Nonlinear-dynamical arrhythmia control in humans

A training pipeline of an arrhythmia classifier for atrial fibrillation detection using Photoplethysmography signal

Photoplethysmography (PPG) signal is potentially suitable in atrial fibrillation (AF) detection for its convenience in use and similarity in physiological origin to electrocardiogram (ECG). There are a few preceding studies that have shown the possibility of using the peak-to-peak interval of the PPG signal (PPIp) in AF detection. However, as a generalized model, the accuracy of an AF detector should be pursued on the one hand; on the other hand, its generalizability should be paid attention to in view of the individual differences in PPG manifestation of even the same arrhythmia and the existence of sub-types. Moreover, a binary classifier for atrial fibrillation and normal sinus rhythm is not convincing enough for the similarity between AF and ectopic beats. In this study, we project the atrial fibrillation detection as a multiple-class classification and try to propose a training pipeline that is advantageous both to the accuracy and generalizability of the classifier by designing and determining the configurable options of the pipeline, in terms of input format, deep learning model (with hyperparameter optimization), and scheme of transfer learning. With a rigorous comparison of the possible combinations of the configurable components in the pipeline, we confirmed that first-order difference of heartbeat sequence as the input format, a 2-layer CNN-1-layer Transformer hybrid^R model as the learning model and the whole model fine-tuning as the implementing scheme of transfer learning is the best combination for the pipeline (F1 value: 0.80, overall accuracy: 0.87)^R.

Interpretable machine learning of action potential duration restitution kinetics in single-cell models of atrial cardiomyocytes

Action potential duration (APD) restitution curve and its maximal slope (Smax) reflect single cell-level dynamic instability for inducing chaotic heart rhythms. However, conventional parameter sensitivity analysis often fails to describe nonlinear relationships between ion channel parameters and electrophysiological phenotypes, such as Smax. We explored the parameter-phenotype mapping in a population of 5000 single-cell atrial cell models through interpretable machine learning (ML) approaches. Parameter sensitivity analyses could explain the linear relationships between parameters and electrophysiological phenotypes, including APD₉₀, resting membrane potential, Vmax, refractory period, and APD/calcium alternans threshold, but not for Smax. However, neural network models had better prediction performance for Smax. To interpret the ML model, we evaluated the parameter importance at the global and local levels by computing the permutation feature importance and the local interpretable model-agnostic explanations (LIME) values, respectively. Increases in I_CaL, I_NCX, and I_Kr, and decreases in I_K1, I_b,Cl, I_Kur, I_SERCA, and I_to are correlated with higher Smax values. The LIME algorithm determined that I_NaK plays a significant role in determining Smax as well as I_to and I_Kur. The atrial cardiomyocyte population was hierarchically clustered into three distinct groups based on the LIME values and the single-cell simulation confirmed that perturbations in I_NaK resulted in different behaviors of APD restitution curves in three clusters. Our combined top-down interpretable ML and bottom-up mechanistic simulation approaches uncovered the role of I_NaK in heterogeneous behaviors of Smax in the atrial cardiomyocyte population.

Advances in Cardiac Pacing: Arrhythmia Prediction, Prevention and Control Strategies

Cardiac arrhythmias constitute a tremendous burden on healthcare and are the leading cause of mortality worldwide. An alarming number of people have been reported to manifest sudden cardiac death as the first symptom of cardiac arrhythmias, accounting for about 20% of all deaths annually. Furthermore, patients prone to atrial tachyarrhythmias such as atrial flutter and fibrillation often have associated comorbidities including hypertension, ischemic heart disease, valvular cardiomyopathy and increased risk of stroke. Technological advances in electrical stimulation and sensing modalities have led to the proliferation of medical devices including pacemakers and implantable defibrillators, aiming to restore normal cardiac rhythm. However, given the complex spatiotemporal dynamics and non-linearity of the human heart, predicting the onset of arrhythmias and preventing the transition from steady state to unstable rhythms has been an extremely challenging task. Defibrillatory shocks still remain the primary clinical intervention for lethal ventricular arrhythmias, yet patients with implantable cardioverter defibrillators often suffer from inappropriate shocks due to false positives and reduced quality of life. Here, we aim to present a comprehensive review of the current advances in cardiac arrhythmia prediction, prevention and control strategies. We provide an overview of traditional clinical arrhythmia management methods and describe promising potential pacing techniques for predicting the onset of abnormal rhythms and effectively suppressing cardiac arrhythmias. We also offer a clinical perspective on bridging the gap between basic and clinical science that would aid in the assimilation of promising anti-arrhythmic pacing strategies.

Controllability and state feedback control of a cardiac ionic cell model

A phenomenon called alternans, which is a beat-to-beat alternation in action potential (AP) duration, sometimes precedes fatal cardiac arrhythmias. Alternans-suppressing electrical stimulus protocols are often represented as perturbations to the dynamics of membrane potential or AP duration variables in nonlinear models of cardiac tissue. Controllability analysis has occasionally been applied to cardiac AP models to determine whether different control or perturbation strategies are capable of suppressing alternans or other unwanted behavior. Since almost all previous cardiac controllability studies have focused on low-dimensional models, we conducted the present study to assess controllability of a higher-dimensional model, specifically the Luo Rudy dynamic (LRd) model of a cardiac ventricular myocyte. Higher-dimensional models are of interest because they provide information on the influence of a wider range of measurable quantities, including ionic concentrations, on controllability. After computing modal controllability measures, we found that larger eigenvalues of a linearized LRd model were on average more strongly controllable through perturbations to calcium-ion concentrations compared with perturbations to other variables. When only membrane potential was adjusted, the best time to apply perturbations (in the sense of maximizing controllability of the largest alternans eigenvalue) was near the AP peak time for shorter cycle lengths. Controllability results were found to be similar for both the default model parameters and for an alternans-promoting parameter set. Additionally, we developed several alternans-suppressing state feedback controllers that were tested in simulations. For the scenarios examined, our controllability measures correctly predicted which strategies and perturbation timings would lead to better feedback controller performance.

