Restricted feedback control of one-dimensional maps

Beyond Alternans: Detection of Higher-Order Periodicity in Ex-Vivo Human Ventricles Before Induction of Ventricular Fibrillation

Background

Repolarization alternans, defined as period-2 oscillation in the repolarization phase of the action potentials, is one of the cornerstones of cardiac electrophysiology as it provides a mechanistic link between cellular dynamics and ventricular fibrillation (VF). Theoretically, higher-order periodicities (e.g., period-4, period-8,...) are expected but have very limited experimental evidence.

Methods

We studied explanted human hearts, obtained from the recipients of heart transplantation at the time of surgery, using optical mapping technique with transmembrane voltage-sensitive fluorescent dyes. The hearts were stimulated at an increasing rate until VF was induced. The signals recorded from the right ventricle endocardial surface just before the induction of VF and in the presence of 1:1 conduction were processed using the Principal Component Analysis and a combinatorial algorithm to detect and quantify higher-order dynamics.

Results

A prominent and statistically significant 1:4 peak (corresponding to period-4 dynamics) was seen in three of the six studied hearts. Local analysis revealed the spatiotemporal distribution of higher-order periods. Period-4 was localized to temporally stable islands. Higher-order oscillations (period-5, 6, and 8) were transient and primarily occurred in arcs parallel to the activation isochrones.

Discussion

We present evidence of higher-order periodicities and the co-existence of such regions with stable non-chaotic areas in ex-vivo human hearts before VF induction. This result is consistent with the period-doubling route to chaos as a possible mechanism of VF initiation, which complements the concordant to discordant alternans mechanism. The presence of higher-order regions may act as niduses of instability that can degenerate into chaotic fibrillation.

Voltage and calcium dynamics both underlie cellular alternans in cardiac myocytes

Cardiac alternans, a putative trigger event for cardiac reentry, is a beat-to-beat alternation in membrane potential and calcium transient. Alternans was originally attributed to instabilities in transmembrane ion channel dynamics (i.e., the voltage mechanism). As of this writing, the predominant view is that instabilities in subcellular calcium handling are the main underlying mechanism. That being said, because the voltage and calcium systems are bidirectionally coupled, theoretical studies have suggested that both mechanisms can contribute. To date, to our knowledge, no experimental evidence of such a dual role within the same cell has been reported. Here, a combined electrophysiological and calcium imaging approach was developed and used to illuminate the contributions of voltage and calcium dynamics to alternans. An experimentally feasible protocol, quantification of subcellular calcium alternans and restitution slope during cycle-length ramping alternans control, was designed and validated. This approach allows simultaneous illumination of the contributions of voltage and calcium-driven instability to total cellular instability as a function of cycle-length. Application of this protocol in in vitro guinea-pig left-ventricular myocytes demonstrated that both voltage- and calcium-driven instabilities underlie alternans, and that the relative contributions of the two systems change as a function of pacing rate.

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.

Control of action potential duration alternans in canine cardiac ventricular tissue

Cardiac electrical alternans, characterized by a beat-to-beat alternation in action potential waveform, is a naturally occurring phenomenon, which can occur at sufficiently fast pacing rates. Its presence has been putatively linked to the onset of cardiac reentry, which is a precursor to ventricular fibrillation. Previous studies have shown that closed-loop alternans control techniques that apply a succession of externally administered cycle perturbations at a single site provide limited spatially-extended alternans elimination in sufficiently large cardiac substrates. However, detailed experimental investigations into the spatial dynamics of alternans control have been restricted to Purkinje fiber studies. A complete understanding of alternans control in the more clinically relevant ventricular tissue is needed. In this paper, we study the spatial dynamics of alternans and alternans control in arterially perfused canine right ventricular preparations using an optical mapping system capable of high-resolution fluorescence imaging. Specifically, we quantify the spatial efficacy of alternans control along 2.5 cm of tissue, focusing on differences in spatial control between different subregions of tissue. We demonstrate effective control of spatially-extended alternans up to 2.0 cm, with control efficacy attenuating as a function of distance. Our results provide a basis for future investigations into electrode-based control interventions of alternans in cardiac tissue.

