduction of a high-dimensional atrial cell model to a much lower-dimensional model. Through a systematic series of reduction steps, they arrive at a model that retains only 26% of the original model variables and 14% of the original parameters while retaining key electrophysiological properties. Detailed models, especially those employing Markov descriptions of ion channel kinetics, often suffer from computational stiffness along with their formulation complexity. Starý and Biktashev (2017) propose an approach to make reductions through careful identification of small model parameters and singular perturbation analysis. They show that their technique can lead to a reduction in the number of states required in a Markov model of the sodium channel while also improving computational stiffness. Simplified models also offer the opportunity to produce analytical results that often are not possible for very complex models. Here, Bezekci and Biktashev (2017) focus on initiation of propagating waves using the FitzHugh-Nagumo model. By applying singular perturbation theory involving two small parameters, the time scale separation of the two variations and the excitation threshold, they derive an analytical expression for the model's strength-duration curve.

Other authors discuss the consequences of choosing different types of models. Given that cardiac tissue is made of discrete cells surrounded by extracellular matrix, an important question to address is when a discrete formulation is needed rather than the commonly utilized cable equation and its variations. Toward that end, Gokhale et al. (2017) compare a discrete microstructural model for fibrosis with continuous models. They find that as fibrosis is increased, the discrete model displays a significant lengthening of reentrant cycle length resulting from changes to the spiral wave trajectory and restitution properties. The same behavior does not occur in a homogeneous continuous model, but an appropriately tuned hybrid model that includes discrete decoupling septa can overcome many of these limitations while avoiding the high computational costs of discrete approaches. An important question on the electrophysiology side is how models can treat the fact that cells exhibit significant variability even when showing similar behavior, which can complicate reaching conclusions about differences in healthy and diseased states. In an application to understand electrophysiological differences in atrial fibrillation (AF) patients, Vagos et al. (2017) developed two populations of model parameterizations calibrated to represent variability for separate sinus rhythm and chronic atrial fibrillation groups. Through a sensitivity analysis, they identified correlations between values of model parameters and pro-arrhythmic biomarkers. Ultimately, they found a smaller variability in action potential duration (APD) and more stable restitution properties for the AF population and, as a result, suggest that tissue-level effects may be more important in generating and sustaining reentrant arrhythmias.

Rather than focusing on the modeling equations themselves, some papers focus on using data to find model parameters or to reconstruct states that are observed partially. For example, conductivity parameters used in monodomain and bidomain cardiac models have not been established with great certainty. Barone et al. (2017) seek to identify cardiac conductivity parameters from experimental data using a variational data assimilation approach. They arrive at a parameter estimation procedure by choosing conductivity parameters to minimize the difference between a numerical simulation and recorded data located at specific measurement sites. They find a range of values for the number of measurement sites that can provide accurate identification of the conductivity parameters. Cairns et al. (2017) take a different approach toward parameter estimation: specifically, they use a genetic algorithm to find parameter values to fit two phenomenological models to simulated data and microelectrode voltage recordings. Although they are able to find relatively good matches with both fit and unfit data in an efficient manner, they also observe significant variability in the parameter values obtained from different runs of the algorithm, a result that highlights issues of parameter sensitivity and identifiability for the models. In another data assimilation application, LaVigne et al. (2017) study the impact of discrepancies between model formulation and the dynamics demonstrated by data on obtaining accurate state estimates during discordant alternans using a Kalman filter-based method. They find that their approach can be tuned to produce high-quality estimates even when the dynamics of the model and the true system differ significantly and even in the presence of an underlying spatial heterogeneity not represented in the model.

The dynamics of the reentrant waves that underlie many cardiac arrhythmias are critical to gaining fundamental insights into arrhythmogenesis and arrhythmia treatment. As to how reentrant activity is initiated, our understanding goes back 100 years to the seminal work of Mines, who established unidirectional conduction block due to spatial heterogeneity in refractoriness as the underlying mechanism. In terms of nonlinear dynamics, we now understand many aspects of how different bifurcations can cause such heterogeneity. One example is a period-doubling bifurcation causing spatially heterogeneous discordant alternans. In terms of bifurcations, transitions from, say, a stable alternans solution to conduction block, reentry, and possibly spiral wave breakup and chaos are less well understood. The article by Garzón and Grigoriev (2017) investigates the transition from normal rhythm to spiral wave breakup with increased pacing frequency in a simulated tissue sheet. The authors use different pacing protocols to demonstrate that the transition to spiral wave chaos may occur due to growth of disturbances during discordant alternans, even when this alternans state is stable. Rather than a bifurcation, the transition from alternans appears to be a switch between solutions in a multistable system.

