Firing dynamics 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.

Altered Cerebellar Response to Somatosensory Stimuli in the Cntnap2 Mouse Model of Autism

Atypical sensory processing is currently included within the diagnostic criteria of autism. The cerebellum is known to integrate sensory inputs of different modalities through its connectivity to the cerebral cortex. Interestingly, cerebellar malformations are among the most replicated features found in postmortem brain of individuals with autism. We studied sensory processing in the cerebellum in a mouse model of autism, knock-out (KO) for the Cntnap2 gene. Cntnap2 is widely expressed in Purkinje cells (PCs) and has been recently reported to regulate their morphology. Further, individuals with CNTNAP2 mutations display cerebellar malformations and CNTNAP2 antibodies are associated with a mild form of cerebellar ataxia. Previous studies in the Cntnap2 mouse model show an altered cerebellar sensory learning. However, a physiological analysis of cerebellar function has not been performed yet. We studied sensory evoked potentials in cerebellar Crus I/II region on electrical stimulation of the whisker pad in alert mice and found striking differences between wild-type and Cntnap2 KO mice. In addition, single-cell recordings identified alterations in both sensory-evoked and spontaneous firing patterns of PCs. These changes were accompanied by altered intrinsic properties and morphologic features of these neurons. Together, these results indicate that the Cntnap2 mouse model could provide novel insight into the pathophysiological mechanisms of autism core sensory deficits.

A biochemical mechanism for time-encoding memory formation within individual synapses of Purkinje cells

Within the classical eye-blink conditioning, Purkinje cells within the cerebellum are known to suppress their tonic firing rates for a well defined time period in response to the conditional stimulus after training. The temporal profile of the drop in tonic firing rate, i.e., the onset and the duration, depend upon the time interval between the onsets of the conditional and unconditional training stimuli. Direct stimulation of parallel fibers and climbing fiber by electrodes was found to be sufficient to reproduce the same characteristic drop in the firing rate of the Purkinje cell. In addition, the specific metabotropic glutamate-based receptor type 7 (mGluR₇) was found responsible for the initiation of the response, suggesting an intrinsic mechanism within the Purkinje cell for the temporal learning. In an attempt to look for a mechanism for time-encoding memory formation within individual Purkinje cells, we propose a biochemical mechanism based on recent experimental findings. The proposed mechanism tries to answer key aspects of the “Coding problem” of Neuroscience by focusing on the Purkinje cell’s ability to encode time intervals through training. According to the proposed mechanism, the time memory is encoded within the dynamics of a set of proteins—mGluR₇, G-protein, G-protein coupled Inward Rectifier Potassium ion channel, Protein Kinase A, Protein Phosphatase 1 and other associated biomolecules—which self-organize themselves into a protein complex. The intrinsic dynamics of these protein complexes can differ and thus can encode different time durations. Based on their amount and their collective dynamics within individual synapses, the Purkinje cell is able to suppress its own tonic firing rate for a specific time interval. The time memory is encoded within the effective dynamics of the biochemical reactions and altering these dynamics means storing a different time memory. The proposed mechanism is verified by both a minimal and a more comprehensive mathematical model of the conditional response behavior of the Purkinje cell and corresponding dynamical simulations of the involved biomolecules, yielding testable experimental predictions.

Bursting in cerebellar stellate cells induced by pharmacological agents: Non-sequential spike adding

Cerebellar stellate cells (CSCs) are spontaneously active, tonically firing (5-30 Hz), inhibitory interneurons that synapse onto Purkinje cells. We previously analyzed the excitability properties of CSCs, focusing on four key features: type I excitability, non-monotonic first-spike latency, switching in responsiveness and runup (i.e., temporal increase in excitability during whole-cell configuration). In this study, we extend this analysis by using whole-cell configuration to show that these neurons can also burst when treated with certain pharmacological agents separately or jointly. Indeed, treatment with 4-Aminopyridine (4-AP), a partial blocker of delayed rectifier and A-type K⁺ channels, at low doses induces a bursting profile in CSCs significantly different than that produced at high doses or when it is applied at low doses but with cadmium (Cd²⁺), a blocker of high voltage-activated (HVA) Ca²⁺ channels. By expanding a previously revised Hodgkin–Huxley type model, through the inclusion of Ca²⁺-activated K⁺ (K(Ca)) and HVA currents, we explain how these bursts are generated and what their underlying dynamics are. Specifically, we demonstrate that the expanded model preserves the four excitability features of CSCs, as well as captures their bursting patterns induced by 4-AP and Cd²⁺. Model investigation reveals that 4-AP is potentiating HVA, inducing square-wave bursting at low doses and pseudo-plateau bursting at high doses, whereas Cd²⁺ is potentiating K(Ca), inducing pseudo-plateau bursting when applied in combination with low doses of 4-AP. Using bifurcation analysis, we show that spike adding in square-wave bursts is non-sequential when gradually changing HVA and K(Ca) maximum conductances, delayed Hopf is responsible for generating the plateau segment within the active phase of pseudo-plateau bursts, and bursting can become “chaotic” when HVA and K(Ca) maximum conductances are made low and high, respectively. These results highlight the secondary effects of the drugs applied and suggest that CSCs have all the ingredients needed for bursting.

Excitable cells, including neurons, fire action potentials (APs) in their membrane voltage that allow them to communicate with each other and to serve certain physiological purposes. They do so either tonically by firing APs periodically, or episodically by repeatedly firing clusters of APs (called bursts) separated by quiescent periods. Each one of those firing patterns can be neuron-specific and dependent on synaptic inputs and/or their physiological environment. Cerebellar stellate cells (CSCs) that synapse onto Purkinje cells, the sole output of the cerebellum responsible for motor control, are spontaneously active inhibitory interneurons that fire APs tonically. We previously studied the excitability properties of these neurons and showed that they possess several important key features, including type I excitability, runup, non-monotonic first spike latency and switching in responsiveness. In this study, we show that CSCs can also exhibit two modes of burst firing, called square-wave and pseudo-plateau, when treated with certain pharmacological agents. Using bifurcation theory, we demonstrate that spike adding in the square-wave burst is non-sequential, changing by several spikes when certain conductances are altered gradually. This study thus sheds lights onto the overall effects of the pharmacological agents and highlights the ability of CSCs to burst in certain biological conditions.

Switching in Cerebellar Stellate Cell Excitability in Response to a Pair of Inhibitory/Excitatory Presynaptic Inputs: A Dynamical System Perspective

Cerebellar stellate cells form inhibitory synapses with Purkinje cells, the sole output of the cerebellum. Upon stimulation by a pair of varying inhibitory and fixed excitatory presynaptic inputs, these cells do not respond to excitation (i.e., do not generate an action potential) when the magnitude of the inhibition is within a given range, but they do respond outside this range. We previously used a revised Hodgkin–Huxley type of model to study the nonmonotonic first-spike latency of these cells and their temporal increase in excitability in whole cell configuration (termed run-up). Here, we recompute these latency profiles using the same model by adapting an efficient computational technique, the two-point boundary value problem, that is combined with the continuation method. We then extend the study to investigate how switching in responsiveness, upon stimulation with presynaptic inputs, manifests itself in the context of run-up. A three-dimensional reduced model is initially derived from the original six-dimensional model and then analyzed to demonstrate that both models exhibit type 1 excitability possessing a saddle-node on an invariant cycle (SNIC) bifurcation when varying the amplitude of [Formula: see text]. Using slow-fast analysis, we show that the original model possesses three equilibria lying at the intersection of the critical manifold of the fast subsystem and the nullcline of the slow variable [Formula: see text] (the inactivation of the A-type K[Formula: see text] channel), the middle equilibrium is of saddle type with two-dimensional stable manifold (computed from the reduced model) acting as a boundary between the responsive and non-responsive regimes, and the (ghost of) SNIC is formed when the [Formula: see text]-nullcline is (nearly) tangential to the critical manifold. We also show that the slow dynamics associated with (the ghost of) the SNIC and the lower stable branch of the critical manifold are responsible for generating the nonmonotonic first-spike latency. These results thus provide important insight into the complex dynamics of stellate cells.

