Nonlinear and Stochastic Dynamics in the Heart

Role of ryanodine receptor cooperativity in Ca2+-wave-mediated triggered activity in cardiomyocytes

Ca2+ waves are known to trigger delayed after-depolarizations that can cause malignant cardiac arrhythmias. However, modelling Ca2+ waves using physiologically realistic models has remained a major challenge. Existing models with low Ca2+ sensitivity of ryanodine receptors (RyRs) necessitate large release currents, leading to an unrealistically large Ca2+ transient amplitude incompatible with the experimental observations. Consequently, current physiologically detailed models of delayed after-depolarizations resort to unrealistic cell architectures to produce Ca2+ waves with a normal Ca2+ transient amplitude. Here, we address these challenges by incorporating RyR cooperativity into a physiologically detailed model with a realistic cell architecture. We represent RyR cooperativity phenomenologically through a Hill coefficient within the sigmoid function of RyR open probability. Simulations in permeabilized myocytes with high Ca2+ sensitivity reveal that a sufficiently large Hill coefficient is required for Ca2+ wave propagation via the fire-diffuse-fire mechanism. In intact myocytes, propagating Ca2+ waves can occur only within an intermediate Hill coefficient range. Within this range, the spark rate is neither too low, enabling Ca2+ wave propagation, nor too high, allowing for the maintenance of a high sarcoplasmic reticulum load during diastole of the action potential. Moreover, this model successfully replicates other experimentally observed manifestations of Ca2+-wave-mediated triggered activity, including phase 2 and phase 3 early after-depolarizations and high-frequency voltage-Ca2+ oscillations. These oscillations feature an elevated take-off potential with depolarization mediated by the L-type Ca2+ current. The model also sheds light on the roles of luminal gating of RyRs and the mobile buffer ATP in the genesis of these arrhythmogenic phenomena. KEY POINTS: Existing mathematical models of Ca2+ waves use an excessively large Ca2+-release current or unrealistic diffusive coupling between release units. Our physiologically realistic model, using a Hill coefficient in the ryanodine receptor (RyR) gating function to represent RyR cooperativity, addresses these limitations and generates organized Ca2+ waves at Hill coefficients ranging from ∼5 to 10, as opposed to the traditional value of 2. This range of Hill coefficients gives a spark rate neither too low, thereby enabling Ca2+ wave propagation, nor too high, allowing for the maintenance of a high sarcoplasmic reticulum load during the plateau phase of the action potential. Additionally, the model generates Ca2+-wave-mediated phase 2 and phase 3 early after-depolarizations, and coupled membrane voltage with Ca2+ oscillations mediated by the L-type Ca2+ current. This study suggests that pharmacologically targeting RyR cooperativity could be a promising strategy for treating cardiac arrhythmias linked to Ca2+-wave-mediated triggered activity.

Ultrastructure and cardiac impulse propagation: scaling up from microscopic to macroscopic conduction

The standard conception of cardiac conduction is based on the cable theory of nerve conduction, which treats cardiac tissue as a continuous syncytium described by the Hodgkin-Huxley equations. However, cardiac tissue is composed of discretized cells with microscopic and macroscopic heterogeneities and discontinuities, such as subcellular localizations of sodium channels and connexins. In addition to this, there are heterogeneities in the distribution of sympathetic and parasympathetic nerves, which powerfully regulate impulse propagation. In the continuous models, the ultrastructural details, i.e. the microscopic heterogeneities and discontinuities, are ignored by 'coarse graining' or 'smoothing'. However, these ultrastructural components may play crucial roles in cardiac conduction and arrhythmogenesis, particularly in disease states. We discuss the current progress of modelling the effects of ultrastructural components on electrical conduction, the issues and challenges faced by the cardiac modelling community, and how to scale up conduction properties at the subcellular (microscopic) scale to the tissue and whole-heart (macroscopic) scale in future modelling and experimental studies, i.e. how to link the ultrastructure at different scales to impulse conduction and arrhythmogenesis in the heart.

From resonance to chaos by modulating spatiotemporal patterns through a synthetic optogenetic oscillator

Oscillations are a recurrent phenomenon in biological systems across scales, but deciphering their fundamental principles is very challenging. Here, we tackle this challenge by redesigning the wellcharacterised synthetic oscillator known as "repressilator" in Escherichia coli and controlling it using optogenetics, creating the "optoscillator". Bacterial colonies manifest oscillations as spatial ring patterns. When we apply periodic light pulses, the optoscillator behaves as a forced oscillator and we systematically investigate the properties of the rings under various light conditions. Combining experiments with mathematical modeling, we demonstrate that this simple oscillatory circuit can generate complex dynamics that are transformed into distinct spatial patterns. We report the observation of synchronisation, resonance, subharmonic resonance and period doubling. Furthermore, we present evidence of a chaotic regime. This work highlights the intricate spatiotemporal patterns accessible by synthetic oscillators and underscores the potential of our approach in revealing fundamental principles of biological oscillations.

Evaluation of the responsiveness pattern to caffeine through a smart data-driven ECG non-linear multi-band analysis

This study aimed to explore more efficient ways of administering caffeine to the body by investigating the impact of caffeine on the modulation of the nervous system's activity through the analysis of electrocardiographic signals (ECG). An ECG non-linear multi-band analysis using Discrete Wavelet Transform (DWT) was employed to extract various features from healthy individuals exposed to different caffeine consumption methods: expresso coffee (EC), decaffeinated coffee (ED), Caffeine Oral Films (OF_caffeine), and placebo OF (OF_placebo). Non-linear feature distributions representing every ECG minute time series have been selected by PCA with different variance percentages to serve as inputs for 23 machine learning models in a leave-one-out cross-validation process for analyzing the behavior differences between ED/EC and OF_placebo/OF_caffeine groups, respectively, over time. The study generated 50-point accuracy curves per model, representing the discrimination power between groups throughout the 50 min. The best model accuracies for ED/EC varied between 30 and 70 %, (using the decision tree classifier) and OF_placebo/OF_caffeine ranged from 62 to 84 % (using Fine Gaussian). Notably, caffeine delivery through OFs demonstrated effective capacity compared to its placebo counterpart, as evidenced by significant differences in accuracy curves between OF_placebo/OF_caffeine. Caffeine delivery via OFs also exhibited rapid dissolution efficiency and controlled release rate over time, distinguishing it from EC. The study supports the potential of caffeine delivery through Caffeine OFs as a superior technology compared to traditional methods by means of ECG analysis. It highlights the efficiency of OFs in controlling the release of caffeine and underscores their promise for future caffeine delivery systems.

Intracellular ion accumulation in the genesis of complex action potential dynamics under cardiac diseases

Intracellular ions, including sodium (Na^{+}), calcium (Ca^{2+}), and potassium (K^{+}), etc., accumulate slowly after a change of the state of the heart, such as a change of the heart rate. The goal of this study is to understand the roles of slow ion accumulation in the genesis of cardiac memory and complex action-potential duration (APD) dynamics that can lead to lethal cardiac arrhythmias. We carry out numerical simulations of a detailed action potential model of ventricular myocytes under normal and diseased conditions, which exhibit memory effects and complex APD dynamics. We develop a low-dimensional iterated map (IM) model to describe the dynamics of Na^{+}, Ca^{2+}, and APD and use it to uncover the underlying dynamical mechanisms. The development of the IM model is informed by simulation results under the normal condition. We then use the IM model to perform linear stability analyses and computer simulations to investigate the bifurcations and complex APD dynamics, which depend on the feedback loops between APD and intracellular Ca^{2+} and Na^{+} concentrations and the steepness of the APD response to the ion concentrations. When the feedback between APD and Ca^{2+} concentration is positive, a Hopf bifurcation leading to periodic oscillatory behavior occurs as the steepness of the APD response to the ion concentrations increases. The negative feedback loop between APD and Na^{+} concentration is required for the Hopf bifurcation. When the feedback between APD and Ca^{2+} concentration is negative, period-doubling bifurcations leading to high periodicity and chaos occurs. In this case, Na^{+} accumulation plays little role in the dynamics. Finally, we carry out simulations of the detailed action potential model under two diseased conditions, which exhibit steep APD responses to ion concentrations. Under both conditions, Hopf bifurcations leading to slow oscillations or period-doubling bifurcations leading to high periodicity and chaotic APD dynamics occur, depending on the strength of the ion pump-Na^{+}-Ca^{2+} exchanger. Using functions reconstructed from the simulation data, the IM model accurately captures the bifurcations and dynamics under the two diseased conditions. In conclusion, besides using computer simulations of a detailed high-dimensional action-potential model to investigate the effects of slow ion accumulation and short-term memory on bifurcations and genesis of complex APD dynamics in cardiac myocytes under diseased conditions, this study also provides a low-dimensional mathematical tool, i.e., the IM model, to allow stability analyses for uncovering the underlying mechanisms.

