Spatiotemporal control of cardiac alternans

Electrical alternans are believed to be linked to the onset of life-threatening ventricular arrhythmias and sudden cardiac death. Recent studies have shown that alternans can be suppressed temporally by dynamic feedback control of the pacing interval. Here we investigate theoretically whether control can suppress alternans both temporally and spatially in homogeneous tissue paced at a single site. We first carry out ionic model simulations in a one-dimensional cable geometry which show that control is only effective up to a maximum cable length that decreases sharply away from the alternans bifurcation point. We then explain this finding by a linear stability analysis of an amplitude equation that describes the spatiotemporal evolution of alternans. This analysis reveals that control failure above a critical cable length is caused by the formation of standing wave patterns of alternans that are eigenfunctions of a forced Helmholtz equation, and therefore remarkably analogous to sound harmonics in an open pipe. We discuss the implications of these results for using control to suppress alternans in the human ventricles as well as to probe fundamental aspects of alternans morphogenesis. (c) 2002 American Institute of Physics.     

Bifurcation analysis of a normal form for excitable media: are stable dynamical alternans on a ring possible?

We present a bifurcation analysis of a normal form for traveling waves in one-dimensional excitable media. The normal form that has been recently proposed on phenomenological grounds is given in the form of a differential delay equation. The normal form exhibits a symmetry-preserving Hopf bifurcation that may coalesce with a saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf bifurcation for the propagation of a single pulse in a ring by means of a center manifold reduction, and for a wave train by means of a multiscale analysis leading to a real Ginzburg-Landau equation as the corresponding amplitude equation. Both the center manifold reduction and the multiscale analysis show that the Hopf bifurcation is always subcritical independent of the parameters. This may have links to cardiac alternans, which have so far been believed to be stable oscillations emanating from a supercritical bifurcation. We discuss the implications for cardiac alternans and revisit the instability in some excitable media where the oscillations had been believed to be stable. In particular, we show that our condition for the onset of the Hopf bifurcation coincides with the well known restitution condition for cardiac alternans.     

Mechanisms for initiation of cardiac discordant alternans

Electrical alternans, defined as a beat-to-beat change in the duration of the excited phase of cardiac cells, is among the known precursors of sudden cardiac death. It may appear as concordant (all the tissue presenting the same phase of oscillation) or discordant (with out-of-phase regions distributed among tissue). Spatially discordant alternans can lead to unidirectional block that initiates reentry and ventricular fibrillation. The role played by tissue heterogeneities and heart rate changes in their initiation remains, however, unclear. We study the mechanisms for initiation of spatially discordant alternans by numerical simulations of an ionic model spatially distributed in a one-dimensional cable and in an anatomical model of the rabbit heart. The effects of CV-restitution, ectopic beats, and the role of spatial gradients of electrical restitution properties are investigated. In homogeneous tissue, the origin of discordant alternans may be dynamical, through CV-restitution, or due to a localized change in the pacing period. We also find that a sudden change of stimulation rate can initiate discordant alternans in the presence of a spatial gradient of APD-restitution without necessitating CV-restitution. The mechanism of, and the conditions for, initiation are determined based on an iterated map analysis of beat to beat changes of APD. This analysis leads to the definition of a vulnerable window for initiation of discordant alternans. Moreover, the pattern of spatially discordant alternans is found to change slowly over several beats following initiation, as reflected in ECG recordings.     

Instability and spatiotemporal dynamics of alternans in paced cardiac tissue

We derive an equation that governs the spatiotemporal dynamics of small amplitude alternans in paced cardiac tissue. We show that a pattern-forming linear instability leads to the spontaneous formation of stationary or traveling waves whose nodes divide the tissue into regions with opposite phase of oscillation of action potential duration. This instability is important because it creates dynamically a heterogeneous electrical substrate for the formation of conduction blocks and the induction of fibrillation if the tissue size exceeds a fraction of the pattern wavelength. We derive an analytical expression for this wavelength as a function of three basic length scales related to dispersion and intercellular electrical coupling.     