Once developed, what are the mechanisms that maintain the complex spatiotemporal dynamics associated with cardiac fibrillation? A body of experimental and numerical studies suggest that this depends on the substrate but can involve spiral wave reentry, anatomical reentry, mother rotors spawning new spiral waves, and spiral wave breakup. In their study, Marcotte and Grigoriev (2017) provide a topological approach to define and analyze spiral waves and their organizing centers. The authors use this approach to evaluate complex patterns during spatiotemporal multi-spiral chaotic states and to describe transitions that change the number of wavelets.

Heterogeneity is a characteristic feature of the myocardium and is of key importance in initiation, maintenance, and termination of arrhythmias, both in the atria and in the ventricles. Heterogeneity of some sort is almost always increased in diseased hearts. The majority of previous work in this area has focused mainly on effects of heterogeneity (such as slow or non-conducting regions) on the initiation of reentrant waves or on the stability and dynamics of a single reentrant wave. In their article, Zykov et al. (2017) analyze conditions for conduction block and initiation of wave breaks due to spatial heterogeneities in one- and two-dimensional excitable media. The authors establish quantitative relationships between propagation block, tissue excitability and coupling strength, and heterogeneity geometry and border steepness. In particular, the authors demonstrate how a fast conducting region of appropriate size and shape may cause conduction block and initiate reentrant waves. Bittihn et al. (2017) present a study on the effects of added heterogeneity to a medium that exhibits chaotic reentrant waves. The authors focused on heterogeneities as regions of reduced excitability and investigated a range of settings for their spatial scale and distributions. The study demonstrates that in the setting of spatiotemporal chaotic activity, added heterogeneities may have a range of effects depending on their features, including the contrasting outcomes of increased turbulence versus stabilized wave activity with phase singularities becoming trapped in the heterogeneous zones.

In recent biophysically detailed cardiac cell models, slow (with respect to the cardiac action potential) accumulation of intracellular ionic concentrations can regulate electrical activity. Krogh-Madsen and Christini (2017) demonstrate how such slow accumulation also alters bifurcations at the single-cell level as well the dynamics of a reentrant wave in a two-dimensional tissue, where the reentry is accelerated by a shortened action potential duration. In an otherwise homogeneous two-dimensional model, spatial heterogeneity in the intracellular sodium concentration can arise dynamically and anchor the reentrant activity.

One way to characterize the response of a spiral wave to localized heterogeneities is to calculate the so-called response functions. Traditionally, the response functions are determined by solving the adjoint linearized problem. Dierckx et al. (2017) present a practical method for finding the response curves based on perturbation-induced drift responses and demonstrate its use for both a circular core and a meandering spiral. Because the method does not make use of the underlying model equations, it is applicable to experimental data.

The clinical therapy of ablating regions of the atria to cure atrial fibrillation was developed on the premise of isolating particular problem areas associated with increased reentry initiation. Several articles in the Focus Issue pertain to this process. The study by Iravanian and Langberg (2017) focuses on spiral-wave dynamics during atrial fibrillation ablation. The authors use a realistic human three-dimensional atrial geometry to simulate cases of ablation and find that in many such cases, the organization of the wave dynamics from a more disorganized state occurs very suddenly, resembling a phase transition. This result suggests that novel clinical ablation strategies aimed at facilitating phase transitions may be advantageous.

Recently, spiral waves have been directly targeted as ablation goals for atrial fibrillation therapy, based on detection of their cores using multiple electrodes. The paper by You et al. (2017) uses optical mapping in cardiac monolayers, as well as model simulations, to investigate effects of spatial undersampling on the ability to correctly identify rotors. The authors show how decreased spatial resolution impairs the identification and calls for further analysis and validation of algorithms used for rotor detection.

