Low-threshold potassium channels and a low-threshold calcium channel regulate Ca2+ spike firing in the dendrites of cerebellar Purkinje neurons: a modeling study

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.

Kinesin family member 2A gates nociception

Nociceptive axons undergo remodeling as they innervate their targets during development and in response to environmental insults and pathological conditions. How is nociceptive morphogenesis regulated? Here, we show that the microtubule destabilizer kinesin family member 2A (Kif2a) is a key regulator of nociceptive terminal structures and pain sensitivity. Ablation of Kif2a in sensory neurons causes hyperinnervation and hypersensitivity to noxious stimuli in young adult mice, whereas touch sensitivity and proprioception remain unaffected. Computational modeling predicts that structural remodeling is sufficient to explain the phenotypes. Furthermore, Kif2a deficiency triggers a transcriptional response comprising sustained upregulation of injury-related genes and homeostatic downregulation of highly specific channels and receptors at the late stage. The latter effect can be predicted to relieve the hyperexcitability of nociceptive neurons, despite persisting morphological aberrations, and indeed correlates with the resolution of pain hypersensitivity. Overall, we reveal a critical control node defining nociceptive terminal structure, which is regulating nociception.

mTORC2 mediates structural plasticity in distal nociceptive endings that contributes to pain hypersensitivity following inflammation

The encoding of noxious stimuli into action potential firing is largely mediated by nociceptive free nerve endings. Tissue inflammation, by changing the intrinsic properties of the nociceptive endings, leads to nociceptive hyperexcitability and thus to the development of inflammatory pain. Here, we showed that tissue inflammation–induced activation of the mammalian target of rapamycin complex 2 (mTORC2) triggers changes in the architecture of nociceptive terminals and leads to inflammatory pain. Pharmacological activation of mTORC2 induced elongation and branching of nociceptor peripheral endings and caused long-lasting pain hypersensitivity. Conversely, nociceptor-specific deletion of the mTORC2 regulatory protein rapamycin-insensitive companion of mTOR (Rictor) prevented inflammation-induced elongation and branching of cutaneous nociceptive fibers and attenuated inflammatory pain hypersensitivity. Computational modeling demonstrated that mTORC2-mediated structural changes in the nociceptive terminal tree are sufficient to increase the excitability of nociceptors. Targeting mTORC2 using a single injection of antisense oligonucleotide against Rictor provided long-lasting alleviation of inflammatory pain hypersensitivity. Collectively, we showed that tissue inflammation–induced activation of mTORC2 causes structural plasticity of nociceptive free nerve endings in the epidermis and inflammatory hyperalgesia, representing a therapeutic target for inflammatory pain.

Do Biological Constraints Impair Dendritic Computation?

Computations on the dendritic trees of neurons have important constraints. Voltage dependent conductances in dendrites are not similar to arbitrary direct-current generation, they are the basis for dendritic nonlinearities and they do not allow converting positive currents into negative currents. While it has been speculated that the dendritic tree of a neuron can be seen as a multi-layer neural network and it has been shown that such an architecture could be computationally strong, we do not know if that computational strength is preserved under these biological constraints. Here we simulate models of dendritic computation with and without these constraints. We find that dendritic model performance on interesting machine learning tasks is not hurt by these constraints but may benefit from them. Our results suggest that single real dendritic trees may be able to learn a surprisingly broad range of tasks.

NeuroGPU: Accelerating multi-compartment, biophysically detailed neuron simulations on GPUs

BACKGROUND

The membrane potential of individual neurons depends on a large number of interacting biophysical processes operating on spatial-temporal scales spanning several orders of magnitude. The multi-scale nature of these processes dictates that accurate prediction of membrane potentials in specific neurons requires the utilization of detailed simulations. Unfortunately, constraining parameters within biologically detailed neuron models can be difficult, leading to poor model fits. This obstacle can be overcome partially by numerical optimization or detailed exploration of parameter space. However, these processes, which currently rely on central processing unit (CPU) computation, often incur orders of magnitude increases in computing time for marginal improvements in model behavior. As a result, model quality is often compromised to accommodate compute resources.

NEW METHOD

Here, we present a simulation environment, NeuroGPU, that takes advantage of the inherent parallelized structure of the graphics processing unit (GPU) to accelerate neuronal simulation.