Cardiovascular Diseases Diagnosis Using an ECG Multi-Band Non-Linear Machine Learning Framework Analysis

BACKGROUND

cardiovascular diseases (CVDs), which encompass heart and blood vessel issues, stand as the leading cause of global mortality for many people.

METHODS

the present study intends to perform discrimination between seven well-known CVDs (bundle branch block, cardiomyopathy, myocarditis, myocardial hypertrophy, myocardial infarction, valvular heart disease, and dysrhythmia) and one healthy control group, respectively, by feeding a set of machine learning (ML) models with 10 non-linear features extracted every 1 s from electrocardiography (ECG) lead signals of a well-known ECG database (PTB diagnostic ECG database) using multi-band analysis performed by discrete wavelet transform (DWT). The ML models were trained and tested using a leave-one-out cross-validation approach, assessing the individual and combined capabilities of features, per each lead or combined, to distinguish between pairs of study groups and for conducting a comprehensive all vs. all analysis.

RESULTS

the Accuracy discrimination results ranged between 73% and 100%, the Recall between 68% and 100%, and the AUC between 0.42 and 1.

CONCLUSIONS

the results suggest that our method is a good tool for distinguishing CVDs, offering significant advantages over other studies that used the same dataset, including a multi-class comparison group (all vs. all), a wider range of binary comparisons, and the use of classical non-linear analysis under ECG multi-band analysis performed by DWT.

Bistable spiral wave dynamics in electrically excitable media

We show that a positive feedback loop between sodium current inactivation and wave-front ramp-up speed causes a saddle-node bifurcation to result in bistable planar and spiral waves in electrically excitable media, in which both slow and fast waves are triggered by different stimulation protocols. Moreover, the two types of spiral wave conduction may interact to give rise to more complex spiral wave dynamics. The transitions between different spiral wave behaviors via saddle-node bifurcation can be a candidate mechanism for transitions widely seen in cardiac arrhythmias and neural diseases.

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.

Theoretical prediction of early afterdepolarization-evoked triggered activity formation initiating ventricular reentrant arrhythmias

BACKGROUND AND OBJECTIVE

Excessive prolongation of QT interval on ECGs in patients with congenital/acquired long QT syndrome and heart failure is a sign suggesting the development of early afterdepolarization (EAD), an abnormal repolarization in the action potential of ventricular cardiomyocytes. The development of EAD has been believed to be a trigger for fatal tachyarrhythmia, which can be a risk for sudden cardiac death. The role of EAD in triggering ventricular tachycardia (VT) remains unclear. The aim of this study was to elucidate the mechanism of EAD-induced triggered activity formation that leads to the VT such as Torsades de Pointes.

METHODS

We investigated the relationship between EAD and tachyarrhythmia initiation by constructing homogeneous myocardial sheet models consisting of the mid-myocardial cell version of a human ventricular myocyte model and performing simulations of excitation propagation.

RESULTS

A solitary island-like (clustering) occurrence of EADs in the homogeneous myocardial sheet could induce a focal excitation wave. However, reentrant excitation, an entity of tachyarrhythmia, was not able to be triggered regardless of the EAD cluster size when the focal excitation wave formed a repolarization potential difference boundary consisting of only a convex surface. The discontinuous distribution of multiple EAD clusters in the ventricular tissue formed a specific repolarization heterogeneity due to the repolarization potential difference, the shape of which depended on EAD cluster size and placed intervals. We found that the triggered activity was formed in such a manner that the repolarization potential difference boundary included a concave surface.

CONCLUSIONS

The formation of triggered activity that led to tachyarrhythmia required not only the occurrence of EAD onset-mediated focal excitation wave but also a repolarization heterogeneity-based specific repolarization potential difference boundary shape formed within the tissue.

Inpainting surgical occlusion from laparoscopic video sequences for robot-assisted interventions

Purpose

Medical technology for minimally invasive surgery has undergone a paradigm shift with the introduction of robot-assisted surgery. However, it is very difficult to track the position of the surgical tools in a surgical scene, so it is crucial to accurately detect and identify surgical tools. This task can be aided by deep learning-based semantic segmentation of surgical video frames. Furthermore, due to the limited working and viewing areas of these surgical instruments, there is a higher chance of complications from tissue injuries (e.g., tissue scars and tears).

Approach

With the aid of digital inpainting algorithms, we present an application that uses image segmentation to remove surgical instruments from laparoscopic/endoscopic video. We employ a modified U-Net architecture (U-NetPlus) to segment the surgical instruments. It consists of a redesigned decoder and a pre-trained VGG11 or VGG16 encoder. The decoder was modified by substituting an up-sampling operation based on nearest-neighbor interpolation for the transposed convolution operation. Furthermore, these interpolation weights do not need to be learned to perform upsampling, which eliminates the artifacts generated by the transposed convolution. In addition, we use a very fast and adaptable data augmentation technique to further enhance performance. The instrument segmentation mask is filled in (i.e., inpainted) by the tool removal algorithms using the previously acquired tool segmentation masks and either previous instrument-containing frames or instrument-free reference frames.

Results

We have shown the effectiveness of the proposed surgical tool segmentation/removal algorithms on a robotic instrument dataset from the MICCAI 2015 and 2017 EndoVis Challenge. We report a 90.20% DICE for binary segmentation, a 76.26% DICE for instrument part segmentation, and a 46.07% DICE for instrument type (i.e., all instruments) segmentation on the MICCAI 2017 challenge dataset using our U-NetPlus architecture, outperforming the results of earlier techniques used and tested on these data. In addition, we demonstrated the successful execution of the tool removal algorithm from surgical tool-free videos that contained moving surgical tools that were generated artificially.

Conclusions

Our application successfully separates and eliminates the surgical tool to reveal a view of the background tissue that was otherwise hidden by the tool, producing results that are visually similar to the actual data.

Development of a novel low-order model for atrial function and a study of atrial mechano-electric feedback

Numerical models of the cardiovascular system have largely focused on the function of the ventricles, with atrial function often neglected. Furthermore, the time-varying elastance method that prescribes the pressure-volume relationship rather than calculating it consistently is frequently used for the ventricles and atrium. This method has yet to be validated however, so its applicability for cardiac modelling is frequently questioned. To overcome this challenge, we propose a synergistic model of left atrium (LA) and left ventricle (LV) by self-consistently integrating various feedback mechanisms among the electro-mechanical and chemical functions of the micro-scale myofiber, the macro-scale dynamics of the LA and LV, the atrioventricular node (AV), and circulation. The model is tested and shown to reproduce the essential features of the atrium cycling, such as the characteristic figure of eight pressure-volume loops. Our model is further developed to investigate the effect of dysfunctions of the mechanical-electric feedback (MEF) in the atrium. Our model not only successfully reproduces key experimental MEF observations such as prolonged action-potential and increases in action-potential magnitude induced by atrial stretch but also shows how MEF and arrhythmia of the atrium lead to a degradation of cardiac output and pumping power with significant consequences. In particular, MEF reproduces arrhythmia such as ectopic and erratic cycling, missed heart beats and restricted function.

