Fig. 8

Singular limit spiking activity transitions. Singular solution trajectories from the layer problem, (7), and the reduced problem, (15), are concatenated to produce singular global trajectories (black). The singular limit predicts a range of ${\tau }_s$ values for which rebound spiking occurs; ${\tau }_s\in [5,24]$. The layer problem dictates that the trajectory has a base point on $S_a^{-}$ independent of ${\tau }_s$. Once on the manifold, the reduced problem dictates that the trajectory remains to one side of the canard separatrix. a The singular trajectory for ${\tau }_s=24$ in $(v,w)$-space. Since this trajectory lies to the right of the separatrix, it evolves in time toward the fold curve, $F^{-}$ (gray dashed), at which point the layer problem describes the onset of oscillatory behavior. This singular prediction corresponds to a successful rebound spike. b The singular trajectory for ${\tau }_s=25$ in $(v,w)$-space. This trajectory lies to the left of the separatrix and evolves in time toward ${\mathbf{eq}}_3$ (green). This singular prediction corresponds to an unsuccessful rebound spike. An animation of this figure under variation of ${\tau }_s$ is given within Additional file 1