Localization of response functions of spiral waves in the FitzHugh-Nagumo system

Dynamics of spiral waves in perturbed, e. g. slightly inhomogeneous or subject to a small periodic external force, two-dimensional autowave media can be described asymptotically in terms of Aristotelean dynamics, so that the velocities of the spiral wave drift in space and time are proportional to the forces caused by the perturbation. The forces are defined as a convolution of the perturbation with the spiral's Response Functions, which are eigenfunctions of the adjoint linearised problem. In this paper we find numerically the Response Functions of a spiral wave solution in the classic excitable FitzHugh-Nagumo model, and show that they are effectively localised in the vicinity of the spiral core.Comment: 11 pages, 2 figure

Asymptotic properties of mathematical models of excitability

We analyse small parameters in selected models of biological excitability, including Hodgkin-Huxley (1952) model of nerve axon, Noble (1962) model of heart Purkinje fibres, and Courtemanche et al. (1998) model of human atrial cells. Some of the small parameters are responsible for differences in the characteristic timescales of dynamic variables, as in the traditional singular perturbation approaches. Others appear in a way which makes the standard approaches inapplicable. We apply this analysis to study the behaviour of fronts of excitation waves in spatially-extended cardiac models. Suppressing the excitability of the tissue leads to a decrease in the propagation speed, but only to a certain limit; further suppression blocks active propagation and leads to a passive diffusive spread of voltage. Such a dissipation may happen if a front propagates into a tissue recovering after a previous wave, e.g. re-entry. A dissipated front does not recover even when the excitability restores. This has no analogy in FitzHugh-Nagumo model and its variants, where fronts can stop and then start again. In two spatial dimensions, dissipation accounts for break-ups and self-termination of re-entrant waves in excitable media with Courtemanche et al. (1998) kinetics.Comment: 15 pages, 8 figures, to appear in Phil Trans Roy Soc London

Orbital Motion of Spiral Waves in Excitable Media

Spiral waves in active media react to small perturbations as particle-like objects. Here we apply the asymptotic theory to the interaction of spiral waves with a localized inhomogeneity, which leads to a novel prediction: drift of the spiral rotation centre along circular orbits around the inhomogeneity. The stationary orbits have alternating stability and fixed radii, determined by the properties of the bulk medium and the type of inhomogeneity, while the drift speed along an orbit depends on the strength of the inhomogeneity. Direct simulations confirm the validity and robustness of the theoretical predictions and show that these unexpected effects should be observable in experiment

Drift of scroll waves in thin layers caused by thickness features: asymptotic theory and numerical simulations

Copyright © 2015 American Physical SocietyA scroll wave in a very thin layer of excitable medium is similar to a spiral wave, but its behavior is affected by the layer geometry. We identify the effect of sharp variations of the layer thickness, which is separate from filament tension and curvature-induced drifts described earlier. We outline a two-step asymptotic theory describing this effect, including asymptotics in the layer thickness and calculation of the drift of so-perturbed spiral waves using response functions. As specific examples, we consider drift of scrolls along thickness steps, ridges, ditches, and disk-shaped thickness variations. Asymptotic predictions agree with numerical simulations.FWO-Flanders (Belgium)Engineering and Physical Sciences Research Council (EPSRC

Localization of response functions of spiral waves in the FitzHugh-Nagumo system

Preprint of an article submitted for consideration and subsequently published in International Journal of Bifurcation and Chaos © 2005 copyright World Scientific Publishing Company. http://www.worldscientific.com/worldscinet/ijbcDynamics of spiral waves in perturbed, e. g. slightly inhomogeneous or subject to a small periodic external force, two-dimensional autowave media can be described asymptotically in terms of Aristotelean dynamics, so that the velocities of the spiral wave drift in space and time are proportional to the forces caused by the perturbation. The forces are defined as a convolution of the perturbation with the spiral's Response Functions, which are eigenfunctions of the adjoint linearised problem. In this paper we find numerically the Response Functions of a spiral wave solution in the classic excitable FitzHugh-Nagumo model, and show that they are effectively localised in the vicinity of the spiral core

Filament tension and phase-locked drift of meandering scroll waves

Rotating scroll waves are self-organising patterns which are found in many oscillating or excitable systems. Here we show that quasi-periodic (meandering) scroll waves, which include the rotors that organise cardiac arrhythmias, exhibit filament tension when averaged over the meander cycle. With strong filament curvature or medium thickness gradients, however, scroll wave dynamics are governed by phase-locked drift instead of filament tension. Our results are validated in computational models of cycloidal meander and a cardiac tissue model with linear core.Comment: accepted for publication in Physical Review Letters (December 2017

Low energy defibrillation in human cardiac tissue: a simulation study.

Copyright © 2009 Biophysical SocietyJournal ArticleWe aim to assess the effectiveness of feedback-controlled resonant drift pacing as a method for low energy defibrillation. Antitachycardia pacing is the only low energy defibrillation approach to have gained clinical significance, but it is still suboptimal. Low energy defibrillation would avoid adverse side effects associated with high voltage shocks and allow the application of implantable cardioverter defibrillator (ICD) therapy, in cases where such therapy is not tolerated today. We present results of computer simulations of a bidomain model of cardiac tissue with human atrial ionic kinetics. Reentry was initiated and low energy shocks were applied with the same period as the reentry, using feedback to maintain resonance. We demonstrate that such stimulation can move the core of reentrant patterns, in the direction that depends on the location of the electrodes and the time delay in the feedback. Termination of reentry is achieved with shock strength one-order-of-magnitude weaker than in conventional single-shock defibrillation. We conclude that resonant drift pacing can terminate reentry at a fraction of the shock strength currently used for defibrillation and can potentially work where antitachycardia pacing fails, due to the feedback mechanisms. Success depends on a number of details that these numerical simulations have uncovered

Control of scroll wave turbulence using resonant perturbations

Turbulence of scroll waves is a sort of spatio-temporal chaos that exists in three-dimensional excitable media. Cardiac tissue and the Belousov-Zhabotinsky reaction are examples of such media. In cardiac tissue, chaotic behaviour is believed to underlie fibrillation which, without intervention, precedes cardiac death. In this study we investigate suppression of the turbulence using stimulation of two different types, "modulation of excitability" and "extra transmembrane current". With cardiac defibrillation in mind, we used a single pulse as well as repetitive extra current with both constant and feedback controlled frequency. We show that turbulence can be terminated using either a resonant modulation of excitability or a resonant extra current. The turbulence is terminated with much higher probability using a resonant frequency perturbation than a non-resonant one. Suppression of the turbulence using a resonant frequency is up to fifty times faster than using a non-resonant frequency, in both the modulation of excitability and the extra current modes. We also demonstrate that resonant perturbation requires strength one order of magnitude lower than that of a single pulse, which is currently used in clinical practice to terminate cardiac fibrillation. Our results provide a robust method of controlling complex chaotic spatio-temporal processes. Resonant drift of spiral waves has been studied extensively in two dimensions, however, these results show for the first time that it also works in three dimensions, despite the complex nature of the scroll wave turbulence.Comment: 13 pages, 12 figures, submitted to Phys Rev E 2008/06/13. Last version: 2008/09/18, after revie

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

This work deals with the position control of selected patterns in reaction-diffusion systems. Exemplarily, the Schl\"{o}gl 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.Comment: 22 pages, 3 figures, 2 table