Bifurcations to transient and oscillatory excitations in inhomogeneous excitable media: Insights into arrhythmogenesis in long QT syndrome

Ventricular arrhythmias are the leading cause of sudden cardiac death. Understanding the mechanisms of arrhythmia initiation is important for developing effective therapeutics for prevention. Arrhythmias can be induced via premature external stimuli or occur spontaneously via dynamical instabilities. Computer simulations have shown that a large repolarization gradient due to regional prolongation of the action potential duration can result in instabilities leading to premature excitations and arrhythmias, but the bifurcation remains to be elucidated. In this study we carry out numerical simulations and linear stability analyses using a one-dimensional heterogeneous cable consisting of the FitzHugh-Nagumo model. We show that a Hopf bifurcation leads to local oscillations, which, once their amplitudes are large enough, lead to spontaneous propagating excitations. Depending on the degree of heterogeneities, these excitations can range from one to many and to be sustained oscillations, manifesting as premature ventricular contractions (PVCs) and sustained arrhythmias. The dynamics depends on the repolarization gradient and the length of the cable. Complex dynamics is also induced by the repolarization gradient. The mechanistic insights from the simple model may help in the understanding of the genesis of PVCs and arrhythmias in long QT syndrome.

Computational and experimental modulation of a noisy chaotic neuronal system

In this work, we study the interplay between chaos and noise in neuronal state transitions involving period doubling cascades. Our approach involves the implementation of a neuronal mathematical model under the action of neuromodulatory input, with and without noise, as well as equivalent experimental work on a biological neuron in the stomatogastric ganglion of the crab Cancer borealis. Our simulations show typical transitions between tonic and bursting regimes that are mediated by chaos and period doubling cascades. While this transition is less evident when intrinsic noise is present in the model, the noisy computational output displays features akin to our experimental results. The differences and similarities observed in the computational and experimental approaches are discussed.

Toward a Physiologically Relevant 3D Helicoidal-Oriented Cardiac Model: Simultaneous Application of Mechanical Stimulation and Surface Topography

Myocardium consists of cardiac cells that interact with their environment through physical, biochemical, and electrical stimulations. The physiology, function, and metabolism of cardiac tissue are affected by this dynamic structure. Within the myocardium, cardiomyocytes' orientations are parallel, creating a dominant orientation. Additionally, local alignments of fibers, along with a helical organization, become evident at the macroscopic level. For the successful development of a reliable in vitro cardiac model, evaluation of cardiac cells' behavior in a dynamic microenvironment, as well as their spatial architecture, is mandatory. In this study, we hypothesize that complex interactions between long-term contraction boundary conditions and cyclic mechanical stimulation may provide a physiological mechanism to generate off-axis alignments in the preferred mechanical stretch direction. This off-axis alignment can be engineered in vitro and, most importantly, mirrors the helical arrangements observed in vivo. For this purpose, uniaxial mechanical stretching of dECM-fibrin hydrogels was performed on pre-aligned 3D cultures of cardiac cells. In view of the potential development of helical structures similar to those in native hearts, the possibility of generating oblique alignments ranging between 0° and 90° was explored. Indeed, our investigations of cell alignment in 3D, employing both mechanical stimulation and groove constraint, provide a reliable mechanism for the generation of helicoidal structures in the myocardium. By combining cyclic stretch and geometric alignment in grooves, an intermediate angle toward favored direction can be achieved experimentally: while cyclic stretch produces a perpendicular orientation, geometric alignment is associated with a parallel one. In our 2D and 3D culture conditions, nonlinear cellular addition of the strains and strain avoidance concept reliably predicted the preferred cellular alignment. The 3D dECM-fibrin model system in this study shows that cyclical stretching supports cell survival and development. Using mechanical stimulation of pre-aligned heart cells, maturation markers are augmented in neonatal cardiomyocytes, while the beating culture period is prolonged, indicating an improved model function. We propose a simplified theoretical model based on numerical simulation and nonlinear strain avoidance by cells to explain oblique alignment angles. Thus, this work lays a possible rational basis for understanding and engineering oblique cellular alignments, such as the helicoidal layout of the heart, using approaches that simultaneously enhance maturation and function.

Validation of a non-linear index of heart rate variability to determine aerobic and anaerobic thresholds during incremental cycling exercise in women

Studies highlight the usage of non-linear time series analysis of heart rate variability (HRV) using the short-term scaling exponent alpha1 of Detrended Fluctuation Analysis (DFA-alpha1) during exercise to determine aerobic and anaerobic thresholds. The present study aims to further verify this approach in women. Gas exchange and HRV data were collected from 26 female participants with different activity levels. Oxygen uptake (VO2) and heart rate (HR) at first (VT1) and second ventilatory thresholds (VT2) were compared with DFA-alpha1-based thresholds 0.75 (HRVT1) and 0.50 (HRVT2). Results: VO2 at VT1 and VT2 were 25.2 ml/kg/min (± 2.8) and 31.5 ml/kg/min (± 3.6) compared with 26.5 ml/kg/min (± 4.0) and 31.9 ml/kg/min (± 4.5) for HRVT1 and HRVT2, respectively (ICC3,1 = 0.77, 0.84; r = 0.81, 0.86, p < 0.001). The mean HR at VT1 was 147 bpm (± 15.6) and 167 bpm (± 12.7) for VT2, compared with 152 bpm (± 15.5) and 166 bpm (± 13.2) for HRVT1 and HRVT2, respectively (ICC3,1 = 0.87, 0.90; r = 0.87, 0.90, p < 0.001). Bland-Altman analysis for VT1 vs. HRVT1 showed a mean difference of - 1.3 ml/kg/min (± 2.4; LoA: 3.3, - 6.0 ml/kg/min) for VO2 and of - 4.7 bpm (± 7.8; LoA: 10.6, - 20.0 bpm) for HR. VT2 vs. HRVT2 showed a mean difference of - 0.4 ml/kg/min (± 2.3; LoA: 4.1, - 4.9 ml/kg/min) for VO2 and 0.5 bpm (± 5.7; LoA: 11.8, - 10.8 bpm) for HR. DFA-alpha1-based thresholds showed good agreement with traditionally used thresholds and could be used as an alternative approach for marking organismic transition zones for intensity distribution in women.

Cardiac Alternans: From Bedside to Bench and Back

Cardiac alternans arises from dynamical instabilities in the electrical and calcium cycling systems of the heart, and often precedes ventricular arrhythmias and sudden cardiac death. In this review, we integrate clinical observations with theory and experiment to paint a holistic portrait of cardiac alternans: the underlying mechanisms, arrhythmic manifestations and electrocardiographic signatures. We first summarize the cellular and tissue mechanisms of alternans that have been demonstrated both theoretically and experimentally, including 3 voltage-driven and 2 calcium-driven alternans mechanisms. Based on experimental and simulation results, we describe their relevance to mechanisms of arrhythmogenesis under different disease conditions, and their link to electrocardiographic characteristics of alternans observed in patients. Our major conclusion is that alternans is not only a predictor, but also a causal mechanism of potentially lethal ventricular and atrial arrhythmias across the full spectrum of arrhythmia mechanisms that culminate in functional reentry, although less important for anatomic reentry and focal arrhythmias.

Modeling Calcium Cycling in the Heart: Progress, Pitfalls, and Challenges

Intracellular calcium (Ca) cycling in the heart plays key roles in excitation-contraction coupling and arrhythmogenesis. In cardiac myocytes, the Ca release channels, i.e., the ryanodine receptors (RyRs), are clustered in the sarcoplasmic reticulum membrane, forming Ca release units (CRUs). The RyRs in a CRU act collectively to give rise to discrete Ca release events, called Ca sparks. A cell contains hundreds to thousands of CRUs, diffusively coupled via Ca to form a CRU network. A rich spectrum of spatiotemporal Ca dynamics is observed in cardiac myocytes, including Ca sparks, spark clusters, mini-waves, persistent whole-cell waves, and oscillations. Models of different temporal and spatial scales have been developed to investigate these dynamics. Due to the complexities of the CRU network and the spatiotemporal Ca dynamics, it is challenging to model the Ca cycling dynamics in the cardiac system, particularly at the tissue sales. In this article, we review the progress of modeling of Ca cycling in cardiac systems from single RyRs to the tissue scale, the pros and cons of the current models and different modeling approaches, and the challenges to be tackled in the future.