Voltage-Dependent Membrane Properties Shape the Size But Not the Frequency Content of Spontaneous Voltage Fluctuations in Layer 2/3 Somatosensory Cortex

Under awake and idling conditions, spontaneous intracellular membrane voltage is characterized by large, synchronous, low-frequency fluctuations. Although these properties reflect correlations in synaptic inputs, intrinsic membrane properties often indicate voltage-dependent changes in membrane resistance and time constant values that can amplify and help to generate low-frequency voltage fluctuations. The specific contribution of intrinsic and synaptic factors to the generation of spontaneous fluctuations, however, remains poorly understood. Using visually guided intracellular recordings of somatosensory layer 2/3 pyramidal cells and interneurons in awake male and female mice, we measured the spectrum and size of voltage fluctuation and intrinsic cellular properties at different voltages. In both cell types, depolarizing neurons increased the size of voltage fluctuations. Amplitude changes scaled with voltage-dependent changes in membrane input resistance. Because of the small membrane time constants observed in both pyramidal cells and interneuron cell bodies, the low-frequency content of membrane fluctuations reflects correlations in the synaptic current inputs rather than significant filtering associated with membrane capacitance. Further, blocking synaptic inputs minimally altered somatic membrane resistance and time constant values. Overall, these results indicate that spontaneous synaptic inputs generate a low-conductance state in which the amplitude, but not frequency structure, is influenced by intrinsic membrane properties.SIGNIFICANCE STATEMENT In the absence of sensory drive, cortical activity in awake animals is associated with self-generated and seemingly random membrane voltage fluctuations characterized by large amplitude and low frequency. Partially, these properties reflect correlations in synaptic input. Nonetheless, neurons express voltage-dependent intrinsic properties that can potentially influence the amplitude and frequency of spontaneous activity. Using visually guided intracellular recordings of cortical neurons in awake mice, we measured the voltage dependence of spontaneous voltage fluctuations and intrinsic membrane properties. We show that voltage-dependent changes in membrane resistance amplify synaptic activity, whereas the frequency of voltage fluctuations reflects correlations in synaptic inputs. Last, synaptic activity has a small impact on intrinsic membrane properties in both pyramidal cells and interneurons.

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.

Propensity for Bistability of Bursting and Silence in the Leech Heart Interneuron

The coexistence of neuronal activity regimes has been reported under normal and pathological conditions. Such multistability could enhance the flexibility of the nervous system and has many implications for motor control, memory, and decision making. Multistability is commonly promoted by neuromodulation targeting specific membrane ionic currents. Here, we investigated how modulation of different ionic currents could affect the neuronal propensity for bistability. We considered a leech heart interneuron model. It exhibits bistability of bursting and silence in a narrow range of the leak current parameters, conductance (g_leak) and reversal potential (E_leak). We assessed the propensity for bistability of the model by using bifurcation diagrams. On the diagram (g_leak, E_leak), we mapped bursting and silent regimes. For the canonical value of E_leak we determined the range of g_leak which supported the bistability. We use this range as an index of propensity for bistability. We investigated how this index was affected by alterations of ionic currents. We systematically changed their conductances, one at a time, and built corresponding bifurcation diagrams in parameter planes of the maximal conductance of a given current and the leak conductance. We found that conductance of only one current substantially affected the index of propensity; the increase of the maximal conductance of the hyperpolarization-activated cationic current increased the propensity index. The second conductance with the strongest effect was the conductance of the low-threshold fast Ca²⁺ current; its reduction increased the propensity index although the effect was about two times smaller in magnitude. Analyzing the model with both changes applied simultaneously, we found that the diagram (g_leak, E_leak) showed a progressively expanded area of bistability of bursting and silence.

Oscillations, Timing, Plasticity, and Learning in the Cerebellum

The highly stereotyped, crystal-like architecture of the cerebellum has long served as a basis for hypotheses with regard to the function(s) that it subserves. Historically, most clinical observations and experimental work have focused on the involvement of the cerebellum in motor control, with particular emphasis on coordination and learning. Two main models have been suggested to account for cerebellar functioning. According to Llinás's theory, the cerebellum acts as a control machine that uses the rhythmic activity of the inferior olive to synchronize Purkinje cell populations for fine-tuning of coordination. In contrast, the Ito-Marr-Albus theory views the cerebellum as a motor learning machine that heuristically refines synaptic weights of the Purkinje cell based on error signals coming from the inferior olive. Here, we review the role of timing of neuronal events, oscillatory behavior, and synaptic and non-synaptic influences in functional plasticity that can be recorded in awake animals in various physiological and pathological models in a perspective that also includes non-motor aspects of cerebellar function. We discuss organizational levels from genes through intracellular signaling, synaptic network to system and behavior, as well as processes from signal production and processing to memory, delegation, and actual learning. We suggest an integrative concept for control and learning based on articulated oscillation templates.

Phase-Dependent Modulation of Oscillatory Phase and Synchrony by Long-Lasting Depolarizing Inputs in Central Neurons

Oscillatory neural activities have been implicated in various types of information processing in the CNS. The procerebral (PC) lobe of the land mollusk Limax valentianus shows an ongoing oscillatory local field potential (LFP). Olfactory input increases both the frequency and spatial synchrony of the LFP oscillation by a nitric oxide (NO)-mediated mechanism, but how NO modulates the activity in a specific manner has been unclear. In the present study, we used electrical stimulation and NO uncaging to systematically analyze the response of the LFP oscillation and found phase-dependent effects on phase shifting and synchrony. The neurons that presumably release NO in the PC lobe preferentially fired at phases in which NO has a synchronizing effect, suggesting that the timing of NO release is regulated to induce a stereotyped response to natural sensory stimuli. The phase-response curve (PRC) describes the timing dependence of responses of an oscillatory system to external input. PRCs are usually constructed by recording the temporal shifts of the neural activity in response to brief electrical pulses. However, NO evokes a long-lasting depolarization persisting for several cycles of oscillation. The phase-response relationship obtained by NO stimulation was approximately the integral of the PRC. A similar relationship was also shown for regular firing of mouse cerebellar Purkinje cells receiving step depolarization, suggesting the generality of the results to oscillatory neural systems with highly distinct properties. These results indicate novel dynamic effects of long-lasting inputs on network oscillation and synchrony, which are based on simple and ubiquitous mechanisms.

