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

Activation of small conductance Ca2+ -activated K+ channels suppresses Ca2+ transient and action potential alternans in ventricular myocytes

At the cellular level, cardiac alternans is observed as beat-to-beat alternations in contraction strength, action potential (AP) morphology and Ca2+ transient (CaT) amplitude, and is a risk factor for cardiac arrhythmia. The (patho)physiological roles of small conductance Ca2+ -activated K+ (SK) channels in ventricles are poorly understood. We tested the hypothesis that in single rabbit ventricular myocytes pharmacological modulation of SK channels plays a causative role for the development of pacing-induced CaT and AP duration (APD) alternans. SK channel blockers (apamin, UCL1684) had only a minor effect on AP repolarization. However, SK channel activation by NS309 resulted in significant APD shortening, demonstrating that functional SK channels are well expressed in ventricular myocytes. The effects of NS309 were prevented or reversed by apamin and UCL1684, indicating that NS309 acted on SK channels. SK channel activation abolished or reduced the degree of pacing-induced CaT and APD alternans. Inhibition of KV 7.1 (with HMR1556) and KV 11.1 (with E4031) channels was used to mimic conditions of long QT syndromes type-1 and type-2, respectively. Both HMR1556 and E4031 enhanced CaT alternans that was prevented by SK channel activation. In AP voltage-clamped cells the SK channel activator had no effect on CaT alternans, confirming that suppression of CaT alternans was caused by APD shortening. APD shortening contributed to protection from alternans by lowering sarcoplasmic reticulum Ca2+ content and curtailing Ca2+ release. The data suggest that SK activation could be a potential intervention to avert development of alternans with important ramifications for arrhythmia prevention and therapy for patients with long QT syndrome. KEY POINTS: At the cellular level, cardiac alternans is observed as beat-to-beat alternations in contraction strength, action potential (AP) morphology and intracellular Ca2+ release amplitude, and is a risk factor for cardiac arrhythmia. The (patho)physiological roles of small conductance Ca2+ -activated K+ (SK) channels in ventricles are poorly understood. We investigated whether pharmacological modulation of SK channels affects the development of cardiac alternans in normal ventricular cells and in cells with drug-induced long QT syndrome (LQTS). While SK channel blockers have only a minor effect on AP morphology, their activation leads to AP shortening and abolishes or reduces the degree of pacing-induced Ca2+ and AP alternans. AP shortening contributed to protection against alternans by lowering sarcoplasmic reticulum Ca2+ content and curtailing Ca2+ release. The data suggest SK activation as a potential intervention to avert the development of alternans with important ramifications for arrhythmia prevention for patients with LQTS.

Clinical Potential of Beat-to-Beat Diastolic Interval Control in Preventing Cardiac Arrhythmias

Life‐threatening ventricular arrhythmias and sudden cardiac death are often preceded by cardiac alternans, a beat‐to‐beat oscillation in the T‐wave morphology or duration. However, given the spatiotemporal and structural complexity of the human heart, designing algorithms to effectively suppress alternans and prevent fatal rhythms is challenging. Recently, an antiarrhythmic constant diastolic interval pacing protocol was proposed and shown to be effective in suppressing alternans in 0‐, 1‐, and 2‐dimensional in silico studies as well as in ex vivo whole heart experiments. Herein, we provide a systematic review of the electrophysiological conditions and mechanisms that enable constant diastolic interval pacing to be an effective antiarrhythmic pacing strategy. We also demonstrate a successful translation of the constant diastolic interval pacing protocol into an ECG‐based real‐time control system capable of modulating beat‐to‐beat cardiac electrical activity and preventing alternans. Furthermore, we present evidence of the clinical utility of real‐time alternans suppression in reducing arrhythmia susceptibility in vivo. We provide a comprehensive overview of this promising pacing technique, which can potentially be translated into a clinically viable device that could radically improve the quality of life of patients experiencing abnormal cardiac rhythms.

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.