Bistable nerve conduction

It has been demonstrated experimentally that slow and fast conduction waves with distinct conduction velocities can occur in the same nerve system depending on the strength or the form of the stimulus, which give rise to two modes of nerve functions. However, the mechanisms remain to be elucidated. In this study, we use computer simulations of the cable equation with modified Hodgkin-Huxley kinetics and analytical solutions of a simplified model to show that stimulus-dependent slow and fast waves recapitulating the experimental observations can occur in the cable, which are the two stable conduction states of a bistable conduction behavior. The bistable conduction is caused by a positive feedback loop of the wavefront upstroke speed, mediated by the sodium channel inactivation properties. Although the occurrence of bistable conduction only requires the presence of the sodium current, adding a calcium current to the model further promotes bistable conduction by potentiating the slow wave. We also show that the bistable conduction is robust, occurring for sodium and calcium activation thresholds well within the experimentally determined ones of the known sodium and calcium channel families. Since bistable conduction can occur in the cable equation of Hodgkin-Huxley kinetics with a single inward current, i.e., the sodium current, it can be a generic mechanism applicable to stimulus-dependent fast and slow conduction not only in the nerve systems but also in other electrically excitable systems, such as cardiac muscles.

Removal of spiral turbulence by virtual electrodes through the use of a circularly polarized electric field

Heart disease is the leading cause of death and is often accompanied by cardiac fibrillation. Defibrillation using the virtual electrode effects is a promising alternative compared to using the high-voltage electric shock in the clinic. Our earlier works [S. L. Murphy, K. D. Kochanek, J. Xu, and E. Arias, NCHS Data Brief 427 (2021); R. A. Gray, A. M. Pertsov, and J. Jalife, Nature 392, 75-78 (1998); F. X. Witkowski, L. J. Leon, P. A. Penkoske, W. R. Giles, M. L. Spano, W. L. Ditto, and A. T. Winfree, Nature 392, 78-82 (1998); M. Santini, C. Pandozi, G. Altamura, G. Gentilucci, M. Villani, M. C. Scianaro, A. Castro, F. Ammirati, and B. Magris, J. Interv. Card. Electrophysiol. 3, 45-51 (1999).] prove that, compared with other external electric fields, a low voltage circularly polarized electric field is more efficient in turning non-excitable defects in cardiac tissue into virtual electrodes. It, therefore, needs lower voltage to stimulate the excitation waves and causes less harm to reset the spiral turbulence of cardiac excitation for defibrillation. In this paper, we investigate the virtual electrode effect of multiple defects by the circularly polarized electric field for the removal of spiral turbulence. Considering different shapes, sizes, and distributions of multiple defects, we reveal the phase locking of stimulated excitations around multiple virtual electrodes. Furthermore, the circularly polarized electric field parameters are optimized to remove the spiral turbulence.

Synchronization of spatially discordant voltage and calcium alternans in cardiac tissue

The heart is an excitable medium which is excited by membrane potential depolarization and propagation. Membrane potential depolarization brings in calcium (Ca) through the Ca channels to trigger intracellular Ca release for contraction of the heart. Ca also affects voltage via Ca-dependent ionic currents, and thus, voltage and Ca are bidirectionally coupled. It has been shown that the voltage subsystem or the Ca subsystem can generate its own dynamical instabilities which are affected by their bidirectional couplings, leading to complex dynamics of action potential and Ca cycling. Moreover, the dynamics become spatiotemporal in tissue in which cells are diffusively coupled through voltage. A widely investigated spatiotemporal dynamics is spatially discordant alternans (SDA) in which action potential duration (APD) or Ca amplitude exhibits temporally period-2 and spatially out-of-phase patterns, i.e., APD-SDA and Ca-SDA patterns, respectively. However, the mechanisms of formation, stability, and synchronization of APD-SDA and Ca-SDA patterns remain incompletely understood. In this paper, we use cardiac tissue models described by an amplitude equation, coupled iterated maps, and reaction-diffusion equations with detailed physiology (the ionic model) to perform analytical and computational investigations. We show that, when the Ca subsystem is stable, the Ca-SDA pattern always follows the APD-SDA pattern, and thus, they are always synchronized. When the Ca subsystem is unstable, synchronization of APD-SDA and Ca-SDA patterns depends on the stabilities of both subsystems, their coupling strengths, and the spatial scales of the initial Ca-SDA patterns. Spontaneous (initial condition-independent) synchronization is promoted by enhancing APD instability and reducing Ca instability as well as stronger Ca-to-APD and APD-to-Ca coupling, a pattern formation caused by dynamical instabilities. When Ca is more unstable and APD is less unstable or APD-to-Ca coupling is weak, synchronization of APD-SDA and Ca-SDA patterns is promoted by larger initially synchronized Ca-SDA clusters, i.e., initial condition-dependent synchronization. The synchronized APD-SDA and Ca-SDA patterns can be locked in-phase, antiphase, or quasiperiodic depending on the coupling relationship between APD and Ca. These theoretical and simulation results provide mechanistic insights into the APD-SDA and Ca-SDA dynamics observed in experimental studies.

Kramers Rate Theory of Pacemaker Dynamics in Noisy Excitable Media

Rhythmic activities, which are usually driven by pacemakers, are common in biological systems. In noisy excitable media, pacemakers are self-organized firing clusters, but the underlying dynamics remains to be elucidated. Here we develop a Kramers rate theory of coupled cells to describe the firing properties of pacemakers and their dependence on coupling strength and system size and dimension. The theory captures accurately the simulation results of tissue models with stochastic Hodgkin-Huxley equations except when transitions from pacemakers to spiral waves occur under weak coupling.

Bifurcation, chaos, multistability, and organized structures in a predator-prey model with vigilance

There is not a single species that does not strive for survival. Every species has crafted specialized techniques to avoid possible dangers that mostly come from the side of their predators. Survival instincts in nature led prey populations to develop many anti-predator strategies. Vigilance is a well-observed effective antipredator strategy that influences predator-prey dynamics significantly. We consider a simple discrete-time predator-prey model assuming that vigilance affects the predation rate and the growth rate of the prey. We investigate the system dynamics by constructing isoperiodic and Lyapunov exponent diagrams with the simultaneous variation of the prey's growth rate and the strength of vigilance. We observe a series of different types of organized periodic structures with different kinds of period-adding phenomena. The usual period-bubbling phenomenon is shown near a shrimp-shaped periodic structure. We observe the presence of double and triple heterogeneous attractors. We also notice Wada basin boundaries in the system, which is quite rare in ecological systems. The complex dynamics of the system in biparameter space are explored through extensive numerical simulations.

The functional role of cardiac activity in perception and action

SKORA, L.I., J.J.A. LIVERMORE and K. Roelofs. The functional role of cardiac activity in perception and action. NEUROSCI BIOBEHAV REV X(X) XXX-XXX, 2022. Patterns of cardiac activity continuously vary with environmental demands, accelerating or decelerating depending on circumstances. Simultaneously, cardiac cycle affects a host of higher-order processes, where systolic baroreceptor activation largely impairs processing. However, a unified functional perspective on the role of cardiac signal in perception and action has been lacking. Here, we combine the existing strands of literature and use threat-, anticipation-, and error-related cardiac deceleration to show that deceleration is an adaptive mechanism dynamically attenuating the baroreceptor signal associated with each heartbeat to minimise its impact on exteroceptive processing. This mechanism allows to enhance attention afforded to external signal and prepare an appropriate course of action. Conversely, acceleration is associated with a reduced need to attend externally, enhanced action tendencies and behavioural readjustment. This novel account demonstrates that dynamic adjustments in heart rate serve the purpose of regulating the level of precision afforded to internal versus external evidence in order to optimise perception and action. This highlights that the importance of cardiac signal in adaptive behaviour lies in its dynamic regulation.