Unidirectional pinning and hysteresis of spatially discordant alternans in cardiac tissue

Spatially discordant alternans is a widely observed pattern of voltage and calcium signals in cardiac tissue that can precipitate lethal cardiac arrhythmia. Using spatially coupled iterative maps of the beat-to-beat dynamics, we explore this pattern's dynamics in the regime of a calcium-dominated period-doubling instability at the single-cell level. We find a novel nonlinear bifurcation associated with the formation of a discontinuous jump in the amplitude of calcium alternans at nodes separating discordant regions. We show that this jump unidirectionally pins nodes by preventing their motion away from the pacing site following a pacing rate decrease but permitting motion towards this site following a rate increase. This unidirectional pinning leads to strongly history-dependent node motion that is strongly arrhythmogenic.     

Control of spiral waves and turbulent states in a cardiac model by travelling—wave perturbations

We propose a travelling-wave perturbation method to control the spatiotemporal dynamics in a cardiac model.It is numerically demonstrated that the method can successfully suppress the wave instability(alternans in action potential duration) in the one-dimensional case and converty spiral waves and turbulent states to the normal travelling wave states in the two-dimensional case.An experimental scheme is suggested which may provide a new design for a cardiac defibrillator.     

Hysteresis and bistability in the direct transition from 1:1 to 2:1 rhythm in periodically driven single ventricular cells

The transmembrane potential of a single quiescent cell isolated from rabbit ventricular muscle was recorded using a suction electrode in whole-cell recording mode. The cell was then driven with a periodic train of current pulses injected into the cell through the same recording electrode. When the interpulse interval or basic cycle length (BCL) was sufficiently long, 1:1 rhythm resulted, with each stimulus pulse producing an action potential. Gradual decrease in BCL invariably resulted in loss of 1:1 synchronization at some point. When the pulse amplitude was set to a fixed low level and BCL gradually decreased, N+1:N rhythms (N>/=2) reminiscent of clinically observed Wenckebach rhythms were seen. Further decrease in BCL then yielded a 2:1 rhythm. In contrast, when the pulse amplitude was set to a fixed high level, a period-doubled 2:2 rhythm resembling alternans rhythm was seen before a 2:1 rhythm occurred. With the pulse amplitude set to an intermediate level (i.e., to a level between those at which Wenckebach and alternans rhythms were seen), there was a direct transition from 1:1 to 2:1 rhythm as the BCL was decreased: Wenckebach and alternans rhythms were not seen. When at that point the BCL was increased, the transition back to 1:1 rhythm occurred at a longer BCL than that at which the {1:1-->2:1} transition had initially occurred, demonstrating hysteresis. With the BCL set to a value within the hysteresis range, injection of a single well-timed extrastimulus converted 1:1 rhythm into 2:1 rhythm or vice versa, providing incontrovertible evidence of bistability (the coexistence of two different periodic rhythms at a fixed set of stimulation parameters). Hysteresis between 1:1 and 2:1 rhythms was also seen when the stimulus amplitude, rather than the BCL, was changed. Simulations using numerical integration of an ionic model of a single ventricular cell formulated as a nonlinear system of differential equations provided results that were very similar to those found in the experiments. The steady-state action potential duration restitution curve, which is a plot of the duration of the action potential during 1:1 rhythm as a function of the recovery time or diastolic interval immediately preceding that action potential, was determined. Iteration of a finite-difference equation derived using the restitution curve predicted the direct {1:1<-->2:1} transition, as well as bistability, in both the experimental and modeling work. However, prediction of the action potential duration during 2:1 rhythm was not as accurate in the experiments as in the model. Finally, we point out a few implications of our findings for cardiac arrhythmias (e.g., Mobitz type II block, ischemic alternans). (c) 1999 American Institute of Physics.     