Feedback control of calcium driven alternans in cardiac myocytes

Cardiac alternans is a beat-to-beat alternation of the action potential duration (APD), which has been implicated as a possible cause of ventricular fibrillation. Previous studies have shown that alternans can originate via a period doubling bifurcation caused by the nonlinear dependence of the APD on the previous diastolic interval. In this case, it has been demonstrated that alternans can be eliminated by applying feedback control on the pacing cycle length. However, studies have shown that alternans can also originate due to unstable calcium (Ca) cycling in cardiac myocytes. In this study, we explore the effectiveness of APD feedback control to suppress alternans when the underlying instability is due to unstable Ca cycling. In particular, we explore the role of the bi-directional coupling between Ca and voltage and determine the effectiveness of feedback control under a wide range of conditions. We also analyze the applicability of feedback control on a coupled two cell system and show that APD control induces spatially out-of-phase alternans. We analyze the onset and the necessary conditions for the emergence of these out-of-phase patterns and assess the effectiveness of feedback control to suppress Ca driven alternans in a multi-cellular system.

Action potential shortening rescues atrial calcium alternans

KEY POINTS

Cardiac alternans refers to a beat-to-beat alternation in contraction, action potential (AP) morphology and Ca²⁺ transient (CaT) amplitude, and represents a risk factor for cardiac arrhythmia, including atrial fibrillation. We developed strategies to pharmacologically manipulate the AP waveform with the goal to reduce or eliminate the occurrence of CaT and contraction alternans in atrial tissue. With combined patch-clamp and intracellular Ca²⁺ measurements we investigated the effect of specific ion channel inhibitors and activators on alternans. In single rabbit atrial myocytes, suppression of Ca²⁺ -activated Cl^- channels eliminated AP duration alternans, but prolonged the AP and failed to eliminate CaT alternans. In contrast, activation of K⁺ currents (I_Ks and I_Kr ) shortened the AP and eliminated both AP duration and CaT alternans. As demonstrated also at the whole heart level, activation of K⁺ conductances represents a promising strategy to suppress alternans, and thus reducing a risk factor for atrial fibrillation.

ABSTRACT

At the cellular level alternans is observed as beat-to-beat alternations in contraction, action potential (AP) morphology and magnitude of the Ca²⁺ transient (CaT). Alternans is a well-established risk factor for cardiac arrhythmia, including atrial fibrillation. This study investigates whether pharmacological manipulation of AP morphology is a viable strategy to reduce the risk of arrhythmogenic CaT alternans. Pacing-induced AP and CaT alternans were studied in rabbit atrial myocytes using combined Ca²⁺ imaging and electrophysiological measurements. Increased AP duration (APD) and beat-to-beat alternations in AP morphology lowered the pacing frequency threshold and increased the degree of CaT alternans. Inhibition of Ca²⁺ -activated Cl^- channels reduced beat-to-beat AP alternations, but prolonged APD and failed to suppress CaT alternans. In contrast, AP shortening induced by activators of two K⁺ channels (ML277 for Kv7.1 and NS1643 for Kv11.1) abolished both APD and CaT alternans in field-stimulated and current-clamped myocytes. K⁺ channel activators had no effect on the degree of Ca²⁺ alternans in AP voltage-clamped cells, confirming that suppression of Ca²⁺ alternans was caused by the changes in AP morphology. Finally, activation of Kv11.1 channel significantly attenuated or even abolished atrial T-wave alternans in isolated Langendorff perfused hearts. In summary, AP shortening suppressed or completely eliminated both CaT and APD alternans in single atrial myocytes and atrial T-wave alternans at the whole heart level. Therefore, we suggest that AP shortening is a potential intervention to avert development of alternans with important ramifications for arrhythmia prevention and therapy.

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.