Fractal Correlation Properties of Heart Rate Variability as a Biomarker for Intensity Distribution and Training Prescription in Endurance Exercise: An Update

While established methods for determining physiologic exercise thresholds and intensity distribution such as gas exchange or lactate testing are appropriate for the laboratory setting, they are not easily obtainable for most participants. Data over the past two years has indicated that the short-term scaling exponent alpha1 of Detrended Fluctuation Analysis (DFA a1), a heart rate variability (HRV) index representing the degree of fractal correlation properties of the cardiac beat sequence, shows promise as an alternative for exercise load assessment. Unlike conventional HRV indexes, it possesses a dynamic range throughout all intensity zones and does not require prior calibration with an incremental exercise test. A DFA a1 value of 0.75, reflecting values midway between well correlated fractal patterns and uncorrelated behavior, has been shown to be associated with the aerobic threshold in elite, recreational and cardiac disease populations and termed the heart rate variability threshold (HRVT). Further loss of fractal correlation properties indicative of random beat patterns, signifying an autonomic state of unsustainability (DFA a1 of 0.5), may be associated with that of the anaerobic threshold. There is minimal bias in DFA a1 induced by common artifact correction methods at levels below 3% and negligible change in HRVT even at levels of 6%. DFA a1 has also shown value for exercise load management in situations where standard intensity targets can be skewed such as eccentric cycling. Currently, several web sites and smartphone apps have been developed to track DFA a1 in retrospect or in real-time, making field assessment of physiologic exercise thresholds and internal load assessment practical. Although of value when viewed in isolation, DFA a1 tracking in combination with non-autonomic markers such as power/pace, open intriguing possibilities regarding athlete durability, identification of endurance exercise fatigue and optimization of daily training guidance.

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.

Bifurcations and Proarrhythmic Behaviors in Cardiac Electrical Excitations

The heart is a hierarchical dynamic system consisting of molecules, cells, and tissues, and acts as a pump for blood circulation. The pumping function depends critically on the preceding electrical activity, and disturbances in the pattern of excitation propagation lead to cardiac arrhythmia and pump failure. Excitation phenomena in cardiomyocytes have been modeled as a nonlinear dynamical system. Because of the nonlinearity of excitation phenomena, the system dynamics could be complex, and various analyses have been performed to understand the complex dynamics. Understanding the mechanisms underlying proarrhythmic responses in the heart is crucial for developing new ways to prevent and control cardiac arrhythmias and resulting contractile dysfunction. When the heart changes to a pathological state over time, the action potential (AP) in cardiomyocytes may also change to a different state in shape and duration, often undergoing a qualitative change in behavior. Such a dynamic change is called bifurcation. In this review, we first summarize the contribution of ion channels and transporters to AP formation and our knowledge of ion-transport molecules, then briefly describe bifurcation theory for nonlinear dynamical systems, and finally detail its recent progress, focusing on the research that attempts to understand the developing mechanisms of abnormal excitations in cardiomyocytes from the perspective of bifurcation phenomena.

Multi-Scale Computational Modeling of Spatial Calcium Handling From Nanodomain to Whole-Heart: Overview and Perspectives

Regulation of intracellular calcium is a critical component of cardiac electrophysiology and excitation-contraction coupling. The calcium spark, the fundamental element of the intracellular calcium transient, is initiated in specialized nanodomains which co-locate the ryanodine receptors and L-type calcium channels. However, calcium homeostasis is ultimately regulated at the cellular scale, by the interaction of spatially separated but diffusively coupled nanodomains with other sub-cellular and surface-membrane calcium transport channels with strong non-linear interactions; and cardiac electrophysiology and arrhythmia mechanisms are ultimately tissue-scale phenomena, regulated by the interaction of a heterogeneous population of coupled myocytes. Recent advances in imaging modalities and image-analysis are enabling the super-resolution reconstruction of the structures responsible for regulating calcium homeostasis, including the internal structure of nanodomains themselves. Extrapolating functional and imaging data from the nanodomain to the whole-heart is non-trivial, yet essential for translational insight into disease mechanisms. Computational modeling has important roles to play in relating structural and functional data at the sub-cellular scale and translating data across the scales. This review covers recent methodological advances that enable image-based modeling of the single nanodomain and whole cardiomyocyte, as well as the development of multi-scale simulation approaches to integrate data from nanometer to whole-heart. Firstly, methods to overcome the computational challenges of simulating spatial calcium dynamics in the nanodomain are discussed, including image-based modeling at this scale. Then, recent whole-cell models, capable of capturing a range of different structures (such as the T-system and mitochondria) and cellular heterogeneity/variability are discussed at two different levels of discretization. Novel methods to integrate the models and data across the scales and simulate stochastic dynamics in tissue-scale models are then discussed, enabling elucidation of the mechanisms by which nanodomain remodeling underlies arrhythmia and contractile dysfunction. Perspectives on model differences and future directions are provided throughout.

Understanding the origins of the basic equations of statistical fibrillatory dynamics

The mechanisms governing cardiac fibrillation remain unclear; however, it most likely represents a form of spatiotemporal chaos with conservative system dynamics. Renewal theory has recently been suggested as a statistical formulation with governing equations to quantify the formation and destruction of wavelets and rotors in fibrillatory dynamics. In this perspective Review, we aim to explain the origin of the renewal theory paradigm in spatiotemporal chaos. The ergodic nature of pattern formation in spatiotemporal chaos is demonstrated through the use of three chaotic systems: two classical systems and a simulation of cardiac fibrillation. The logistic map and the baker's transformation are used to demonstrate how the apparently random appearance of patterns in classical chaotic systems has macroscopic parameters that are predictable in a statistical sense. We demonstrate that the renewal theory approach developed for cardiac fibrillation statistically predicts pattern formation in these classical chaotic systems. Renewal theory provides governing equations to describe the apparently random formation and destruction of wavelets and rotors in atrial fibrillation (AF) and ventricular fibrillation (VF). This statistical framework for fibrillatory dynamics provides a holistic understanding of observed rotor and wavelet dynamics and is of conceptual significance in informing the clinical and mechanistic research of the rotor and multiple-wavelet mechanisms of AF and VF.

Voltage-mediated mechanism for calcium wave synchronization and arrhythmogenesis in atrial tissue

A wide range of atrial arrythmias are caused by molecular defects in proteins that regulate calcium (Ca) cycling. In many cases, these defects promote the propagation of subcellular Ca waves in the cell, which can perturb the voltage time course and induce dangerous perturbations of the action potential (AP). However, subcellular Ca waves occur randomly in cells and, therefore, electrical coupling between cells substantially decreases their effect on the AP. In this study, we present evidence that Ca waves in atrial tissue can synchronize in-phase owing to an order-disorder phase transition. In particular, we show that, below a critical pacing rate, Ca waves are desynchronized and therefore do not induce substantial AP fluctuations in tissue. However, above this critical pacing rate, Ca waves gradually synchronize over millions of cells, which leads to a dramatic amplification of AP fluctuations. We exploit an underlying Ising symmetry of paced cardiac tissue to show that this transition exhibits universal properties common to a wide range of physical systems in nature. Finally, we show that in the heart, phase synchronization induces spatially out-of-phase AP duration alternans which drives wave break and reentry. These results suggest that cardiac tissue exhibits a phase transition that is required for subcellular Ca cycling defects to induce a life-threatening arrhythmia.

Creating Ion Channel Kinetic Models Using Cloud Computing

Computational modeling of ion channels provides key insight into experimental electrophysiology results and can be used to connect channel dynamics to emergent phenomena observed at the tissue and organ levels. However, creation of these models requires substantial mathematical and computational background. This tutorial seeks to lower the barrier to creating these models by providing an automated pipeline for creating Markov models of an ion channel kinetics dataset. We start by detailing how to encode sample voltage-clamp protocols and experimental data into the program and its implementation in a cloud computing environment. We guide the reader on how to build a containerized instance, push the machine image, and finally run the routine on cluster nodes. While providing open-source code has become more standard in computational studies, this tutorial provides unprecedented detail on the use of the program and the creation of channel models, starting from inputting the raw experimental data. © 2022 Wiley Periodicals LLC. Basic Protocol: Creation of ion channel kinetic models with a cloud computing environment Alternate Protocol: Instructions for use in a standard high-performance compute cluster.