Controlling nonlinear dynamical systems into arbitrary states using machine learning

Controlling nonlinear dynamical systems is a central task in many different areas of science and engineering. Chaotic systems can be stabilized (or chaotified) with small perturbations, yet existing approaches either require knowledge about the underlying system equations or large data sets as they rely on phase space methods. In this work we propose a novel and fully data driven scheme relying on machine learning (ML), which generalizes control techniques of chaotic systems without requiring a mathematical model for its dynamics. Exploiting recently developed ML-based prediction capabilities, we demonstrate that nonlinear systems can be forced to stay in arbitrary dynamical target states coming from any initial state. We outline and validate our approach using the examples of the Lorenz and the Rössler system and show how these systems can very accurately be brought not only to periodic, but even to intermittent and different chaotic behavior. Having this highly flexible control scheme with little demands on the amount of required data on hand, we briefly discuss possible applications ranging from engineering to medicine.

The feasibility of predicting impending malignant ventricular arrhythmias by using nonlinear features of short heartbeat intervals

BACKGROUND AND OBJECTIVE

Malignant ventricular arrhythmias (MAs) occur unpredictably and lead to emergencies. A new approach that uses a timely tracking device e.g., photoplethysmogram (PPG) solely to predict MAs would be irreplaceably valuable and it is natural to expect the approach can predict the occurrence as early as possible.

METHOD

We assumed that with an appropriate metric based on signal complexity, the heartbeat interval time series (HbIs) can be used to manifest the intrinsic characteristics of the period immediately precedes the MAs (preMAs). The approach first characterizes the patterns of preMAs by a new complexity metric (the refined composite multi-scale entropy). The MAs detector is then constructed by checking the discriminability of the MAs against the sinus rhythm and other prevalent arrhythmias (atrial fibrillation and premature ventricular contraction) of three machine-learning models (SVM, Random Forest, and XGboost).

RESULTS

Two specifications are of interest: the length of the HbIs needed to delineate the preMAs patterns sufficiently (l_spec) and how long before the occurrence of MAs will the HbIs manifest specific patterns that are distinct enough to predict the impending MAs (t_spec). Our experimental results confirmed the best performance came from a Random-Forest model with an average precision of 99.99% and recall of 88.98% using a HbIs of 800 heartbeats (the l_spec), 108 seconds (the t_spec) before the occurrence of MAs.

CONCLUSION

By experimental validation of the unique pattern of the preMAs in HbIs and using it in the machine learning model, we showed the high possibility of MAs prediction in a broader circumstance, which may cover daily healthcare using the alternative sensor in HbIs monitoring. Therefore, this research is theoretically and practically significant in cardiac arrest prevention.

Controllability of voltage- and calcium-driven cardiac alternans in a map model

Certain cardiac arrhythmias are preceded by electrical alternans, a state characterized by beat-to-beat alternation in cellular action potential duration. Cardiac alternans may arise from different mechanisms including instabilities in voltage or intracellular calcium cycling. Although a number of techniques have been proposed to suppress alternans, these methods have mainly been tested using models that do not support calcium-driven alternans. Therefore, it is important to understand how control methods may perform when alternans is driven by instabilities in calcium cycling. In this study, we applied controllability analysis to a discrete map of alternans dynamics in a cardiac cell. We compared two different controllability measures to determine to what extent different control strategies could suppress alternans and tested these predictions using three feedback controllers. We found a modal controllability measure, unlike the minimum singular value of the controllability matrix, consistently indicated the control strategies requiring the least control effort and yielding the smallest closed-loop eigenvalue. In addition, action potential duration was identified as the most effective variable through which control can be applied, regardless of alternans mechanism, although sarcoplasmic reticulum calcium load was also useful for the calcium-driven alternans cases.

Effect of constant-DI pacing on single cell pacing dynamics

Cardiac alternans, beat-to-beat alternations in action potential duration, is a precursor to fatal arrhythmias such as ventricular fibrillation. Previous research has shown that voltage driven alternans can be suppressed by application of a constant diastolic interval (DI) pacing protocol. However, the effect of constant-DI pacing on cardiac cell dynamics and its interaction with the intracellular calcium cycle remains to be determined. Therefore, we aimed to examine the effects of constant-DI pacing on the dynamical behavior of a single-cell numerical model of cardiac action potential and the influence of voltage-calcium (V-Ca) coupling on it. Single cell dynamics were analyzed in the vicinity of the bifurcation point using a hybrid pacing protocol, a combination of constant-basic cycle length (BCL) and constant-DI pacing. We demonstrated that in a small region beneath the bifurcation point, constant-DI pacing caused the cardiac cell to remain alternans-free after switching to the constant-BCL pacing, thus introducing a region of bistability (RB). The size of the RB increased with stronger V-Ca coupling and was diminished with weaker V-Ca coupling. Overall, our findings demonstrate that the application of constant-DI pacing on cardiac cells with strong V-Ca coupling may induce permanent changes to cardiac cell dynamics increasing the utility of constant-DI pacing.