Memory-induced nonlinear dynamics of excitation in cardiac diseases

Excitable cells, such as cardiac myocytes, exhibit short-term memory, i.e., the state of the cell depends on its history of excitation. Memory can originate from slow recovery of membrane ion channels or from accumulation of intracellular ion concentrations, such as calcium ion or sodium ion concentration accumulation. Here we examine the effects of memory on excitation dynamics in cardiac myocytes under two diseased conditions, early repolarization and reduced repolarization reserve, each with memory from two different sources: slow recovery of a potassium ion channel and slow accumulation of the intracellular calcium ion concentration. We first carry out computer simulations of action potential models described by differential equations to demonstrate complex excitation dynamics, such as chaos. We then develop iterated map models that incorporate memory, which accurately capture the complex excitation dynamics and bifurcations of the action potential models. Finally, we carry out theoretical analyses of the iterated map models to reveal the underlying mechanisms of memory-induced nonlinear dynamics. Our study demonstrates that the memory effect can be unmasked or greatly exacerbated under certain diseased conditions, which promotes complex excitation dynamics, such as chaos. The iterated map models reveal that memory converts a monotonic iterated map function into a nonmonotonic one to promote the bifurcations leading to high periodicity and chaos.

Real-Time Closed Loop Diastolic Interval Control Prevents Cardiac Alternans in Isolated Whole Rabbit Hearts

Cardiac alternans, a beat-to-beat alternation in action potential duration (APD), can lead to fatal arrhythmias. During periodic pacing, changes in diastolic interval (DI) depend on subsequent changes in APD, thus enhancing cardiac instabilities through a 'feedback' mechanism. Recently, an anti-arrhythmic Constant DI pacing protocol was proposed and shown to be effective in suppressing alternans in 0D and 1D in silico studies. However, previous experimental validation of Constant DI pacing in the heart has been unsuccessful due to the spatio-temporal complexity of 2D cardiac tissue and the technical challenges in its real-time implementation. Here, we developed a novel closed loop system to detect T-waves from real-time ECG data, enabling successful implementation of Constant DI pacing protocol, and performed high-resolution optical mapping experiments on isolated whole rabbit hearts to validate its anti-arrhythmic effects. The results were compared with: (1) Periodic pacing (feedback inherent) and (2) pacing with heart rate variability (HRV) (feedback modulation) introduced by using either Gaussian or Physiological patterns. We observed that Constant DI pacing significantly suppressed alternans in the heart, while maintaining APD spatial dispersion and flattening the slope of the APD restitution curve, compared to traditional Periodic pacing. In addition, introduction of HRV in Periodic pacing failed to prevent cardiac alternans, and was arrhythmogenic.

Theory of the development of alternans in the heart during controlled diastolic interval pacing

The beat-to-beat alternation in action potential durations (APDs) in the heart, called APD alternans, has been linked to the development of serious cardiac rhythm disorders, including ventricular tachycardia and fibrillation. The length of the period between action potentials, called the diastolic interval (DI), is a key dynamical variable in the standard theory of alternans development. Thus, methods that control the DI may be useful in preventing dangerous cardiac rhythms. In this study, we examine the dynamics of alternans during controlled-DI pacing using a series of single-cell and one-dimensional (1D) fiber models of alternans dynamics. We find that a model that combines a so-called memory model with a calcium cycling model can reasonably explain two key experimental results: the possibility of alternans during constant-DI pacing and the phase lag of APDs behind DIs during sinusoidal-DI pacing. We also find that these results can be replicated by incorporating the memory model into an amplitude equation description of a 1D fiber. The 1D fiber result is potentially concerning because it seems to suggest that constant-DI control of alternans can only be effective over only a limited region in space.

Alternans in atria: Mechanisms and clinical relevance

Atrial fibrillation is the most common sustained arrhythmia and its prevalence is rapidly rising with the aging of the population. Cardiac alternans, defined as cyclic beat-to-beat alternations in contraction force, action potential (AP) duration and intracellular Ca²⁺ release at constant stimulation rate, has been associated with the development of ventricular arrhythmias. Recent clinical data also provide strong evidence that alternans plays a central role in arrhythmogenesis in atria. The aim of this article is to review the mechanisms that are responsible for repolarization alternans and contribute to the transition from spatially concordant alternans to the more arrhythmogenic spatially discordant alternans in atria.

