Figure 1

Phenomenological sketch (a) and matching numerical portrait (b) of the model (1) at the blue-sky catastrophe. A saddle-node periodic orbit $L_{\mathrm{b}\mathrm{s}}$ is depicted in (a) for the slow-fast system (2) in the $(z,x)$ phase space combined with the bifurcation diagram of its fast subsystem, in (b) for the neuron system (1) in $(m_{\mathrm{K}2},V)$ phase space for $V_{\mathrm{K}2}^s=24.5\mathrm{m}\mathrm{V}$. The blue Z-shaped line, $M_{\mathrm{e}\mathrm{q}}$, consists of the equilibrium states of the fast subsystem (dotted and solid segments represent unstable and stable ones). The point of its intersection with the regular nullcline $\dot{z}=0$ in (a) and $m_{\mathrm{K}2}^{\prime }=0$ in (b) is an equilibrium state of the system. The green cylinder-shaped surface $M_{\mathrm{L}\mathrm{C}}=M_{\mathrm{L}\mathrm{C}}^s\cup M_{\mathrm{L}\mathrm{C}}^u$ is composed of the stable and unstable limit cycles of the fast subsystem. The line $⟨x⟩$ shows the dependence of the $x$ coordinate of the limit cycle averaged over its period on $z$, and $⟨V⟩$ vs $m_{\mathrm{K}2}$ in (b). The dashed, blue line is the average nullcline $⟨\dot{z}⟩=0$ in (a) and $⟨m_{\mathrm{K}2}^{\prime }⟩=0$ in (b). The contact point between $⟨V⟩$ and $⟨m_{\mathrm{K}2}^{\prime }⟩=0$ corresponds to the saddle-node periodic orbit, $L_{\mathrm{b}\mathrm{s}}$. The gray disk $W^{ss}$ in (a) is its strongly stable manifold. The part of $M_{\mathrm{L}\mathrm{C}}^s$ to the right of $L_{\mathrm{b}\mathrm{s}}$ is the unstable manifold, $W^u$ of the saddle-node periodic orbit. In (a), the red line outlines rapid transitions of the phase point between the hyperpolarized phase and tonic spiking phase of bursting. In (b), the red curve represents a trajectory homoclinic to $L_{\mathrm{b}\mathrm{s}}$. This trajectory transforms into a closed periodic orbit representing bursting as parameter $V_{\mathrm{K}2}^s$ passes a bifurcation value and $L_{\mathrm{b}\mathrm{s}}$ disappears.Reuse & Permissions

Figure 2

Samples of oscillatory waveforms generated by the neuron model (1) for decreasing values of the bifurcation parameter $V_{\mathrm{K}2}^s$. The bursting regime (the three top traces) is continuously transformed into tonic spiking (the bottom trace). The burst duration increases as $V_{\mathrm{K}2}^s$ approaches the blue-sky bifurcation's value ($V_{\mathrm{K}2}^s=24.25\mathrm{m}\mathrm{V}$). The bottom chart samples the spiking oscillations from the traces above.Reuse & Permissions

Figure 3

Dependence of the period of bursting on the control parameter $V_{\mathrm{K}2}^s$. The numerically obtained points are marked by $\times$'s. The curve is given by $0.31/\sqrt{|(V_{\mathrm{K}2}^s+24.25)|}$, where $24.25\mathrm{m}\mathrm{V}$ is the critical value of the transition.Reuse & Permissions