An active membrane model of the cerebellar Purkinje cell I. Simulation of current clamps in slice

1. A detailed compartmental model of a cerebellar Purkinje cell with active dendritic membrane was constructed. The model was based on anatomic reconstructions of single Purkinje cells and included 10 different types of voltage-dependent channels described by Hodgkin-Huxley equations, derived from Purkinje cell-specific voltage-clamp data where available. These channels included a fast and persistent Na⁺ channel, three voltage-dependent K⁺ channels. T-type and P-type Ca²⁺ channels, and two types of Ca²⁺- activated K⁺ channels. 2. The ionic channels were distributed differentially over three zones of the model, with Na⁺ channels in the soma, fast K⁺ channels in the soma and main dendrite, and Ca²⁺ channels and Ca²⁺- activated K⁺ channels in the entire dendrite. Channel densities in the model were varied until it could reproduce Purkinje cell responses to current injections in the soma or dendrite, as observed in slice recordings. 3. As in real Purkinje cells, the model generated two types of spiking behavior. In response to small current injections the model fired exclusively fast somatic spikes. These somatic spikes were caused by Na⁺ channels and repolarized by the delayed rectifier. When higher-amplitude current injections were given, sodium spiking increased in frequency until the model generated large dendritic Ca²⁺ spikes. Analysis of membrane currents underlying this behavior showed that these Ca²⁺ spikes were caused by the P-type Ca²⁺ channel and repolarized by the BK-type Ca²⁺-activated K⁺ channel. As in pharmacological blocking experiments, removal of Na⁺ channels abolished the fast spikes and removal of Ca²⁺ channels removed Ca²⁺ spiking. 4. In addition to spiking behavior, the model also produced slow plateau potentials in both the dendrite and soma. These longer-duration potentials occurred in response to both short and prolonged current steps. Analysis of the model demonstrated that the plateau potentials in the soma were caused by the window current component of the fast Na⁺ current, which was much larger than the current through the persistent Na⁺ channels. Plateau potentials in the dendrite were carried by the same P-type Ca²⁺ channel that was also responsible for Ca²⁺ spike generation. The P channel could participate in both model functions because of the low-threshold K2-type Ca²⁺-activated K⁺ channel, which dynamically changed the threshold for dendritic spike generation through a negative feedback loop with the activation kinetics of the P-type Ca²⁺ channel. 5. These model responses were robust to changes in the densities of all of the ionic channels. For most of the channels, modifying their densities by factors of ≥2 resulted only in left or right shifts of the frequency-current curve. However, changes of >20% to the amount of P-type Ca²⁺ channels or of one of the Ca²⁺-activated K⁺ channels in the model either suppressed dendritic spikes or caused the model to always fire Ca²⁺ spikes. Modeling results were also robust to variations in Purkinje cell morphology. We simulated models of two other anatomically reconstructed Purkinje cells with the same channel distributions and got similar responses to current injections. 6. The model was used to compare the electrotonic length of the Purkinje cell in the presence and absence of active dendritic conductances. The electrotonic distance from soma to the tip of the most distal dendrite increased from 0.57 λ in a passive model to 0.95 λ in a quiet model with active membrane. During a dendritic spike generated by current injection the distance increased even more, to 1.57 λ. 7. Finally, the model was used to study the probable accuracy of experimental voltage-clamp data. Whole-cell patch-clamp conditions were simulated by blocking most of the K⁺ currents in the model. The increased electrotonic length due to the active dendritic membrane caused space clamp to fail, resulting in membrane potentials in proximal and distal dendrites that differed critically from the holding potential in the soma.