Strength-duration relationship in an excitable medium

This is the final version. Available on open access from Elsevier via the DOI in this recordWe consider the strength-duration relationship in one-dimensional spatially extended excitable media. In a previous study [36] set out to separate initial (or boundary) conditions leading to propagation wave solutions from those leading to decay solutions, an analytical criterion based on an approximation of the (center-)stable manifold of a certain critical solution was presented. The theoretical prediction in the case of strength-extent curve was later on extended to cover a wider class of excitable systems including multicomponent reaction-diffusion systems, systems with non-self-adjoint linearized operators and in particular, systems with moving critical solutions (critical fronts and critical pulses) [7]. In the present work, we consider extension of the theory to the case of strength-duration curve.Engineering and Physical Sciences Research Council (EPSRC)Ministry of National Education of the Republic of TurkeyNational Science FoundationNational Institutes of Health (NIH)Gordon and Betty Moore Foundatio

Orbital movement of spiral waves

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

Evolution of spiral and scroll waves of excitation in a mathematical model of ischaemic border zone

Abnormal electrical activity from the boundaries of ischemic cardiac tissue is recognized as one of the major causes in generation of ischemia-reperfusion arrhythmias. Here we present theoretical analysis of the waves of electrical activity that can rise on the boundary of cardiac cell network upon its recovery from ischaemia-like conditions. The main factors included in our analysis are macroscopic gradients of the cell-to-cell coupling and cell excitability and microscopic heterogeneity of individual cells. The interplay between these factors allows one to explain how spirals form, drift together with the moving boundary, get transiently pinned to local inhomogeneities, and finally penetrate into the bulk of the well-coupled tissue where they reach macroscopic scale. The asymptotic theory of the drift of spiral and scroll waves based on response functions provides explanation of the drifts involved in this mechanism, with the exception of effects due to the discreteness of cardiac tissue. In particular, this asymptotic theory allows an extrapolation of 2D events into 3D, which has shown that cells within the border zone can give rise to 3D analogues of spirals, the scroll waves. When and if such scroll waves escape into a better coupled tissue, they are likely to collapse due to the positive filament tension. However, our simulations have shown that such collapse of newly generated scrolls is not inevitable and that under certain conditions filament tension becomes negative, leading to scroll filaments to expand and multiply leading to a fibrillation-like state within small areas of cardiac tissue.Comment: 26 pages, 13 figures, appendix and 2 movies, as accepted to PLoS ONE 2011/08/0

Exact propagating wave solutions in reaction cross-diffusion system

This is the final version. Available on open access from Elsevier via the DOI in this recordReaction-diffusion systems with cross-diffusion terms in addition to, or instead of, the usual self-diffusion demonstrate interesting features which motivate their further study. The present work is aimed at designing a toy reaction-cross-diffusion model with exact solutions in the form of propagating fronts. We propose a minimal model of this kind which involves two species linked by cross-diffusion, one of which governed by a linear equation and the other having a polynomial kinetic term. We classify the resulting exact propagating front solutions. Some of them have some features of the Fisher-KPP fronts and some features of the ZFK-Nagumo fronts.Prince Sattam Bin Abdulaziz UniversityEngineering and Physical Sciences Research Council (EPSRC)National Science Foundation (NSF)National Institutes of Health (NIH)Gordon and Betty and Gordon Moore Foundatio