Control of voltage-driven instabilities in cardiac myocytes with memory

Sudden cardiac death is known to be associated with dynamical instabilities in the heart, and thus control of dynamical instabilities is considered a potential therapeutic strategy. Different control methods were developed previously, including time-delayed feedback pacing control and constant diastolic interval pacing control. Experimental, theoretical, and simulation studies have examined the efficacy of these control methods in stabilizing action potential dynamics. In this study, we apply these control methods to control complex action potential (AP) dynamics under two diseased conditions: early repolarization syndrome and long QT syndrome, in which voltage-driven instabilities occur in the presence of short-term cardiac memory. In addition, we also develop a feedback pacing method to stabilize these instabilities. We perform theoretical analyses using iterated map models and carry out numerical simulations of AP models. We show that under the normal condition where the memory effect is minimal, all three methods can effectively control the action potential duration (APD) dynamics. Under the two diseased conditions where the memory effect is exacerbated, constant diastolic pacing control is least effective, while the feedback pacing control is most effective. Under a very strong memory effect, all three methods fail to stabilize the voltage-driven instabilities. The failure of effective control is due to memory and the all-or-none AP dynamics which results in very steep changes in APD.

A computational framework for testing arrhythmia marker sensitivities to model parameters in functionally calibrated populations of atrial cells

Models of cardiac cell electrophysiology are complex non-linear systems which can be used to gain insight into mechanisms of cardiac dynamics in both healthy and pathological conditions. However, the complexity of cardiac models can make mechanistic insight difficult. Moreover, these are typically fitted to averaged experimental data which do not incorporate the variability in observations. Recently, building populations of models to incorporate inter- and intra-subject variability in simulations has been combined with sensitivity analysis (SA) to uncover novel ionic mechanisms and potentially clarify arrhythmogenic behaviors. We used the Koivumäki human atrial cell model to create two populations, representing normal Sinus Rhythm (nSR) and chronic Atrial Fibrillation (cAF), by varying 22 key model parameters. In each population, 14 biomarkers related to the action potential and dynamic restitution were extracted. Populations were calibrated based on distributions of biomarkers to obtain reasonable physiological behavior, and subjected to SA to quantify correlations between model parameters and pro-arrhythmia markers. The two populations showed distinct behaviors under steady state and dynamic pacing. The nSR population revealed greater variability, and more unstable dynamic restitution, as compared to the cAF population, suggesting that simulated cAF remodeling rendered cells more stable to parameter variation and rate adaptation. SA revealed that the biomarkers depended mainly on five ionic currents, with noted differences in sensitivities to these between nSR and cAF. Also, parameters could be selected to produce a model variant with no alternans and unaltered action potential morphology, highlighting that unstable dynamical behavior may be driven by specific cell parameter settings. These results ultimately suggest that arrhythmia maintenance in cAF may not be due to instability in cell membrane excitability, but rather due to tissue-level effects which promote initiation and maintenance of reentrant arrhythmia.

Constant DI pacing suppresses cardiac alternans formation in numerical cable models

Cardiac repolarization alternans describe the sequential alternation of the action potential duration (APD) and can develop during rapid pacing. In the ventricles, such alternans may rapidly turn into life risking arrhythmias under conditions of spatial heterogeneity. Thus, suppression of alternans by artificial pacing protocols, or alternans control, has been the subject of numerous theoretical, numerical, and experimental studies. Yet, previous attempts that were inspired by chaos control theories were successful only for a short spatial extent (<2 cm) from the pacing electrode. Previously, we demonstrated in a single cell model that pacing with a constant diastolic interval (DI) can suppress the formation of alternans at high rates of activation. We attributed this effect to the elimination of feedback between the pacing cycle length and the last APD, effectively preventing restitution-dependent alternans from developing. Here, we extend this idea into cable models to study the extent by which constant DI pacing can control alternans during wave propagation conditions. Constant DI pacing was applied to ventricular cable models of up to 5 cm, using human kinetics. Our results show that constant DI pacing significantly shifts the onset of both cardiac alternans and conduction blocks to higher pacing rates in comparison to pacing with constant cycle length. We also demonstrate that constant DI pacing reduces the propensity of spatially discordant alternans, a precursor of wavebreaks. We finally found that the protective effect of constant DI pacing is stronger for increased electrotonic coupling along the fiber in the sense that the onset of alternans is further shifted to higher activation rates. Overall, these results support the potential clinical applicability of such type of pacing in improving protocols of implanted pacemakers, in order to reduce the risk of life-threatening arrhythmias. Future research should be conducted in order to experimentally validate these promising results.

Dynamiceuticals: The Next Stage in Personalized Medicine

The surge in the interest in personalized medicine necessitates a corresponding rational approach for implementing such individualized therapies. Dynamiceuticals represents a natural extension of the Pharmaceutical and Electroceutical fields, where the precise determination of the dynamical regimes of the pathophysiology will guide to devise therapies that ameliorate the pathology in a well-controlled manner, thus being precisely tailored toward the implementation of individualized medicine. This approach foretells to lessen side-effects and achieve superior efficacy as compared with current trial-and-error or open-loop strategies. But does the current state of knowledge and technology allow this scheme to offer what it claims?

Migraine — New Perspectives from Chaos Theory

Converging from a number of disciplines, non-linear systems theory and in particular chaos theory offer new descriptive and prescriptive insights into physiological systems. This paper briefly reviews an approach to physiological systems from these perspectives and outlines how these concepts can be applied to the study of migraine. It suggests a wide range of potential applications including new approaches to classification, treatment and pathophysiological mechanisms. A hypothesis is developed that suggests that dysfunctional consequences can result from a mismatch between the complexity of the environment and the system that is seeking to regulate it and that the migraine phenomenon is caused by an incongruity between the complexity of mid brain sensory integration and cortical control networks. Chaos theory offers a new approach to the study of migraine that complements existing frameworks but may more accurately reflect underlying physiological mechanisms.