Inverse Stochastic Resonance in Cerebellar Purkinje Cells

Purkinje neurons play an important role in cerebellar computation since their axons are the only projection from the cerebellar cortex to deeper cerebellar structures. They have complex internal dynamics, which allow them to fire spontaneously, display bistability, and also to be involved in network phenomena such as high frequency oscillations and travelling waves. Purkinje cells exhibit type II excitability, which can be revealed by a discontinuity in their f-I curves. We show that this excitability mechanism allows Purkinje cells to be efficiently inhibited by noise of a particular variance, a phenomenon known as inverse stochastic resonance (ISR). While ISR has been described in theoretical models of single neurons, here we provide the first experimental evidence for this effect. We find that an adaptive exponential integrate-and-fire model fitted to the basic Purkinje cell characteristics using a modified dynamic IV method displays ISR and bistability between the resting state and a repetitive activity limit cycle. ISR allows the Purkinje cell to operate in different functional regimes: the all-or-none toggle or the linear filter mode, depending on the variance of the synaptic input. We propose that synaptic noise allows Purkinje cells to quickly switch between these functional regimes. Using mutual information analysis, we demonstrate that ISR can lead to a locally optimal information transfer between the input and output spike train of the Purkinje cell. These results provide the first experimental evidence for ISR and suggest a functional role for ISR in cerebellar information processing.

How neurons generate output spikes in response to various combinations of inputs is a central issue in contemporary neuroscience. Due to their large dendritic tree and complex intrinsic properties, cerebellar Purkinje cells are an important model system to study this input-output transformation. Here we examine how noise can change the parameters of this transformation. In experiments we found that spike generation in Purkinje cells can be efficiently inhibited by noise of a particular amplitude. This effect is called inverse stochastic resonance (ISR) and has previously been described only in theoretical models of neurons. We explain the mechanism underlying ISR using a simple model matching the properties of experimentally characterized Purkinje cells. We found that ISR is present in Purkinje cells when the mean input current is near threshold for spike generation. ISR can be explained by the co-existence of resting and spiking solutions of the simple model. Changes of the input noise variance change the lifetime of these resting and spiking states, suggesting a mechanism for a tunable filter with long time constants implemented by a Purkinje cell population in the cerebellum. Finally, ISR leads to locally optimal information transfer from the input to the output of a Purkinje cell.

Modeling the Cerebellar Microcircuit: New Strategies for a Long-Standing Issue

The cerebellar microcircuit has been the work bench for theoretical and computational modeling since the beginning of neuroscientific research. The regular neural architecture of the cerebellum inspired different solutions to the long-standing issue of how its circuitry could control motor learning and coordination. Originally, the cerebellar network was modeled using a statistical-topological approach that was later extended by considering the geometrical organization of local microcircuits. However, with the advancement in anatomical and physiological investigations, new discoveries have revealed an unexpected richness of connections, neuronal dynamics and plasticity, calling for a change in modeling strategies, so as to include the multitude of elementary aspects of the network into an integrated and easily updatable computational framework. Recently, biophysically accurate “realistic” models using a bottom-up strategy accounted for both detailed connectivity and neuronal non-linear membrane dynamics. In this perspective review, we will consider the state of the art and discuss how these initial efforts could be further improved. Moreover, we will consider how embodied neurorobotic models including spiking cerebellar networks could help explaining the role and interplay of distributed forms of plasticity. We envisage that realistic modeling, combined with closed-loop simulations, will help to capture the essence of cerebellar computations and could eventually be applied to neurological diseases and neurorobotic control systems.

Purkinje cell intrinsic excitability increases after synaptic long term depression

Coding in cerebellar Purkinje cells not only depends on synaptic plasticity but also on their intrinsic membrane excitability. We performed whole cell patch-clamp recordings of Purkinje cells in sagittal cerebellar slices in mice. We found that inducing long-term depression (LTD) in the parallel fiber to Purkinje cell synapses results in an increase in the gain of the firing rate response. This increase in excitability is accompanied by an increase in the input resistance and a decrease in the amplitude of the hyperpolarization-activated cyclic nucleotide-gated (HCN) channel-mediated voltage sag. Application of a HCN channel blocker prevents the increase in input resistance and excitability without blocking the expression of synaptic LTD. We conclude that the induction of parallel fiber-Purkinje cell LTD is accompanied by an increase in excitability of Purkinje cells through downregulation of the HCN-mediated h current. We suggest that HCN downregulation is linked to the biochemical pathway that sustains synaptic LTD. Given the diversity of information carried by the parallel fiber system, we suggest that changes in intrinsic excitability enhance the coding capacity of the Purkinje cell to specific input sources.

Long Pauses in Cerebellar Interneurons in Anesthetized Animals

Are long pauses in the firing of cerebellar interneurons (CINs) related to Purkinje cell (PC) pauses? If PC pauses affect the larger network, then we should find a close relationship between CIN pauses and those in PCs. We recorded activity of 241 cerebellar cortical neurons (206 CINs and 35 PCs) in three anesthetized cats. One fifth of the CINs and more than half of the PCs were identified as pausing. Pauses in CINs and PCs showed some differences: CIN mean pause length was shorter, and, after pauses, only CINs had sustained reduction in their firing rate (FR). Almost all pausing CINs fell into same cluster when we used different methods of clustering CINs by their spontaneous activity. The mean spontaneous firing rate of that cluster was approximately 53 Hz. We also examined cross-correlations in simultaneously recorded neurons. Of 39 cell pairs examined, 14 (35 %) had cross-correlations significantly different from those expected by chance. Almost half of the pairs with two CINs showed statistically significant negative correlations. In contrast, PC/CIN pairs did not often show significant effects in the cross-correlation (12/15 pairs). However, for both CIN/CIN and PC/CIN pairs, pauses in one unit tended to correspond to a reduction in the firing rate of the adjacent unit. In our view, our results support the possibility that previously reported PC bistability is part of a larger network response and not merely a biophysical property of PCs. Any functional role for PC bistability should probably be sought in the context of the broader network.

Cerebellar Nuclear Neurons Use Time and Rate Coding to Transmit Purkinje Neuron Pauses

Neurons of the cerebellar nuclei convey the final output of the cerebellum to their targets in various parts of the brain. Within the cerebellum their direct upstream connections originate from inhibitory Purkinje neurons. Purkinje neurons have a complex firing pattern of regular spikes interrupted by intermittent pauses of variable length. How can the cerebellar nucleus process this complex input pattern? In this modeling study, we investigate different forms of Purkinje neuron simple spike pause synchrony and its influence on candidate coding strategies in the cerebellar nuclei. That is, we investigate how different alignments of synchronous pauses in synthetic Purkinje neuron spike trains affect either time-locking or rate-changes in the downstream nuclei. We find that Purkinje neuron synchrony is mainly represented by changes in the firing rate of cerebellar nuclei neurons. Pause beginning synchronization produced a unique effect on nuclei neuron firing, while the effect of pause ending and pause overlapping synchronization could not be distinguished from each other. Pause beginning synchronization produced better time-locking of nuclear neurons for short length pauses. We also characterize the effect of pause length and spike jitter on the nuclear neuron firing. Additionally, we find that the rate of rebound responses in nuclear neurons after a synchronous pause is controlled by the firing rate of Purkinje neurons preceding it.

Neurons can transmit information by two different coding strategies: Rate coding, where the firing rate of the neuron is vital, and time coding where timing of individual spikes carries relevant information. In this study we analyze the importance of brief cessations in firing of the presynaptic neuron (pauses) on the spiking of the postsynaptic neuron. We perform this analysis on the inhibitory synaptic connection between Purkinje neurons (presynaptic) and nuclear neurons (postsynaptic) of the cerebellum. We employ a computational model of nuclear neurons and “synthetic” Purkinje neuron spike trains to study the effect of synchronous pauses on the spiking responses of nuclear neurons. We find that synchronous pauses can cause both well-timed spikes and increased firing rate in the nuclear neuron. In addition, we characterize the effect of pause length, amount and type of pause synchrony, and spike jitter. As such, we conclude that nuclear cells use both rate and time coding to relay upstream spiking information.

