Predicting filament drift in twisted anisotropy

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.

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.

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 particular 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 Belousov-Zhabotinsky 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.

Frustrated drift of an anchored scroll-wave filament and the geodesic principle

We investigate anchored scroll-wave filaments in an excitable medium whose diffusivity matrix, including its determinant, is spatially nonuniform. The study is motivated by cardiological applications where scroll-wave behavior in the presence of diffusivity gradients is believed to play an important role in the development of severe arrhythmias. A diffusivity gradient is expected to make the filament drift, unless drift is prevented ("frustrated") by anchoring to localized defects in the propagation medium. The resulting stationary filament is a geodesic curve, as demonstrated here in the case of a nonzero but constant gradient. That is, the diffusivity matrix has a determinant that varies in space, in contrast to what was assumed in earlier work. Here, we show that the filament shape results from a metric tensor of the form (det D)D{-1} , where D is the diffusivity tensor. The filament's shape is solely determined by the diffusivity tensor and is independent of the equation's reaction terms. We derive the analytic solution for the filament and determine conditions for the existence of that solution. The theory is in excellent agreement with numerical simulations.

Scroll waves meandering in a model of an excitable medium

We study numerically the dynamics of a scroll wave in a three-dimensional (3D) excitable medium in the presence of substantial meandering of the corresponding 2D spiral wave in the Aliev-Panfilov model. We identify three types of dynamics of the scroll wave filament--quasi-2D, periodic, and aperiodic meandering--and we study their dependence on parameter settings and thickness of the medium.

Molecular mechanisms and global dynamics of fibrillation: an integrative approach to the underlying basis of vortex-like reentry

Art Winfree's scientific legacy has been particularly important to our laboratory whose major goal is to understand the mechanisms of ventricular fibrillation (VF). Here, we take an integrative approach to review recent studies on the manner in which nonlinear electrical waves organize to result in VF. We describe the contribution of specific potassium channel proteins and of the myocardial fiber structure to such organization. The discussion centers on data derived from a model of stable VF in the Langendorff-perfused guinea pig heart that demonstrates distinct patterns of organization in the left (LV) and right (RV) ventricles. Analysis of optical mapping data reveals that VF excitation frequencies are distributed throughout the ventricles in clearly demarcated domains. The highest frequency domains are found on the anterior wall of the LV at a location where sustained reentrant activity is present. The optical data suggest that a high frequency rotor that remains stationary in the LV is the mechanism that sustains VF in this model. Computer simulations predict that the inward rectifying potassium current (IK1) is an essential determinant of rotor stability and frequency, and patch-clamp results strongly suggest that the outward component of IK1 of cells in the LV is significantly larger than in the RV. Additional computer simulations and analytical procedures predict that the filaments of the reentrant activity (scroll waves) adopt a non-random configuration depending on fiber organization within the ventricular wall. Using the minimal principle we have concluded that filaments align with the trajectory of least resistance (i.e. the geodesic) between their endpoints. Overall, the data discussed have opened new and potentially exciting avenues of research on the possible role played by inward rectifier channels in the mechanism of VF, as well as the organization of its reentrant sources in three-dimensional cardiac muscle. Such an integrative approach may lead us toward an understanding of the molecular and structural basis of VF and hopefully to new preventative approaches.

Eikonal formulation of the minimal principle for scroll wave filaments

Recently, Wellner et al. [Proc. Natl. Acad. Sci. U.S.A. 99, 8015 (2002)]] proposed a principle for predicting a stable scroll wave filament shape as a geodesic in a 3D space with a metric determined by the inverse diffusivity tensor of the medium. Using the Hamilton-Jacobi theory we show that this geodesic is the shortest path for a wave propagating through the medium. This allows the use of shortest path algorithms to predict filament shapes, which we confirm numerically for a medium with orthotropic anisotropy. Our method can be used in cardiac tissue experiments since it does not require knowledge of the tissue anisotropy.

Spiral wave stability in cardiac tissue with biphasic restitution

Human ventricular tissue as well as several animal ventricular preparations show a biphasic shape of the action potential duration restitution curve, with a local maximum at low diastolic intervals. We study numerically how the location and properties of this nonmonotonicity affect the stability of spiral waves. We find that, depending on the slopes of the ascending and of the descending parts of the restitution curve, we can have either stable rotation of the spiral wave or spiral breakup. We identify two types of spiral breakup: one due to a steep positive slope and another due to a steep negative slope in the restitution curve. We discuss the differences in their manifestation and possible implications. We also find that increasing the slope of the descending part of the restitution curve increases the meandering of the spiral wave, due to the repeated occurrence of conduction blocks near the spiral wave tip.

