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An attempt is made to understand in what manner, and to what extent, the notion of phase, useful in extended media exhibiting cyclical local dynamics, is applicable to the behavior of waves propagating in excitable media. Computer simulations of two situations are examined in each of the two types of media, and the results analyzed. One of these two situations involves a new scroll wave generation mechanism which produces two scroll waves rotating in the same direction. For the simulations conducted for this study, topological arguments involving the phase are relevant for exctable media, but applicaion of the arguments requires careful consideration of the details of the dynamics. Several examples of these principles, as observed in the simulations, are described. One of these examples also leads to an additional observation--that crossing scroll wave filaments probably can not "reconnect" to form essentially parallel filaments in the case of FizHugh-Nagumo dynamics.
The purpose of this study was to determine whether biphasic restitution of action potential duration (APD) in ventricular muscle permits the development of complex dynamic behavior. Such behavior is expected because of the steep ascending slope of restitution and the presence of a maximum. Action potentials recorded from strips of epicardial muscle in which biphasic APD restitution occurred demonstrated a characteristic pattern of phase locking during progressive shortening of the pacing cycle length. 1:1 locking was replaced by irregular dynamics, which in turn was replaced by higher order periodic behavior (eg, 8:8 locking), then by 2:2 locking, and finally by 2:1 locking. Similar patterns of dynamic behavior were produced in a computer model by using a piecewise linear approximation of biphasic APD restituion. Features of APD restitution that were critical determinants of irregular dynamics included the slopes of the ascending and the nonmonotonic regions. These results suggest that rate-related alterations of APD and refractoriness may be affected signicantly by small nonmonotonicities in APD restitution.
The contribution of cumulative changes in action potential duration (memory) to complex cellular electrophysiological behavior was investigated in canine cardiac Purkinje fibers. Complex behavior induced during constant pacing was caused by reciprocal interactions between the the time to full repolarization (TFR), where TFR = response duration + latency, and the diastolic interval (DI). The relationship between TFR and the preceding DI during complex behavior differed from that obtained using a standard restitution protocol. In particular, higher-order periodicities and chaos were produced in fibers in which the restitution curve lacked the prerequisites for such behavior. To investigate whether shifts in the restitution curve might be expected during rapid pacing, the relationship betweeen TFR of a test response (TFRn+1) and the immediately preceding response (TFRn) was determined. For any fixed DIn, reduction of TFRn from 240 to 130 ms was accompanied by a corresponding reducion of TFRn+1, whereas as TFRn was reduced further to 120 msec, TFRn+1 increased. Because of the dependence of TFRn+1 on TFRn (memory) and on the preceding DIn (restitution), the slope of the low-dimensional relationship between TFRn+1 and DIn at a constant pacing cycle length depended on the slopes of the restitution and memory functions. These results suggest that rapid accumulation and dissipation of memory may contribute importantly to complex electrical behavior in cardiac tissue.
A series of related new models for the local dynamics of cardiac tissue is introduced. The models are based on a simple memory-like quantity that is used to determine the relationship among the durations and amplitudes of the stimulated action potentials. The first of these models produces period-doubling and chaos, consistent with constant pacing experiments, when standard restitution dynamics would predict stability of the primary 1:1 pattern. Analysis of the associated one-dimensional map suggests how various physiological parameters affect the period-doubling sequence. Many of these relationships have been observed in experiments. The remaining models extend the formalism of the first to account for the Hopf bifurcation of 2:2 patterns observed in experiments. One of these models reproduces the bifurcation sequence, 1:1, 2:2, Hopf bifurcation of 2:2, 2:2, and 2:1 seen in experiments as the pacing interval is decreased. The models clarify the dynamics involved in determining the amplitudes and durations of successive action potentials. Results from these models together with comparison with the experiment strongly suggest that quantities with time constants on the order of 50 and 400 msec exist and affect action potential formation in heart tissue.
See some animations accompanying the paper.
See a Java applet of the two-memory model and its reproduction of the 2:2 Hopf bifurcation.
Background: Previous studies have suggested that irregular T wave morphologies
are associated with an increased risk of sudden death. However, automated characterization
of T wave abnormalities has been hampered by the lack of suitable analysis techniques.
In this study, we tested a new method of T wave analysis in German shepherd dogs
with inherited ventricular arrhythmias and sudden death.
Methods: Sets of 24-hour ambulatory ECG recordings obtained from unaffected
(n=6) and affected (n=5) dogs were digitized, automatically annotated to label each
R wave, and placed into a matrix with the R waves aligned. A vector quantization
algorithm separated the QRS-T complexes into classes according to T wave morphology.