Controlling spatiotemporal chaos in active dissipative-dispersive nonlinear systems

We present an alternative methodology for the stabilization and control of infinite-dimensional dynamical systems exhibiting low-dimensional spatiotemporal chaos. We show that with an appropriate choice of time-dependent controls we are able to stabilize and/or control all stable or unstable solutions, including steady solutions, traveling waves (single and multipulse ones or bound states), and spatiotemporal chaos. We exemplify our methodology with the generalized Kuramoto-Sivashinsky equation, a paradigmatic model of spatiotemporal chaos, which is known to exhibit a rich spectrum of wave forms and wave transitions and a rich variety of spatiotemporal structures.

Control of cardiac alternans in an electromechanical model of cardiac tissue

Electrical alternations in cardiac action potential duration have been shown to be a precursor to arrhythmias and sudden cardiac death. Through the mechanism of excitation-contraction coupling, the presence of electrical alternans induces alternations in the heart muscle contractile activity. Also, contraction of cardiac tissue affects the process of cardiac electric wave propagation through the mechanism of the so-called mechanoelectrical feedback. Electrical excitation and contraction of cardiac tissue can be linked by an electromechanical model such as the Nash-Panfilov model. In this work, we explore the feasibility of suppressing cardiac alternans in the Nash-Panfilov model which is employed for small and large deformations. Several electrical pacing and mechanical perturbation feedback strategies are considered to demonstrate successful suppression of alternans on a one-dimensional cable. This is the first attempt to combine electrophysiologically relevant cardiac models of electrical wave propagation and contractility of cardiac tissue in a synergistic effort to suppress cardiac alternans. Numerical examples are provided to illustrate the feasibility and the effects of the proposed algorithms to suppress cardiac alternans.

Nonlinear and Stochastic Dynamics in the Heart

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

Dynamics of violence

RATIONALE, AIMS AND OBJECTIVES

Three behavioural models suggest different dynamic patterns of intimate partner violence (IPV). However, few studies permit assessment of IPV dynamics. The purpose of this study was to estimate the degree of non-linearity in daily violence between partners over a 3-month period, identify their specific dynamic patterns and determine whether measures of violence severity and dynamics are interrelated.

METHODS

From six primary care clinics, we enrolled 200 adult women who experienced violence in the previous month and asked them to complete daily telephone assessments of household environment, marital relationship and violence using Interactive Verbal Response. To assess non-linearity of violence, algorithmic complexity was measured by LZ complexity and lack of regularity was measured by approximate entropy. Lyapunov exponents and correlation dimension saturation were used to approximate dynamic patterns.

RESULTS

Of the 9618 daily reports, women reported experiencing abuse on 39% of days, while perpetrating violence themselves on 23% of days. Most (59%) displayed random dynamics, 30% showed chaotic and 12% showed periodic dynamics. All three measures of non-linearity consistently demonstrated non-linear patterns of violence. Using multivariate analysis of variance, neither episode severity for men or women showed significant differences across dynamic types, but chaotic dynamics had the lowest frequencies of violence in men and women while random dynamics had the highest frequencies. Approximate entropy was positively correlated with violence frequency and burden in men and women, but Lyapunov exponent was inversely related to violence. LZ complexity correlated positively with wife-perpetrated violence only.

CONCLUSIONS

IPV is rarely a predictable, periodic phenomenon; no behavioural model describes the violence dynamics for all violent relationships. Yet, the measures of non-linearity and specific dynamic patterns correlate with different violent features of these relationships.

A translational approach to probe the proarrhythmic potential of cardiac alternans: a reversible overture to arrhythmogenesis?

Electrocardiographic alternans, a phenomenon of beat-to-beat alternation in cardiac electrical waveforms, has been implicated in the pathogenesis of ventricular arrhythmias and sudden cardiac death (SCD). In the clinical setting, a positive microvolt T-wave alternans test has been associated with a heightened risk of arrhythmic mortality and SCD during medium- and long-term follow-up. However, rather than merely being associated with an increased risk for SCD, several lines of preclinical and clinical evidence suggest that cardiac alternans may play a causative role in generating the acute electrophysiological substrate necessary for the onset of ventricular arrhythmias. Deficiencies in Ca(2+) transport processes have been implicated in the genesis of alternans at the subcellular and cellular level and are hypothesized to contribute to the conditions necessary for dispersion of refractoriness, wave break, reentry, and onset of arrhythmia. As such, detecting acute surges in alternans may provide a mechanism for predicting the impending onset of arrhythmia and opens the door to delivering upstream antiarrhythmic therapies. In this review, we discuss the preclinical and clinical evidence to support a causative association between alternans and acute arrhythmogenesis and outline the potential clinical implications of such an association.

A novel pacing method to suppress repolarization alternans in vivo: implications for arrhythmia prevention

BACKGROUND

Repolarization alternans (RA), a pattern of ventricular repolarization that repeats on an every other beat basis, has been closely linked with the substrate associated with ventricular tachycardia/ventricular fibrillation.

OBJECTIVE

To evaluate a novel method to suppress RA.

METHODS

We have developed a novel method to dynamically (on R-wave detection) trigger pacing pulses during the absolute refractory period. We have tested the ability of this method to control RA in a structurally normal swine heart in vivo.