Interaction dynamics of a pair of vortex filament rings

Vortex filament-filament interactions are believed to underlie lethal arrhythmias in cardiac tissue but their dynamics remain poorly understood. We numerically replicate an experimentally postulated reentrant filament configuration as a pair of adjacent circular filaments (scroll rings) with common symmetry axes and varying initial radii and separation distances. The interaction properties are quantified in terms of the scroll-ring lifetime T(L) and direction of initial velocity V0. Two cases were examined, differing only in the direction of the wave around the filament, and observed drastic differences in T(L) between the cases as the separation distance between the rings was decreased. We conclude that ring interactions present unexpected behaviors associated with competing interaction and decay mechanisms.

Minimal principle for rotor filaments

Three-dimensional rotors, or scroll waves, provide essential insight into the activity of excitable media. They also are a suspected cause in the formation and maintenance of ventricular fibrillation, whose lethality is well known. It is therefore of considerable interest to find out what configurations can be adopted by such pathologies. A scroll's behavior is embodied in its organizing center or filament, a largely quiescent tube about which the scroll rotates. Predicting filament shape has normally required computer-intensive simulations of the whole scroll in time. We have found a fast and robust principle that yields the prediction for stationary filaments on a purely geometrical basis, blind to the reaction parameters of the medium. The procedure is to calculate the filament shape as a minimal path. We work in singly diffusive media whose diffusivity tensor--and no other feature--varies spatially. Mathematical and numerical evidence is presented for the proposition that a stable filament is a geodesic in a three-dimensional space whose metric is given by the inverse diffusivity tensor of the medium. Away from the boundaries, a stable filament is unaffected by the reaction parameters. The algorithmic aspects of this work are subsidiary to our main purpose of drawing attention to the universal and unexpectedly exact fit of an elementary geodesic principle within reaction-diffusion theories.

Scroll waves in the presence of slowly varying anisotropy with application to the heart

We consider the dynamics of scroll waves in the presence of rotating anisotropy, a model of the left ventricle of the heart in which the orientation of fibers in successive layers of tissue rotates. By choosing a coordinate system aligned with the fiber rotation and studying the phase dynamics of a straight but twisted scroll wave, we derive a Burgers' equation with forcing associated with the fiber rotation rate. We present asymptotic solutions for scroll twist, verified by numerics, using a realistic fiber distribution profile. We make connection with earlier numerical and analytical work on scroll dynamics.

Computational models of normal and abnormal action potential propagation in cardiac tissue: linking experimental and clinical cardiology

Computational models have the potential to make a huge impact on our understanding of normal and abnormal cardiac function. The aim of this article is to review tools that have been developed to simulate the electrophysiology of cardiac cells and tissue, and to show how computational models have been used to gain insight into normal and abnormal action potential propagation. Some of the practical problems experienced in the development and application of these models are described, and examples are given.

Shaping of a scroll wave filament by cardiac fibers

Scroll waves of electrical excitation in heart tissue are implicated in the development of lethal cardiac arrhythmias. Here we study the relation between the geometry of myocardial fibers and the equilibrium shape of a scroll wave filament. Our theory accommodates a wide class of myocardial models with spatially varying diffusivity tensor, adjusted to fit fiber geometry. We analytically predict the exact equilibrium shapes of the filaments. The major conclusion is that the filament shape is a compromise between a straight line and full alignment with the fibers. The degree of alignment increases with the anisotropy ratio. The results, being purely geometrical, are independent of details of ionic membrane mechanisms. Our theoretical predictions have been verified to excellent accuracy by numerically simulating the stable equilibration of a scroll filament in a model of the FitzHugh-Nagumo type.

Twist localizes three-dimensional patterns

A mechanism for the localization of spatially periodic, self-organized patterns in anisotropic media which requires systems extended in all three spatial dimensions is presented: When the anisotropy axis is twisted, the pattern becomes localized in planes parallel to the anisotropy axis. An analytical description of the effect is developed, and used to interpret recent experiments in the high-frequency regime of electroconvection by Bohatsch and Stannarius [Phys. Rev. E 60, 5591 (1999)]. The localization width is found to be of the order of magnitude of the geometrical average of the pattern wavelength and the inverse twist.