Memory-Induced Chaos in Cardiac Excitation

Excitable systems display memory, but how memory affects the excitation dynamics of such systems remains to be elucidated. Here we use computer simulation of cardiac action potential models to demonstrate that memory can cause dynamical instabilities that result in complex excitation dynamics and chaos. We develop an iterated map model that correctly describes these dynamics and show that memory converts a monotonic first return map of action potential duration into a nonmonotonic one, resulting in a period-doubling bifurcation route to chaos.

Predicting the onset of period-doubling bifurcations in noisy cardiac systems

Biological, physical, and social systems often display qualitative changes in dynamics. Developing early warning signals to predict the onset of these transitions is an important goal. The current work is motivated by transitions of cardiac rhythms, where the appearance of alternating features in the timing of cardiac events is often a precursor to the initiation of serious cardiac arrhythmias. We treat embryonic chick cardiac cells with a potassium channel blocker, which leads to the initiation of alternating rhythms. We associate this transition with a mathematical instability, called a period-doubling bifurcation, in a model of the cardiac cells. Period-doubling bifurcations have been linked to the onset of abnormal alternating cardiac rhythms, which have been implicated in cardiac arrhythmias such as T-wave alternans and various tachycardias. Theory predicts that in the neighborhood of the transition, the system's dynamics slow down, leading to noise amplification and the manifestation of oscillations in the autocorrelation function. Examining the aggregates' interbeat intervals, we observe the oscillations in the autocorrelation function and noise amplification preceding the bifurcation. We analyze plots--termed return maps--that relate the current interbeat interval with the following interbeat interval. Based on these plots, we develop a quantitative measure using the slope of the return map to assess how close the system is to the bifurcation. Furthermore, the slope of the return map and the lag-1 autocorrelation coefficient are equal. Our results suggest that the slope and the lag-1 autocorrelation coefficient represent quantitative measures to predict the onset of abnormal alternating cardiac rhythms.

The mechanisms of calcium cycling and action potential dynamics in cardiac alternans

RATIONALE

Alternans is a risk factor for cardiac arrhythmia, including atrial fibrillation. At the cellular level alternans manifests as beat-to-beat alternations in contraction, action potential duration (APD), and magnitude of the Ca(2+) transient (CaT). Electromechanical and CaT alternans are highly correlated, however, it has remained controversial whether the primary cause of alternans is a disturbance of cellular Ca(2+) signaling or electrical membrane properties.

OBJECTIVE

To determine whether a primary failure of intracellular Ca(2+) regulation or disturbances in membrane potential and AP regulation are responsible for the occurrence of alternans in atrial myocytes.

METHODS AND RESULTS

Pacing-induced APD and CaT alternans were studied in single rabbit atrial and ventricular myocytes using combined [Ca(2+)]i and electrophysiological measurements. In current-clamp experiments, APD and CaT alternans strongly correlated in time and magnitude. CaT alternans was observed without alternation in L-type Ca(2+) current, however, elimination of intracellular Ca(2+) release abolished APD alternans, indicating that [Ca(2+)]i dynamics have a profound effect on the occurrence of CaT alternans. Trains of 2 distinctive voltage commands in form of APs recorded during large and small alternans CaTs were applied to voltage-clamped cells. CaT alternans was observed with and without alternation in the voltage command shape. During alternans AP-clamp large CaTs coincided with both long and short AP waveforms, indicating that CaT alternans develop irrespective of AP dynamics.

CONCLUSIONS

The primary mechanism underlying alternans in atrial cells, similarly to ventricular cells, resides in a disturbance of Ca(2+) signaling, whereas APD alternans are a secondary consequence, mediated by Ca(2+)-dependent AP modulation.

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

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

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.