Finally, a different proposed approach to atrial fibrillation ablation, aimed towards patients with a high degree of structural heterogeneity (fibrosis), consists of predicting target ablation regions using computational models individualized to imaging data. The contribution by Deng et al. (2017) uses such models of fibrotic regions in anatomical models to investigate the sensitivity of ablation targets to variations in electrophysiological features not obtainable with clinical imaging. The somewhat sobering conclusion presented by Deng et al. is that the simulated spiral wave dynamics depends on electrophysiological features to such an extent that for personalized modeling to be successful in this context, it is likely that both electrophysiology and structure must be established in a given patient.

Although early studies on the genesis of arrhythmias using mathematical models of cardiac cells focused on the dynamics of ion channels and spatiotemporal coupling, the discovery of complexity in intracellular calcium dynamics has led to new detailed studies focusing on intracellular calcium models and calcium-mediated arrhythmias. For example, premature ventricular beats (PVCs) have been associated with spontaneous calcium release (SCR) events at the cellular level and, while mostly benign in healthy individuals, under some conditions, they can be the trigger of more complex and deadly arrhythmias. In their article, Campos et al. (2017) investigate some of the pathological conditions necessary for premature ventricular beats (PVCs) by developing a multiscale computational model of rabbit ventricles at an average discretization of about 300 μm that accounts for fiber rotation, transmural cell heterogeneity, and a Purkinje network based on a rabbit ventricular cell model with a stochastic model of spontaneous calcium release (SCR). They found that SCR-mediated delayed after depolarizations (DADs) led to PVC formation at the Purkinje fibers, where electrotonic load was lower, leading to conduction block and initiation of reentry in the 3D muscle only when sodium channelopathies produced a negative shift in sodium channel steady-state inactivation. This is observed in cases such as ischemia, heart failure, and Brugada Syndrome as well as with Class I antiarrhythmic drugs (Na⁺ channel blockers).

Another known trigger for arrhythmias associated with abnormal calcium handling can occur during bradycardia and long-QT type 2 (LQT2) syndrome, where abnormal systolic calcium oscillations caused by sarcoplasmic reticulum (SR) overload lead to early afterdepolarizations (EADs), which are thought to promote reentrant arrhythmias such as Torsade de Pointes. To gain insights into this possible mechanism, Wilson et al. (2017) modified a rabbit ventricular model so that the ryanodine receptor (RyR2) is controlled by three gates: activation and inactivation gates for cytosolic calcium and a gate for luminal junctional SR calcium. They show that this model, when simulated in space and under conditions of bradycardia and low extracellular potassium, was able not only to predict EADs but also to reproduce experimentally observed calcium waves and systolic cell-synchronous and multifocal diastolic calcium releases. Their simulations thus support the role of SR calcium overload, abnormal SR calcium release, and the subsequent activation of the electrogenic sodium-calcium exchanger as a mechanism of Torsade de Pointes in tissue.

Alternans in cardiac tissue is yet another known trigger for arrhythmias and results from a period-doubling bifurcation that can be driven by either voltage or calcium dynamics or a combination of both with positive or negative feedback. Because of its complexity, there are different mechanisms at play during alternans, including the effect of memory, which can be included in a cardiac model in many ways. For example, Comlekoglu and Weinberg (2017) consider that the membrane capacitance does not have an ideal behavior and that in fact the capacitive current-voltage relationship of the cardiac membrane follows a fractional-order derivative, which mathematically induces a history dependence. They thus study, using a minimal cardiac cell model, the effects of a fractional-order time diffusion for the voltage and for the ionic currents. They found that when only voltage had a fractional time dependence, the action potential duration typically decreased and suppressed alternans. However, when fractional diffusion was included in addition for the gating variables, APD increased, alternans was promoted, and even spontaneous DADs could be produced. Delay differential equations also can be used to include history effects and can promote formation of alternans. Gomes et al. (2017) introduced delay differential equations into the gating dynamics of five different cardiac models whose alternans had been suppressed and showed that the use of delays could re-introduce alternans in these settings. Not all gating variables promoted alternans when delayed, and the authors identified two criteria to guide selection of variables to delay: an effective candidate gate should affect action potential duration significantly and the delay must impair the gate's recovery.

