A spontaneous excitable event has two phases: the initiation phase and the excitation phase. During the initiation phase, ion channel fluctuations push the voltage and \(\text{K}^{{+}}\) channel population to a threshold. The MPP taken during the initiation phase is most likely to follow the most probable exit path (MPEP), which is very different from any deterministic trajectory. The excitation phase takes the system from the threshold through a transient spike in voltage and ultimately back to the stable fixed point. The MPP taken during the excitation phase follows (at least in part) closely to a deterministic trajectory.
For the basic type I system, there is a single MPP that characterizes the spontaneous action potential. During the initiation phase, the MPP moves from the stable fixed point to the saddle along the MPEP (red curve Fig. \ref{fig:type1a_pp2}). During the excitation phase, the MPP follows the deterministic unstable manifold around the unstable fixed point back toward the stable fixed point (green curve Fig. \ref{fig:type1a_pp2}). The initiation phase for the type I system with bursting is similar (red curve Fig. \ref{fig:type1b_qp}(a)). Unlike the previous case, the excitation phase has multiple parts. The excitation phase starts with the right branch of the unstable manifold until reaching the stable limit cycle (green curve Fig. \ref{fig:type1b_pp1}(a)). The number of oscillations around the unstable fixed point, and therefore the length of the burst, is not described by a MPP, but after fluctuations push the trajectory far enough away from the limit cycle, an MPP describes its approach back to the saddle (blue curve Fig. \ref{fig:type1b_pp2}). After returning to the saddle, the trajectory is most likely to return to the stable fixed point along the left branch of the unstable manifold (green curve Fig. \ref{fig:type1b_pp1}(a)).
The analysis of the type II system is less tractable due to the lack of a saddle. However, the presence of a caustic provides an effective energy barrier for initiation. Even without the saddle, MPPs follow closely along a single path (dashed red curve Fig. \ref{fig:type2a_qp}(a)) as they approach the energy barrier. Hence, a single MPP characterizes the initiation phase similar to the type I case described above. However, during the excitation phase, the amplitude of the spontaneous action potential is not described by a single MPP because there are many MPPs that have approximately equal likelihood (blue region in Fig. \ref{fig:type2a_qp}(a)).
The biggest difference between type I and type II spontaneous action potentials is the behavior during the excitation phase. Type I action potentials can exhibit voltage oscillations, especially when there is a stable limit cycle surrounding the unstable fixed point. Type II action potentials on the other hand do not show this behavior.
Interestingly, the behavior of type I and type II spontaneous action potentials are similar during the initiation phase. There are two ion channel species that contribute to spontaneous initiation. The number of open channels determines the net current \({I_{\rm ion}}(v, m, n)\), and if the net current is increased for enough time, the voltage rises above threshold, generating an action potential. There are two ways to increase the net current: by opening \(\text{Na}^{{+}}\) channels or by closing \(\text{K}^{{+}}\) channels.
Clearly, the maximum increase in the net current from closing \(\text{K}^{{+}}\) channels occurs when all of the \(\text{K}^{{+}}\) channels are closed. If we fix \(m = 0\) to be constant, removing \(\text{K}^{{+}}\) channel fluctuations and dynamics, the deterministic system becomes \[v' = {I_{\rm ion}}(v, 0, Nx_{\infty}(v)).\] The function \({I_{\rm ion}}(v, 0, Nx_{\infty}(v))\) has cubic-like shape. For \(I_{{\mathrm{app}}} < I_{*}\), there are three fixed points where \(v' = {I_{\rm ion}}(v, 0, Nx_{\infty}(v)) = 0\): two stable separated by one unstable. Only the third fixed point is above the threshold. To generate an action potential, \(\text{Na}^{{+}}\) channel fluctuations are required to increase the voltage from the first fixed point past the second. At \(I_{{\mathrm{app}}} = I_{*}\), the first two fixed points vanish. For \(I_{{\mathrm{app}}} > I_{*}\), only the third fixed point remains, which means that \(\text{K}^{{+}}\) channels alone are capable of initiating an action potential. That is, once all of the \(\text{K}^{{+}}\) channels close, the voltage can deterministically increase above threshold. Fig. \ref{fig:type1a_pp1} shows the \(I_{{\mathrm{app}}}<I_{*}\) case while Fig. \ref{fig:type2a_pp1} shows the \(I_{{\mathrm{app}}}>I_{*}\) case.
Recall that the parameter \({\varphi}= 1/(\epsilon M)\) controls the relative strength of \(\text{Na}^{{+}}\) and \(\text{K}^{{+}}\) channel fluctuation. As \({\varphi}\) decreases (equivalently \(M\) increases with \(\epsilon\) fixed) the \(\text{K}^{{+}}\) channel fluctuations become less significant than \(\text{Na}^{{+}}\) channel fluctuations. For \({\varphi}\) small enough, \(\text{Na}^{{+}}\) channels provide the dominant contribution to spontaneous initiation. It is natural to conclude then that for \({\varphi}\) large enough, \(\text{K}^{{+}}\) channels provide the dominant contribution to spontaneous initiation. Indeed, this is the case when \(I_{{\mathrm{app}}} > I_{*}\). Notice that MPPs leading to action potentials drop below the \(v\)-nullcline in Fig. \ref{fig:type2a_pp2}. If instead we have \(I_{{\mathrm{app}}} < I_{*}\) then \(\text{K}^{{+}}\) channels alone cannot induce an action potential, regardless of how large \({\varphi}\) is.