Fig. \ref{fig:type1a_pp2} shows a several MPPs (orange curves, computed using the GMAM) that start at the stable fixed point. MPPs can only directly lead to the threshold (dashed line) below the saddle, where they cross from left to right. Above the saddle, MPPs first travel through the saddle and then cross the threshold from right to left. Hence, MPPs that cross the threshold above the saddle are most likely to be returning action potential trajectories. That is, action potential trajectories at the end of the excitation phase that are returning to the stable resting potential. Spontaneous action potentials are most likely initiated below the saddle.

Above the stable fixed point many characteristic projections overlap. Uniqueness of the solution is recovered using the minimum action principle (see Section \ref{sec:mpp}). At any given point \({\mathbf{x}}\) through which two or more characteristics cross, the value of \(W({\mathbf{x}})\) is given by the characteristic that has the smallest action. A caustic is a curve along which each point is the terminus of two ore more MPPs and the gradient of the quasipotential is discontinuous. Fig. \ref{fig:type1a_pp2} shows two branches (yellow curves) of the caustic. To the left of the threshold, transient excursions from the attractor reach the caustic from below, while returning action potentials reach the caustic from above. The upper branch of the caustic crosses the separatrix above the saddle, separating excitation-phase MPPs into those that travel around the unstable fixed point and those that do not.