RESULTS & COMPARISON WITH EXISTING METHODS

NeuroGPU can simulate most biologically detailed models 10-200 times faster than NEURON simulation running on a single core and 5 times faster than GPU simulators (CoreNEURON). NeuroGPU is designed for model parameter tuning and best performs when the GPU is fully utilized by running multiple (> 100) instances of the same model with different parameters. When using multiple GPUs, NeuroGPU can reach to a speed-up of 800 fold compared to single core simulations, especially when simulating the same model morphology with different parameters. We demonstrate the power of NeuoGPU through large-scale parameter exploration to reveal the response landscape of a neuron. Finally, we accelerate numerical optimization of biophysically detailed neuron models to achieve highly accurate fitting of models to simulation and experimental data.

CONCLUSIONS

Thus, NeuroGPU is the fastest available platform that enables rapid simulation of multi-compartment, biophysically detailed neuron models on commonly used computing systems accessible by many scientists.

Computational epidemiology study of homeostatic compensation during sensorimotor aging

The vestibulo-ocular reflex (VOR) stabilizes vision during head motion. Age-related changes of vestibular neuroanatomical properties predict a linear decay of VOR function. Nonetheless, human epidemiological data show a stable VOR function across the life span. In this study, we model cerebellum-dependent VOR adaptation to relate structural and functional changes throughout aging. We consider three neurosynaptic factors that may codetermine VOR adaptation during aging: the electrical coupling of inferior olive neurons, the long-term spike timing-dependent plasticity at parallel fiber - Purkinje cell synapses and mossy fiber - medial vestibular nuclei synapses, and the intrinsic plasticity of Purkinje cell synapses Our cross-sectional aging analyses suggest that long-term plasticity acts as a global homeostatic mechanism that underpins the stable temporal profile of VOR function. The results also suggest that the intrinsic plasticity of Purkinje cell synapses operates as a local homeostatic mechanism that further sustains the VOR at older ages. Importantly, the computational epidemiology approach presented in this study allows discrepancies among human cross-sectional studies to be understood in terms of interindividual variability in older individuals. Finally, our longitudinal aging simulations show that the amount of residual fibers coding for the peak and trough of the VOR cycle constitutes a predictive hallmark of VOR trajectories over a lifetime.

Inferring phenomenological models of first passage processes

Biochemical processes in cells are governed by complex networks of many chemical species interacting stochastically in diverse ways and on different time scales. Constructing microscopically accurate models of such networks is often infeasible. Instead, here we propose a systematic framework for building phenomenological models of such networks from experimental data, focusing on accurately approximating the time it takes to complete the process, the First Passage (FP) time. Our phenomenological models are mixtures of Gamma distributions, which have a natural biophysical interpretation. The complexity of the models is adapted automatically to account for the amount of available data and its temporal resolution. The framework can be used for predicting behavior of FP systems under varying external conditions. To demonstrate the utility of the approach, we build models for the distribution of inter-spike intervals of a morphologically complex neuron, a Purkinje cell, from experimental and simulated data. We demonstrate that the developed models can not only fit the data, but also make nontrivial predictions. We demonstrate that our coarse-grained models provide constraints on more mechanistically accurate models of the involved phenomena.

Data-driven reduction of dendritic morphologies with preserved dendro-somatic responses

Dendrites shape information flow in neurons. Yet, there is little consensus on the level of spatial complexity at which they operate. Through carefully chosen parameter fits, solvable in the least-squares sense, we obtain accurate reduced compartmental models at any level of complexity. We show that (back-propagating) action potentials, Ca²⁺ spikes, and N-methyl-D-aspartate spikes can all be reproduced with few compartments. We also investigate whether afferent spatial connectivity motifs admit simplification by ablating targeted branches and grouping affected synapses onto the next proximal dendrite. We find that voltage in the remaining branches is reproduced if temporal conductance fluctuations stay below a limit that depends on the average difference in input resistance between the ablated branches and the next proximal dendrite. Furthermore, our methodology fits reduced models directly from experimental data, without requiring morphological reconstructions. We provide software that automatizes the simplification, eliminating a common hurdle toward including dendritic computations in network models.