Restricted feedback control in discrete-time dynamical systems with memory

When an equilibrium state of a physical or biological system suffers a loss of stability (e.g., via a bifurcation), it may be both possible and desirable to stabilize the equilibrium via closed-loop feedback control. Significant effort has been devoted towards using such control to prevent oscillatory or chaotic behavior in dynamical systems, both continuous-time and discrete-time. Regarding control in discrete-time systems, most prior attempts to stabilize unstable equilibria require that the system be perturbed once during each time step. However, there are examples of systems for which this is neither feasible nor possible. In this paper, we analyze a restricted feedback control method for discrete-time systems (restricted in the sense that the controller's perturbations may be applied only in every other time step). We apply our theoretical analysis to a specific example from cardiac electrophysiology in which this sort of restricted feedback control is especially relevant. The example is a useful test case for the theory, and one for which an experimental setup is rather straightforward.

Non-linear dynamics of cardiac alternans: subcellular to tissue-level mechanisms of arrhythmia

Cardiac repolarization alternans is a rhythm disturbance of the heart in which rapid stimulation elicits a beat-to-beat alternation in the duration of action potentials and magnitude of intracellular calcium transients in individual cardiac myocytes. Although this phenomenon has been identified as a potential precursor to dangerous reentrant arrhythmias and sudden cardiac death, significant uncertainty remains regarding its mechanism and no clinically practical means of halting its occurrence or progression currently exists. Cardiac alternans has well-characterized tissue, cellular, and subcellular manifestations, the mechanisms and interplay of which are an active area of research.

Short-term memory and electrical restitution in the canine transmural ventricle

Cardiac short-term memory is an intrinsic property of paced myocardium that reflects the influence of pacing history. Using an optical mapping method to record membrane voltage and intracellular calcium (Ca(2 +)(i)), this study investigated the properties and mechanisms of short-term memory in isolated and perfused canine wedge preparations. In addition to the dynamic and S1S2 pacing protocols, a perturbed downsweep pacing protocol was used to get a complete overview of the restitution portrait. Abrupt changes in basic cycle length (BCL) were applied to investigate the accommodation process of action potential duration (APD). The results showed unobvious differences of memory among the epi-, mid- and endo-myocytes, implying an insignificant memory-induced transient heterogeneity in APD across the transmural canine hearts. With the decrease of pacing rate S1, memory gradually elevated and achieved a maximum around 400 ms, and then reduced as S1 decreased further, indicating a non-monotonic relationship between memory and the pacing rate. After suppressing the Ca(2 +)(i) transient with ryanodine (3 µmol l(-1)), the accommodation process of APD to a new BCL significantly abbreviated (τ = 37.41 ± 4.42 stimuli before ryanodine, τ = 15.84 ± 4.74 stimuli after ryanodine, p < 0.01). Therefore, Ca(2 +)(i) cycling was suggested to play an important role in memory during dynamic pacing.

Control of action potential duration alternans in canine ventricular tissue

Cardiac action potential duration alternans is characterized by a beat-to-beat alternation in action potential waveform. Its presence has been putatively linked to the onset of lethal cardiac arrhythmias. Previous studies, which have been limited to cardiac Purkinje fibers, have shown that closed-loop alternans control techniques, which apply a succession of externally administered cycle perturbations, provide ineffectual spatial alternans elimination. A more complete understanding of alternans control in the more clinically relevant ventricular tissue is needed. Here, we study the spatial dynamics of alternans and alternans control in arterially perfused canine right ventricular preparations using optical mapping. We quantified the spatial efficacy of alternans control across 2.5 cm of tissue, focusing primarily on differences in spatial control within several sub-regions of tissue. Our results provide a basis for future investigations into multi-electrode-based control interventions of alternans in cardiac tissue.