The existence of notched T waves was determined by assessing the number of zero crossings
of the first derivative during the T wave.
Results: The duration of the QT interval was similar in affected and unaffected
dogs (182 +- 14 ms vs 176 +- 16 ms, respectively). However, T wave morphology differed
between the two groups. Specifically, affected dogs had a higher percentage of notched
T waves than unaffected dogs (41.6% +- 10.8% vs 5.0% +- 1.2%, respectively). Notched
T waves did not appear at all times of day, nor were they present in all leads.
Conclusion: Vector quantization and first derivative analyses were feasible
and effective methods for detecting T wave abnormalities associated with the development
of ventricular arrhythmias. These methods ultimately may be useful for risk stratification
of patients susceptible to ventricular arrhythmias and sudden death.
Cardiac action potential propagation is modeled with a two-dimensional simulation that includes full Luo-Rudy (LRd) ion channel dynamics. Preliminary results show that the L-type calcium current is dependent on the curvature of the propagating wavefront, and that, in the case of spiral wave reentry, calcium-induced calcium release exhibits complex temporal behavior including occasional large releases which occur on the boundary of the spiral wave core. The core region also exhibits significantly lower intracellular calcium concentrations. The computer model uses a simple forward Euler method in combination with a variable, multiple timestep method. The algorithm thereby allows the desktop modeling of a reasonable size spatial region while simultaneously providing for a detailed representation of the action potential upstroke. Attempts to reduce the runtime of the code, including parallelization issues, are also discussed.
(No abstract)
The skills of a computational physicist are shown to be useful in the seemingly distant field of cardiac electrophysiology. The propagation of cardiac action potentials and their ability to form spiral waves are easily understood at a basic level when standard concepts from the theory of partial differential equations are applied. Certain types of local behavior in cardiac cells may be characterized in terms of familiar ideas from nonlinear dynamics. The design of computer simulations of action potential propagation is facilitated by adapting numerical concepts routinely applied by computational physicists. These concepts must be modified and combined with computer structures standard in computer science to handle the timescale and spatial scale problems unique to the action potential propagation problem. The resulting simulation shows how details of ion channel dynamics leads to modification of the wavefront in regions of wavefront curvature and complex dynamics in spiral wave core regions.
In a sufficiently short reentry pathway, the excitation wavefront (head) propagates into tissue which is partially refractory (tail) from the previous action potential (AP). We incorporate a detailed mathematical model of the ventricular myocyte into a one-dimensional closed pathway to investigate effects of head-tail interaction and ion accumulation on the dynamics of reentry. Results: (1) A high degree of head-tail interaction produces oscillations in AP properties. (2) Ca2+ transient oscillations are in phase with AP duration (APD) oscillations. (3) As the wavefront propagates around the pathway, AP properties undergo periodic oscillations which produce complicated oscillations at a single site. (4) Depending on the degree of head-tail interaction, [Na+]i accumulation either stabilizes reentry or increases the CL oscillations which promote termination of reentry.
Electrical alternans, the alternation in action potential morphology, has been suggested as an important cause of potentially dangerous cardiac rhythm disorders. Previous studies have developed alternans control strategies based on the dynamics of the relationship between action potential duration and the previous diastolic interval.We demonstrate that alternans in a single cardiac cell can also be controlled by directly modifying the underlying ion channel dynamics. Surprisingly, we find that, for a detailed canine ventricular cell model, the best time to apply the control stimulus is not during the repolarization phase of the action potential, but rather during the early plateau phase, when the charge requirements are two orders of magnitude smaller. Computer simulations show a single control stimulus applied during the early plateau can completely eliminate small-amplitude alternans, while a small number of stimuli can rapidly extinguish large-amplitude alternans. We have also developed an effective control algorithm that uses only the membrane potential as control input and requires no prior detailed knowledge of the cell dynamics. The study suggests that control strategies based on ion channel dynamics can provide newdirections for the development of algorithms intended to control dangerous cardiac rhythm disorders.