The Input-Output Relation of Primary Nociceptive Neurons is Determined by the Morphology of the Peripheral Nociceptive Terminals

The output from the peripheral terminals of primary nociceptive neurons, which detect and encode the information regarding noxious stimuli, is crucial in determining pain sensation. The nociceptive terminal endings are morphologically complex structures assembled from multiple branches of different geometry, which converge in a variety of forms to create the terminal tree. The output of a single terminal is defined by the properties of the transducer channels producing the generation potentials and voltage-gated channels, translating the generation potentials into action potential (AP) firing. However, in the majority of cases, noxious stimuli activate multiple terminals; thus, the output of the nociceptive neuron is defined by the integration and computation of the inputs of the individual terminals. Here, we used a computational model of nociceptive terminal tree to study how the architecture of the terminal tree affects the input-output relation of the primary nociceptive neurons. We show that the input-output properties of the nociceptive neurons depend on the length, the axial resistance (Ra), and location of individual terminals. Moreover, we show that activation of multiple terminals by a capsaicin-like current allows summation of the responses from individual terminals, thus leading to increased nociceptive output. Stimulation of the terminals in simulated models of inflammatory or neuropathic hyperexcitability led to a change in the temporal pattern of AP firing, emphasizing the role of temporal code in conveying key information about changes in nociceptive output in pathologic conditions, leading to pain hypersensitivity.SIGNIFICANCE STATEMENT Noxious stimuli are detected by terminal endings of primary nociceptive neurons, which are organized into morphologically complex terminal trees. The information from multiple terminals is integrated along the terminal tree, computing the neuronal output, which propagates toward the CNS, thus shaping the pain sensation. Here, we revealed that the structure of the nociceptive terminal tree determines the output of nociceptive neurons. We show that the integration of noxious information depends on the morphology of the terminal trees and how this integration and, consequently, the neuronal output change under pathologic conditions. Our findings help to predict how nociceptive neurons encode noxious stimuli and how this encoding changes in pathologic conditions, leading to pain.

Active membrane conductances and morphology of a collision detection neuron broaden its impedance profile and improve discrimination of input synchrony

How neurons filter and integrate their complex patterns of synaptic inputs is central to their role in neural information processing. Synaptic filtering and integration are shaped by the frequency-dependent neuronal membrane impedance. Using single and dual dendritic recordings in vivo, pharmacology, and computational modeling, we characterized the membrane impedance of a collision detection neuron in the grasshopper Schistocerca americana. This neuron, the lobula giant movement detector (LGMD), exhibits consistent impedance properties across frequencies and membrane potentials. Two common active conductances g_H and g_M, mediated respectively by hyperpolarization-activated cyclic nucleotide-gated (HCN) channels and by muscarine-sensitive M-type K⁺ channels, promote broadband integration with high temporal precision over the LGMD's natural range of membrane potentials and synaptic input frequencies. Additionally, we found that a model based on the LGMD's branching morphology increased the gain and decreased the delay associated with the mapping of synaptic input currents to membrane potential. More generally, this was true for a wide range of model neuron morphologies, including those of neocortical pyramidal neurons and cerebellar Purkinje cells. These findings show the unexpected role played by two widespread active conductances and by dendritic morphology in shaping synaptic integration.NEW & NOTEWORTHY Neuronal filtering and integration of synaptic input patterns depend on the electrochemical properties of dendrites. We used an identified collision detection neuron in grasshoppers to examine how its morphology and two conductances affect its membrane impedance in relation to the computations it performs. The neuronal properties examined are ubiquitous and therefore promote a general understanding of neuronal computations, including those in the human brain.

Coding Capacity of Purkinje Cells With Different Schemes of Morphological Reduction

The brain as a neuronal system has very complex structures with a large diversity of neuronal types. The most basic complexity is seen from the structure of neuronal morphology, which usually has a complex tree-like structure with dendritic spines distributed in branches. To simulate a large-scale network with spiking neurons, the simple point neuron, such as the integrate-and-fire neuron, is often used. However, recent experimental evidence suggests that the computational ability of a single neuron is largely enhanced by its morphological structure, in particular, by various types of dendritic dynamics. As the morphology reduction of detailed biophysical models is a classic question in systems neuroscience, much effort has been taken to simulate a neuron with a few compartments to include the interaction between the soma and dendritic spines. Yet, novel reduction methods are still needed to deal with the complex dendritic tree. Here, using 10 individual Purkinje cells of the cerebellum from three species of guinea-pig, mouse and rat, we consider four types of reduction methods and study their effects on the coding capacity of Purkinje cells in terms of firing rate, timing coding, spiking pattern, and modulated firing under different stimulation protocols. We found that there is a variation of reduction performance depending on individual cells and species, however, all reduction methods can preserve, to some degree, firing activity of the full model of Purkinje cell. Therefore, when stimulating large-scale network of neurons, one has to choose a proper type of reduced neuronal model depending on the questions addressed. Among these reduction schemes, Branch method, that preserves the geometrical volume of neurons, can achieve the best balance among different performance measures of accuracy, simplification, and computational efficiency, and reproduce various phenomena shown in the full morphology model of Purkinje cells. Altogether, these results suggest that the Branch reduction scheme seems to provide a general guideline for reducing complex morphology into a few compartments without the loss of basic characteristics of the firing properties of neurons.

