Adjoint eigenfunctions of temporally recurrent single-spiral solutions in a simple model of atrial fibrillation

Annihilation dynamics during spiral defect chaos revealed by particle models

Pair-annihilation events are ubiquitous in a variety of spatially extended systems and are often studied using computationally expensive simulations. Here, we develop an approach in which we simulate the pair-annihilation of spiral wave tips in cardiac models using a computationally efficient particle model. Spiral wave tips are represented as particles with dynamics governed by diffusive behavior and short-ranged attraction. The parameters for diffusion and attraction are obtained by comparing particle motion to the trajectories of spiral wave tips in cardiac models during spiral defect chaos. The particle model reproduces the annihilation rates of the cardiac models and can determine the statistics of spiral wave dynamics, including its mean termination time. We show that increasing the attraction coefficient sharply decreases the mean termination time, making it a possible target for pharmaceutical intervention.

Optimal entrainment for removal of pinned spiral waves

Cardiac fibrillation is caused by self-sustaining spiral waves that occur in the myocardium, some of which can be pinned to anatomical obstacles, making them more difficult to eliminate. A small electrical stimulation is often sufficient to unpin these spirals but only if it is applied during the vulnerable unpinning window. Even if these unpinning windows can be inferred from data, when multiple pinned spirals exist, their unpinning windows will not generally overlap. Using phase-based reduction techniques, we formulate and solve an optimal control problem to yield a time-varying external voltage gradient that can synchronize a collection of spiral waves that are pinned to a collection of heterogeneous obstacles. Upon synchronization, the unpinning windows overlap so that they can be simultaneously unpinned by applying an external voltage gradient pulse at an appropriate moment. Numerical validation is presented in bidomain model simulations. Results represent a proof-of-concept illustration of the proposed unpinning strategy which explicitly incorporates heterogeneity in the problem formulation and requires no real-time feedback about the system state.

High-frequency pacing of scroll waves in a three-dimensional slab model of cardiac tissue

Vortices in excitable media underlie dangerous cardiac arrhythmias. One way to eliminate them is by stimulating the excitable medium with a period smaller than the period of the vortex. So far, this phenomenon has been studied mostly for two-dimensional vortices known as spiral waves. Here we present a first study of this phenomenon for three-dimensional vortices, or scroll waves, in a slab. We consider two main types of scroll waves dynamics: with positive filament tension and with negative filament tension and show that such elimination is possible for some values of the period in all cases. However, in the case of negative filament tension for relatively long stimulation periods, three-dimensional instabilities occur and make elimination impossible. We derive equations of motion for the drift of paced filaments and identify a bifurcation parameter that determines whether the filaments orient themselves perpendicular to the impeding wave train or not.

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

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

Sensitivity of long periodic orbits of chaotic systems

The properties of long, numerically determined periodic orbits of two low-dimensional chaotic systems, the Lorenz equations and the Kuramoto-Sivashinsky system in a minimal-domain configuration, are examined. The primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite-amplitude parameter perturbations. To address this question, an inventory of thousands of orbits at least two orders of magnitude longer than the shortest admissible cycles is constructed. The expectation of period averages, Floquet exponents, and sensitivities over such set is then obtained. It is shown that all these quantities converge to a limiting value as the orbit period is increased. However, while period averages and Floquet exponents appear to converge to analogous quantities computed from chaotic trajectories, the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state, similar to observations made with other shadowing algorithms.

Observability analysis and state observer design for a cardiac ionic cell model

To gain insights into cardiac arrhythmias, researchers have developed and employed various measurement techniques, such as electrocardiography, optical mapping, and patch clamping. However, there are no measurement methods that allow simultaneous recording of all cellular quantities, including intracellular ionic concentrations and gating states, that may play an important role in arrhythmia formation. To help address this shortcoming, we applied observability analysis, a method from control theory, to the Luo-Rudy dynamic (LRd) model of a cardiac ventricular myocyte. After linearizing the time-integrated LRd model about selected periodic orbits, we computed the observability properties of the model to determine whether past system states could be reconstructed from different hypothetical sets of measurements. Under the simplifying assumption that only one dynamical variable could be measured periodically, we found that intracellular potassium concentration generally yielded the largest observability values and thus contained the most information about the dominant modes of the system. The impacts on observability of measurement timings, inter-stimulus interval length, and an alternans-promoting parameter shift were also studied. Pole-placement state observer algorithms were designed and tested in simulations for several scenarios, and we found that it is possible to infer unmeasured variables from potassium-concentration measurements, and to an extent from membrane-potential measurements, both for longer periods that represent normal rhythms and shorter periods associated with tachyarrhythmias. Our results could lead to improved data assimilation algorithms that combine model predictions with measurements to estimate quantities that are difficult or impossible to measure during in vitro experiments.