RESULTS

RA induced by triggered pacing can be measured from both intracardiac and body surface leads and the amplitude of R-wave triggered pacing-induced alternans can be locally modulated by varying the amplitude and width of the pacing pulse. We have estimated that to induce a 1 μV change in alternans voltage on the body surface, coronary sinus, and left ventricle leads, a triggered pacing pulse delivered in the right ventricle of 0.04±0.02, 0.05±0.025, and 0.06±0.033 μC, respectively, is required. Similarly, to induce a 1 unit change in Kscore (ratio of alternans peak to noise), a pacing stimulus of 0.93±0.73, 0.32±0.29, and 0.33±0.37 μC, respectively, is required. We have been able to demonstrate that RA can be suppressed by R-wave triggered pacing from a site that is within or across ventricles. Lastly, we have demonstrated that the proposed method can be used to suppress spontaneously occurring alternans in the diseased heart.

CONCLUSION

We have developed a novel method to suppress RA in vivo.

Oscillation in cycle length induces transient discordant and steady-state concordant alternans in the heart

Alternans is a beat-to-beat alternation of the cardiac action potential duration (APD) or intracellular calcium (Ca_i) transient. In cardiac tissue, alternans can be spatially concordant or discordant, of which the latter has been shown to increase dispersion of repolarization and promote a substrate for initiation of ventricular fibrillation. Alternans has been studied almost exclusively under constant cycle length pacing conditions. However, heart rate varies greatly on a beat-by-beat basis in normal and pathological conditions. The purpose of this study was to determine if applying a repetitive but non-constant pacing pattern, specifically cycle length oscillation (CLO), promotes or suppresses a proarrhythmic substrate. We performed computational simulations and optical mapping experiments to investigate the potential consequences of CLO. In a single cell computational model, CLO induced APD and Ca_i alternans, which became “phase-matched” with the applied oscillation. As a consequence of the phase-matching, in one-dimensional cable simulations, neonatal rat ventricular myocyte monolayers, and isolated adult guinea pig hearts CLO could transiently induce spatial and electromechanical discordant alternans followed by a steady-state of concordance. Our results demonstrated that under certain conditions, CLO can initiate ventricular fibrillation in the isolated hearts. On the other hand, CLO can also exert an antiarrhythmic effect by converting an existing state of discordant alternans to concordant alternans.

Nonlinear dynamics in cardiology

The dynamics of many cardiac arrhythmias, as well as the nature of transitions between different heart rhythms, have long been considered evidence of nonlinear phenomena playing a direct role in cardiac arrhythmogenesis. In most types of cardiac disease, the pathology develops slowly and gradually, often over many years. In contrast, arrhythmias often occur suddenly. In nonlinear systems, sudden changes in qualitative dynamics can, counterintuitively, result from a gradual change in a system parameter-this is known as a bifurcation. Here, we review how nonlinearities in cardiac electrophysiology influence normal and abnormal rhythms and how bifurcations change the dynamics. In particular, we focus on the many recent developments in computational modeling at the cellular level that are focused on intracellular calcium dynamics. We discuss two areas where recent experimental and modeling work has suggested the importance of nonlinearities in calcium dynamics: repolarization alternans and pacemaker cell automaticity.

Repolarization alternans reveals vulnerability to human atrial fibrillation

BACKGROUND

The substrates for human atrial fibrillation (AF) are poorly understood, but involve abnormal repolarization (action potential duration [APD]). We hypothesized that beat-to-beat oscillations in APD may explain AF substrates, and why vulnerability to AF forms a spectrum from control subjects without AF to patients with paroxysmal then persistent AF.

METHODS AND RESULTS

In 33 subjects (12 with persistent AF, 13 with paroxysmal AF, and 8 controls without AF), we recorded left (n=33) and right (n=6) atrial APD on pacing from cycle lengths 600 to 500 ms (100 to 120 bpm) up to the point where AF initiated. Action potential duration alternans required progressively faster rates for patients with persistent AF, patients with paroxysmal AF, and controls (cycle length 411±94 versus 372±72 versus 218±33 ms; P<0.01). In AF patients, APD alternans occurred at rates as slow as 100 to 120 bpm, unrelated to APD restitution (P>0.10). In this milieu, spontaneous ectopy initiated AF. At fast rates, APD alternans disorganized to complex oscillations en route to AF. Complex oscillations also arose at progressively faster rates for persistent AF, paroxysmal AF, and controls (cycle length: 316±99 versus 266±19 versus 177±16 ms; P=0.02). In paroxysmal AF, APD oscillations amplified before AF (P<0.001). In controls, APD alternans arose only at very fast rates (cycle length <250 ms; P<0.001 versus AF groups) just preceding AF. In 4 AF patients in whom rapid pacing did not initiate AF, APD alternans arose transiently then extinguished.

CONCLUSIONS

Atrial APD alternans reveals dynamic substrates for AF, arising most readily (at lower rates and higher magnitudes) in persistent AF then paroxysmal AF, and least readily in controls. APD alternans preceded all AF episodes and was absent when AF did not initiate. The cellular mechanisms for APD alternans near resting heart rates require definition.

Applications of control theory to the dynamics and propagation of cardiac action potentials

Sudden cardiac arrest is a widespread cause of death in the industrialized world. Most cases of sudden cardiac arrest are due to ventricular fibrillation (VF), a lethal cardiac arrhythmia. Electrophysiological abnormalities such as alternans (a beat-to-beat alternation in action potential duration) and conduction block have been suspected to contribute to the onset of VF. This study focuses on the use of control-systems techniques to analyze and design methods for suppressing these precursor factors. Control-systems tools, specifically controllability analysis and Lyapunov stability methods, were applied to a two-variable Karma model of the action-potential (AP) dynamics of a single cell, to analyze the effectiveness of strategies for suppressing AP abnormalities. State-feedback-integral (SFI) control was then applied to a Purkinje fiber simulated with the Karma model, where only one stimulating electrode was used to affect the system. SFI control converted both discordant alternans and 2:1 conduction block back toward more normal patterns, over a wider range of fiber lengths and pacing intervals compared with a Pyragas-type chaos controller. The advantages conferred by using feedback from multiple locations in the fiber, and using integral (i.e., memory) terms in the controller, are discussed.