Alternans has also been shown in several animal experiments to be originated by complex intracellular calcium dynamics. In many cases where systems are complex, it is useful to derive simple minimum models that allow easier quantification of the mechanisms responsible and identification of their parameters. For example, Cantalapiedra et al. (2017) show that calcium alternans due to refractoriness of the RyR2 can be understood via reduction of a complex model to a simple two state variable model composed of a nonlinear calcium concentration of the dyadic space and a relaxation equation for the number of recovered RyR2s. Using this model, they found that the onset of alternans requires a well-defined value of diffusion and is less sensitive to inactivation or conductance release.

Another mechanism known to enhance alternans and to lead to even more arrhythmogenic substrates is ischemia. Bai et al. (2017) study the impact of ischemia-induced dysfunction of the sinoatrial node (SAN). Specifically, they use an action potential model for rabbit atrial and sinoatrial nodal (SAN) cells with modifications that incorporate changes to membrane ion channels and intracellular ion homeostasis due to ischemia and simulate an anatomically detailed two-dimensional model of the intact SAN-atrium structure. Their model shows that compared to normal function, ischemia slowed both pacemaking rate and conduction velocity, which was mainly attributable to altered Na⁺-Ca²⁺exchanger current and the ATP-sensitive potassium current. They further show arrhythmic effects of acetylcholine in the model, as it amplified the effects of ischemia, leading to pacemaking arrest and/or conduction exit block to the atria.

Studies have focused on understanding the mechanisms that can not only initiate arrhythmias but also terminate them. For example, in the continuing search for and development of new drugs that are not only safer but also more effective in treating and terminating fibrillation, mathematical modeling of cardiac cells has become an important way to test the effects of drugs under various conditions. Such studies, however, require the development of physiologically accurate models for the particular species and type of cardiac tissue studied. For example, Ellinwood et al. (2017) have developed a six-state Markov chain model for the ultra-rapid delayed-rectifier potassium current (I_Kur) based on experimental data obtained from human right atrial myocytes and have incorporated it into one of the most detailed human atrial models to study drug interactions. Since I_Kur is predominantly expressed in the atria and not in the ventricles, it represents a viable target to alter atrial fibrillation without impacting ventricular function. This study presents one of the first detailed analyses of ion-channel gating and channel-drug interactions (binding and unbinding) to determine optimal binding state dependence and kinetic properties in silico. Their simulations indicate that for this model, I_Kur inhibitors that bind to open or open-inactivated states of the channels have the potential to display positive rate dependence such that no effects on action potential dynamics are observed at normal sinus rhythm but anti-arrhythmic modification of the action potential at fast periods occurs through refractory period prolongation. This study lays a solid theoretical ground to identify possible therapeutic properties of I_Kur blockers in the treatment of atrial fibrillation.

As stated above, heterogeneity in the multicellular cardiac substrate (due to scars, fatty tissue, or fibrotic tissue) may cause the onset of cardiac arrhythmias. However, there is also increasing evidence that heterogeneity may play an unexpected role by stabilizing or even suppressing spatial-temporal complexity in the heart. In particular, the pinning of electrical excitation waves to heterogeneities in electrical conductance has been studied in detail. Zhang et al. (2017) investigate the formation and interaction of three-dimensional scroll waves with spatially extended, macroscopic heterogeneities in electrical conductance. They use the Fenton-Karma model and study the interaction of scroll waves with cylindrical heterogeneities (radius: 0.1–2 cm) with transmural or perpendicular alignment. For thick cylinders, multi-armed spirals are observed. For long cylinders, the pinned scroll waves show complex transient self-wrapping dynamics.

The role of heterogeneity in stabilizing electrical conduction is an important aspect in the development of novel far-field pacing strategies aiming for low-energy termination of cardiac fibrillation. In these approaches, weak pulsed electric fields are used to recruit virtual electrodes or wave-emitting sites from heterogeneities in electrical conductance. A detailed understanding of this phenomenon is key to the development and optimization of low-energy defibrillation. The numerical study by Connolly et al. (2017) investigates the response of the highly trabeculated structure of human endocardium on low-energy monophasic electric pulses. The study uses a bi-ventricular human computational model constructed from high-resolution MRI that provides anatomically accurate endocardial structures. Monophasic shocks were applied in an electrode configuration mimicking implantable cardioverter/defibrillators [right-ventricular (RV) electrode and an exterior ground]. Shocks with both polarities (anodal = positive RV electrode; cathodal = negative RV electrode) and varying field strengths were applied during induced cardiac arrhythmias. The simulations indicate the role of trabecular structures and polarity-dependent activation dynamics and emphasize the important role of computer models for the optimization and translation of low-energy defibrillation methods towards clinical application.

