Negative Filament Tension at High Excitability in a Model of Cardiac Tissue

One of the fundamental mechanisms for the onset of turbulence in 3D excitable media is negative filament tension. Thus far, negative tension has always been obtained in media under low excitability. For this reason, its application to normal (nonischemic) cardiac tissue has been questionable, as such cardiac turbulence typically occurs at high excitability. Here, we report expansion of scroll rings (low curvature negative filament tension) in a medium with high excitability by numerical integration of the Luo-Rudy model of cardiac tissue. We discuss the relation between negative tension and the meandering of 2D spiral waves and the possible applications to cardiac modeling.

Figure 1

Expansion of a quarter of scroll ring in the LR1 model at high excitability ($4\times 4\times 4{\mathrm{cm}}^3$). (a) Wave pattern (gray) at $t=0$, the white line represents the filament. (b) Sketch of the successive filament positions (a,b,c) in the course of time. (c) Transmembrane potential in the course of time at point ($0.2,0.2,0.04$). $G_{\mathrm{Na}}=15.0\mathrm{mS}/{\mathrm{cm}}^2$ and $G_{\mathrm{si}}=0.0\mathrm{mS}/{\mathrm{cm}}^2$.

Figure 2

(a) Radius of the scroll ring filament in the course of time for the 3D simulation from Fig. 1. (b) Evolution of the radius ($R$) for different initial conditions. (c),(d) Dependence of the radial velocity on the parameters. Simulations in panels (c) and (d) are done with 2D cylindrical coordinates with $R_0=3.5\mathrm{cm}$ and $j=1$.

Figure 3

Filament tension in 3D and 2D meandering patterns in parametric space for the LR1 model with $j$ clamped to 1. The gray zones correspond to expansion of 3D scroll rings. Different regimes of spiral dynamics are indicated by the corresponding legends and are shown in the insets. The horizontal lines next to the meandering patterns are 2 mm long. 3D dynamics were computed for $R_0=3.5\mathrm{cm}$.

Figure 4

Meandering of 2D spiral waves in the LR1 model. Transition between inward and outward meandering at low excitability for increasing $G_{\mathrm{Na}}$, at $G_{\mathrm{si}}=0\mathrm{mS}/{\mathrm{cm}}^2$ (a) and under high excitability conditions for increasing $G_{\mathrm{si}}$ at $G_{\mathrm{Na}}=10\mathrm{mS}/{\mathrm{cm}}^2$ (b). Gate variable $j$ is clamped to 1.

Figure 5

Radial velocity of scroll rings for different values of the parameter $G_{\mathrm{Na}}$ at $G_{\mathrm{si}}=0\mathrm{mS}/{\mathrm{cm}}^2$ for the full LR1 model (gate variable $j$ is unclamped). Such simulations have been done with cylindrical coordinates. Panels with trajectories of the spiral tip for several values of $G_{\mathrm{Na}}$ accompany the curve.