Feedback-control induced pattern formation in cardiac myocytes: a mathematical modeling study

Cardiac alternans is a dangerous rhythm disturbance of the heart, in which rapid stimulation elicits a beat-to-beat alternation in the action potential duration (APD) and calcium (Ca) transient amplitude of individual myocytes. Recently, "subcellular alternans", in which the Ca transients of adjacent regions within individual myocytes alternate out-of-phase, has been observed. A previous theoretical study suggested that subcellular alternans may result during static pacing from a Turing-type symmetry breaking instability, but this was only predicted in a subset of cardiac myocytes (with negative Ca to voltage (Ca-->V(m)) coupling) and has never been directly verified experimentally. A recent experimental study, however, showed that subcellular alternans is dynamically induced in the remaining subset of myocytes during pacing with a simple feedback control algorithm ("alternans control"). Here we show that alternans control pacing changes the effective coupling between the APD and the Ca transient (V(m)-->Ca coupling), such that subcellular alternans is predicted to occur by a Turing instability in cells with positive Ca-->V(m) coupling. In addition to strengthening the understanding of the proposed mechanism for subcellular alternans formation, this work (in concert with previous theoretical and experimental results) illuminates subcellular alternans as a striking example of a biological Turing instability in which the diffusing morphogens can be clearly identified.

Dynamical mechanism for subcellular alternans in cardiac myocytes

RATIONALE

Cardiac repolarization alternans is an arrhythmogenic rhythm disturbance, manifested in individual myocytes as a beat-to-beat alternation of action potential durations and intracellular calcium transient magnitudes. Recent experimental studies have reported "subcellular alternans," in which distinct regions of an individual cell are seen to have counterphase calcium alternations, but the mechanism by which this occurs is not well understood. Although previous theoretical work has proposed a possible dynamical mechanism for subcellular alternans formation, no direct evidence for this mechanism has been reported in vitro. Rather, experimental studies have generally invoked fixed subcellular heterogeneities in calcium-cycling characteristics as the mechanism of subcellular alternans formation.

OBJECTIVE

In this study, we have generalized the previously proposed dynamical mechanism to predict a simple pacing algorithm by which subcellular alternans can be induced in isolated cardiac myocytes in the presence or absence of fixed subcellular heterogeneity. We aimed to verify this hypothesis using computational modeling and to confirm it experimentally in isolated cardiac myocytes. Furthermore, we hypothesized that this dynamical mechanism may account for previous reports of subcellular alternans seen in statically paced, intact tissue.

METHODS AND RESULTS

Using a physiologically realistic computational model of a cardiac myocyte, we show that our predicted pacing algorithm induces subcellular alternans in a manner consistent with theoretical predictions. We then use a combination of real-time electrophysiology and fluorescent calcium imaging to implement this protocol experimentally and show that it robustly induces subcellular alternans in isolated guinea pig ventricular myocytes. Finally, we use computational modeling to demonstrate that subcellular alternans can indeed be dynamically induced during static pacing of 1D fibers of myocytes during tissue-level spatially discordant alternans.

CONCLUSION

Here we provide the first direct experimental evidence that subcellular alternans can be dynamically induced in cardiac myocytes. This proposed mechanism may contribute to subcellular alternans formation in the intact heart.

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.

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.

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.

Determining the effects of memory and action potential duration alternans on cardiac restitution using a constant-memory restitution protocol

Restitution, the dependence of action potential duration (APD) on diastolic interval, may be causally linked to the vulnerability of cardiac tissue to certain types of arrhythmias. While a number of pacing protocols are commonly used to quantify the restitution relation, one of these, the dynamic protocol, may result in the occurrence of APD alternans. However, the effects of APD alternans, and the concomitant alternation in cardiac memory, on the restitution curve are currently not well understood. Alternans preceding a given action potential may cause that action potential to have a different duration from one preceded by action potentials of identical duration. This interaction of alternans and memory can result in a dynamic restitution curve that is not unique. To address this, we have developed a constant-memory restitution protocol that enables the experimenter or modeller to obtain unique, constant-memory restitution curves at all diastolic intervals. Using this protocol, we obtained unique restitution curves for two ionic models of the cardiac action potential in the absence of alternans at all diastolic intervals. A comparison of the unique constant-memory and non-unique dynamic restitution curves for the two models shows that the presence of alternans can significantly alter the shape of the restitution curve compared to when alternans is absent.