Off-site control of repolarization alternans in cardiac fibers

Repolarization alternans, a beat-to-beat alternation in action potential duration, has been putatively linked to the onset of cardiac reentry. Anti-alternans control strategies can eliminate alternans in individual cells by exploiting the rate dependence of action potential duration. The same approach, when applied to a common measuring/stimulating site at one end of a cardiac fiber, has been shown to have limited spatial efficacy. As a first step toward spatially distributed electrode control systems, we investigated "off-site" control in canine Purkinje fibers, in which the recording and control sites are different. We found experimentally that alternans can be eliminated at, or very near, the recording site, and that varying the location of the recording site along the fiber causes the node (the location with no alternans) to move along the fiber in close proximity to the recording site. Theoretical predictions based on an amplitude equation [B. Echebarria and A. Karma, Chaos 12, 923 (2002)] show that those findings follow directly from the wave nature of alternans: the most unstable mode of alternans along the fiber is a wave solution of a one-dimensional Helmholtz equation with a node position that only deviates slightly from the recording site by an amount dependent on electrotonic coupling. Computer simulations using a Purkinje fiber model confirm these theoretical and experimental results. Although off-site alternans control does not suppress alternans along the entire fiber, our results indicate that placing the node away from the stimulus site reduces alternans amplitude along the fiber, and may therefore have implications for antiarrhythmic strategies based on alternans termination.

Optimization of feedback pacing for defibrillation

A mathematical model of multisite feedback pacing for defibrillation is optimized for electrode spacing and stimulus period. For four electrodes, the defibrillation success rate is highest at 88% when the electrodes are spaced as far apart as possible. For a single electrode, the optimum success rate was 26%.

Control of electrical alternans in simulations of paced myocardium using extended time-delay autosynchronization

Experimental studies have linked alternans, an abnormal beat-to-beat alternation of cardiac action potential duration, to the genesis of lethal arrhythmias such as ventricular fibrillation. Prior studies have considered various closed-loop feedback control algorithms for perturbing interstimulus intervals in such a way that alternans is suppressed. However, some experimental cases are restricted in that the controller's stimuli must preempt those of the existing waves that are propagating in the tissue, and therefore only shortening perturbations to the underlying pacing are allowed. We present results demonstrating that a technique known as extended time-delay autosynchronization (ETDAS) can effectively control alternans locally while operating within the above constraints. We show that ETDAS, which has already been used to control chaos in physical systems, has numerous advantages over previously proposed alternans control schemes.

Optical mapping system with real-time control capability

Real-time, closed-loop intervention is an emerging experiment-control method that promises to provide invaluable new insight into cardiac electrophysiology. One example is the investigation of closed-loop feedback control of cardiac activity (e.g., alternans) as a possible method of preventing arrhythmia onset. To date, such methods have been investigated only in vitro using microelectrode systems, which are hindered by poor spatial resolution and are not well suited for atrial or ventricular tissue preparations. We have developed a system that uses optical mapping techniques and an electrical stimulator as the sensory and effector arms, respectively, of a closed-loop, real-time control system. The system consists of a 2,048 x 1 pixel line-scan charge-coupled device camera that records optical signals from the tissue. Custom-image processing and control software, which is implemented on top of a hard real-time operation system (RTAI Linux), process the data and make control decisions with a deterministic delay of <1 ms. The system is tested in two ways: 1) it is used to control, in real time, simulated optical signals of electrical alternans; and 2) it uses precisely timed, feedback-controlled initiation of antitachycardia pacing to terminate reentrant arrhythmias in an arterially perfused swine right ventricle stained with voltage-sensitive fluorescent dye 4{beta-[2-(di-n-butylamino)-6-napathy]vinyl}pyridinium (di-4-ANEPPS). Thus real-time control of cardiac activity using optical mapping techniques is feasible. Such a system is attractive because it offers greater measurement resolution than the electrode-based systems with which real-time control has been used previously.

Study of the restitution of action potential duration using the artificial neural network

It is widely accepted that the APD (action potential duration) restitution plays a key role in the initializing and maintaining of the reentry arrhythmias. The Luo-Rudy II models paced with different protocols showed that the current APD had a complex relation with the previous APDs and diastole intervals (DIs). This relation could not be accurately described by a single exponential function. We used an artificial neural network to formularize this relation. The results suggested that back-propagation (BP) network could predict the current APD from the information of the first three previous beats. This would help provide a target for potential anti-arrhythmic therapies.

A model for multi-site pacing of fibrillation using nonlinear dynamics feedback

Traditionally, cardiac defibrillation requires a strong electric shock. Many unwanted side effects of this shock could be eliminated if defibrillation were performed using weak stimuli applied to several locations throughout the heart. Such multi-site pacing algorithms have been shown to defibrillate both experimentally (Pak et al., Am J Physiol 285:H2704-H2711, 2003) and theoretically (Puwal and Roth, J Biol Systems 14:101-112, 2006). Gauthier et al. (Chaos, 12:952-961, 2002) proposed a method to pace the heart using an algorithm based on nonlinear dynamics feedback applied through a single electrode. Our study applies a related but simpler algorithm, which essentially configures each electrode as a demand pacemaker, to simulate the multi-site pacing of fibrillating cardiac tissue. We use the numerical model developed by Fenton et al. (Chaos, 12:852-892, 2002) as the reaction term in a reaction-diffusion equation that we solve over a two-dimensional sheet of tissue. The defibrillation rate after pacing for 3 s is about 30%, which is significantly higher than the spontaneous defibrillation rate and is higher than observed in previous experimental and theoretical studies. Tuning the algorithm period can increase this rate to 45%.

