A biophysical model of nonlinear dynamics underlying plateau potentials and calcium spikes in purkinje cell dendrites

Computational capabilities of Purkinje cells (PCs) are central to the cerebellum function. Information originating from the whole nervous system converges on their dendrites, and their axon is the sole output of the cerebellar cortex. PC dendrites respond to weak synaptic activation with long-lasting, low-amplitude plateau potentials, but stronger synaptic activation can generate fast, large amplitude calcium spikes. Pharmacological data have suggested the involvement of only the P-type of Ca channels in both of these electric responses. However, the mechanism allowing this Ca current to underlie responses with such different dynamics is still unclear. This mechanism was explored by constraining a biophysical model with electrophysiological, Ca-imaging, and single ion channel data. A model is presented here incorporating a simplified description of [Ca](i) regulation and three ionic currents: 1) the P-type Ca current, 2) a delayed-rectifier K current, and 3) a generic class of K channels activating sharply in the sub-threshold voltage range. This model sustains fast spikes and long-lasting plateaus terminating spontaneously with recovery of the resting potential. Small depolarizing, tonic inputs turn plateaus into a stable membrane state and endow the dendrite with bistability properties. With larger tonic inputs, the plateau remains the unique equilibrium state, showing long traces of transient inhibitory inputs that are called "valley potentials" because their dynamics mirrors that of inverted, finite-duration plateaus. Analyzing the slow subsystem obtained by assuming instantaneous activation of the delayed-rectifier reveals that the time course of plateaus and valleys is controlled by the slow [Ca](i) dynamics, which arises from the high Ca-buffering capacity of PCs. A bifurcation analysis shows that tonic currents modulate sub-threshold dynamics by displacing the resting state along a hysteresis region edged by two saddle-node bifurcations; these bifurcations mark transitions from finite-duration plateaus to bistability and from bistability to valley potentials, respectively. This low-dimensionality model may be introduced into large-scale models to explore the role of PC dendrite computations in the functional capabilities of the cerebellum.