Studies on Feedback Control of Cardiac Alternans

A beat-to-beat variation in the electric wave propagation morphology in myocardium is referred to as cardiac alternans and it has been linked to the onset of life threatening arrhythmias and sudden cardiac death. Experimental studies have demonstrated that alternans can be annihilated by the feedback modulation of the basic pacing interval in a small piece of cardiac tissue. In this work, we study the capability of feedback control to suppress alternans both spatially and temporally in an extracted rabbit heart and in a cable of cardiac cells. This work demonstrates real-time control of cardiac alternans in an extracted rabbit heart and provides an analysis of the control methodology applied in the case of a one-dimensional (1D) cable of cardiac cells. The real-time system control is realized through feedback by proportional perturbation of the basic pacing cycle length (PCL). The measurements of the electric wave propagation are obtained by optical mapping of fluorescent dye from the surface of the heart and are fed into a custom-designed software that provides the control action signal that perturbs the basic pacing cycle length. In addition, a novel pacing protocol that avoids conduction block is applied. A numerical analysis, complementary to the experimental study is also carried out, by the ionic model of a 1D cable of cardiac cells under a self-referencing feedback protocol, which is identical to the one applied in the experimental study. Further, the amplitude of alternans linear parabolic PDE that is associated with the 1D ionic cardiac cell cable model under full state feedback control is analyzed. We provide an analysis of the amplitude of alternans parabolic PDE which admits a standard evolutionary form in a well defined functional space. Standard modal decomposition techniques are used in the analysis and the controller synthesis is carried out through pole-placement. State and output feedback controller realizations are developed and the important issue of measurement noise in the controller implementation is addressed. The analysis of stabilization of the amplitude of alternans PDE is in agreement with the experimental results and numerical results produced by the ionic 1D cable of cardiac cells model. Finally, a discussion is provided in light of these results in order to use control to suppress alternans in the human myocardium.

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.

The rate- and species-dependence of short-term memory in cardiac myocytes

Short-term memory is an intrinsic property of paced cardiac myocytes that reflects the influence of pacing history, and not just the immediately preceding diastolic interval (DI), on the action potential duration (APD). Although it is recognized that short-term memory affects the dynamics of cardiac myocytes in general, and the onset of irregular cardiac rhythm in particular, its has never been adequately quantified or measured directly in experiments or numerical simulations, mainly due to the absence of appropriate techniques. As a result, very little is known about the rate- and species dependent behavior of short-term memory. In this study, we introduce a new approach that allows one to estimate how much short-term memory, M(S), is present in the cardiac myocyte at different pacing rates. The new quantification is based on the fact that pacing history affects not only the APD, but the entire dynamics of paced cardiac myocytes, in particular the restitution curve. Using the patch clamp technique and numerical simulations, we measured short-term memory restitution-the dependence of M(S) on the cycle length-in isolated rabbit and guinea pig ventricular myocytes. In both species, M(S) is rate- and species-dependent, displaying a biphasic behavior as a function of cycle length. Moreover, our results indicate that there is a significant difference in M(S) measured between both species at small cycle lengths. Numerical simulations suggest that the kinetics of the rapidly activating delayed rectifier potassium current I(Kr) is partially responsible for this difference.

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.

Alternans and the influence of ionic channel modifications: Cardiac three-dimensional simulations and one-dimensional numerical bifurcation analysis