Deriving the Bidomain Model of Cardiac Electrophysiology From a Cell-Based Model; Properties and Comparisons

The bidomain model is considered to be the gold standard for numerical simulation of the electrophysiology of cardiac tissue. The model provides important insights into the conduction properties of the electrochemical wave traversing the cardiac muscle in every heartbeat. However, in normal resolution, the model represents the average over a large number of cardiomyocytes, and more accurate models based on representations of all individual cells have therefore been introduced in order to gain insight into the conduction properties close to the myocytes. The more accurate model considered here is referred to as the EMI model since both the extracellular space (E), the cell membrane (M) and the intracellular space (I) are explicitly represented in the model. Here, we show that the bidomain model can be derived from the cell-based EMI model and we thus reveal the close relation between the two models, and obtain an indication of the error introduced in the approximation. Also, we present numerical simulations comparing the results of the two models and thereby highlight both similarities and differences between the models. We observe that the deviations between the solutions of the models become larger for larger cell sizes. Furthermore, we observe that the bidomain model provides solutions that are very similar to the EMI model when conductive properties of the tissue are in the normal range, but large deviations are present when the resistance between cardiomyocytes is increased.

Finding type and location of the source of cardiac arrhythmias from the averaged flow velocity field using the determinant-trace method

Life threatening cardiac arrhythmias result from abnormal propagation of nonlinear electrical excitation waves in the heart. Finding the locations of the sources of these waves remains a challenging problem. This is mainly due to the low spatial resolution of electrode recordings of these waves. Also, these recordings are subjected to noise. In this paper, we develop a different approach: the AFV-DT method based on an averaged flow velocity (AFV) technique adopted from the analysis of optical flows and the determinant-trace (DT) method used for vector field analysis of dynamical systems. This method can find the location and determine all important types of sources found in excitable media such as focal activity, spiral waves, and waves rotating around obstacles. We test this method on in silico data of various wave excitation patterns obtained using the Luo-Rudy model for cardiac tissue. We show that the method works well for data with low spatial resolutions (up to 8×8) and is stable against noise. Finally, we apply it to two clinical cases and show that it can correctly identify the arrhythmia type and location. We discuss further steps on the development and improvement of this approach.

Mitochondrial Contributions in the Genesis of Delayed Afterdepolarizations in Ventricular Myocytes

Mitochondria fulfill the cell's energy demand and affect the intracellular calcium (Ca2+) dynamics via direct Ca2+ exchange, the redox effect of reactive oxygen species (ROS) on Ca2+ handling proteins, and other signaling pathways. Recent experimental evidence indicates that mitochondrial depolarization promotes arrhythmogenic delayed afterdepolarizations (DADs) in cardiac myocytes. However, the nonlinear interactions among the Ca2+ signaling pathways, ROS, and oxidized Ca2+/calmodulin-dependent protein kinase II (CaMKII) pathways make it difficult to reveal the mechanisms. Here, we use a recently developed spatiotemporal ventricular myocyte computer model, which consists of a 3-dimensional network of Ca2+ release units (CRUs) intertwined with mitochondria and integrates mitochondrial Ca2+ signaling and other complex signaling pathways, to study the mitochondrial regulation of DADs. With a systematic investigation of the synergistic or competing factors that affect the occurrence of Ca2+ waves and DADs during mitochondrial depolarization, we find that the direct redox effect of ROS on ryanodine receptors (RyRs) plays a critical role in promoting Ca2+ waves and DADs under the acute effect of mitochondrial depolarization. Furthermore, the upregulation of mitochondrial Ca2+ uniporter can promote DADs through Ca2+-dependent opening of mitochondrial permeability transition pores (mPTPs). Also, due to much slower dynamics than Ca2+ cycling and ROS, oxidized CaMKII activation and the cytosolic ATP do not appear to significantly impact the genesis of DADs during the acute phase of mitochondrial depolarization. However, under chronic conditions, ATP depletion suppresses and enhanced CaMKII activation promotes Ca2+ waves and DADs.

Pulsed low-energy stimulation initiates electric turbulence in cardiac tissue

Interruptions in nonlinear wave propagation, commonly referred to as wave breaks, are typical of many complex excitable systems. In the heart they lead to lethal rhythm disorders, the so-called arrhythmias, which are one of the main causes of sudden death in the industrialized world. Progress in the treatment and therapy of cardiac arrhythmias requires a detailed understanding of the triggers and dynamics of these wave breaks. In particular, two very important questions are: 1) What determines the potential of a wave break to initiate re-entry? and 2) How do these breaks evolve such that the system is able to maintain spatiotemporally chaotic electrical activity? Here we approach these questions numerically using optogenetics in an in silico model of human atrial tissue that has undergone chronic atrial fibrillation (cAF) remodelling. In the lesser studied sub-threshold illumination régime, we discover a new mechanism of wave break initiation in cardiac tissue that occurs for gentle slopes of the restitution characteristics. This mechanism involves the creation of conduction blocks through a combination of wavefront-waveback interaction, reshaping of the wave profile and heterogeneous recovery from the excitation of the spatially extended medium, leading to the creation of re-excitable windows for sustained re-entry. This finding is an important contribution to cardiac arrhythmia research as it identifies scenarios in which low-energy perturbations to cardiac rhythm can be potentially life-threatening.

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.

Reconceptualising Atrial Fibrillation Using Renewal Theory: A Novel Approach to the Assessment of Atrial Fibrillation Dynamics

Despite a century of research, the mechanisms of AF remain unresolved. A universal motif within AF research has been unstable re-entry, but this remains poorly characterised, with competing key conceptual paradigms of multiple wavelets and more driving rotors. Understanding the mechanisms of AF is clinically relevant, especially with regard to treatment and ablation of the more persistent forms of AF. Here, the authors outline the surprising but reproducible finding that unstable re-entrant circuits are born and destroyed at quasi-stationary rates, a finding based on a branch of mathematics known as renewal theory. Renewal theory may be a way to potentially unify the multiple wavelet and rotor theories. The renewal rate constants are potentially attractive because they are temporally stable parameters of a defined probability distribution (the exponential distribution) and can be estimated with precision and accuracy due to the principles of renewal theory. In this perspective review, this new representational architecture for AF is explained and placed into context, and the clinical and mechanistic implications are discussed.

A computational method for identifying an optimal combination of existing drugs to repair the action potentials of SQT1 ventricular myocytes

Mutations are known to cause perturbations in essential functional features of integral membrane proteins, including ion channels. Even restricted or point mutations can result in substantially changed properties of ion currents. The additive effect of these alterations for a specific ion channel can result in significantly changed properties of the action potential (AP). Both AP shortening and AP prolongation can result from known mutations, and the consequences can be life-threatening. Here, we present a computational method for identifying new drugs utilizing combinations of existing drugs. Based on the knowledge of theoretical effects of existing drugs on individual ion currents, our aim is to compute optimal combinations that can ‘repair’ the mutant AP waveforms so that the baseline AP-properties are restored. More specifically, we compute optimal, combined, drug concentrations such that the waveforms of the transmembrane potential and the cytosolic calcium concentration of the mutant cardiomyocytes (CMs) becomes as similar as possible to their wild type counterparts after the drug has been applied. In order to demonstrate the utility of this method, we address the question of computing an optimal drug for the short QT syndrome type 1 (SQT1). For the SQT1 mutation N588K, there are available data sets that describe the effect of various drugs on the mutated K⁺ channel. These published findings are the basis for our computational analysis which can identify optimal compounds in the sense that the AP of the mutant CMs resembles essential biomarkers of the wild type CMs. Using recently developed insights regarding electrophysiological properties among myocytes from different species, we compute optimal drug combinations for hiPSC-CMs, rabbit ventricular CMs and adult human ventricular CMs with the SQT1 mutation. Since the ‘composition’ of ion channels that form the AP is different for the three types of myocytes under consideration, so is the composition of the optimal drug.