Resetting and annihilation of reentrant activity in a model of a one-dimensional loop of ventricular tissue

Resetting and annihilation of reentrant activity by a single stimulus pulse (S1) or a pair (S1-S2) of coupled pulses are studied in a model of one-dimensional loop of cardiac tissue using a Beeler-Reuter-type ionic model. Different modes of reentry termination are described. The classical mode of termination by unidirectional block, in which a stimulus produces only a retrograde front that collides with the activation front of the reentry, can be obtained for both S1 and S1-S2 applied over a small vulnerable window. We demonstrate that another scenario of termination-that we term collision block-can also be induced by the S1-S2 protocol. This scenario is obtained over a much wider range of S1-S2 coupling intervals than the one leading to a unidirectional block. In the collision block, S1 produces a retrograde front, colliding with the activation front of the pre-existing reentry, and an antegrade front propagating in the same direction as the initial reentry. Then, S2 also produces an antegrade and a retrograde front. However, the propagation of these fronts in the spatial profile of repolarization left by S1 leads to a termination of the reentrant activity. More complex behaviors also occur in which the antegrade fronts produced by S1 and S2 both persist for several turns, displaying a growing alternation in action potential duration ("alternans amplification") that may lead to the termination of the reentrant activity. The hypothesis that both collision block and alternans amplification depend on the interaction between the action potential duration restitution curve and the recovery curve of conduction velocity is supported by the fact that the dynamical behaviors were reproduced using an integro-delay equation based on these two properties. We thus describe two new mechanisms (collision block and alternans amplification) whereby electrical stimulation can terminate reentrant activity. (c) 2002 American Institute of Physics.     

When more means less: a paradox BOLD response in human visual cortex

The predictions of the 'Linear Transfer Model' (LTM) have been tested only by modulating the frequency of the action potentials while keeping the size of the activated neuronal population constant. The LTM states that the blood oxygenation level-dependent contrast (BOLD) signal is directly proportional to the neuronal activity averaged over milliseconds or seconds. We examined the influence on the BOLD response, of manipulating the size of the activated neuronal population while maintaining the electrical discharge activity constant. We performed functional MR measurements on 30 awake, healthy adult volunteers (15 male and 15 female) using a flashed and reversing checkerboard. These stimuli induced the same vascular response and the same increase in the electrical discharge activity but varied in the size of the neuronal population being activated. The BOLD response measured by the extent of activation and the BOLD signal amplitude, was larger for the flashed than to the reversing checkerboard. An assessment of the local deoxyhemoglobin (HbR) concentration indicated that the neuronal activity was lower during the flashed checkerboard than the reversing checkerboard. Because the checkerboard associated with the lower neuronal activity yielded the larger number of activated voxels and the larger BOLD signal, our results run contrary to the predictions of the 'Linear Transfer Model' and for this reason we refer to them as paradoxical. Stimuli defined by luminance contrast or a chromatic contrast yielded identical results. We conclude that the 'LTM' may apply to stimuli that modulate the electrical discharge activity but not to stimuli that modulate the size of the activated neuronal population.     

Experimental control of cardiac muscle alternans

We demonstrate that alternans in small pieces of in vitro paced bullfrog (Rana Catesbeiana) myocardium can be suppressed by making minute adjustments to the pacing period in response to real time measurements of the action potential duration. Control is possible over a large range of physiological conditions over many animals and the self-referencing control protocol can automatically adjust to changes in the pacing interval. Our results suggest the feasibility of developing low-energy methods for maintaining normal cardiac function.     

Triggered alternans in an ionic model of ischemic cardiac ventricular muscle

It has been known for several decades that electrical alternans occurs during myocardial ischemia in both clinical and experimental work. There are a few reports showing that this alternans can be triggered into existence by a premature ventricular contraction. Detriggering of alternans by a premature ventricular contraction, as well as pause-induced triggering and detriggering, have also been reported. We conduct a search for triggered alternans in an ionic model of ischemic ventricular muscle in which alternans has been described recently: a one-dimensional cable of length 3 cm, containing a central ischemic zone 1 cm long, with 1 cm segments of normal (i.e., nonischemic) tissue at each end. We use a modified form of the Luo-Rudy [Circ. Res. 68, 1501-1526 (1991)] ionic model to represent the ventricular tissue, modeling the effect of ischemia by raising the external potassium ion concentration ([K(+)](o)) in the central ischemic zone. As [K(+)](o) is increased at a fixed pacing cycle length of 400 ms, there is first a transition from 1:1 rhythm to alternans or 2:2 rhythm, and then a transition from 2:2 rhythm to 2:1 block. There is a range of [K(+)](o) over which there is coexistence of 1:1 and 2:2 rhythms, so that dropping a stimulus from the periodic drive train during 1:1 rhythm can result in the conversion of 1:1 to 2:2 rhythm. Within the bistable range, the reverse transition from 2:2 to 1:1 rhythm can be produced by injection of a well-timed extrastimulus. Using a stimulation protocol involving delivery of pre- and post-mature stimuli, we derive a one-dimensional map that captures the salient features of the results of the cable simulations, i.e., the {1:1-->2:2-->2:1} transitions with {1:1<-->2:2} bistability. This map uses a new index of the global activity in the cable, the normalized voltage integral. Finally, we put forth a simple piecewise linear map that replicates the {1:1<-->2:2} bistability observed in the cable simulations and in the normalized voltage integral map. (c) 2002 American Institute of Physics.     