Cardiac propagation is investigated by simulations using a realistic three-dimensional (3D) geometry including muscle fiber orientation of the ventricles of a rabbit heart and the modified Beeler-Reuter ionic model. Electrical excitation is introduced by a periodic pacing of the lower septum. Depending on the pacing frequency, qualitatively different dynamics are observed, namely, normal heart beat, T-wave alternans, and 2:1 conduction block at small, intermediate, and large pacing frequencies, respectively. In a second step, we performed a numerical stability and bifurcation analysis of a pulse propagating in a one-dimensional (1D) ring of cardiac tissue. The precise onset of the alternans instability is obtained from computer-assisted linear stability analysis of the pulse and computation of the associated spectrum. The critical frequency at the onset of alternans and the profiles of the membrane potential agree well with the ones obtained in the 3D simulations. Next, we computed changes in the wave profiles and in the onset of alternans for the Beeler-Reuter model with modifications of the sodium, calcium, and potassium channels, respectively. For this purpose, we employ the method of numerical bifurcation and stability analysis. While blocking of calcium channels has a stabilizing effect, blocked sodium or potassium channels lead to the occurrence of alternans at lower pacing frequencies. The findings regarding channel blocking are verified within three-dimensional simulations. Altogether, we have found T-wave alternans and conduction block in 3D simulations of a realistic rabbit heart geometry. The onset of alternans has been analyzed by numerical bifurcation and stability analysis of 1D wave trains. By comparing the results of the two approaches, we find that alternans is not strongly influenced by ingredients such as 3D geometry and propagation anisotropy, but depends mostly on the frequency of pacing (frequency of subsequent action potentials). In addition, we have introduced numerical bifurcation and stability analysis as a tool into heart modeling and demonstrated its efficiency in scanning a large set of parameters in the case of models with reduced conductivity. Bifurcation analysis also provides an accurate test for analytical theories of alternans as is demonstrated for the case of the restitution hypothesis.

An ionically based mapping model with memory for cardiac restitution

Many features of the sequence of action potentials produced by repeated stimulation of a patch of cardiac muscle can be modeled by a 1D mapping, but not the full behavior included in the restitution portrait. Specifically, recent experiments have found that (i) the dynamic and S1-S2 restitution curves are different (rate dependence) and (ii) the approach to steady state, which requires many action potentials (accommodation), occurs along a curve distinct from either restitution curve. Neither behavior can be produced by a 1D mapping. To address these shortcomings, ad hoc 2D mappings, where the second variable is a "memory" variable, have been proposed; these models exhibit qualitative features of the relevant behavior, but a quantitative fit is not possible. In this paper we introduce a new 2D mapping and determine a set of parameters for it that gives a quantitatively accurate description of the full restitution portrait measured from a bullfrog ventricle. The mapping can be derived as an asymptotic limit of an idealized ionic model in which a generalized concentration acts as a memory variable. This ionic basis clarifies how the present model differs from previous models. The ionic basis also provides the foundation for more extensive cardiac modeling: e.g., constructing a PDE model that may be used to study the effect of memory on propagation. The fitting procedure for the mapping is straightforward and can easily be applied to obtain a mathematical model for data from other experiments, including experiments on different species.

Mechanism of Repolarization Alternans Has Restitution of Action Potential Duration Dependent and Independent Components

INTRODUCTION

Investigation of relationship between diastolic-interval (DI)-dependent restitution of action potential duration (APD) and alternans of APD has produced conflicting results. We used a novel pacing protocol to determine the role of restitution in alternans by minimizing changes in DI preceding each activation.

METHODS

Transmembrane potentials were recorded from right ventricular endocardial tissue isolated from five dogs. We used three pacing sequences: (i) The tissue was paced at a constant DI for 100 beats. (ii) The DIs were changed randomly between two sequences of constant DI. (iii) Each constant DI trial was followed by constant cycle length trial where pacing cycle length was equal to average cycle length during previous constant DI trial.

RESULTS

Alternans of APD occurred even when DIs preceding each activation were invariant. Slopes of restitution during constant DI pacing were both negative and positive and were much larger than unity. Alternans amplitude during constant cycle length pacing was larger than during constant DI, 32.2 +/- 12.3 versus 7.5 +/- 2.8 msec, P < 0.01. Random perturbation of DI decreased alternans amplitude during constant DI pacing from 14.7 +/- 4.8 to 10.5 +/- 3.4 msec, P < 0.01.

CONCLUSION

Our results indicate that mechanism of repolarization alternans has restitution-dependent and restitution-independent components. However, our results also provide direct evidence that shows that DI-dependent restitution of APD is not a necessary mechanism for the alternans to exist. Ability to pace with explicit control of DI provides a novel approach to dissect mechanisms of alternans into restitution-dependent and restitution-independent effects.

