Figure 8. The Shilnikov's homoclinic bifurcation scenario. *A1*, Trajectory of the reduced model during a typical period in the subprimary range (SPR). The trajectory [*V*(*t*), *W*(*t*), *z*(*t*)] is plotted in the three-dimensional *V*, *W*, *z* space. *V* is the voltage, *W* the recovery variable (see Materials and Methods), and *z* the activation variable of the AHP conductance. The arrows show motion direction on the three-dimensional trajectory. *A2*, Magnification of the trajectory (box in *A1*) when it revisits the unstable focus on the *V*, *W* plane. Note the spiral made by the trajectory around this focus. *A3*, Time evolution of the voltage (bottom trace) and of the AHP activation (*z*, top trace). Note the MMOs on the voltage. Note also that *z* fully relaxes to 0 during the interspike interval well before the next spike is fired. *A4*, The two nullclines *dV*/*dt* = 0 (dashed line) and *dW*/*dt* = 0 (gray line) are drawn in the *V*, *W* plane (i.e., the *z* = 0 plane). The trajectory spirals around the intersection of the two nullclines, i.e., the fixed point of the model. *B1*, Typical trajectory of the model in the primary firing range (PR). The trajectory stays away from the *V*, *W* plane where the unstable fixed point is located. *B2*, Time evolution of the voltage and *z*. Given the high discharge frequency, *z* has no time to relax between spikes. *C*, Bifurcation diagram of the model: the voltage of the stationary solution, both stable and unstable, is displayed as a function of the injected current. In the quiescent regime, the model displays a stable fixed point (solid line). This fixed point becomes unstable at 4.1 nA (first vertical line) through a subcritical Hopf bifurcation (HB) when it merges with an unstable, and thus not experimentally observable, periodic solution (thin dashed line). The model then displays MMOs with an alternation of subthreshold oscillations (the dots near the unstable fixed point indicate their minimum and maximum voltage) and full blown spikes (the series of upper and lower dots show their minimum and maximum voltage). The variations in the peak amplitude of spikes correspond to the successive frequency plateaus in the SPR. At 7.3 nA (second vertical line), the subthreshold oscillations and the unstable periodic solution disappear, and the model enters the PR.