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

Predicting critical ignition in slow-fast excitable models

This is the final version. Available from American Physical Society via the DOI in this record.Linearization around unstable travelling waves in excitable systems can be used to approximate strength-extent curves in the problem of initiation of excitation waves for a family of spatially confined perturbations to the rest state. This theory relies on the knowledge of the unstable travelling wave solution as well as the leading left and right eigenfunctions of its linearization. We investigate the asymptotics of these ingredients, and utility of the resulting approximations of the strength-extent curves, in the slow-fast limit in two-component excitable systems of FitzHugh-Nagumo type, and test those on four illustrative models. Of these, two are with degenerate dependence of the fast kinetic on the slow variable, a feature which is motivated by a particular model found in the literature. In both cases, the unstable travelling wave solution converges to a stationary ``critical nucleus'' of the corresponding one-component fast subsystem. We observe that in the full system, the asymptotics of the left and right eigenspaces are distinct. In particular, the slow component of the left eigenfunction corresponding to the translational symmetry does not become negligible in the asymptotic limit. This has a significant detrimental effect on the critical curve predictions. The theory as formulated previously uses an heuristic to address a difficulty related to the translational invariance. We describe two alternatives to that heuristic, which do not use the misbehaving eigenfunction component. These new heuristics show much better predictive properties, including in the asymptotic limit, in all four examples.Engineering and Physical Sciences Research Council (EPSRC)National Science Foundation (NSF)National Institute for Health Research (NIHR)Gordon and Betty Moore Foundatio

Filament tension and phase-locked drift of meandering scroll waves

This paper was subsequently published in Physical Review Letters vol. 119, article 258101 (DOI: http://hdl.handle.net/10871/31612). The author accepted manuscript of the published article is in ORE: http://hdl.handle.net/10871/31612Rotating 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.This paper was subsequently published in Physical Review Letters, vol. 119, article 258101 (DOI: 10.1103/PhysRevLett.119.258101). The accepted version is in ORE at http://hdl.handle.net/10871/3161

Filament Tension and Phase Locking of Meandering Scroll Waves

This is the author accepted manuscript. The final version is available from American Physical Society via the DOI in this recordThe version of this paper which was originally published at arXiv.org is in ORE: http://hdl.handle.net/10871/25921Meandering 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.H. D. was funded by FWO-Flanders during part of this work. The computational resources (Stevin Supercomputer Infrastructure) and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by Ghent University, FWO and the Flemish Government–department EWI. I. V. B. and V. N. B. gratefully acknowledge EPSRC (UK) support via Grant No. EP/D074789/1. I. V. B. acknowledges EPSRC (UK) support via Grant No. EP/P008690/1. V. N. B. acknowledges EPSRC (UK) current support via Grant No. EP/N014391/1 (UK)

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

This is the author accepted manuscript. The final version is available from the American Physical Society via the DOI in this record.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 linearised operator (i.e. ‘response functions’). For spiral waves, the finite radius of the spiral tip trajectory as well as spiral wave meander are taken into account. Different coordinates 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 analyse the drift of a meandering spiral wave in a constant external field in different regimes.Engineering and Physical Sciences Research Council (EPSRC

Mechanisms of vortices termination in the cardiac muscle

We propose a solution to a long standing problem: how to terminate multiple vortices in the heart, when the locations of their cores and their critical time windows are unknown. We scan the phases of all pinned vortices in parallel with electric field pulses (E-pulses). We specify a condition on pacing parameters that guarantees termination of one vortex. For more than one vortex with significantly different frequencies, the success of scanning depends on chance, and all vortices are terminated with a success rate of less than one. We found that a similar mechanism terminates also a free (not pinned) vortex. A series of about 500 experiments with termination of ventricular fibrillation by E-pulses in pig isolated hearts is evidence that pinned vortices, hidden from direct observation, are significant in fibrillation. These results form a physical basis needed for the creation of new effective low energy defibrillation methods based on the termination of vortices underlying fibrillation.The research leading to the results has received funding from Max Planck Gesellschaft, the European Community Seventh Framework Pro- gramme FP7/2007-2013 under Grant Agreement 17 No. HEALTH-F2-2009-241526, EUTrigTreat (DH, TB, SB, VIK, SL), and from EPSRC (UK) grant EP/I029664 (VNB).We also acknowledge support from the German Federal Ministry of Education and Research (BMBF) (project FKZ 031A147, GO-Bio), the German Research Foundation (DFG) (Collaborative Research Centres SFB 1002 Project C3 and SFB 937 Project A18), the Ger- man Center for Cardiovascular Research (DZHK e.V.) (DH, TB, SB, VIK, SL), Science & Engineering Research Board of Department of Science & Technology, Govern- ment of India (TKS), EPSRC (UK) grant EP/N014391 (VNB) and U.S. NIH Grant No. R01HL089271 (NFO)