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.

New uses of LFPs: Pathway-specific threads obtained through spatial discrimination

Local field potentials (LFPs) reflect the coordinated firing of functional neural assemblies during information coding and transfer across neural networks. As such, it was proposed that the extraordinary variety of cytoarchitectonic elements in the brain is responsible for the wide range of amplitudes and for the coverage of field potentials, which in most cases receive contributions from multiple pathways and populations. The influence of spatial factors overrides the bold interpretations of customary measurements, such as the amplitude and polarity, to the point that their cellular interpretation is one of the hardest tasks in Neurophysiology. Temporal patterns and frequency bands are not exclusive to pathways but rather, the spatial configuration of the voltage gradients created by each pathway is highly specific and may be used advantageously. Recent technical and analytical advances now make it possible to separate and then reconstruct activity for specific pathways. In this review, we discuss how spatial features specific to cells and populations define the amplitude and extension of LFPs, why they become virtually indecipherable when several pathways are co-activated, and then we present the recent advances regarding their disentanglement using spatial discrimination techniques. The pathway-specific threads of LFPs have a simple cellular interpretation, and the temporal fluctuations obtained can be applied to a variety of new experimental objectives and improve existing approaches. Among others, they facilitate the parallel readout of activity in several populations over multiple time scales correlating them with behavior. Also, they access information contained in irregular fluctuations, facilitating the testing of ongoing plasticity. In addition, they open the way to unravel the synaptic nature of rhythmic oscillations, as well as the dynamic relationships between multiple oscillatory activities. The challenge of understanding which waves belong to which populations, and the pathways that provoke them, may soon be overcome.

Non-linear Membrane Properties in Entorhinal Cortical Stellate Cells Reduce Modulation of Input-Output Responses by Voltage Fluctuations

The presence of voltage fluctuations arising from synaptic activity is a critical component in models of gain control, neuronal output gating, and spike rate coding. The degree to which individual neuronal input-output functions are modulated by voltage fluctuations, however, is not well established across different cortical areas. Additionally, the extent and mechanisms of input-output modulation through fluctuations have been explored largely in simplified models of spike generation, and with limited consideration for the role of non-linear and voltage-dependent membrane properties. To address these issues, we studied fluctuation-based modulation of input-output responses in medial entorhinal cortical (MEC) stellate cells of rats, which express strong sub-threshold non-linear membrane properties. Using in vitro recordings, dynamic clamp and modeling, we show that the modulation of input-output responses by random voltage fluctuations in stellate cells is significantly limited. In stellate cells, a voltage-dependent increase in membrane resistance at sub-threshold voltages mediated by Na⁺ conductance activation limits the ability of fluctuations to elicit spikes. Similarly, in exponential leaky integrate-and-fire models using a shallow voltage-dependence for the exponential term that matches stellate cell membrane properties, a low degree of fluctuation-based modulation of input-output responses can be attained. These results demonstrate that fluctuation-based modulation of input-output responses is not a universal feature of neurons and can be significantly limited by subthreshold voltage-gated conductances.

The membrane voltage of neurons in vivo is dominated by noisy “background” fluctuations generated by network-based synaptic activity from nearby cells. It has been speculated that membrane voltage fluctuations in neurons play an important role in scaling the relationship between input amplitude and spike rate response. For this to be true, neuronal spike input-output behavior must be sensitive to physiological membrane voltage fluctuations. Using a combination of single cell recordings and modeling, we investigated the mechanisms through which voltage fluctuations modulate neuronal input-output responses. We find that neurons that express an increase in membrane input resistance with depolarization show low levels of noise-mediated modulation of input-output responses due, in part, to voltage trajectories that suppress the likelihood of generating a spike in response to random current input fluctuations. Hence, non-linear membrane properties arising from certain types of voltage-gated conductances limit noise-based modulation of neuronal input-output responses.

On the firing rate dependency of the phase response curve of rat Purkinje neurons in vitro

Synchronous spiking during cerebellar tasks has been observed across Purkinje cells: however, little is known about the intrinsic cellular mechanisms responsible for its initiation, cessation and stability. The Phase Response Curve (PRC), a simple input-output characterization of single cells, can provide insights into individual and collective properties of neurons and networks, by quantifying the impact of an infinitesimal depolarizing current pulse on the time of occurrence of subsequent action potentials, while a neuron is firing tonically. Recently, the PRC theory applied to cerebellar Purkinje cells revealed that these behave as phase-independent integrators at low firing rates, and switch to a phase-dependent mode at high rates. Given the implications for computation and information processing in the cerebellum and the possible role of synchrony in the communication with its post-synaptic targets, we further explored the firing rate dependency of the PRC in Purkinje cells. We isolated key factors for the experimental estimation of the PRC and developed a closed-loop approach to reliably compute the PRC across diverse firing rates in the same cell. Our results show unambiguously that the PRC of individual Purkinje cells is firing rate dependent and that it smoothly transitions from phase independent integrator to a phase dependent mode. Using computational models we show that neither channel noise nor a realistic cell morphology are responsible for the rate dependent shift in the phase response curve.

Intracellular calcium dynamics permit a Purkinje neuron model to perform toggle and gain computations upon its inputs

Without synaptic input, Purkinje neurons can spontaneously fire in a repeating trimodal pattern that consists of tonic spiking, bursting and quiescence. Climbing fiber input (CF) switches Purkinje neurons out of the trimodal firing pattern and toggles them between a tonic firing and a quiescent state, while setting the gain of their response to Parallel Fiber (PF) input. The basis to this transition is unclear. We investigate it using a biophysical Purkinje cell model under conditions of CF and PF input. The model can replicate these toggle and gain functions, dependent upon a novel account of intracellular calcium dynamics that we hypothesize to be applicable in real Purkinje cells.

Bidirectional modulation of deep cerebellar nuclear cells revealed by optogenetic manipulation of inhibitory inputs from Purkinje cells

In the mammalian cerebellum, deep cerebellar nuclear (DCN) cells convey all information from cortical Purkinje cells (PCs) to premotor nuclei and other brain regions. However, how DCN cells integrate inhibitory input from PCs with excitatory inputs from other sources has been difficult to assess, in part due to the large spatial separation between cortical PCs and their target cells in the nuclei. To circumvent this problem we have used a Cre-mediated genetic approach to generate mice in which channelrhodopsin-2 (ChR2), fused with a fluorescent reporter, is selectively expressed by GABAergic neurons, including PCs. In recordings from brain slice preparations from this model, mammalian PCs can be robustly depolarized and discharged by brief photostimulation. In recordings of postsynaptic DCN cells, photostimulation of PC axons induces a strong inhibition that resembles these cells' responses to focal electrical stimulation, but without a requirement for the glutamate receptor blockers typically applied in such experiments. In this optogenetic model, laser pulses as brief as 1 ms can reliably induce an inhibition that shuts down the spontaneous spiking of a DCN cell for ∼50 ms. If bursts of such brief light pulses are delivered, a fixed pattern of bistable bursting emerges. If these pulses are delivered continuously to a spontaneously bistable cell, the immediate response to such photostimulation is inhibitory in the cell's depolarized state and excitatory when the membrane has repolarized; a less regular burst pattern then persists after stimulation has been terminated. These results indicate that the spiking activity of DCN cells can be bidirectionally modulated by the optically activated synaptic inhibition of cortical PCs.