Spike burst-pause dynamics of Purkinje cells regulate sensorimotor adaptation

Cerebellar Purkinje cells mediate accurate eye movement coordination. However, it remains unclear how oculomotor adaptation depends on the interplay between the characteristic Purkinje cell response patterns, namely tonic, bursting, and spike pauses. Here, a spiking cerebellar model assesses the role of Purkinje cell firing patterns in vestibular ocular reflex (VOR) adaptation. The model captures the cerebellar microcircuit properties and it incorporates spike-based synaptic plasticity at multiple cerebellar sites. A detailed Purkinje cell model reproduces the three spike-firing patterns that are shown to regulate the cerebellar output. Our results suggest that pauses following Purkinje complex spikes (bursts) encode transient disinhibition of target medial vestibular nuclei, critically gating the vestibular signals conveyed by mossy fibres. This gating mechanism accounts for early and coarse VOR acquisition, prior to the late reflex consolidation. In addition, properly timed and sized Purkinje cell bursts allow the ratio between long-term depression and potentiation (LTD/LTP) to be finely shaped at mossy fibre-medial vestibular nuclei synapses, which optimises VOR consolidation. Tonic Purkinje cell firing maintains the consolidated VOR through time. Importantly, pauses are crucial to facilitate VOR phase-reversal learning, by reshaping previously learnt synaptic weight distributions. Altogether, these results predict that Purkinje spike burst-pause dynamics are instrumental to VOR learning and reversal adaptation.

Cerebellar Purkinje cells regulate accurate eye movement coordination. However, it remains unclear how cerebellar-dependent oculomotor adaptation depends on the interplay between Purkinje cell characteristic response patterns: tonic, high frequency bursting, and post-complex spike pauses. We explore the role of Purkinje spike burst-pause dynamics in VOR adaptation. A biophysical model of Purkinje cell is at the core of a spiking network model, which captures the cerebellar microcircuit properties and incorporates spike-based synaptic plasticity mechanisms at different cerebellar sites. We show that Purkinje spike burst-pause dynamics are critical for (1) gating the vestibular-motor response association during VOR acquisition; (2) mediating the LTD/LTP balance for VOR consolidation; (3) reshaping synaptic efficacy distributions for VOR phase-reversal adaptation; (4) explaining the reversal VOR gain discontinuities during sleeping.

VOR Adaptation on a Humanoid iCub Robot Using a Spiking Cerebellar Model

We embed a spiking cerebellar model within an adaptive real-time (RT) control loop that is able to operate a real robotic body (iCub) when performing different vestibulo-ocular reflex (VOR) tasks. The spiking neural network computation, including event- and time-driven neural dynamics, neural activity, and spike-timing dependent plasticity (STDP) mechanisms, leads to a nondeterministic computation time caused by the neural activity volleys encountered during cerebellar simulation. This nondeterministic computation time motivates the integration of an RT supervisor module that is able to ensure a well-orchestrated neural computation time and robot operation. Actually, our neurorobotic experimental setup (VOR) benefits from the biological sensory motor delay between the cerebellum and the body to buffer the computational overloads as well as providing flexibility in adjusting the neural computation time and RT operation. The RT supervisor module provides for incremental countermeasures that dynamically slow down or speed up the cerebellar simulation by either halting the simulation or disabling certain neural computation features (i.e., STDP mechanisms, spike propagation, and neural updates) to cope with the RT constraints imposed by the real robot operation. This neurorobotic experimental setup is applied to different horizontal and vertical VOR adaptive tasks that are widely used by the neuroscientific community to address cerebellar functioning. We aim to elucidate the manner in which the combination of the cerebellar neural substrate and the distributed plasticity shapes the cerebellar neural activity to mediate motor adaptation. This paper underlies the need for a two-stage learning process to facilitate VOR acquisition.

Bottom-up approach to torus bifurcation in neuron models

We study the quasi-periodicity phenomena occurring at the transition between tonic spiking and bursting activities in exemplary biologically plausible Hodgkin-Huxley type models of individual cells and reduced phenomenological models with slow and fast dynamics. Using the geometric slow-fast dissection and the parameter continuation approach, we show that the transition is due to either the torus bifurcation or the period-doubling bifurcation of a stable periodic orbit on the 2D slow-motion manifold near a characteristic fold. Various torus bifurcations including stable and saddle torus-canards, resonant tori, the co-existence of nested tori, and the torus breakdown leading to the onset of complex and bistable dynamics in such systems are examined too.