Poly-pharmacology (using multiple drugs to treat disease) has been proposed for improving cardiac anti-arrhythmic therapy for at least two decades. However, the specific arrhythmia contexts in which polytherapy is likely to be both safe and effective have remained elusive. Type 1 short QT syndrome (SQT1) is a rare form of cardiac arrhythmia that results from mutations to the human Ether-á-go-go Related Gene (hERG) potassium channel. Functionally, these mutations are remarkably consistent in that they permit the channel to open earlier during each heart beat. While hundreds of compounds are known to inhibit hERG channels, the specific effect of SQT1 mutations that allows for early channel opening also limits the ability of most of those compounds to correct SQT1 dysfunction. Here, we have applied a suite of ventricular cardiomyocyte computational models to ask whether polytherapy may offer a more effective therapeutic strategy in SQT1, and if so, what the likely characteristics of that strategy are. Our analyses suggest that simultaneous induction of late sodium current and partial hERG blockade offers a promising strategy. While no activators of late sodium current have been clinically approved, several experimental compounds are available and may provide a basis for interrogating this strategy. The method presented here can be used to compute optimal drug combinations provided that the effect of each drug on every relevant ion channel is known.

A space-fractional bidomain framework for cardiac electrophysiology: 1D alternans dynamics

Cardiac electrophysiology modeling deals with a complex network of excitable cells forming an intricate syncytium: the heart. The electrical activity of the heart shows recurrent spatial patterns of activation, known as cardiac alternans, featuring multiscale emerging behavior. On these grounds, we propose a novel mathematical formulation for cardiac electrophysiology modeling and simulation incorporating spatially non-local couplings within a physiological reaction-diffusion scenario. In particular, we formulate, a space-fractional electrophysiological framework, extending and generalizing similar works conducted for the monodomain model. We characterize one-dimensional excitation patterns by performing an extended numerical analysis encompassing a broad spectrum of space-fractional derivative powers and various intra- and extracellular conductivity combinations. Our numerical study demonstrates that (i) symmetric properties occur in the conductivity parameters' space following the proposed theoretical framework, (ii) the degree of non-local coupling affects the onset and evolution of discordant alternans dynamics, and (iii) the theoretical framework fully recovers classical formulations and is amenable for parametric tuning relying on experimental conduction velocity and action potential morphology.

Dynamical analysis of early afterdepolarization patterns in a biophysically detailed cardiac model

Arrhythmogenic early afterdepolarizations (EADs) are investigated in a biophysically detailed mathematical model of a rabbit ventricular myocyte, providing their location in the parameter phase space and describing their dynamical mechanisms. Simulations using the Sato model, defined by 27 state variables and 177 parameters, are conducted to generate electrical action potentials (APs) for different values of the pacing cycle length and other parameters related to sodium and calcium concentrations. A detailed study of the different AP patterns with or without EADs is carried out, showing the presence of a high variety of temporal AP configurations with chaotic and quasiperiodic behaviors. Regions of bistability are identified and, importantly, linked to transitions between different behaviors. Using sweeping techniques, one-, two-, and three-parameter phase spaces are provided, allowing ascertainment of the role of the selected parameters as well as location of the transition regions. A Devil's staircase, with symbolic sequence analysis, is proposed to describe transitions in the ratio between the number of voltage (EAD and AP) peaks and the number of APs. To conclude, the obtained results are linked to recent studies for low-dimensional models and a conjecture is made for the internal dynamical structure of the transition region from non-EAD to EAD behavior using fold and cusp bifurcations and maximal canards.

Modelling the cardiac response to a mechanical stimulation using a low-order model of the heart

Heart diseases are one of the leading causes of death worldwide, and a dysfunction of the cardiac electrical mechanisms is responsible for a significant portion of these deaths. One of these mechanisms, the mechano-electric feedback (MEF), is the electrical response of the heart to local mechanical changes in the environment. This electrical response, in turn, leads to macroscopic changes in heart function. In this paper, we demonstrate that the MEF plays a crucial role in mechanical generation and recovery from arrhythmia which has been observed in experimental studies. To this end, we investigate the cardiac response to a mechanical stimulation using a minimal, multiscale model of the heart which couples the organ level dynamics (left ventricular pressure and volume) and contractile dynamics. By including a mechanical stimulation into the model as a (short, sudden) impulse in the muscle microscale stress, we investigate how the timing, amplitude and duration of the impulse affect the cardiac cycle. In particular, when introduced in the diastolic period of the cardiac cycle, the pulse rate can be stabilised, and ectopic beats and bifurcation can be eliminated, either temporarily or permanently. The stimulation amplitude is a key indicator to this response. We find an optimal value of the impulse amplitude above or below which the impulse maximises the stabilisation. As a result a dysfunction of the MEF can be helped using a mechanical stimulation, by allowing the heart to recover its pumping power. On the other hand, when the mechanical stimulation is introduced towards the end of systole, arrhythmia can be generated.

A three-compartment non-linear model of myocardial cell conduction block during photosensitization

This study constructed a new non-linear model of myocardial electrical conduction block during photosensitization reaction to identify the vulnerable cell population and generate an index for recurrent risk following catheter ablation for tachyarrhythmia. A three-compartment model of conductive, vulnerable, and blocked cells was proposed. To determine the non-linearity of the rate parameter for the change from vulnerable cells to conductive cells, we compared a previously reported non-linear model and our newly proposed model with non-linear rate parameters in the modeling of myocardial cell electrical conduction block during photosensitization reaction. The rate parameters were optimized via a bi-nested structure using measured synchronicity data during the photosensitization reaction of myocardial cell wires. The newly proposed model had a better fit to the measured data than the conventional model. The sum of the error until the time where the measured value was higher than 0.6, was 0.22 in the conventional model and 0.07 in our new model. The non-linear rate parameter from the vulnerable cell to the conductive cell compartment may be the preferred structure of the electrical conduction block model induced by photosensitization reaction. This simulation model provides an index to evaluate recurrent risk after tachyarrhythmia catheter ablation by photosensitization reaction. A three-compartment non-linear model of myocardial cell conduction block during photosensitization.

Segmentation and Removal of Surgical Instruments for Background Scene Visualization from Endoscopic / Laparoscopic Video

Surgical tool segmentation is becoming imperative to provide detailed information during intra-operative execution. These tools can obscure surgeons' dexterity control due to narrow working space and visual field-of-view, which increases the risk of complications resulting from tissue injuries (e.g. tissue scars and tears). This paper demonstrates a novel application of segmenting and removing surgical instruments from laparoscopic/endoscopic video using digital inpainting algorithms. To segment the surgical instruments, we use a modified U-Net architecture (U-NetPlus) composed of a pre-trained VGG11 or VGG16 encoder and redesigned decoder. The decoder is modified by replacing the transposed convolution operation with an up-sampling operation based on nearest-neighbor (NN) interpolation. This modification removes the artifacts generated by the transposed convolution, and, furthermore, these new interpolation weights require no learning for the upsampling operation. The tool removal algorithms use the tool segmentation mask and either the instrument-free reference frames or previous instrument-containing frames to fill-in (i.e., inpaint) the instrument segmentation mask with the background tissue underneath. We have demonstrated the performance of the proposed surgical tool segmentation/removal algorithms on a robotic instrument dataset from the MICCAI 2015 EndoVis Challenge. We also showed successful performance of the tool removal algorithm from synthetically generated surgical instruments-containing videos obtained by embedding a moving surgical tool into surgical tool-free videos. Our application successfully segments and removes the surgical tool to unveil the background tissue view otherwise obstructed by the tool, producing visually comparable results to the ground truth.