Emergence of a 600-Hz buzz UP state Purkinje cell firing in alert mice

Purkinje cell (PC) firing represents the sole output from the cerebellar cortex onto the deep cerebellar and vestibular nuclei. Here, we explored the different modes of PC firing in alert mice by extracellular recording. We confirm the existence of a tonic and/or bursting and quiescent modes corresponding to UP and DOWN state, respectively. We demonstrate the existence of a novel 600-Hz buzz UP state of firing characterized by simple spikes (SS) of very small amplitude. Climbing fiber (CF) input is able to switch the 600-Hz buzz to the DOWN state, as for the classical UP-to-DOWN state transition. Conversely, the CF input can initiate a typical SS pattern terminating into 600-Hz buzz. The 600-Hz buzz was transiently suppressed by whisker pad stimulation demonstrating that it remained responsive to peripheral input. It must not be mistaken for a DOWN state or the sign of PC inhibition. Complex spike (CS) frequency was increased during the 600-Hz buzz, indicating that this PC output actively contributes to the cerebello-olivary loop by triggering a disinhibition of the inferior olive. During the 600-Hz buzz, the first depolarizing component of the CS was reduced and the second depolarizing component was suppressed. Consistent with our experimental observations, using a 559-compartment single-PC model - in which PC UP state (of about -43mV) was obtained by the combined action of large tonic AMPA conductances and counterbalancing GABAergic inhibition - removal of this inhibition produced the 600-Hz buzz; the simulated buzz frequency decreased following an artificial CS.

Signal processing by T-type calcium channel interactions in the cerebellum

T-type calcium channels of the Cav3 family are unique among voltage-gated calcium channels due to their low activation voltage, rapid inactivation, and small single channel conductance. These special properties allow Cav3 calcium channels to regulate neuronal processing in the subthreshold voltage range. Here, we review two different subthreshold ion channel interactions involving Cav3 channels and explore the ability of these interactions to expand the functional roles of Cav3 channels. In cerebellar Purkinje cells, Cav3 and intermediate conductance calcium-activated potassium (IKCa) channels form a novel complex which creates a low voltage-activated, transient outward current capable of suppressing temporal summation of excitatory postsynaptic potentials (EPSPs). In large diameter neurons of the deep cerebellar nuclei, Cav3-mediated calcium current (I T) and hyperpolarization-activated cation current (I H) are activated during trains of inhibitory postsynaptic potentials. These currents have distinct, and yet synergistic, roles in the subthreshold domain with I T generating a rebound burst and I H controlling first spike latency and rebound spike precision. However, by shortening the membrane time constant the membrane returns towards resting value at a faster rate, allowing I H to increase the efficacy of I T and increase the range of burst frequencies that can be generated. The net effect of Cav3 channels thus depends on the channels with which they are paired. When expressed in a complex with a KCa channel, Cav3 channels reduce excitability when processing excitatory inputs. If functionally coupled with an HCN channel, the depolarizing effect of Cav3 channels is accentuated, allowing for efficient inversion of inhibitory inputs to generate a rebound burst output. Therefore, signal processing relies not only on the activity of individual subtypes of channels but also on complex interactions between ion channels whether based on a physical complex or by indirect effects on membrane properties.

Translational approach to behavioral learning: lessons from cerebellar plasticity

The role of cerebellar plasticity has been increasingly recognized in learning. The privileged relationship between the cerebellum and the inferior olive offers an ideal circuit for attempting to integrate the numerous evidences of neuronal plasticity into a translational perspective. The high learning capacity of the Purkinje cells specifically controlled by the climbing fiber represents a major element within the feed-forward and feedback loops of the cerebellar cortex. Reciprocally connected with the basal ganglia and multimodal cerebral domains, this cerebellar network may realize fundamental functions in a wide range of behaviors. This review will outline the current understanding of three main experimental paradigms largely used for revealing cerebellar functions in behavioral learning: (1) the vestibuloocular reflex and smooth pursuit control, (2) the eyeblink conditioning, and (3) the sensory envelope plasticity. For each of these experimental conditions, we have critically revisited the chain of causalities linking together neural circuits, neural signals, and plasticity mechanisms, giving preference to behaving or alert animal physiology. Namely, recent experimental approaches mixing neural units and local field potentials recordings have demonstrated a spike timing dependent plasticity by which the cerebellum remains at a strategic crossroad for deciphering fundamental and translational mechanisms from cellular to network levels.

An integrator circuit in cerebellar cortex

The brain builds dynamic models of the body and the outside world to predict the consequences of actions and stimuli. A well-known example is the oculomotor integrator, which anticipates the position-dependent elasticity forces acting on the eye ball by mathematically integrating over time oculomotor velocity commands. Many models of neural integration have been proposed, based on feedback excitation, lateral inhibition or intrinsic neuronal nonlinearities. We report here that a computational model of the cerebellar cortex, a structure thought to implement dynamic models, reveals a hitherto unrecognized integrator circuit. In this model, comprising Purkinje cells, molecular layer interneurons and parallel fibres, Purkinje cells were able to generate responses lasting more than 10 s, to which both neuronal and network mechanisms contributed. Activation of the somatic fast sodium current by subthreshold voltage fluctuations was able to maintain pulse-evoked graded persistent activity, whereas lateral inhibition among Purkinje cells via recurrent axon collaterals further prolonged the responses to step and sine wave stimulation. The responses of Purkinje cells decayed with a time-constant whose value depended on their baseline spike rate, with integration vanishing at low (< 1 per s) and high rates (> 30 per s). The model predicts that the apparently fast circuit of the cerebellar cortex may control the timing of slow processes without having to rely on sensory feedback. Thus, the cerebellar cortex may contain an adaptive temporal integrator, with the sensitivity of integration to the baseline spike rate offering a potential mechanism of plasticity of the response time-constant.

Probabilistic identification of cerebellar cortical neurones across species

Despite our fine-grain anatomical knowledge of the cerebellar cortex, electrophysiological studies of circuit information processing over the last fifty years have been hampered by the difficulty of reliably assigning signals to identified cell types. We approached this problem by assessing the spontaneous activity signatures of identified cerebellar cortical neurones. A range of statistics describing firing frequency and irregularity were then used, individually and in combination, to build Gaussian Process Classifiers (GPC) leading to a probabilistic classification of each neurone type and the computation of equi-probable decision boundaries between cell classes. Firing frequency statistics were useful for separating Purkinje cells from granular layer units, whilst firing irregularity measures proved most useful for distinguishing cells within granular layer cell classes. Considered as single statistics, we achieved classification accuracies of 72.5% and 92.7% for granular layer and molecular layer units respectively. Combining statistics to form twin-variate GPC models substantially improved classification accuracies with the combination of mean spike frequency and log-interval entropy offering classification accuracies of 92.7% and 99.2% for our molecular and granular layer models, respectively. A cross-species comparison was performed, using data drawn from anaesthetised mice and decerebrate cats, where our models offered 80% and 100% classification accuracy. We then used our models to assess non-identified data from awake monkeys and rabbits in order to highlight subsets of neurones with the greatest degree of similarity to identified cell classes. In this way, our GPC-based approach for tentatively identifying neurones from their spontaneous activity signatures, in the absence of an established ground-truth, nonetheless affords the experimenter a statistically robust means of grouping cells with properties matching known cell classes. Our approach therefore may have broad application to a variety of future cerebellar cortical investigations, particularly in awake animals where opportunities for definitive cell identification are limited.