Because of the pro-arrhythmic nature of cardiac alternans, elimination and prevention of alternans have been pursued as anti-arrhythmic strategies. The diastolic interval (DI), i.e., the time between two consecutive action potentials, has been identified as a key dynamical variable in the development of alternans. Consequently, the control of DI by specific pacing protocols could potentially prevent the onset of alternans and subsequent life-threatening cardiac arrhythmias. However, experiments have demonstrated that alternans may persist while applying constant-DI pacing protocols. Furthermore, even in cases where constant-DI pacing was successful, the spatial extent from the pacing electrode remained relatively small (2 cm). The studies by Cherry (2017), Zlochiver et al. (2017), and Otani (2017) elucidate the mechanisms underlying DI control and highlight the current scientific challenges. Cherry focuses on the role of intracellular calcium to explain alternans persistence during DI control using models that are capable of producing voltage-driven and calcium-driven alternans. She finds that while during voltage-driven alternans action potentials arising from DI control protocols are always the same, the same protocol may induce alternans if the underlying mechanism is calcium-driven. Otani investigates the dynamics of alternans during DI control in single cell and one-dimensional numerical models. Otani combines a so-called memory model with a calcium cycling model to explain the possibility of alternans during constant-DI pacing and the phase lag of APDs behind DIs during sinusoidal-DI pacing. Zlochiver et al. develop a cable model using human kinetics and apply DI control up to 5 cm. They demonstrate that DI pacing may significantly shift the onset of both cardiac alternans and conduction blocks to higher pacing rates in comparison to pacing with constant cycle length and reduces the likelihood of occurrence of spatially discordant alternans. These advanced numerical and theoretical studies clearly stimulate further experimental work in the field of DI control.

Unipolar stimulation is widely used for pacemakers due to low current requirements. However, at shorter coupling intervals when the tissue near the pacing electrode is relatively refractory, the advantages of unipolar stimulation compared to bipolar stimulation may not be obvious. Therefore, the work by Galappaththige et al. (2017) models bipolar stimulation of cardiac tissue. While the activation patterns and spatial-temporal dynamics of bipolar stimulation are inherently complex, systematic analysis of strength-interval curves shows potential for a substantial decrease in stimulation threshold, which may have a significant impact on pacemaker and defibrillator design.

Contraction of cardiomyocytes due to intracellular calcium release is triggered by electrical stimuli. The reverse of this excitation-contraction coupling of cardiac mechanics, a process called mechano-electric feedback, plays a major role in cardiac arrhythmias and other pathological states of the heart and may also have an impact on the electrophysiology of the cell. Therefore, several contributions to this Focus Issue studied such electro-mechanical interactions and their consequences for cardiac dynamics on the organ level. In a pumping heart, the electro-mechanics of the cardiac muscle is also interacting with the laminar or turbulent blood flow inside the heart, adding another dimension of complexity that must be taken into account when trying to understand and model the full system.

Colli Franzone et al. (2017) investigate the influence of cardiac tissue deformation on the stability and potential break-up of reentrant wave dynamics. Using a full three-dimensional electro-mechanic bidomain model including fiber-directed anisotropy and stretch-activated channels, they show that the stability of scroll waves is mainly affected by mechano-electric feedback, while the presence of a deformation gradient in the diffusion coefficients only results in an increase of meandering of the waves but does not lead to a loss of stability.

The impact of mechano-electric feedback is also studied by Collet et al. (2017), with a focus on temperature effects and dynamical features like bifurcations, chaos, and co-existing attractors. For higher temperatures, the investigated qualitative model exhibits an increased parameter range of pro-arrhythmic effects, including self-sustained oscillations and a higher dynamical diversity.

Simulations of the blood flow within the pumping heart are presented by Tagliabue et al. (2017). The flow in a three-dimensional idealized ventricle due to muscle contraction is modeled by means of the Navier-Stokes equations with time-varying boundary conditions describing the opening and closing of the mitral and aortic valves. This numerical study shows the existence of different dynamical regimes during the heartbeat cycle ranging from laminar to turbulent flows.