An effect of temporal asymmetry on loudness

A set of experiments was conducted to examine the loudness of sounds with temporally asymmetric amplitude envelopes. Envelopes were generated with fast-attack/slow-decay characteristics to produce F-S (or "fast-slow") stimuli, while temporally reversed versions of these same envelopes produced corresponding S-F ("slow-fast") stimuli. For sinusoidal (330-6000 Hz) and broadband noise carriers, S-F stimuli were louder than F-S stimuli of equal energy. The magnitude of this effect was sensitive to stimulus order, with the largest differences between F-S and S-F loudness occurring after exposure to a preceding F-S stimulus. These results are not compatible with automatic gain control, power-spectrum models of loudness, or predictions obtained using the auditory image model [Patterson et al., J. Acoust. Soc. Am. 98, 1890-1894 (1995)]. Rather, they are comparable to phenomena of perceptual constancy, and may be related to the parsing of auditory input into direct and reverberant sound.     

Understanding cardiac alternans: a piecewise linear modeling framework

Cardiac alternans is a beat-to-beat alternation in action potential duration (APD) and intracellular calcium (Ca(2+)) cycling seen in cardiac myocytes under rapid pacing that is believed to be a precursor to fibrillation. The cellular mechanisms of these rhythms and the coupling between cellular Ca(2+) and voltage dynamics have been extensively studied leading to the development of a class of physiologically detailed models. These have been shown numerically to reproduce many of the features of myocyte response to pacing, including alternans, and have been analyzed mathematically using various approximation techniques that allow for the formulation of a low dimensional map to describe the evolution of APDs. The seminal work by Shiferaw and Karma is of particular interest in this regard [Shiferaw, Y. and Karma, A., "Turing instability mediated by voltage and calcium diffusion in paced cardiac cells," Proc. Natl. Acad. Sci. U.S.A. 103, 5670-5675 (2006)]. Here, we establish that the key dynamical behaviors of the Shiferaw-Karma model are arranged around a set of switches. These are shown to be the main elements for organizing the nonlinear behavior of the model. Exploiting this observation, we show that a piecewise linear caricature of the Shiferaw-Karma model, with a set of appropriate switching manifolds, can be constructed that preserves the physiological interpretation of the original model while being amenable to a systematic mathematical analysis. In illustration of this point, we formulate the dynamics of Ca(2+) cycling (in response to pacing) and compute the properties of periodic orbits in terms of a stroboscopic map that can be constructed without approximation. Using this, we show that alternans emerge via a period-doubling instability and track this bifurcation in terms of physiologically important parameters. We also show that when coupled to a spatially extended model for Ca(2+) transport, the model supports spatially varying patterns of alternans. We analyze the onset of this instability with a generalization of the master stability approach to accommodate the nonsmooth nature of our system.     

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.     

A method for estimating the parameters of electrodynamic drivers in thermoacoustic coolers

The electroacoustic efficiency of high-power actuators used in thermoacoustic coolers may be estimated using a linear model involving a combination of six parameters. A method to identify these equivalent driver parameters from measured total electrical impedance and velocity-voltage transfer function data was developed. A commercially available, moving-magnet driver coupled to a functional thermoacoustic cooler was used to demonstrate the procedure experimentally. The method, based on linear electrical circuit theory, allowed for the possible frequency and amplitude dependence of the driver parameters to be estimated. The results demonstrated that driver parameters measured in vacuo using this method can be used to predict the driver efficiency and performance for operating conditions which may be encountered under load.     