Robust approach for rotor mapping in cardiac tissue

The motion of and interaction between phase singularities that lie at the centers of spiral waves capture many qualitative and, in some cases, quantitative features of complex dynamics in excitable systems. Being able to accurately reconstruct their position is thus quite important, even if the data are noisy and sparse, as in electrophysiology studies of cardiac arrhythmias, for instance. A recently proposed global topological approach [Marcotte and Grigoriev, Chaos 27, 093936 (2017)] promises to meaningfully improve the quality of the reconstruction compared with traditional, local approaches. Indeed, we found that this approach is capable of handling noise levels exceeding the range of the signal with minimal loss of accuracy. Moreover, it also works successfully with data sampled on sparse grids with spacing comparable to the mean separation between the phase singularities for complex patterns featuring multiple interacting spiral waves.

Response function framework for the dynamics of meandering or large-core spiral waves and modulated traveling waves

In many oscillatory or excitable systems, dynamical patterns emerge which are stationary or periodic in a moving frame of reference. Examples include traveling waves or spiral waves in chemical systems or cardiac tissue. We present a unified theoretical framework for the drift of such patterns under small external perturbations, in terms of overlap integrals between the perturbation and the adjoint critical eigenfunctions of the linearized operator (i.e., response functions). For spiral waves, the finite radius of the spiral tip trajectory and spiral wave meander are taken into account. Different coordinate systems can be chosen, depending on whether one wants to predict the motion of the spiral-wave tip, the time-averaged tip path, or the center of the meander flower. The framework is applied to analyze the drift of a meandering spiral wave in a constant external field in different regimes.

Filament Tension and Phase Locking of Meandering Scroll Waves

Meandering spiral waves are often observed in excitable media such as the Belousov-Zhabotinsky reaction and cardiac tissue. We derive a theory for drift dynamics of meandering rotors in general reaction-diffusion systems and apply it to two types of external disturbances: an external field and curvature-induced drift in three dimensions. We find two distinct regimes: with small filament curvature, meandering scroll waves exhibit filament tension, whose sign determines the stability and drift direction. In the regimes of strong external fields or meandering motion close to resonance, however, phase locking of the meander pattern is predicted and observed.

Measurement and structure of spiral wave response functions

The rotating spiral waves that emerge in diverse natural and man-made systems typically exhibit a particle-like behaviour since their adjoint critical eigenmodes (response functions) are often seen to be localised around the spiral core. We present a simple method to numerically compute response functions for circular-core and meandering spirals by recording their drift response to many elementary perturbations. Although our method is computationally more expensive than solving the adjoint system, our technique is fully parallellisable, does not suffer from memory limitations and can be applied to experiments. For a cardiac tissue model with the linear spiral core, we find that the response functions are localised near the turning points of the trajectory.

Memory effects, transient growth, and wave breakup in a model of paced atrium

The mechanisms underlying cardiac fibrillation have been investigated for over a century, but we are still finding surprising results that change our view of this phenomenon. The present study focuses on the transition from normal rhythm to spiral wave chaos associated with a gradual increase in the pacing rate. While some of our findings are consistent with existing experimental, numerical, and theoretical studies of this problem, one result appears to contradict the accepted picture. Specifically we show that, in a two-dimensional model of paced homogeneous atrial tissue, transition from discordant alternans to conduction block, wave breakup, reentry, and spiral wave chaos is associated with the transient growth of finite amplitude disturbances rather than a conventional instability. It is mathematically very similar to subcritical, or bypass, transition from laminar fluid flow to turbulence, which allows many of the tools developed in the context of fluid turbulence to be used for improving our understanding of cardiac arrhythmias.

Dynamical mechanism of atrial fibrillation: A topological approach

While spiral wave breakup has been implicated in the emergence of atrial fibrillation, its role in maintaining this complex type of cardiac arrhythmia is less clear. We used the Karma model of cardiac excitation to investigate the dynamical mechanisms that sustain atrial fibrillation once it has been established. The results of our numerical study show that spatiotemporally chaotic dynamics in this regime can be described as a dynamical equilibrium between topologically distinct types of transitions that increase or decrease the number of wavelets, in general agreement with the multiple wavelets' hypothesis. Surprisingly, we found that the process of continuous excitation waves breaking up into discontinuous pieces plays no role whatsoever in maintaining spatiotemporal complexity. Instead, this complexity is maintained as a dynamical balance between wave coalescence-a unique, previously unidentified, topological process that increases the number of wavelets-and wave collapse-a different topological process that decreases their number.