Detection of T-wave alternans using an implantable cardioverter-defibrillator

BACKGROUND

Microvolt T-wave alternans (TWA) increases acutely prior to ventricular tachycardia (VT) or ventricular fibrillation (VF) in computer simulations and animal models, suggesting that TWA may provide a warning for VT/VF in patients with an implantable cardioverter-defibrillator (ICD).

OBJECTIVES

The purposes of this study were to develop a method for analyzing TWA recorded from ICD electrograms (EGMs) and to evaluate the degree of concordance between EGM TWA and TWA recorded from the surface ECG.

METHODS

We developed a software program to measure EGM TWA in the frequency domain and then used simulated EGMs to determine the effects of ICD signal processing, electrical noise, and variation in the EGM fiducial point on the recorded amplitude and K score (signal-to-noise ratio) of TWA. We then applied this method to analyze TWA simultaneously using both surface ECGs and ICD EGMs during incremental pacing in 25 ICD patients. Pacing modes and EGM sources were varied in repeated trials. EGMs with dynamic range adjusted to achieve a large T wave were telemetered to a digital Holter recorder and measured offline. ECG TWA was analyzed using a commercial system. A positive (+) ECG test had sustained alternans >or=1.9 microV with K score >or=3. Stored EGMs were reviewed for VT/VF during a 6-month follow-up period.

RESULTS

Simulations demonstrated that the EGM method accurately identified TWA >or=10 microV. Overall, 10 (40%) patients had at least one ECG TWA+ test and 15 patients (60%) had no ECG TWA+ tests. The maximum value of TWA was greater in EGMs than in ECGs (median 64 microV vs 2.2 microV, P <.0001). EGM TWA was greater in ECG TWA+ tests than in ECG TWA- tests (169 +/- 175 microV vs 71 +/- 61 microV, P <.001). Using a sustained EGM TWA threshold of 30 microV, EGM TWA was concordant with ECG TWA in 63 (84%) of 75 analyzed tests (P <.0001) and predicted ECG TWA results with 85% sensitivity and 84% specificity. Both ECG and EGM TWA predicted VT/VF during follow-up (ECG: P = .006; EGM: P = .035).

CONCLUSION

The amplitude of TWA is at least 10 times greater on ICD EGMs than on surface ECGs. EGM and ECG TWA have substantial concordance and comparable predictive value for spontaneous VT/VF. These observations support the hypothesis that ECG and EGM TWA detect the same electrical alternans phenomenon.

Control of electrical alternans in canine cardiac purkinje fibers

Alternation in the duration of consecutive cardiac action potentials (electrical alternans) may precipitate conduction block and the onset of arrhythmias. Consequently, suppression of alternans using properly timed premature stimuli may be antiarrhythmic. To determine the extent to which alternans control can be achieved in cardiac tissue, isolated canine Purkinje fibers were paced from one end using a feedback control method. Spatially uniform control of alternans was possible when alternans amplitude was small. However, control became attenuated spatially as alternans amplitude increased. The amplitude variation along the cable was well described by a theoretically expected standing wave profile that corresponds to the first quantized mode of the one-dimensional Helmholtz equation. These results confirm the wavelike nature of alternans and may have important implications for their control using electrical stimuli.

Detection of Repolarization Alternans With an Implantable Cardioverter Defibrillator Lead in a Porcine Model

Mechanistic links have been suggested between repolarization alternans (RPA) and the onset of ventricular tachycardia (VT) and/or fibrillation. Endocardial detection of RPA may, therefore, be an important step in future device-based treatments of arrhythmias. Here, we investigate if RPA could be detected during acute ischemia using an implantable cardioverter defibrillator (ICD) lead (tip to distal coil) located in the right ventricular apex. In 18 pigs, the right coronary (n = 10) or left anterior descending coronary (n = 8) artery was occluded for 10 min using a balloon catheter, followed by reperfusion for 30 min, and re-occlusion for 30 min. RPA magnitude, computed using the modified moving average (MMA) method, showed a sharp increase in all 18 animals, from a mean baseline level of 1.9 +/- 1.3 mV to 3.0 +/- 1.3 mV during first occlusion (p < 0.001). RPA magnitude showed a prominent increase in 10 animals during re-occlusion, from a mean baseline level of 1.7 +/- 1.0 mV to 3.3 +/- 1.5 mV (p < 0.001). The protocol was terminated during the first two stages of occlusion and reperfusion for the remaining 8 animals due to the occurrence of ventricular fibrillation (VF). These results confirm that RPA increases under ischemic conditions and that it is possible to detect and track RPA dynamics with an ICD lead that is positioned in a clinically realistic location. Such an approach may be useful in formulating improved arrhythmia detection and control algorithms.

Adaptive Diastolic Interval Control of Cardiac Action Potential Duration Alternans

INTRODUCTION

Recent experimental and computational studies have shown that beat-to-beat alternation in action potential duration can trigger cardiac reentry, suggesting that such "alternans" is a mechanistic precursor to arrhythmias. Given such a link, termination of alternans may help prevent the onset of arrhythmias. To this end, recent efforts have shown that chaos control methods can modulate the timing of electrical stimulation to eliminate alternans.

