Validation and Calibration of Models for Reaction-Diffusion Systems

Space and time scales are not independent in diffusion. In fact, numerical simulations show that different patterns are obtained when space and time steps ($\Delta x$ and $\Delta t$) are varied independently. On the other hand, anisotropy effects due to the symmetries of the discretization lattice prevent the quantitative calibration of models. We introduce a new class of explicit difference methods for numerical integration of diffusion and reaction-diffusion equations, where the dependence on space and time scales occurs naturally. Numerical solutions approach the exact solution of the continuous diffusion equation for finite $\Delta x$ and $\Delta t$, if the parameter $\gamma_N=D \Delta t/(\Delta x)^2$ assumes a fixed constant value, where $N$ is an odd positive integer parametrizing the alghorithm. The error between the solutions of the discrete and the continuous equations goes to zero as $(\Delta x)^{2(N+2)}$ and the values of $\gamma_N$ are dimension independent. With these new integration methods, anisotropy effects resulting from the finite differences are minimized, defining a standard for validation and calibration of numerical solutions of diffusion and reaction-diffusion equations. Comparison between numerical and analytical solutions of reaction-diffusion equations give global discretization errors of the order of $10^{-6}$ in the sup norm. Circular patterns of travelling waves have a maximum relative random deviation from the spherical symmetry of the order of 0.2%, and the standard deviation of the fluctuations around the mean circular wave front is of the order of $10^{-3}$.Comment: 33 pages, 8 figures, to appear in Int. J. Bifurcation and Chao

Motion by curvature of a three-dimensional filament: similarity solutions

We systematically classify and investigate fully three-dimensional similarity solutions to a system of equations describing the motion of a filament moving in the direction of its principle normal with velocity proportional to its curvature, ν = κn, where n is the principle normal and κ the curvature of the filament. Such formulations are relevant to superconducting vortices and disclinations

The evolution of space curves by curvature and torsion

We apply Lie group based similarity methods to the study of a new, and widely relevant, class of objects, namely motions of a space curve. In particular, we consider the motion of a curve evolving with a curvature kappa and torsion tau dependent velocity law. We systematically derive the Lie point symmetries of all such laws of motion and use these to catalogue all their possible similarity reductions. This calculation reveals special classes of law with high degrees of symmetry (and a correspondingly large number of similarity reductions). Of particular note is one class which is invariant under general linear transformations in space. This has potential applications in pattern and signal recognition

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

Stable and unstable vortex knots in excitable media

We study the dynamics of knotted vortices in a bulk excitable medium using the FitzHugh-Nagumo model. From a systematic survey of all knots of at most eight crossings we establish that the generic behavior is of unsteady, irregular dynamics, with prolonged periods of expansion of parts of the vortex. The mechanism for the length expansion is a long-range “wave-slapping” interaction, analogous to that responsible for the annihilation of small vortex rings by larger ones. We also show that there are stable vortex geometries for certain knots; in addition to the unknot, trefoil, and figure-eight knots reported previously, we have found stable examples of the Whitehead link and 6 2 knot. We give a thorough characterization of their geometry and steady-state motion. For the unknot, trefoil, and figure-eight knots we greatly expand previous evidence that FitzHugh-Nagumo dynamics untangles initially complex geometries while preserving topolog

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

New concepts for use in low-energy cardiac defibrillation

The presence of three-dimensional rotating action potential waves, called scroll waves, in the heart causes ventricular fibrillation. Recently, there has been interest in developing low-energy methods, consisting of applying an electric field to terminate these waves, as a means of defibrillation. The success of these methods often depends on the orientation of the waves. We present computer simulations of a method that applies multiple electrical fields in a hemispherical shell system representative of the ventricles of the heart. Scroll waves in this system persist when the filament (the curve around which the wave rotates) connects the inside and outside surfaces. Our scheme for applying electric fields aims to disconnect these filaments from the surfaces. Once the filaments no longer connect the inside and outside surfaces, they contract and disappear, terminating the scroll wave. Importantly, as opposed to most existing schemes, the idea on which this scheme is based is applicable irrespective of how many scroll waves are present, where they are located, or where they are in their rotation. We discuss the success of this scheme both for different numbers of waves and for different wave orientations and present potential failure mechanisms. The effects of other conditions, such as the stability of the waves and heart geometry, remain to be studied. In the future, the presented low-energy method for termination of scroll waves may be a useful means of cardiac defibrillation