Fig. 5

Bifurcation diagrams of the fast subsystem. Bifurcation diagrams of the fast subsystem with the slow variables \(na_{i}\) and \(ca_{i}\) taken as static parameters. The yellow star marks the start point of the SB solution. (A) The effect of \(ca_{i}\) on the bifurcation diagram for the fast subsystem, projected into \((na_{i}, v)\)-space, along with the \(na_{i}\)-nullcline (cyan). Increasing \(ca_{i}\) from \(8\mathrm{e}{-}3\) to \(3\mathrm{e}{-}2\) to \(5\mathrm{e}{-}2\) results in a shift of the bifurcation diagram to the right (black to blue to green). (B) Projection of the first small burst in the SB solution of (5a)–(5g) onto the bifurcation diagram (with \(ca_{i}=8\mathrm{e}{-}3\)) in \((na_{i}, v)\)-space, along with the \(na_{i}\)-nullcline (cyan). The blue and green dashed lines indicate the \(na_{i}\) values where the lower fold and homoclinic bifurcations occur, respectively. (C) The curve of saddle-node bifurcations corresponding to the lower fold of the bifurcation diagram (blue), homoclinic bifurcation curve (green) and part of the trajectory (black) generated by (5a)–(5g) in \((na_{i}, ca_{i})\)-space. The HC curve splits the \((na_{i}, ca_{i})\)-space into two regions labeled as ‘Active’ and ‘Silent’, respectively. The part of the trajectory corresponding to the first burst, as shown in (B), is magenta. (D) A zoomed-in and enlarged view of (C)