Bistability in Purkinje neurons: ups and downs in cerebellar research

The output of cerebellar Purkinje cells has been characterized extensively and theories regarding the role of simple spike (SS) and complex spike (CS) patterns have evolved through many different studies. A bistable pattern of SS output can be observed in vitro; however, differing views exist regarding the occurrence of bistable SS output in vivo. Bistability in Purkinje cell output is characterized by abrupt transitions between tonic firing and quiescence, usually evoked by synaptic inputs to the neuron. This is in contrast to the trimodal pattern of activity which has been found in vitro and in vivo when climbing fiber input to Purkinje cells is removed. The mechanisms underlying bistable membrane properties in Purkinje cells have been determined through in vitro studies and computational analysis. In vitro studies have further established that Purkinje cells possess the ability to toggle between firing states, but in vivo studies in both awake and anesthetized animals have found conflicting results as to the presence of toggling in the intact circuit. Here, we provide an overview of the current state of research on bistability, examining the mechanisms underlying bistability and current findings from in vivo studies. We also suggest possible reasons for discrepancies between in vivo studies and propose future studies which would aid in clarifying the role of bistability in the cerebellar circuit.

Dendritic calcium signaling in cerebellar Purkinje cell

The Purkinje cells in the cerebellum are unique neurons that generate local and global Ca(2+) signals in response to two types of excitatory inputs, parallel fiber and climbing fiber, respectively. The spatiotemporal distribution and interaction of these synaptic inputs produce complex patterns of Ca(2+) dynamics in the Purkinje cell dendrites. The Ca(2+) signals originate from Ca(2+) influx through voltage-gated Ca(2+) channels and Ca(2+) release from intracellular stores that are mediated by the metabotropic glutamate receptor signaling pathway. These Ca(2+) signals are essential for the induction of various forms of synaptic plasticity and for controlling the input-output relationship of Purkinje cells. In this article we review Ca(2+) signaling in Purkinje cell dendrites.

Storage of correlated patterns in standard and bistable Purkinje cell models

The cerebellum has long been considered to undergo supervised learning, with climbing fibers acting as a 'teaching' or 'error' signal. Purkinje cells (PCs), the sole output of the cerebellar cortex, have been considered as analogs of perceptrons storing input/output associations. In support of this hypothesis, a recent study found that the distribution of synaptic weights of a perceptron at maximal capacity is in striking agreement with experimental data in adult rats. However, the calculation was performed using random uncorrelated inputs and outputs. This is a clearly unrealistic assumption since sensory inputs and motor outputs carry a substantial degree of temporal correlations. In this paper, we consider a binary output neuron with a large number of inputs, which is required to store associations between temporally correlated sequences of binary inputs and outputs, modelled as Markov chains. Storage capacity is found to increase with both input and output correlations, and diverges in the limit where both go to unity. We also investigate the capacity of a bistable output unit, since PCs have been shown to be bistable in some experimental conditions. Bistability is shown to enhance storage capacity whenever the output correlation is stronger than the input correlation. Distribution of synaptic weights at maximal capacity is shown to be independent on correlations, and is also unaffected by the presence of bistability.

Tunable oscillations in the Purkinje neuron

In this paper, we experimentally study the dynamics of slow oscillations in Purkinje neurons in vitro, and derive a strong association with a forced parametric oscillator model. We observed the precise rhythmicity of these oscillations in Purkinje neurons, as well as a dynamic tunability of this oscillation using a photoswitchable compound. We found that this slow oscillation can be induced in every Purkinje neuron measured, having periods ranging between 10 and 25 s. Starting from a Hodgkin-Huxley model, we demonstrate that this oscillation can be externally modulated, and that the neurons will return to their intrinsic firing frequency after the forced oscillation is concluded. These findings signify an additional timing functional role of tunable oscillations within the cerebellum, as well as a dynamic control of a time scale in the brain in the range of seconds.

Dendritic calcium signaling triggered by spontaneous and sensory-evoked climbing fiber input to cerebellar Purkinje cells in vivo

Cerebellar Purkinje cells have one of the most elaborate dendritic trees in the mammalian CNS, receiving excitatory synaptic input from a single climbing fiber (CF) and from ∼200,000 parallel fibers. The dendritic Ca(2+) signals triggered by activation of these inputs are crucial for the induction of synaptic plasticity at both of these synaptic connections. We have investigated Ca(2+) signaling in Purkinje cell dendrites in vivo by combining targeted somatic or dendritic patch-clamp recording with simultaneous two-photon microscopy. Both spontaneous and sensory-evoked CF inputs triggered widespread Ca(2+) signals throughout the dendritic tree that were detectable even in individual spines of the most distal spiny branchlets receiving parallel fiber input. The amplitude of these Ca(2+) signals depended on dendritic location and could be modulated by membrane potential, reflecting modulation of dendritic spikes triggered by the CF input. Furthermore, the variability of CF-triggered Ca(2+) signals was regulated by GABAergic synaptic input. These results indicate that dendritic Ca(2+) signals triggered by sensory-evoked CF input can act as associative signals for synaptic plasticity in Purkinje cells in vivo and may differentially modulate plasticity at parallel fiber synapses depending on the location of synapses, firing state of the Purkinje cell, and ongoing GABAergic synaptic input.

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.

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

Dendrites of cerebellar Purkinje cells (PCs) respond to brief excitations from parallel fibers with lasting plateau depolarizations. It is unknown whether these plateaus are local events that boost the synaptic signals or they propagate to the soma and directly take part in setting the cell firing dynamics. To address this issue, we analyzed a likely mechanism underlying plateaus in three representations of a reconstructed PC with increasing complexity. Analysis in an infinite cable suggests that Ca plateaus triggered by direct excitatory inputs from parallel fibers and their mirror signals, valleys (putatively triggered by the local feed forward inhibitory network), cannot propagate. However, simulations of the model in electrotonic equivalent cables prove that Ca plateaus (resp. valleys) are conducted over the entire cell with velocities typical of passive events once they are triggered by threshold synaptic inputs that turn the membrane current inward (resp. outward) over the whole cell surface. Bifurcation analysis of the model in equivalent cables, and simulations in a fully reconstructed PC both indicate that dendritic Ca plateaus and valleys, respectively, command epochs of firing and silencing of PCs.

Network bistability mediates spontaneous transitions between normal and pathological brain states

Little is known about how cortical networks support the emergence of remarkably different activity patterns. Physiological activity interspersed with epochs of pathological hyperactivity in the epileptic brain represents a clinically relevant yet poorly understood case of such rich dynamic repertoire. Using a realistic computational model, we demonstrate that physiological sparse and pathological tonic-clonic activity may coexist in the same cortical network for identical afferent input level. Transient perturbations in the afferent input were sufficient to switch the network between these two stable states. The effectiveness of the potassium regulatory apparatus determined the stability of the physiological state and the threshold for seizure initiation. Our findings contrast with the common notions of (1) pathological brain activity representing dynamic instabilities and (2) necessary adjustments of experimental conditions to elicit different network states. Rather, we propose that the rich dynamic repertoire of cortical networks may be based on multistabilities intrinsic to the network.

