Dendritic signals command firing dynamics in a mathematical model of cerebellar Purkinje cells

Purkinje cell models: past, present and future

The investigation of the dynamics of Purkinje cell (PC) activity is crucial to unravel the role of the cerebellum in motor control, learning and cognitive processes. Within the cerebellar cortex (CC), these neurons receive all the incoming sensory and motor information, transform it and generate the entire cerebellar output. The relatively homogenous and repetitive structure of the CC, common to all vertebrate species, suggests a single computation mechanism shared across all PCs. While PC models have been developed since the 70's, a comprehensive review of contemporary models is currently lacking. Here, we provide an overview of PC models, ranging from the ones focused on single cell intracellular PC dynamics, through complex models which include synaptic and extrasynaptic inputs. We review how PC models can reproduce physiological activity of the neuron, including firing patterns, current and multistable dynamics, plateau potentials, calcium signaling, intrinsic and synaptic plasticity and input/output computations. We consider models focusing both on somatic and on dendritic computations. Our review provides a critical performance analysis of PC models with respect to known physiological data. We expect our synthesis to be useful in guiding future development of computational models that capture real-life PC dynamics in the context of cerebellar computations.

The role of relative membrane capacitance and time delay in cerebellar Purkinje cells

The membrane capacitance of a neuron can influence the synaptic efficacy and the speed of electrical signal propagation. Exploring the role of membrane capacitance will help facilitate a deeper understanding of the electrical properties of neurons. Thus, in this paper, we investigated the neuronal firing behaviors of a two-compartment model in Purkinje cells. We evaluated the influence of membrane capacitance under two different circumstances: in the absence of time delay and in the presence of time delay. Firstly, we separately studied the influence of somatic membrane capacitance Cs and dendritic membrane capacitance Cd on neuronal firing patterns. Through numerical simulation, we observed that they had two different types of period-adding scenarios, i.e. with and without chaotic bursting. Secondly, our results indicated that when the time delay was included in the model, periodic motions were more inclined to be destroyed, while at the same time, corresponding new chaotic motions were induced. These findings suggested that membrane capacitance and time delay play a pivotal functional role in modulating dynamical firing properties of neurons, especially aspects which lead to behaviors which result in changes to bursting patterns.

Conditional Bistability, a Generic Cellular Mnemonic Mechanism for Robust and Flexible Working Memory Computations

Persistent neural activity, the substrate of working memory, is thought to emerge from synaptic reverberation within recurrent networks. However, reverberation models do not robustly explain the fundamental dynamics of persistent activity, including high-spiking irregularity, large intertrial variability, and state transitions. While cellular bistability may contribute to persistent activity, its rigidity appears incompatible with persistent activity labile characteristics. Here, we unravel in a cellular model a form of spike-mediated conditional bistability that is robust and generic. and provides a rich repertoire of mnemonic computations. Under asynchronous synaptic inputs of the awakened state, conditional bistability generates spiking/bursting episodes, accounting for the irregularity, variability, and state transitions characterizing persistent activity. This mechanism has likely been overlooked because of the subthreshold input it requires, and we predict how to assess it experimentally. Our results suggest a reexamination of the role of intrinsic properties in the collective network dynamics responsible for flexible working memory.SIGNIFICANCE STATEMENT This study unravels a novel form of intrinsic neuronal property: conditional bistability. We show that, thanks to its conditional character, conditional bistability favors the emergence of flexible and robust forms of persistent activity in PFC neural networks, in opposition to previously studied classical forms of absolute bistability. Specifically, we demonstrate for the first time that conditional bistability (1) is a generic biophysical spike-dependent mechanism of layer V pyramidal neurons in the PFC and that (2) it accounts for essential neurodynamical features for the organization and flexibility of PFC persistent activity (the large irregularity and intertrial variability of the discharge and its organization under discrete stable states), which remain unexplained in a robust fashion by current models.

The 40-year history of modeling active dendrites in cerebellar Purkinje cells: emergence of the first single cell "community model"

The subject of the effects of the active properties of the Purkinje cell dendrite on neuronal function has been an active subject of study for more than 40 years. Somewhat unusually, some of these investigations, from the outset have involved an interacting combination of experimental and model-based techniques. This article recounts that 40-year history, and the view of the functional significance of the active properties of the Purkinje cell dendrite that has emerged. It specifically considers the emergence from these efforts of what is arguably the first single cell "community" model in neuroscience. The article also considers the implications of the development of this model for future studies of the complex properties of neuronal dendrites.

Long-lasting small-amplitude TRP-mediated dendritic depolarizations in CA1 pyramidal neurons are intrinsically stable and originate from distal tuft regions

In several regions of the nervous system, neurons display bi- or multistable intrinsic properties. Such stable states may be subthreshold and long-lasting, and can appear as a sustained afterdepolarization. In hippocampal CA1 pyramidal neurons, small-amplitude (1 mV) long-lasting (seconds) afterdepolarizations have been reported and are thought to depend on calcium-activated nonselective (CAN) currents recently identified as transient receptor potential (TRP) channels. Continuing our previous experimental and computational work on synaptically metabotropic glutamate receptor (mGluR)-activated TRP currents, we here explore small-amplitude long-lasting depolarizations in a detailed multicompartmental model of a CA1 pyramidal neuron. We confirm a previous hypothesis suggesting that the depolarization results from an interplay of TRP and voltage-gated calcium channels, and contribute to the understanding of the depolarization in several ways. Specifically, we show that: (i) the long-lasting depolarization may be intrinsically stable to weak excitatory and inhibitory input, (ii) the phenomenon is essentially located in distal apical dendrites, (iii) induction is facilitated if simultaneous input arrives at several dendritic branches, and if calcium- and/or mGluR-evoked signals undergo summation, suggesting that both spatial and temporal synaptic summation might be required for the depolarization to occur and (iv) we also show that the integration of inputs to oblique dendrites is strongly modulated by the presence of small-amplitude long-lasting depolarizations in distal tuft dendrites. To conclude, we suggest that small-amplitude long-lasting dendritic depolarizations may contribute to sustaining neural information during behavioural tasks in cases where information is separated in time, as in trace conditioning and delay tasks.

An elementary model of torus canards

We study the recently observed phenomena of torus canards. These are a higher-dimensional generalization of the classical canard orbits familiar from planar systems and arise in fast-slow systems of ordinary differential equations in which the fast subsystem contains a saddle-node bifurcation of limit cycles. Torus canards are trajectories that pass near the saddle-node and subsequently spend long times near a repelling branch of slowly varying limit cycles. In this article, we carry out a study of torus canards in an elementary third-order system that consists of a rotated planar system of van der Pol type in which the rotational symmetry is broken by including a phase-dependent term in the slow component of the vector field. In the regime of fast rotation, the torus canards behave much like their planar counterparts. In the regime of slow rotation, the phase dependence creates rich torus canard dynamics and dynamics of mixed mode type. The results of this elementary model provide insight into the torus canards observed in a higher-dimensional neuroscience model.