Feedback-mediated control of spiral waves

Zykov, V. S.; Engel, Harald

doi:10.1016/j.physd.2004.10.001

Cited by 50 publications

(24 citation statements)

References 61 publications

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“…For example, in a square domain, equilibria of both systems are found on a lattice of spacing Λ 2. Furthermore, works by Zykov et al [54][55][56] revealed that the velocity fields for the spiral wave drift in reaction-diffusion systems in many cases have atypical phase portraits. For example, in [56], unusual equilibrium manifolds of attracting lines have been observed for a system with two-point feedback control.…”

Section: Discussionmentioning

confidence: 99%

A theory for spiral wave drift in reaction-diffusion-mechanics systems

Dierckx

Arens

et al. 2015

New J. Phys.

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Reaction-diffusion mechanics (RDM) systems describe a wide range of practically important phenomena where deformation substantially affects wave and vortex dynamics. Here, we develop the first theory to describe the dynamics of rotating spiral waves in RDM systems, combining response function theory with a mechanical Green's function. This theory explains the mechanically-induced drift of spiral waves as a resonance phenomenon, and it can predict the drift trajectories and the final attractors from measurable characteristics of the system. Theoretical predictions are confirmed by numerical simulations. The results can be applied to cardiac tissue, where the drift of spiral waves is an important factor in determining different types of cardiac arrhythmias.In this work we present an analytical approach to study spiral wave dynamics in RDM systems. We combine response function theory [23] with a Green's function formalism to account for mechanical influence, and we derive an equation for spiral wave drift. The MEF-induced drift is reduced to a resonant forcing problem in the rotating frame of the spiral: during its rotation, the spiral wave perceives a time-varying perturbation since the domain boundaries are not stationary in the spiral's frame of reference. The resonant component of this boundary-induced forcing yields the net spiral drift.Our theoretical predictions are compared to numerical simulations. We find the relative angle and magnitude of the spiral wave's drift and identify the spatial attractors of the system. Some of them, such as the center of the medium, have already been reported in numerical simulations [19]. Using this theory, we also find new regimes and attractors that also are confirmed by simulations. For example, our theory predicts that the center of the domain can also be repulsive, and that multiple stable dynamical attractors may coexist. Although different from previous findings, these predictions are confirmed by numerical simulations.The developed analytical approach allows us to generalize the numerical results, as our analytical findings are based on an Archimedean description of spiral wave geometry, which is common for all types of excitable media. ModelThe reaction and MEF parts of our model for cardiac tissue are as in [18,20,24,25], supplemented here with the Navier-Cauchy equilibrium equations from linear elasticity to facilitate analytical calculations. In a twodimensional medium with Cartesian material coordinates (x, y), the transmembrane voltage u and recovery variable v are evolved according to modified Aliev-Panfilov kinetics [26], as used in [18]:

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Section: Discussionmentioning

confidence: 99%

A theory for spiral wave drift in reaction-diffusion-mechanics systems

Dierckx

Arens

et al. 2015

New J. Phys.

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“…Next, we revisit the task to extinguish a spiral wave by controlling its tip dynamics such that the whole pattern moves out of the spatial domain towards the Neumann boundaries [9,21,54]. To this goal, following Ex.…”

Section: Example 2: Optimal and Sparse Optimal Position Controlmentioning

confidence: 99%

Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems

Ryll

Löber

Martens

et al. 2016

Understanding Complex Systems

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Abstract. This work deals with the position control of selected patterns in reactiondiffusion systems. Exemplarily, the Schlögl and FitzHugh-Nagumo model are discussed using three different approaches. First, an analytical solution is proposed. Second, the standard optimal control procedure is applied. The third approach extends standard optimal control to so-called sparse optimal control that results in very localized control signals and allows the analysis of second order optimality conditions.

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Engineering of Synchronization and Clustering of a Population of Chaotic Chemical Oscillators

et al. 2011

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Description, control, and design of weakly interacting dynamical units are challenging tasks that play a central role in many physical, chemical, and biological systems. [1][2][3][4][5] Control of temporal and spatial variations of reaction rates is especially daunting when the dynamical units exhibit deterministic chaotic oscillations that are sensitive to initial conditions and are long-term unpredictable. Control of spatiotemporal chaos by means of suppression of spiralwave turbulence to standing waves, cluster patterns, and uniform periodic oscillations, in the catalytic CO oxidation on a Pt (110) single-crystal surface has been successfully achieved with linear delayed feedback of the carbon monoxide partial pressure.[6] The same delayed global feedback methodology has also been applied to induce various dynamical clusters in electrochemical corrosion processes [7] and in the light-sensitive Belousov-Zhabotinsky (BZ) reaction.[8] A fundamental problem of tuning the complex structure of chaotic systems is choosing a feedback scheme and appropriate control parameters to steer the system to a desired structure of spatiotemporal reactivity. This objective requires the use of simple yet accurate models of nonlinear processes, their interactions, and responses to external perturbations. When the individual units exhibit periodic oscillatory behavior, the effect of weak feedback and coupling can be described by phase models [4,9] in which the state of each unit is represented by a single variable, the phase of oscillations.Typical phase-synchronized behavior of a population of periodic oscillators is dynamical differentiation (clustering) where the elements form groups (clusters) in which the phases are identical at all times but different from the phases of the elements in the other groups.[9]Phase models successfully described synchronization and dynamical differentiation (clustering) of oscillatory electrochemical [10] and BZ bead experiments.[11] Experiment-based phase models can also be used for synchronization engineering of periodic oscillators [12] where a carefully designed external feedback is applied to dial-up complex dynamical structures such as stable and sequentially visited dynamical cluster patterns and desynchronization.Herein, we consider populations of chaotic oscillators that exhibit strong cycle-to-cycle period variations. We show that the feedback scheme can be designed to obtain dynamical synchronization states of phase coherent chaotic oscillators in a quantitative manner. The method relies on the flexibility and versatility of synchronization engineering (originally developed for periodic oscillators [12] ) and on the observation that the long-term phase dynamics of chaotic oscillators is often similar to that of noisy periodic oscillators. [13] An experimental system of uncoupled, phase-coherent, chaotic oscillators was constructed by using an electrochemical cell consisting of 64 Ni working electrodes (99.98 % pure), a Pt mesh counter electrode, and a Hg/Hg 2 SO 4 /K 2 SO 4 (saturated) r...

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Feedback-mediated control of spiral waves

Cited by 50 publications

References 61 publications

A theory for spiral wave drift in reaction-diffusion-mechanics systems

A theory for spiral wave drift in reaction-diffusion-mechanics systems

Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems

Engineering of Synchronization and Clustering of a Population of Chaotic Chemical Oscillators

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