Figure 1

Simulation results of the soma compartment voltage (black) from a detailed Purkinje cell model [13]. An injected current (gray) depolarizes the cell and produces rapid spiking modulated in amplitude (AM, upper left expanded trace) followed by more complicated activity (upper right expanded trace) between the quiescent intervals ($Q$) of bursts. The horizontal and vertical scales (right) denote 50 ms and 50 mV, respectively.Reuse & Permissions

Figure 2

The transition from bursting to AM spiking to rapid spiking in the reduced model as $J$ decreases. We plot in gray the voltage activity of the neuron for five different values of $J$. In the middle trace ($J=J_{\star }$) we observe both bursting and AM spiking activity. We shade one complete AM cycle in black. Further reductions in $J$ result in AM spiking and (unmodulated) spiking. The vertical bars in the top and bottom traces denote a 100 mV range, extending from $-60$ to 40 mV, with 10 mV between tick marks.Reuse & Permissions

Figure 3

Bifurcation diagram of the fast subsystem (thick and color) and dynamics of the full system (thin and gray/black) for $J=J_{\star }$. In the fast subsystem, the attracting and repelling fixed points and limit cycles meet at folds (yellow circles) labeled Fold FP and Fold LC, respectively. The attracting limit cycles appear in a supercritical Hopf bifurcation (not shown). At the fold of limit cycles the (slow) $M$ current changes direction and the complete dynamics follow the branch of repelling limit cycles temporarily until returning to the curve of attracting fixed points (gray) or attracting limit cycles (black).Reuse & Permissions

Figure 4

Bifurcation diagrams of the fast subsystem (thick and color) and dynamics of the full system (thin and gray/black) for five different values of $J$. In the main figures, the vertical axes are identical; the horizontal axes (identical for top four figures) indicate the value of $M$ and the axis is expanded for the bottom figure. For $J=-23$, only fold-fold cycle bursting occurs (gray). At $J=J_{\star }$, the dynamics follow temporarily the branch of repelling limit cycles, eventually returning to the branch of attracting fixed points (burst in gray) or the branch of attracting limit cycles (AM cycle in black). Reducing $J$ further results in AM spiking alone and rapid, unmodulated spiking. Inset: Poincaré map (voltage versus $M$ current) of the full system for five different values of $J$. Stable fixed points, corresponding to unmodulated rapid spiking, occur for three values of $J$. For $J=\{-32.95,-32.94\}$ stable invariant closed curves indicate amplitude modulation in the complete system.Reuse & Permissions