METHODS AND RESULTS

We have developed an alternative control method founded entirely in cardiac electrophysiology (rather than borrowing techniques from the control of physical systems as with existing control techniques). Using computer simulations, we show that this method, which exploits the rate-dependent behavior of cardiac tissue, can be used to control alternans (and higher-order) rhythms, and is robust to drift and noise. When applied to individual model cells exhibiting alternans, the algorithm converges to the period-1 rhythm over as wide, and in some cases a wider, range of feedback proportionality constant values relative to existing methods. Control success comparable to existing methods is achieved when the algorithm is applied to a simulated one-dimensional Purkinje fiber exhibiting alternans.

CONCLUSION

We have developed a method that adaptively controls the timing of electrical stimulation to rapidly eliminate action potential duration alternans in cardiac tissue. This control method may prove valuable in future arrhythmia prevention therapies.

Preventing alternans-induced spiral wave breakup in cardiac tissue: an ion-channel-based approach

The detailed processes involved in spiral wave breakup, believed to be one major mechanism by which tachycardia evolves into fibrillation, are still poorly understood. This has rendered difficult the proper design of an efficient and practical control stimulus protocol to eliminate such events. In order to gain new insights into the underlying electrophysiological and dynamical mechanisms of breakup, we applied linear perturbation theory to a steadily rotating spiral wave in two spatial dimensions. The tissue was composed of cells modeled using the Fenton-Karma equations whose parameters were chosen to emphasize alternans as a primary mechanism for breakup. Along with one meandering mode, not just one but several unstable alternans modes were found with differing growth rates, frequencies, and spatial structures. As the conductance of the fast inward current was increased, the instability of the modes increased, consistent with increased meandering and propensity for spiral breakup in simulations. We also explored a promising new approach, based on the theory, for the design of an energy efficient electrical stimulus protocol to control spiral wave breakup. The novelty lies in addressing the problem directly at the ion channel level and taking advantage of the inherent two dimensional nature of the rotating wave. With the help of the eigenmode method, we were able to calculate the exact timing and amplitude of the stimulus, and locate it optimally to maximize efficiency. The analysis led to a special-case example that demonstrated that a single, properly timed stimulus can have a global effect, suppressing all growing alternans modes over the entire tissue, thus inhibiting spiral wave breakup.

Delayed feedback as a means of control of noise-induced motion

Time-delayed feedback is exploited for controlling noise-induced motion in coherence resonance oscillators. Namely, under the proper choice of time delay, one can either increase or decrease the regularity of motion. It is shown that in an excitable system, delayed feedback can stabilize the frequency of oscillations against variation of noise strength. Also, for fixed noise intensity, the phenomenon of entrainment of the basic oscillation period by the delayed feedback occurs. This allows one to steer the time scales of noise-induced motion by changing the time delay.

A subharmonic dynamical bifurcation during in vitro epileptiform activity

Epileptic seizures are considered to result from a sudden change in the synchronization of firing neurons in brain neural networks. We have used an in vitro model of status epilepticus (SE) to characterize dynamical regimes underlying the observed seizure-like activity. Time intervals between spikes or bursts were used as the variable to construct first-return interpeak or interburst interval plots, for studying neuronal population activity during the transition to seizure, as well as within seizures. Return maps constructed for a brief epoch before seizures were used for approximating the local system dynamics during that time window. Analysis of the first-return maps suggests that intermittency is a dynamical regime underlying the observed epileptic activity. This type of analysis may be useful for understanding the collective dynamics of neuronal populations in the normal and pathological brain.

Physiological complexity, aging, and the path to frailty

Healthy physiological processes require the complex interaction of multiple control systems operating over multiple time scales. The output of these processes (for example, heart rate, blood pressure, hormonal rhythms, or postural sway) demonstrates complex variability that can be quantified using the concept of fractals, derived from the field of nonlinear dynamics. Complex physiological dynamics enable an organism to rapidly respond to the internal and external perturbations of everyday life. Aging and disease are associated with a loss of complexity in the dynamics of many physiological systems. This loss of complexity may reduce the ability to adapt to stress and lead to the syndrome of frailty.

Condition for alternans and its control in a two-dimensional mapping model of paced cardiac dynamics

We investigate a two-dimensional mapping model of a paced, isolated cardiac cell that relates the duration of the action potential to the two preceding diastolic intervals as well as the preceding action potential duration. The model displays rate-dependent restitution and hence memory. We derive a criterion for the stability of the 1:1 response pattern displayed by the model. This criterion can be written in terms of experimentally measured quantities-the slopes of restitution curves obtained via different pacing protocols. In addition, we analyze the two-dimensional mapping model in the presence of closed-loop feedback control. The control is initiated by making small adjustments to the pacing interval in order to suppress alternans and stabilize the 1:1 pattern. We find that the domain of control does not depend on the functional form of the map, and, in the general case, is characterized by a combination of the slopes. We show that the gain gamma necessary to establish control may vary significantly depending on the value of the slope of the so-called standard restitution curve (herein denoted as S12), but that the product gammaS12 stays approximately in the same range.

Linear and nonlinear approaches to the analysis of R-R interval variability

Analysis techniques derived from linear and non-linear dynamics systems theory qualify and quantify physiological signal variability. Both clinicians and researchers use physiological signals in their scopes of practice. The clinician monitors patients with signal-analysis technology, and the researcher analyzes physiological data with signal-analysis techniques. Understanding the theoretical basis for analyzing physiological signals within one's scope of practice ensures proper interpretation of the relationship between physiolgical function and signal variability. This article explains the concepts of linear and nonlinear signal analysis and illustrates these concepts with descriptions of power spectrum analysis and recurrence quantification analysis. This article also briefly describes the relevance of these 2 techniques to R-to-R wave interval (i.e., heart rate variability) signal analysis and demonstrates their application to R-to-R wave interval data obtained from an isolated rat heart model.