Topological charge-density-vector method of identifying filaments of scroll waves.

Scroll waves have been found in a variety of three-dimensional excitable media, including physical, chemical, and biological origins. Scroll waves in cardiac tissue are of particular significance as they underlie ventricular fibrillation that can cause sudden death. The behavior of a scroll wave is characterized by a line of phase singularity at its organizing center, known as a filament. A thorough investigation into the filament dynamics is the key to further exploration of the general theory of scroll waves in excitable media and the mechanisms of ventricular fibrillation. In this paper, we propose a method to identify filaments of scroll waves in excitable media. From the definition of the topological charge of filaments, we obtain the discrete expression of the topological charge-density vector, which is useful in calculating the topological charge vectors at each grid in the space directly. The set of starting points of these topological charge vectors represents a set of phase singularities, thereby forming a line of phase singularity, that is, a filament of a scroll wave.

Numerical study of the drift of scroll waves by optical feedback in cardiac tissue.

Nonlinear waves were found in various types of physical, chemical, and biological excitable media, e.g., in heart muscle. They can form three-dimensional (3D) vortices, called scroll waves, that are of particular significance in the heart, as they underlie lethal cardiac arrhythmias. Thus controlling the behavior of scroll waves is interesting and important. Recently, the optical feedback control procedure for two-dimensional vortices, called spiral waves, was developed. It can induce directed linear drift of spiral waves in optogenetically modified cardiac tissue. However, the extension of this methodology to 3D scroll waves is nontrivial, as optogenetic signals only penetrate close to the surface of cardiac tissue. Here we present a study of this extension in a two-variable reaction-diffusion model and in a detailed model of cardiac tissue. We show that the success of the control procedure is determined by the tension of the scroll wave filament. In tissue with positive filament tension the control procedure works in all cases. However, in the case of negative filament tension for a sufficiently large medium, instabilities occur and make drift and control of scroll waves impossible. Because in normal cardiac tissue the filament tension is assumed to be positive, we conclude that the proposed optical feedback scheme can be a robust method in inducing the linear drift of scroll waves that can control their positions in cardiac tissue.

Topological charge-density method of identifying phase singularities in cardiac fibrillation.

Spiral waves represent the key motifs of typical self-sustained dynamical patterns in excitable systems such as cardiac tissue. The motion of phase singularities (PSs) that lies at the center of spiral waves captures many qualitative and, in some cases, quantitative features of their complex dynamics. Recent clinical studies suggested that ablating the tissue at PS locations may cure atrial fibrillation. Here, we propose a different method to determine the location of PSs. Starting from the definition of the topological charge of spiral waves, we obtain the expression of the topological charge density in a discrete case. With this expression, we can calculate the topological charge at each grid in the space directly, so as to accurately identify the position of PSs.

Approximating the energy landscape of a two-dimensional bistable gene autoregulation model by separating slow and fast dynamics.

The energy landscape is widely used to quantify the stability of multistable nonlinear systems, such as bistable gene regulation networks. In physics, the potential can be obtained through integration only for gradient systems. However, multidimensional nonlinear systems are often nongradient, for which the potential is calculated by decomposing the dynamics to gradient and nongradient parts. This potential is then called a quasipotential. Given that one-dimensional (1D) systems can be regarded as gradient systems, we attempt to separate the two-dimensional (2D) system into two 1D systems working on distinct timescales, and the potential can be easily calculated for the two 1D systems separately. This method is used in this study to estimate the energy landscape of a two-variable gene autoregulation model. This elegant and comprehensive method is accessible for 2D nonlinear systems in which the dynamics can be divided into slow and fast parts.