148 research outputs found
Effective dynamics of twisted and curved scroll waves using virtual filaments
Scroll waves are three-dimensional excitation patterns that rotate around a
central filament curve; they occur in many physical, biological and chemical
systems. We explicitly derive the equations of motion for scroll wave filaments
in reaction-diffusion systems with isotropic diffusion up to third order in the
filament's twist and curvature. The net drift components define at every
instance of time a virtual filament which lies close to the instantaneous
filament. Importantly, virtual filaments obey simpler, time-independent laws of
motion which we analytically derive here and illustrate with numerical
examples. Stability analysis of scroll waves is performed using virtual
filaments, showing that filament curvature and twist add as quadratic terms to
the nominal filament tension. Applications to oscillating chemical reactions
and cardiac tissue are discussed.Comment: 28 page
Variational principle for non-linear wave propagation in dissipative systems
The dynamics of many natural systems is dominated by non-linear waves
propagating through the medium. We show that the dynamics of non-linear wave
fronts with positive surface tension can be formulated as a gradient system.
The variational potential is simply given by a linear combination of the
occupied volume and surface area of the wave front, and changes monotonically
in time. Finally, we demonstrate that vortex filaments can be written as a
gradient system only if their binormal velocity component vanishes, which
occurs in chemical system with equal diffusion of reactants
Spiral wave chimeras in locally coupled oscillator systems
The recently discovered chimera state involves the coexistence of
synchronized and desynchronized states for a group of identical oscillators.
This fascinating chimera state has until now been found only in non-local or
globally coupled oscillator systems. In this work, we for the first time show
numerical evidence of the existence of spiral wave chimeras in
reaction-diffusion systems where each element is locally coupled by diffusion.
This spiral wave chimera rotates inwardly, i.e., coherent waves propagate
toward the phase randomized core. A continuous transition from spiral waves
with smooth core to spiral wave chimeras is found as we change the local
dynamics of the system. Our findings on the spiral wave chimera in locally
coupled oscillator systems largely improve our understanding of the chimera
state and suggest that spiral chimera states may be found in natural systems
which can be modeled by a set of oscillators indirectly coupled by a diffusive
environment.Comment: 5 pages, 5 figure
Drift Laws for Spiral Waves on Curved Anisotropic Surfaces
Rotating spiral waves organize spatial patterns in chemical, physical and
biological excitable systems. Factors affecting their dynamics such as
spatiotemporal drift are of great interest for par- ticular applications. Here,
we propose a quantitative description for spiral wave dynamics on curved
surfaces which shows that for a wide class of systems, including the BZ
reaction and anisotropic cardiac tissue, the Ricci curvature scalar of the
surface is the main determinant of spiral wave drift. The theory provides
explicit equations for spiral wave drift direction, drift velocity and the
period of rotation. Depending on the parameters, the drift can be directed to
the regions of either maximal or minimal Ricci scalar curvature, which was
verified by direct numerical simulations.Comment: preprint before submission to Physical Review
Generalized minimal principle for rotor filaments
To a reaction-diffusion medium with an inhomogeneous anisotropic diffusion tensor D, we add a fourth spatial dimension such that the determinant of the diffusion tensor is constant in four dimensions. We propose a generalized minimal principle for rotor filaments, stating that the scroll wave filament strives to minimize its surface area in the higher-dimensional space. As a consequence, stationary scroll wave filaments in the original 3D medium are geodesic curves with respect to the metric tensor G = det(D)D-1. The theory is confirmed by numerical simulations for positive and negative filament tension and a model with a non-stationary spiral core. We conclude that filaments in cardiac tissue with positive tension preferentially reside or anchor in regions where cardiac cells are less interconnected, such as portions of the cardiac wall with a large number of cleavage planes
Accurate eikonal-curvature relation for wave fronts in locally anisotropic reaction-diffusion systems
The dependency of wave velocity in reaction-diffusion (RD) systems on the local front curvature determines not only the stability of wave propagation, but also the fundamental properties of other spatial configurations such as vortices. This Letter gives the first derivation of a covariant eikonal-curvature relation applicable to general RD systems with spatially varying anisotropic diffusion properties, such as cardiac tissue. The theoretical prediction that waves which seem planar can nevertheless possess a nonvanishing geometrical curvature induced by local anisotropy is confirmed by numerical simulations, which reveal deviations up to 20% from the nominal plane wave speed
Analysis of cardiac arrhythmia sources using Feynman diagrams
The contraction of the heart muscle is triggered by self-organizing
electrical patterns. Abnormalities in these patterns lead to cardiac
arrhythmias, a prominent cause of mortality worldwide. The targeted treatment
or prevention of arrhythmias requires a thorough understanding of the
interacting wavelets, vortices and conduction block sites within the excitation
pattern. Currently, there is no conceptual framework that covers the elementary
processes during arrhythmogenesis in detail, in particular the transient
pivoting patterns observed in patients, which can be interleaved with periods
of less fragmented waves. Here, we provide such a framework in terms of
quasiparticles and Feynman diagrams, which were originally developed in
theoretical physics. We identified three different quasiparticles in excitation
patterns: heads, tails and pivots. In simulations and experiments, we show that
these basic building blocks can combine into at least four different bound
states. By representing their interactions as Feynman diagrams, the creation
and annihilation of rotor pairs are shown to be sequences of dynamical
creation, annihilation and recombination of the identified quasiparticles. Our
results provide a new theoretical foundation for a more detailed theory,
analysis and mechanistic insights of topological transitions in excitation
patterns, to be applied within and beyond the context of cardiac
electrophysiology
Strings, branes and twistons: topological analysis of phase defects in excitable media such as the heart
Several excitable systems, such as the heart, self-organize into complex
spatio-temporal patterns that involve wave collisions, wave breaks, and
rotating vortices, of which the dynamics are incompletely understood. Recently,
conduction block lines in two-dimensional media were recognized as phase
defects, on which quasi-particles can be defined. These particles also form
bound states, one of which corresponds to the classical phase singularity.
Here, we relate the quasi-particles to the structure of the dynamical attractor
in state space and extend the framework to three spatial dimensions. We reveal
that 3D excitable media are governed by phase defect surfaces, i.e. branes, and
three flavors of topologically preserved curves, i.e. strings: heads, tails,
and pivot curves. We identify previously coined twistons as points of
co-dimension three at the crossing of a head curve and a pivot curve. Our
framework predicts splitting and branching phase defect surfaces that can
connect multiple classical filaments, thereby proposing a new mechanism for the
origin, perpetuation, and control of complex excitation patterns, including
cardiac fibrillation
Chiral selection and frequency response of spiral waves in reaction-diffusion systems under a chiral electric field
Chirality is one of the most fundamental properties of many physical,
chemical and biological systems. However, the mechanisms underlying the onset
and control of chiral symmetry are largely understudied. We investigate
possibility of chirality control in a chemical excitable system (the BZ
reaction) by application of a chiral (rotating) electric field using the
Oregonator model. We find that unlike previous findings, we can achieve the
chirality control not only in the field rotation direction, but also opposite
to it, depending on the field rotation frequency. To unravel the mechanism, we
further develop a comprehensive theory of frequency synchronization based on
the response function approach. We find that this problem can be described by
the Adler equation and show phase-locking phenomena, known as the Arnold
tongue. Our theoretical predictions are in good quantitative agreement with the
numerical simulations and provide a solid basis for chirality control in
excitable media.Comment: 21 pages with 9 figures; update references; to appear in J. Chem.
Phy
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