Action potential morphology influences intracellular calcium handling stability and the occurrence of alternans

Instability in the intracellular Ca2+ handling system leading to Ca2+ alternans is hypothesized to be an underlying cause of electrical alternans. The highly coupled nature of membrane voltage and Ca2+ regulation suggests that there should be reciprocal effects of membrane voltage on the stability of the Ca2+ handling system. We investigated such effects using a mathematical model of the cardiac intracellular Ca2+ handling system. We found that the morphology of the action potential has a significant effect on the stability of the Ca2+ handling system at any given pacing rate, with small changes in action potential morphology resulting in a transition from stable nonalternating Ca2+ transients to stable alternating Ca2+ transients. This bifurcation occurs as the alternans eigen value of the system changes from absolute value <1 to absolute value >1. These results suggest that the stability of the intracellular Ca2+ handling system and the occurrence of Ca2+ alternans are not dictated solely by the Ca2+ handling system itself, but are also modulated to a significant degree by membrane voltage (through its influence on sarcolemmal Ca2+ currents) and, therefore, by all ionic currents that affect membrane voltage.

Restitution in mapping models with an arbitrary amount of memory

Restitution, the characteristic shortening of action potential duration (APD) with increased heart rate, has been studied extensively because of its purported link to the onset of fibrillation. Restitution is often represented in the form of mapping models where APD is a function of previous diastolic intervals (DIs) and/or APDs, A(n+1)=F(D(n),A(n),D(n-1),A(n-1),...), where A(n+1) is the APD following a DI given by D(n). The number of variables previous to D(n) determines the degree of memory in the mapping model. Recent experiments have shown that mapping models should contain at least three variables (D(n),A(n),D(n-1)) to reproduce a restitution portrait (RP) that is qualitatively similar to that seen experimentally, where the RP shows three different types of restitution curves (RCs) [dynamic, S1-S2, and constant-basic cycle length (BCL)] simultaneously. However, an interpretation of the different RCs has only been presented in detail for mapping models of one and two variables. Here we present an analysis of the different RCs in the RP for mapping models with an arbitrary amount of memory. We determine the number of variables necessary to represent the different RCs in the RP. We also present a graphical visualization of these RCs. Our analysis reveals that the dynamic and S1-S2 RCs reside on two-dimensional surfaces, and therefore provide limited information for mapping models with more than two variables. However, constant-BCL restitution is a feature of the RP that depends on higher dimensions and can possibly be used to determine a lower bound on the dimensionality of cardiac dynamics.

The Restitution Portrait:. A New Method for Investigating Rate-Dependent Restitution

INTRODUCTION

Electrical restitution, relating action potential duration (APD) to diastolic interval (DI), was believed to determine the stability of heart rhythm. However, recent studies demonstrate that stability also depends on long-term APD changes caused by memory. This study presents a new method for investigation of rate- and memory-dependent aspects of restitution and for assessment of mapping models of APD.

METHODS AND RESULTS

Bullfrog ventricular myocardium was paced with a "perturbed downsweep protocol." Starting from a basic cycle length (BCL) of 1,000 ms, the tissue was paced until steady state was achieved, followed by single beats of longer and shorter cycle lengths. BCL was decreased by 50 to 100 ms and the process repeated. All APDs were plotted as a function of the preceding DI, which allowed simultaneous observation of dynamic, S1-S2, and two constant-BCL restitution curves in a "restitution portrait." Responses were classified as 1:1 (stimulus:response), transient 2:2, or persistent 2:2 (alternans) and were related to the slopes of the restitution curves. None of these slopes approached unity for the persistent 2:2 response, demonstrating that the traditional restitution condition does not predict alternans. The restitution portrait was used to evaluate three mapping models of APD. The models with no memory and with one-beat memory did not produce restitution portraits similar to the experimental one. A model with two-beat memory produced a qualitatively similar portrait.

CONCLUSION

The restitution portrait allows a more comprehensive assessment of cardiac dynamics than methods used to date. Further study of models with memory may result in a clinical criterion for electrical instability.