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.

Figure 1

FHN model was used to simulate spiral waves with $\varepsilon =0.22$, $\beta =0.70$, and $\gamma =0.80$. (a) The spatial distribution of the fast variable $V$ of a meandering spiral wave. Panels (b)–(f) show the distribution of the topological charge $\rho (i,j)\mathrm{\Delta }{x}^2$ in two-dimensional space. The coordinates of PSs identified by different methods are marked in these figures respectively. (b) Convolution method with Sobel $3\times 3$ convolutional kernels. (c) Convolution method with nabla $3\times 3$ convolutional kernels. (d) Convolution method with nabla $2\times 2$ convolutional kernels. (e) Charge-density method with $3\times 3$ arrays. (f) Charge-density method with $2\times 2$ arrays.

Figure 2

Simple schematics are used to illustrate the charge-density method and the convolution method. (a) Using charge-density method with $3\times 3$ arrays obtains four points marked by black squares with $\rho (i,j)\mathrm{\Delta }{x}^2=0.25$. The center of these four points is identified as a PS (black dot). (b) Using convolution method with nabla $3\times 3$ convolutional kernels identifies four PSs which are marked by black stars and the real PS (black dot) is located in the intersection of these PSs.

Figure 3

One of the four identical trajectories obtained by four different methods. PS points are recorded once per time unit from 140 to 260 time units. All these four methods lead to the exactly same trajectories, and for brevity, it just shows the trajectory obtained by the charge-density method with $3\times 3$ arrays.

Figure 4

The effect of noise. The FHN model's parameters are the same as those in Fig. 1 while the noise amplitude $\eta =0.004$. (a) The spatial distribution of the fast variable $V$ of a meandering spiral wave. Panels (b)–(f) show the distribution of the topological charge $\rho (i,j)\mathrm{\Delta }{x}^2$ in two-dimensional space. The coordinates of PSs identified by different methods are marked in these figures respectively. (b) Convolution method with Sobel $3\times 3$ convolutional kernels. (c) Convolution method with nabla $3\times 3$ convolutional kernels. (d) Convolution method with nabla $2\times 2$ convolutional kernels. (e) Charge-density method with $3\times 3$ arrays. (f) Charge-density method with $2\times 2$ arrays.

Figure 5

PS trajectories obtained by the charge-density method with $3\times 3$ arrays under different noise amplitudes. The initial conditions are the same as those in Fig. 3. PS points are recorded once per time unit and the time duration is the same as that in Fig. 3. (a) $\eta =0.0002$. (b) $\eta =0.0005$. (c) $\eta =0.001$. (d) $\eta =0.004$.

Figure 6

Use different methods to identify PSs for multiple spiral waves. The simulation, in the range of $256\times 256$ grids, used the same model and parameters as those in Fig. 1. (a) The spatial distribution of the fast variable $V$ of multiple spiral waves. Panels (b)–(f) show the distribution of the topological charge $\rho (i,j)\mathrm{\Delta }{x}^2$ in two-dimensional space. The coordinates of PSs identified by different methods are marked in these figures respectively. (b) Convolution method with Sobel $3\times 3$ convolutional kernels. (c) Convolution method with nabla $3\times 3$ convolutional kernels. (d) Convolution method with nabla $2\times 2$ convolutional kernels. (e) Charge-density method with $3\times 3$ arrays. (f) Charge-density method with $2\times 2$ arrays.

Figure 7

The effect of spatial resolution. The FHN model's parameters are the same as those in Fig. 1 while the spatial resolution reduced from $128\times 128$ to $32\times 32$. (a) The spatial distribution of the fast variable $V$ of a meandering spiral wave. Panels (b)–(f) show the distribution of the topological charge $\rho (i,j)\mathrm{\Delta }{x}^2$ in two-dimensional space. The coordinates of PSs identified by different methods are marked in these figures respectively. (b) Convolution method with Sobel $3\times 3$ convolutional kernels. (c) Convolution method with nabla $3\times 3$ convolutional kernels. (d) Convolution method with nabla $2\times 2$ convolutional kernels. (e) Charge-density method with $3\times 3$ arrays. (f) Charge-density method with $2\times 2$ arrays.

Figure 8

One of the four identical trajectories obtained by four different methods with low spatial resolution ($32\times 32$). PS points are recorded once per time unit and the time duration is the same as that in Fig. 3. All these four methods lead to the exactly same trajectories, and for brevity, it just shows the trajectory obtained by the charge-density method with $3\times 3$ arrays.

Figure 9

Luo-Rudy model was used to simulate spiral waves in cardiac tissues. All parameters are chosen the same as those in Fig. 3A(a) in Ref. [35]. (a) The spatial distribution of a spiral wave in cardiac tissues ($64\times 64$). Panels (b)–(f) show the distribution of the topological charge $\rho (i,j)\mathrm{\Delta }{x}^2$ in two-dimensional space. The coordinates of PSs identified by different methods are marked in these figures respectively. (b) Convolution method with Sobel $3\times 3$ convolutional kernels. (c) Convolution method with nabla $3\times 3$ convolutional kernels. (d) Convolution method with nabla $2\times 2$ convolutional kernels. (e) Charge-density method with $3\times 3$ arrays. (f) Charge-density method with $2\times 2$ arrays.

Figure 10

One of the four identical trajectories obtained by four different methods with Luo-Rudy model. All these four methods lead to the exactly same trajectories, and for simplicity, it just shows the trajectory obtained by the charge-density method with $3\times 3$ arrays.