Fig. 7

(3D version of Figure 2) Intersection point of the branch $M_{\mathrm{eq}}$ with the slow nullcline $\dot{z}=0$ yields the equilibrium state of the HR model at given $x_0$. The dark blue point is the center of the gravity of the stable periodic orbit of the HR model, which is depicted on the tonic spiking manifold $M_{\mathrm{lc}}$ at $x_0=1.8$. It is located around the intersection point of the slow nullcline $\dot{z}=0$ with the space curve $〈x〉$ of the average x-values on each periodic orbit that foliate the spiking manifold $M_{\mathrm{lc}}$. The phase point, while turning around $M_{\mathrm{lc}}$, moves slowly toward the homoclinic edge ($\dot{z}>0$) as long as it stays above the slow nullcline, and goes backward ($\dot{z}<0$) after it lowers below the slow nullcline $\dot{z}=0$. When these opposite forces are canceled out over the revolution period, the phase point spins around the ‘center of the gravity’, that is, stays on the same periodic orbit. The variations of $x_0$ move the slow nullcline and thus make the periodic orbit slide along the manifold. When the slow nullcline $\dot{z}=0$ cuts through the unstable section of $M_{\mathrm{eq}}$ between HB, standing for homoclinic bifurcation in the fast subsystem and the fold AH/SN, the model becomes a burster.