FIG. 3.

Dynamics of two identical cells with ephaptic coupling only, varying fraction of junctional membranes $I_{K1}$ current, $f_{K1}$. Increasing $f_{K1}$ leads to automaticity. (a) Bifurcation diagram for the voltage (lateral segment) vs $f_{K1}$, illustrating regimes of stable (red) and unstable (black) fixed points, and stable (red) and unstable (blue) limit cycles. (b) Bifurcation diagrams with various (fixed) cleft volumes $vcl$, with all fixed points and limit cycles shown with a single color to illustrate how Hopf bifurcation point varies. (c)–(f) Bifurcation diagrams with fixed $f_{K1}$ values (0.95, 0.9, 0.85, respectively), varying cleft volume $vcl$. Decreasing the cleft volume $vcl$ leads to automaticity via a Hopf bifurcation. The critical volume $vcl$ decreases as $f_{K1}$ decreases. (f) Summary: two-parameter curve of Hopf bifurcation points computed by XPP-AUTO (black)^29,30 is well-fit by a quadratic function: $f_{K1}^{\ast }=-(8.665\times {10}^{-3}){(vcl)}^2+(6.287\times {10}^{-2})vcl+0.8497$. Note that automaticity occurs in the lower right triangular region of parameter space. Dashed black horizontal line corresponds with $vcl$ value in (a).