Multiple firing patterns in deep dorsal horn neurons of the spinal cord: computational analysis of mechanisms and functional implications

Deep dorsal horn relay neurons (dDHNs) of the spinal cord are known to exhibit multiple firing patterns under the control of local metabotropic neuromodulation: tonic firing, plateau potential, and spontaneous oscillations. This work investigates the role of interactions between voltage-gated channels and the occurrence of different firing patterns and then correlates these two phenomena with their functional role in sensory information processing. We designed a conductance-based model using the NEURON software package, which successfully reproduced the classical features of plateau in dDHNs, including a wind-up of the neuronal response after repetitive stimulation. This modeling approach allowed us to systematically test the impact of conductance interactions on the firing patterns. We found that the expression of multiple firing patterns can be reproduced by changes in the balance between two currents (L-type calcium and potassium inward rectifier conductances). By investigating a possible generalization of the firing state switch, we found that the switch can also occur by varying the balance of any hyperpolarizing and depolarizing conductances. This result extends the control of the firing switch to neuromodulators or to network effects such as synaptic inhibition. We observed that the switch between the different firing patterns occurs as a continuous function in the model, revealing a particular intermediate state called the accelerating mode. To characterize the functional effect of a firing switch on information transfer, we used correlation analysis between a model of peripheral nociceptive afference and the dDHN model. The simulation results indicate that the accelerating mode was the optimal firing state for information transfer.

Geometry and dynamics of activity-dependent homeostatic regulation in neurons

To maintain activity in a functional range, neurons constantly adjust membrane excitability to changing intra- and extracellular conditions. Such activity-dependent homeostatic regulation (ADHR) is critical for normal processing of the nervous system and avoiding pathological conditions. Here, we posed a homeostatic regulation problem for the classical Morris-Lecar (ML) model. The problem was motivated by the phenomenon of the functional recovery of stomatogastric neurons in crustaceans in the absence of neuromodulation. In our study, the regulation of the ionic conductances in the ML model depended on the calcium current or the intracellular calcium concentration. We found an asymptotic solution to the problem under the assumption of slow regulation. The solution provides a full account of the regulation in the case of correlated or anticorrelated changes of the maximal conductances of the calcium and potassium currents. In particular, the solution shows how the target and parameters of the regulation determine which perturbations of the conductances can be compensated by the ADHR. In some cases, the sets of compensated initial perturbations are not convex. On the basis of our analysis we formulated specific questions for subsequent experimental and theoretical studies of ADHR.

A new approach for determining phase response curves reveals that Purkinje cells can act as perfect integrators

Cerebellar Purkinje cells display complex intrinsic dynamics. They fire spontaneously, exhibit bistability, and via mutual network interactions are involved in the generation of high frequency oscillations and travelling waves of activity. To probe the dynamical properties of Purkinje cells we measured their phase response curves (PRCs). PRCs quantify the change in spike phase caused by a stimulus as a function of its temporal position within the interspike interval, and are widely used to predict neuronal responses to more complex stimulus patterns. Significant variability in the interspike interval during spontaneous firing can lead to PRCs with a low signal-to-noise ratio, requiring averaging over thousands of trials. We show using electrophysiological experiments and simulations that the PRC calculated in the traditional way by sampling the interspike interval with brief current pulses is biased. We introduce a corrected approach for calculating PRCs which eliminates this bias. Using our new approach, we show that Purkinje cell PRCs change qualitatively depending on the firing frequency of the cell. At high firing rates, Purkinje cells exhibit single-peaked, or monophasic PRCs. Surprisingly, at low firing rates, Purkinje cell PRCs are largely independent of phase, resembling PRCs of ideal non-leaky integrate-and-fire neurons. These results indicate that Purkinje cells can act as perfect integrators at low firing rates, and that the integration mode of Purkinje cells depends on their firing rate.

By observing how brief current pulses injected at different times between spikes change the phase of spiking of a neuron (and thus obtaining the so-called phase response curve), it should be possible to predict a full spike train in response to more complex stimulation patterns. When we applied this traditional protocol to obtain phase response curves in cerebellar Purkinje cells in the presence of noise, we observed a triangular region devoid of data points near the end of the spiking cycle. This “Bermuda Triangle” revealed a flaw in the classical method for constructing phase response curves. We developed a new approach to eliminate this flaw and used it to construct phase response curves of Purkinje cells over a range of spiking rates. Surprisingly, at low firing rates, phase changes were independent of the phase of the injected current pulses, implying that the Purkinje cell is a perfect integrator under these conditions. This mechanism has not yet been described in other cell types and may be crucial for the information processing capabilities of these neurons.

Regularity, variability and bi-stability in the activity of cerebellar purkinje cells

Recent studies have demonstrated that the membrane potential of Purkinje cells is bi-stable and that this phenomenon underlies bi-modal simple spike firing. Membrane potential alternates between a depolarized state, that is associated with spontaneous simple spike firing (up state), and a quiescent hyperpolarized state (down state). A controversy has emerged regarding the relevance of bi-stability to the awake animal, yet recordings made from behaving cat Purkinje cells have demonstrated that at least 50% of the cells exhibit bi-modal firing. The robustness of the phenomenon in vitro or in anaesthetized systems on the one hand, and the controversy regarding its expression in behaving animals on the other hand suggest that state transitions are under neuronal control. Indeed, we have recently demonstrated that synaptic inputs can induce transitions between the states and suggested that the role of granule cell input is to control the states of Purkinje cells rather than increase or decrease firing rate gradually. We have also shown that the state of a Purkinje cell does not only affect its firing but also the waveform of climbing fiber-driven complex spikes and the associated calcium influx. These findings call for a reconsideration of the role of Purkinje cells in cerebellar function. In this manuscript we review the recent findings on Purkinje cell bi-stability and add some analyses of its effect on the regularity and variability of Purkinje cell activity.

Neocortical networks entrain neuronal circuits in cerebellar cortex

Activity in neocortex is often characterized by synchronized oscillations of neurons and networks, resulting in the generation of a local field potential (LFP) and electroencephalogram. Do the neuronal networks of the cerebellum also generate synchronized oscillations and are they under the influence of those in the neocortex? Here we show that, in the absence of any overt external stimulus, the cerebellar cortex generates a slow oscillation that is correlated with that of the neocortex. Disruption of the neocortical slow oscillation abolishes the cerebellar slow oscillation, whereas blocking cerebellar activity has no overt effect on the neocortex. We provide evidence that the cerebellar slow oscillation results in part from the activation of granule, Golgi, and Purkinje neurons. In particular, we show that granule and Golgi cells discharge trains of single spikes, and Purkinje cells generate complex spikes, during the "up" state of the slow oscillation. Purkinje cell simple spiking is weakly related to the cerebellar and neocortical slow oscillation in a minority of cells. Our results indicate that the cerebellum generates rhythmic network activity that can be recorded as an LFP in the anesthetized animal, which is driven by synchronized oscillations of the neocortex. Furthermore, we show that correlations between neocortical and cerebellar LFPs persist in the awake animal, indicating that neocortical circuits modulate cerebellar neurons in a similar manner in natural behavioral states. Thus, the projection neurons of the neocortex collectively exert a driving and modulatory influence on cerebellar network activity.

The first second: models of short-term memory traces in the brain

Many network models in computational neuroscience rise to the challenge of explaining behavioural phenomena ranging from microseconds to tens of seconds using components operating mostly on a time-scale of milliseconds. These models have in common that the underlying system has a memory, which implies that its output depends on its past input history. In this review we compare how such memory traces or delayed responses may be implemented in different brain areas supporting a diversity of functions.