The detailed processes involved in spiral wave breakup, believed to be one major mechanism by which tachycardia evolves into fibrillation, is still poorly understood. This has rendered difficult the proper design of an efficient and practical control stimulus protocol to eliminate such events. In order to gain new insights into the underlying electrophysiological and dynamical mechanisms of breakup, we applied linear perturbation theory to a steadily rotating spiral wave in two spatial dimensions. The tissue was composed of cells modeled using the Fenton-Karma equations whose parameters were chosen to emphasize alternans as a primary mechanism for breakup. Along with one meandering mode, not just one but several stable and unstable alternans modes were found with differing growth rates, frequencies and spatial structures. As the conductance of the fast inward current was increased, the instability of the modes increased, consistent with increased meandering and propensity for spiral breakup in simulations. We also explored a promising new approach, based on the theory, for the design of an energy efficient electrical stimulus protocol to control spiral wave breakup. The novelty lies in addressing the problem directly at the ion channel level and taking advantage of the inherent two dimensional nature of the rotating wave. With the help of the eigenmode method, we were able to calculate the exact timing and amplitude of the stimulus, and locate it optimally to maximize efficiency. The analysis led to a special-case example that demonstrated that a single, properly timed stimulus can have a global effect, suppressing all growing alternans modes over the entire tissue, thus inhibiting spiral wave breakup.
(No abstract)
N. F. Otani, Theory of action potential wave block at-a-distance in the heart, Physical Review E 75, 021910 (2007).
Propagation failure of an action potential wave at a finite distance from its source (so-called type-II block) may cause spiral wave formation or wave breakup in the heart, phenomena that are believed to underlie lethal and nonlethal heart rhythm disorders. In this study, we develop a sufficient condition for this type of block in a homogeneous, spatially one-dimensional system. Using a topological argument, we find that type-II block of a wave will always occur when launched within a finite range of times if the velocity of the trailing edge of the preceding wave, as measured at the stimulus site, is smaller than the velocity of a wave launched with the minimum diastolic interval (DI) for which propagation is possible. This “blocking condition” is robust, remaining valid even when memory and waveback electrotonic effects are included. The condition suggests that type-II block is greatly facilitated when waves are initiated at irregular intervals in time such that (1) the velocities of consecutive waves are as different as possible and (2) the DIs preceding each wave fall on the steeply sloped portion of the action potential duration restitution curve as often as possible. The set of timing intervals between stimuli that are predicted by the blocking condition to produce block are found to be consistent with these guidelines, and also to agree well with a coupled-maps computer simulation model, for the case of waves launched by four rapidly and irregularly timed stimuli.
R. F. Gilmour Jr., A. R. Gelzer, N. F. Otani, Cardiac electrical dynamics: maximizing dynamical heterogeneity, Journal of Electrocardiology 40, S51-S55 (2007).
The relationships between key features of the cardiac electrical activity, such as electrical restitution, discordant alternans, wavebreak, and reentry, and the onset of ventricular tachyarrhythmias have been characterized extensively under the condition of constant rapid pacing. However, it is unlikely that this scenario applies directly to the clinical situation, where the induction of ventricular tachycardia (VT) typically is associated with the interruption of normal cardiac rhythm by several premature beats. To address this issue, we have developed a general theory to explain why specific patterns of premature stimuli increase dynamic heterogeneity of repolarization and precipitate conduction block. The theory predicts that conduction block is caused by (1) creation of a spatial gradient in diastolic interval (DI) by waves traveling at slightly different velocities (ie, conduction velocity dispersion) and (2) amplification of the spatial gradient in DI over subsequent action potentials, secondary to a strong dependence of action potential duration on the preceding DI (ie, a steep action potential duration restitution function). Tests of this theory have been conducted in computer models of homogeneous tissue, where increased spatial dispersion of repolarization during premature stimulation can be attributed solely to the development of dynamical heterogeneity, and in a canine model exhibiting spontaneously occurring VT and sudden death. Our results thus far indicate that the probability of inducing ventricular fibrillation (VF) in the animal model is highest for those sequences predicted to cause conduction block in the computer model. An understanding of the mechanisms underlying these observations will help to identify key electrical phenomena in the onset of VT and fibrillation. Drug and electrical therapies can then be improved by targeting these specific phenomena.
A fast, variable timestep method for solving parabolic equations is described. For large timesteps, the method has strong stability properties derived from an unusual source: the randomness in the order in which simulation cells are advanced in time. The method is particularly well-suited for excitable systems and other systems exhibiting large temporal and spatial timescale variations. For typical cardiac excitation problems, the method is seven to 11 times as fast, and requires less than 1/115th as many cell updates, as variable- and fixed-timestep forward Euler methods, respectively.
Basic plasma and fusion plasma abstracts.
Ionospheric, magnetospheric, and solar abstracts.
Computational plasma physics abstracts.
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