Clinical diagnosis of heart diseases is still mainly based on the analysis of measured ECG time series. In particular, heart rate variability (HRV) describing beat-to-beat fluctuations of the heart rate is an important indicator for characterizing healthy or pathological states of the heart. Furthermore, the interaction between the beating heart and respiration or other physiological processes may have an impact on HRV and is of particular interest for understanding the “system heart” and for future diagnostic and therapeutic applications.

The main pacemaker in the heart is the sinoatrial (SA) node, which consists of a population of synchronized pacemaker cells. These cells, however, exhibit a wide range of beating frequencies when being isolated. Krogh-Madsen et al. (2017) study small populations of pacemaker cells extracted from embryonic chick hearts and compare the observed dynamics with an ionic model of the cluster describing the stochastic opening and closing of about 20,000 ion channels. Their simulation results are consistent with the hypothesis that experimentally observed beat-to-beat fluctuations are generated by stochastic opening and closing of ion channels in the cell membrane.

Krause et al. (2017) compare two time series-based methods for describing and quantifying the interaction between heart-beat and breathing. They show that a recently introduced concept called Cardiorespiratory Coordination (CRC) provides different information about the interaction than Cardiorespiratory Phase Synchronization (CRS), a characterization which was suggested about 20 years ago. Both concepts are potentially relevant for diagnostic purposes, and their mutual relation is investigated and illustrated using measured data from a pregnant woman and a man suffering from sleep apnea.

Lerma et al. (2017) investigate the different effects of intrinsic (pathological impairment) and extrinsic (physiological adaption) cardiac factors on short-term HRV. In order to identify such differences, beat-to-beat time series are investigated with respect to their (multifractal) scaling properties.

HRV analysis is also the main topic of the contribution by Porta et al. (2017), who propose a multiscale complexity (MSC) method for assessing irregularity of beat-to-beat time series in different frequency bands (quantified by coefficients of autoregressive models). The aim of this approach is to characterize the complexity of cardiac control at temporal time scales of sinus node neural regulation. By focusing on different time scales, the proposed method may separate the influences of cardiac regulation and of physiological mechanisms, both resulting in HRV.

Witt et al. (2017) investigate the statistics of arrhythmia-triggering PVCs, which typically occur in groups followed by a series of normal heart-beats. To characterize features of PVCs, complexity measures and methods from research on extreme events are applied to 24-h ECG time series of post-myocardial-infarcted patients. As a main result, Witt et al. find that post-infarct patients older than 65 years have much more randomly timed PVCs compared to younger patients, for whom arrhythmic beats typically occur in clusters.

Significant challenges in understanding, modeling, and controlling complex cardiac dynamics continue to stimulate fundamental research on spatiotemporal chaotic dynamics in extended excitable multiscale systems. We are also now at the cusp of applying mathematical modeling to drug safety testing and clinical applications, which presents further modeling challenges, including development of rigorous evaluation and validation processes.

For all research topics addressed in this Focus Issue, we anticipate remarkable progress in the years to come, towards both fundamental research and applications. Novel time series analysis methods, applied to high quality multivariate times series and used in combination with advanced machine learning algorithms, may bring forth a new generation of diagnostic tools for assessing individual pathologies and risks. Technological progress in electrical and acoustical imaging modalities will likely provide new “insight” into the beating heart that can be used for immediate diagnosis or model validation. On the theoretical side, increased computational power may enable detailed 3D modeling including disease and/or patient-specific features of the heart. In this context, parameter estimation and data assimilation methods will be crucial for incorporating large amounts of measured data into mathematical models in order to adapt and evaluate them. In this way, models of different levels of complexity are likely to become even more relevant and reliable for predicting the impact of novel therapies, e.g., the response to pharmaceutic interventions, the success of ablation of arrhythmogenic substrates, or the efficacy of dynamically inspired low-energy defibrillation strategies.

We acknowledge the novel and insightful contributions of the authors of the articles included in this Focus Issue as well as the reviewers, who made numerous helpful comments and suggestions that strengthened all the articles. We also appreciate the professionalism of the Chaos editorial staff, especially Deborah Doherty, Eric Mills, Kristen Overstreet, as well as editor-in-chief Jürgen Kurths, all of whom provided unfailing support through every step of the process.