Mitochondrial depolarization promotes calcium alternans: Mechanistic insights from a ventricular myocyte model

Mitochondria are vital organelles inside the cell and contribute to intracellular calcium (Ca2+) dynamics directly and indirectly via calcium exchange, ATP generation, and production of reactive oxygen species (ROS). Arrhythmogenic Ca2+ alternans in cardiac myocytes has been observed in experiments under abnormal mitochondrial depolarization. However, complex signaling pathways and Ca2+ cycling between mitochondria and cytosol make it difficult in experiments to reveal the underlying mechanisms of Ca2+ alternans under abnormal mitochondrial depolarization. In this study, we use a newly developed spatiotemporal ventricular myocyte computer model that integrates mitochondrial Ca2+ cycling and complex signaling pathways to investigate the mechanisms of Ca2+ alternans during mitochondrial depolarization. We find that elevation of ROS in response to mitochondrial depolarization plays a critical role in promoting Ca2+ alternans. Further examination reveals that the redox effect of ROS on ryanodine receptors and sarco/endoplasmic reticulum Ca2+-ATPase synergistically promote alternans. Upregulation of mitochondrial Ca2+ uniporter promotes Ca2+ alternans via Ca2+-dependent mitochondrial permeability transition pore opening. Due to their relatively slow kinetics, oxidized Ca2+/calmodulin-dependent protein kinase II activation and ATP do not play significant roles acutely in the genesis of Ca2+ alternans after mitochondrial depolarization, but their roles can be significant in the long term, mainly through their effects on sarco/endoplasmic reticulum Ca2+-ATPase activity. In conclusion, mitochondrial depolarization promotes Ca2+ alternans acutely via the redox effect of ROS and chronically by ATP reduction. It suppresses Ca2+ alternans chronically through oxidized Ca2+/calmodulin-dependent protein kinase II activation.

Stability of spatially discordant repolarization alternans in cardiac tissue

Cardiac alternans, a period-2 behavior of excitation and contraction of the heart, is a precursor of ventricular arrhythmias and sudden cardiac death. One form of alternans is repolarization or action potential duration alternans. In cardiac tissue, repolarization alternans can be spatially in-phase, called spatially concordant alternans, or spatially out-of-phase, called spatially discordant alternans (SDA). In SDA, the border between two out-of-phase regions is called a node in a one-dimensional cable or a nodal line in a two-dimensional tissue. In this study, we investigate the stability and dynamics of the nodes and nodal lines of repolarization alternans driven by voltage instabilities. We use amplitude equation and coupled map lattice models to derive theoretical results, which are compared with simulation results from the ionic model. Both conduction velocity restitution induced SDA and non-conduction velocity restitution induced SDA are investigated. We show that the stability and dynamics of the SDA nodes or nodal lines are determined by the balance of the tensions generated by conduction velocity restitution, convection due to action potential propagation, curvature of the nodal lines, and repolarization and coupling heterogeneities. Our study provides mechanistic insights into the different SDA behaviors observed in experiments.

Multiplex recurrence networks from multi-lead ECG data

We present an integrated approach to analyze the multi-lead electrocardiogram (ECG) data using the framework of multiplex recurrence networks (MRNs). We explore how their intralayer and interlayer topological features can capture the subtle variations in the recurrence patterns of the underlying spatio-temporal dynamics of the cardiac system. We find that MRNs from ECG data of healthy cases are significantly more coherent with high mutual information and less divergence between respective degree distributions. In cases of diseases, significant differences in specific measures of similarity between layers are seen. The coherence is affected most in the cases of diseases associated with localized abnormality such as bundle branch block. We note that it is important to do a comprehensive analysis using all the measures to arrive at disease-specific patterns. Our approach is very general and as such can be applied in any other domain where multivariate or multi-channel data are available from highly complex systems.

Using mathematics to diagnose, cure, and predict cardiac arrhythmia

Mathematics can be used to analyze and model cardiac arrhythmia. I discuss three different problems. (1) Diagnosis of atrial fibrillation based on the time intervals between subsequent beats. The probability density histograms of the differences of the intervals between consecutive beats have characteristic shapes for atrial fibrillation. (2) Curing atrial fibrillation by ablation of the core of rotors. Recent clinical studies have proposed that ablating the core of rotors in atrial tissue can cure atrial fibrillation. However, the claims are controversial. One problem that arises relates to difficulties associated with developing algorithms to identify the core of rotors. In model tissue culture systems, heterogeneity in the structure makes it difficult to unambiguously locate the core of rotors. (3) Risk stratification for sudden cardiac death (SCD). Despite numerous clinical studies, there is still a need for improved criteria to assess the risk of SCD. I discuss the possibility of using the dynamics of premature ventricular complexes to help make predictions. The development of wearable devices to record and analyze cardiac rhythms offers new prospects for the diagnosis and treatment of cardiac arrhythmia.

Fractal Correlation Properties of Heart Rate Variability: A New Biomarker for Intensity Distribution in Endurance Exercise and Training Prescription?

Exercise and training prescription in endurance-type sports has a strong theoretical background with various practical applications based on threshold concepts. Given the challenges and pitfalls of determining individual training zones on the basis of subsystem indicators (e.g., blood lactate concentration, respiratory parameters), the question arises whether there are alternatives for intensity distribution demarcation. Considering that training in a low intensity zone substantially contributes to the performance outcome of endurance athletes and exceeding intensity targets based on a misleading aerobic threshold can lead to negative performance and recovery effects, it would be desirable to find a parameter that could be derived via non-invasive, low cost and commonly available wearable devices. In this regard, analytics conducted from non-linear dynamics of heart rate variability (HRV) have been adapted to gain further insights into the complex cardiovascular regulation during endurance-type exercise. Considering the reciprocal antagonistic behavior and the interaction of the sympathetic and parasympathetic branch of the autonomic nervous system from low to high exercise intensities, it may be promising to use an approach that utilizes information about the regulation quality of the organismic system to determine training-intensity distribution. Detrended fluctuation analysis of HRV and its short-term scaling exponent alpha1 (DFA-alpha1) seems suitable for applied sport-specific settings including exercise from low to high intensities. DFA-alpha1 may be taken as an indicator for exercise prescription and intensity distribution monitoring in endurance-type sports. The present perspective illustrates the potential of DFA-alpha1 for diagnostic and monitoring purposes as a “global” system parameter and proxy for organismic demands.

Multi-scale Entropy Evaluates the Proarrhythmic Condition of Persistent Atrial Fibrillation Patients Predicting Early Failure of Electrical Cardioversion

Atrial fibrillation (AF) is nowadays the most common cardiac arrhythmia, being associated with an increase in cardiovascular mortality and morbidity. When AF lasts for more than seven days, it is classified as persistent AF and external interventions are required for its termination. A well-established alternative for that purpose is electrical cardioversion (ECV). While ECV is able to initially restore sinus rhythm (SR) in more than 90% of patients, rates of AF recurrence as high as 20-30% have been found after only a few weeks of follow-up. Hence, new methods for evaluating the proarrhythmic condition of a patient before the intervention can serve as efficient predictors about the high risk of early failure of ECV, thus facilitating optimal management of AF patients. Among the wide variety of predictors that have been proposed to date, those based on estimating organization of the fibrillatory (f-) waves from the surface electrocardiogram (ECG) have reported very promising results. However, the existing methods are based on traditional entropy measures, which only assess a single time scale and often are unable to fully characterize the dynamics generated by highly complex systems, such as the heart during AF. The present work then explores whether a multi-scale entropy (MSE) analysis of the f-waves may provide early prediction of AF recurrence after ECV. In addition to the common MSE, two improved versions have also been analyzed, composite MSE (CMSE) and refined MSE (RMSE). When analyzing 70 patients under ECV, of which 31 maintained SR and 39 relapsed to AF after a four week follow-up, the three methods provided similar performance. However, RMSE reported a slightly better discriminant ability of 86%, thus improving the other multi-scale-based outcomes by 3-9% and other previously proposed predictors of ECV by 15-30%. This outcome suggests that investigation of dynamics at large time scales yields novel insights about the underlying complex processes generating f-waves, which could provide individual proarrhythmic condition estimation, thus improving preoperative predictions of ECV early failure.