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.     

Spatiotemporal dynamics of damped propagation in excitable cardiac tissue

Compared to steadily propagating waves (SPW), damped waves (DW), another solution to the nonlinear wave equation, are seldom studied. In cardiac tissue after electrical stimulation in an SPW wake, we observe DW with diminished amplitude and velocity that either gradually decrease as the DW dies, or exhibit a sharp amplitude increase after a delay to become an SPW. The cardiac DW-SPW transition is a key link in understanding defibrillation and stimulation close to the refractory period, and is ideal for a general study of DW dynamics.     

Estimating bottlenose dolphin (Tursiops truncatus) hearing thresholds from single and multiple simultaneous auditory evoked potentials

Hearing thresholds were estimated in four bottlenose dolphins by measuring auditory evoked responses to single and multiple sinusoidal amplitude modulated tones. Subjects consisted of two males and two females with ages from 4 to 22 years. Testing was conducted in air using a "jawphone" transducer to couple sound into each subject's lower right jaw. Carrier frequencies ranged from 10 to 160 kHz in one-half octave steps. Amplitude modulated stimuli were presented individually and as the sum of four, five, and nine simultaneous tones with unique carrier and modulation frequencies. Evoked potentials were noninvasively recorded using surface electrodes embedded in silicon suction cups. The presence or absence of an evoked response at each modulation frequency was assessed by calculating the magnitude-squared coherence from the frequency spectra of the recorded sweeps. All subjects exhibited traditional "U-shaped" audiograms with upper cutoff frequencies above 113 kHz. The time required for threshold estimates ranged from 23 to 37 min for single stimuli to 5-9 min for nine simultaneous stimuli. Agreement between thresholds estimated from single stimuli and multiple, simultaneous stimuli was generally good, indicating that multiple stimuli may be used for quick hearing assessment when time is limited.     

Electrical alternans and spiral wave breakup in cardiac tissue

This paper reports the results of a theoretical investigation of spiral wave breakup in model equations of action potential propagation in cardiac tissue. A general formulation of these equations is described in which arbitrary experimentally determined restitution and dispersion curves can in principle be fitted. Spiral wave behavior is studied in two-dimension as a function of a parameter Re which controls the steepness of the restitution curve at short diastolic intervals. Spiral breakup is found to occur when the minimum period T(min), below which a periodically stimulated tissue exhibits alternans in action potential duration, exceeds by a finite amount the spiral rotation period T(S). At this point, oscillations in action potential duration are of sufficiently large amplitude to cause a spontaneous conduction block to form along the wavefront. The latter occurs closer to the initiation point of reentry (spiral tip) with increasing steepness and, hence, in smaller tissue sizes. Spiral breakup leads to a spatially disorganized wave activity which is always transient, except for tissues larger than some minimum size and within a very narrow range of Re which increases with dispersion.     

On the role of the amplitude envelope for the perception of [b] and [w]

This study investigated the role of the amplitude envelope in the vicinity of consonantal release in the perception of the stop-glide contrast. Three sets of acoustic [b-w] continua, each in the vowel environments [a] and [i], were synthesized using parameters derived from natural speech. In the first set, amplitude, formant frequency, and duration characteristics were interpolated between exemplar stop and glide endpoints. In the second set, formant frequency and duration characteristics were interpolated, but all stimuli were given a stop amplitude envelope. The third set was like the second, except that all stimuli were given a glide amplitude envelope. Subjects were given both forced-choice and free-identification tasks. The results of the forced-choice task indicated that amplitude cues were able to override transition slope, duration, and formant frequency cues in the perception of the stop-glide contrast. However, results from the free-identification task showed that, although presence of a stop amplitude envelope turned all stimuli otherwise labeled as glides to stops, the presence of a glide amplitude envelope changed stimuli labeled otherwise as stops to fricatives rather than to glides. These results support the view that the amplitude envelope in the vicinity of the consonantal release is a critical acoustic property for the continuant / noncontinuant contrast. The results are discussed in relation to a theory of acoustic invariance.