Figure 3

Extracellular calcium controls the emergence of the isola of limit cycles. A–F: bifurcation diagram of the model with \(I_{\text{DC}}\) as the bifurcation parameter for different external Ca²⁺ concentrations (\([ \text{Ca} ]_{0}\)). The color-code is the same as in Fig. 2, in particular for the envelope of the sLC (green), the uLC (blue) and the stable branch of fixed points FP (red). Around 1.3 mM extracellular Ca (panel D), the uLC collides with FP thus creating two Hopf bifurcation points (\(H_{1}\) and \(H_{2}\)). Bifurcation \(F_{2}\) is also altered by the appearance of two additional folds of limit cycles (\(F_{3}\) and \(F_{4}\)). \(F_{1}\) is also eventually modified by the appearance of another fold of limit cycles (\(F_{5}\)). Panels in (G) show the corresponding two-parameter bifurcation diagrams: G₁ is the overall diagram while G₁ and G₂ show magnifications of G₁ around \([ \text{Ca} ]_{0} = 1.3\) and 2.0 mM, respectively. The curves show how the bifurcation points described in A–F change when \([ \text{Ca} ]_{0}\) or \(I_{\text{DC}}\) are varied. The branch of folds \(F_{1}\) and \(F_{2}\) are shown in pink and the two branches corresponding to the Hopf bifurcations \(H_{1}\) and \(H_{2}\) are shown in cyan. The additional folds associated to \(F_{1}\) (\(F_{3}\) and \(F_{4}\)) are shown in purple and the additional fold that is associated to \(F_{2}\) (\(F_{5}\)), is in black. Those Fold branches appear and disappear by collisions with each other at codimension-2 bifurcation points called cusps of limit cycles, show as points C. Thus, \(F_{3}\) and \(F_{4}\) collides at cusp \(C_{1}\), \(F_{3}\) and \(F_{1}\) at \(C_{2}\), and \(F_{2}\) and \(F_{5}\) at \(C_{5}\)