Progress toward controlling in vivo fibrillating sheep atria using a nonlinear-dynamics-based closed-loop feedback method

We describe preliminary experiments on controlling in vivo atrial fibrillation using a closed-loop feedback protocol that measures the dynamics of the right atrium at a single spatial location and applies control perturbations at a single spatial location. This study allows investigation of control of cardiac dynamics in a preparation that is physiologically close to an in vivo human heart. The spatial-temporal response of the fibrillating sheep atrium is measured using a multi-channel electronic recording system to assess the control effectiveness. In an attempt to suppress fibrillation, we implement a scheme that paces occasionally the cardiac muscle with small shocks. When successful, the inter-activation time interval is the same and electrical stimuli are only applied when the controller senses that the dynamics are beginning to depart from the desired periodic rhythm. The shock timing is adjusted in real time using a control algorithm that attempts to synchronize the most recently measured inter-activation interval with the previous interval by inducing an activation at a time projected by the algorithm. The scheme is "single-sided" in that it can only shorten the inter-activation time but not lengthen it. Using probability distributions of the inter-activation time intervals, we find that the feedback protocol is not effective in regularizing the dynamics. One possible reason for the less-than-successful results is that the controller often attempts to stimulate the tissue while it is still in the refractory state and hence it does not induce an activation. (c) 2002 American Institute of Physics.

Control of long-period orbits and arbitrary trajectories in chaotic systems using dynamic limiting

We demonstrate experimental control of long-period orbits and arbitrary chaotic trajectories using a new chaos control technique called dynamic limiting. Based on limiter control, dynamic limiting uses a predetermined sequence of limiter levels applied to the chaotic system to stabilize natural states of the system. The limiter sequence is clocked by the natural return time of the chaotic system such that the oscillator sees a new limiter level for each peak return. We demonstrate control of period-8 and period-34 unstable periodic orbits in a low-frequency circuit and provide evidence that the control perturbations are minimal. We also demonstrate control of an arbitrary waveform by replaying a sequence captured from the uncontrolled oscillator, achieving a form of delayed self-synchronization. Finally, we discuss the use of dynamic limiting for high-frequency chaos communications. (c) 2002 American Institute of Physics.

Controlling dynamics in spatially extended systems

Spatially extended systems exhibit a variety of spatiotemporal dynamics--from stable to chaotic. These dynamics can change under pathological conditions and impair normal functions. Thus, having the ability to control the altered dynamics for improved functioning has the potential for wide ranging applications in real and artificial systems. Here we propose a simple and general method that can be used to target the spatiotemporal dynamics, both globally and in spatially localized regions, in either direction--i.e., towards the stable or unstable manifold-by simply changing the strength and the sign of an externally applied perturbation or pinning. The method is applicable to both chaotic and nonchaotic systems, with discrete and continuous local dynamics, and for different topologies of interactions. We also apply it to simulate an experiment on epileptogenic neuronal activity in rat hippocampal tissue [B. J. Gluckman et al., J. Neurophys. 76, 6202 (1996)]. This unified approach for differential targeting of global and local dynamics promises to be useful for systems spanning large spatial scales and having structural and functional heterogeneity.

Nonlinear-dynamical arrhythmia control in humans

Nonlinear-dynamical control techniques, also known as chaos control, have been used with great success to control a wide range of physical systems. Such techniques have been used to control the behavior of in vitro excitable biological tissue, suggesting their potential for clinical utility. However, the feasibility of using such techniques to control physiological processes has not been demonstrated in humans. Here we show that nonlinear-dynamical control can modulate human cardiac electrophysiological dynamics by rapidly stabilizing an unstable target rhythm. Specifically, in 52/54 control attempts in five patients, we successfully terminated pacing-induced period-2 atrioventricular-nodal conduction alternans by stabilizing the underlying unstable steady-state conduction. This proof-of-concept demonstration shows that nonlinear-dynamical control techniques are clinically feasible and provides a foundation for developing such techniques for more complex forms of clinical arrhythmia.