Reduction of spike afterdepolarization by increased leak conductance alters interspike interval variability

Data from neurons in vivo have shown that spike output can often sustain episodes of high variability. Theoretical studies have indicated that the high conductance state of neurons brought on by synaptic activity can contribute to an increase in the variability of spike output by decreasing the integration timescale of the neuron. In the present work, we were interested in understanding how background synaptic conductance activity alters the interspike interval (ISI) variability of layer III pyramidal cells of the medial entorhinal cortex. We compared ISI variability in pyramidal cells as a result of synaptic current- or conductance-mediated membrane fluctuations. We found that the effects of background synaptic conductance activity on ISI variability depend on the neuron type. In pyramidal cells lacking spike frequency adaptation, the variability increased in relation to a comparable synaptic current stimulus. In contrast, in pyramidal cells displaying spike frequency adaptation, the synaptic conductance stimulus produced lower ISI variability. To understand this result, we constructed a phenomenological model that reproduced the basic properties of these neurons under control and increased leak conductance. We found that leak can change the properties of the neuron by acting as a bifurcation parameter that reduces the afterdepolarization (ADP) and decreases the slope (gain) of the frequency-current relationship, particularly for transient stimuli. A lower gain with the added leak causes a reduction in ISI variability. We conclude that the ability of a high conductance state to increase ISI variability cannot be generalized and can depend on the spike ADP dynamics expressed by the neuron.

Patterns and pauses in Purkinje cell simple spike trains: experiments, modeling and theory

We review our recent experimental and modeling results on how cerebellar Purkinje cells encode information in their simple spike trains and present a theory of the function of pauses and regular spiking patterns. The regular spiking patterns were discovered in extracellular recordings of simple spikes in awake and anesthetized rodents, where it was shown that more than half of the spontaneous activity consists of short epochs of regular spiking. These periods of regular spiking are interrupted by pauses, which can be tightly synchronized among nearby Purkinje cells, while the spikes in the regular patterns are not. Interestingly, pauses are affected by long-term depression of the parallel fiber synapses. Both in modeling and slice experiments it was demonstrated that long-term depression causes a decrease in the duration of pauses, leading to an increase of the spike output of the neuron. Based on these results we propose that pauses in the simple spike train form a temporal code which can lead to a rebound burst in the target deep cerebellar nucleus neurons. Conversely, the regular spike patterns may be a rate code, which presets the amplitude of future rebound bursts.

Pausing purkinje cells in the cerebellum of the awake cat

A recent controversy has emerged concerning the existence of long pauses, presumably reflecting bistability of membrane potential, in the cerebellar Purkinje cells (PC) of awake animals. It is generally agreed that in the anesthetized animals and in vitro, these cells switch between two stable membrane potential states: a depolarized state (the ‘up-state’) characterized by continuous firing of simple spikes (SS) and a hyperpolarized state (the ‘down-state’) characterized by long pauses in the SS activity. To address the existence of long pauses in the neural activity of cerebellar PCs in the awake and behaving animal we used extracellular recordings in cats and find that approximately half of the recorded PCs exhibit such long pauses in the SS activity and transition between activity – periods with uninterrupted SS lasting an average of 1300 ms – and pauses up to several seconds. We called these cells pausing Purkinje cells (PPC) and they can easily be distinguished from continuous firing Purkinje cells. In most PPCs, state transitions in both directions were often associated (25% of state transitions) with complex spikes (CSs). This is consistent with intracellular findings of CS-driven state transitions. In sum, we present proof for the existence of long pauses in the PC SS activity that probably reflect underlying bistability, provide the first in-depth analysis of these pauses and show for the first time that transitions in and out of these pauses are related to CS firing in the awake and behaving animal.

Ionic Factors Governing Rebound Burst Phenotype in Rat Deep Cerebellar Neurons

Large diameter cells in rat deep cerebellar nuclei (DCN) can be distinguished according to the generation of a transient or weak rebound burst and the expression of T-type Ca2+ channel isoforms. We studied the ionic basis for the distinction in burst phenotypes in rat DCN cells in vitro. Following a hyperpolarization, transient burst cells generated a high-frequency spike burst of ≤450 Hz, whereas weak burst cells generated a lower-frequency increase (<140 Hz). Both cell types expressed a low voltage–activated (LVA) Ca2+ current near threshold for rebound burst discharge (−50 mV) that was consistent with T-type Ca2+ current, but on average 7 times more current was recorded in transient burst cells. The number and frequency of spikes in rebound bursts was tightly correlated with the peak Ca2+ current at −50 mV, showing a direct relationship between the availability of LVA Ca2+ current and spike output. Transient burst cells exhibited a larger spike depolarizing afterpotential that was insensitive to blockers of voltage-gated Na+ or Ca2+ channels. In comparison, weak burst cells exhibited larger afterhyperpolarizations (AHPs) that reduced cell excitability and rebound spike output. The sensitivity of AHPs to Ca2+ channel blockers suggests that both LVA and high voltage–activated (HVA) Ca2+ channels trigger AHPs in weak burst compared with only HVA Ca2+ channels in transient burst cells. The two burst phenotypes in rat DCN cells thus derive in part from a difference in the availability of LVA Ca2+ current following a hyperpolarization and a differential activation of AHPs that establish distinct levels of membrane excitability.

In vitro studies of closed-loop feedback and electrosensory processing in Apteronotus leptorhynchus

Electrosensory systems comprise extensive feedback pathways. It is also well known that these pathways exhibit synaptic plasticity on a wide-range of time scales. Recent in vitro brain slice studies have characterized synaptic plasticity in the two main feedback pathways to the electrosensory lateral line lobe (ELL), a primary electrosensory nucleus in Apteronotus leptorhynchus. Currently-used slice preparations, involving networks in open-loop conditions, allow feedback inputs to be studied in isolation, a critical step in determining their synaptic properties. However, to fully understand electrosensory processing, we must understand how dynamic feedback modulates neuronal responses under closed-loop conditions. To bridge the gap between current in vitro approaches and more complex in vivo work, we present two new in vitro approaches for studying the roles of closed-loop feedback in electrosensory processing. The first involves a hybrid-network approach using dynamic clamp, and the second involves a new slice preparation that preserves one of the feedback pathways to ELL in a closed-loop condition.

Computer simulations of neuron-glia interactions mediated by ion flux

Extracellular potassium concentration, [K(+)](o), and intracellular calcium, [Ca(2+)](i), rise during neuron excitation, seizures and spreading depression. Astrocytes probably restrain the rise of K(+) in a way that is only partly understood. To examine the effect of glial K(+) uptake, we used a model neuron equipped with Na(+), K(+), Ca(2+) and Cl(-) conductances, ion pumps and ion exchangers, surrounded by interstitial space and glia. The glial membrane was either "passive", incorporating only leak channels and an ion exchange pump, or it had rectifying K(+) channels. We computed ion fluxes, concentration changes and osmotic volume changes. Increase of [K(+)](o) stimulated the glial uptake by the glial 3Na/2K ion pump. The [K(+)](o) flux through glial leak and rectifier channels was outward as long as the driving potential was outwardly directed, but it turned inward when rising [K(+)](o)/[K(+)](i) ratio reversed the driving potential. Adjustments of glial membrane parameters influenced the neuronal firing patterns, the length of paroxysmal afterdischarge and the ignition point of spreading depression. We conclude that voltage gated K(+) currents can boost the effectiveness of the glial "potassium buffer" and that this buffer function is important even at moderate or low levels of excitation, but especially so in pathological states.