Integrating the Allen Brain Institute Cell Types Database into Automated Neuroscience Workflow

We developed software tools to download, extract features, and organize the Cell Types Database from the Allen Brain Institute (ABI) in order to integrate its whole cell patch clamp characterization data into the automated modeling/data analysis cycle. To expand the potential user base we employed both Python and MATLAB. The basic set of tools downloads selected raw data and extracts cell, sweep, and spike features, using ABI's feature extraction code. To facilitate data manipulation we added a tool to build a local specialized database of raw data plus extracted features. Finally, to maximize automation, we extended our NeuroManager workflow automation suite to include these tools plus a separate investigation database. The extended suite allows the user to integrate ABI experimental and modeling data into an automated workflow deployed on heterogeneous computer infrastructures, from local servers, to high performance computing environments, to the cloud. Since our approach is focused on workflow procedures our tools can be modified to interact with the increasing number of neuroscience databases being developed to cover all scales and properties of the nervous system.

The Role of Kv7/M Potassium Channels in Controlling Ectopic Firing in Nociceptors

Peripheral nociceptive neurons encode and convey injury-inducing stimuli toward the central nervous system. In normal conditions, tight control of nociceptive resting potential prevents their spontaneous activation. However, in many pathological conditions the control of membrane potential is disrupted, leading to ectopic, stimulus-unrelated firing of nociceptive neurons, which is correlated to spontaneous pain. We have investigated the role of K_V7/M channels in stabilizing membrane potential and impeding spontaneous firing of nociceptive neurons. These channels generate low voltage-activating, noninactivating M-type K⁺ currents (M-current, I_M ), which control neuronal excitability. Using perforated-patch recordings from cultured, rat nociceptor-like dorsal root ganglion neurons, we show that inhibition of M-current leads to depolarization of nociceptive neurons and generation of repetitive firing. To assess to what extent the M-current, acting at the nociceptive terminals, is able to stabilize terminals' membrane potential, thus preventing their ectopic activation, in normal and pathological conditions, we built a multi-compartment computational model of a pseudo-unipolar unmyelinated nociceptive neuron with a realistic terminal tree. The modeled terminal tree was based on the in vivo structure of nociceptive peripheral terminal, which we assessed by in vivo multiphoton imaging of GFP-expressing nociceptive neuronal terminals innervating mice hind paw. By modifying the conductance of the K_V7/M channels at the modeled terminal tree (terminal g_KV7/M) we have found that 40% of the terminal g_KV7/M conductance is sufficient to prevent spontaneous firing, while ~75% of terminal g_KV7/M is sufficient to inhibit stimulus induced activation of nociceptive neurons. Moreover, we showed that terminal M-current reduces susceptibility of nociceptive neurons to a small fluctuations of membrane potentials. Furthermore, we simulated how the interaction between terminal persistent sodium current and M-current affects the excitability of the neurons. We demonstrated that terminal M-current in nociceptive neurons impeded spontaneous firing even when terminal Na_(V)1.9 channels conductance was substantially increased. On the other hand, when terminal g_KV7/M was decreased, nociceptive neurons fire spontaneously after slight increase in terminal Na_(V)1.9 conductance. Our results emphasize the pivotal role of M-current in stabilizing membrane potential and hereby in controlling nociceptive spontaneous firing, in normal and pathological conditions.

Action potential initiation in a two-compartment model of pyramidal neuron mediated by dendritic Ca2+ spike

Dendritic Ca²⁺ spike endows cortical pyramidal cell with powerful ability of synaptic integration, which is critical for neuronal computation. Here we propose a two-compartment conductance-based model to investigate how the Ca²⁺ activity of apical dendrite participates in the action potential (AP) initiation to affect the firing properties of pyramidal neurons. We have shown that the apical input with sufficient intensity triggers a dendritic Ca²⁺ spike, which significantly boosts dendritic inputs as it propagates to soma. Such event instantaneously shifts the limit cycle attractor of the neuron and results in a burst of APs, which makes its firing rate reach a plateau steady-state level. Delivering current to two chambers simultaneously increases the level of neuronal excitability and decreases the threshold of input-output relation. Here the back-propagating APs facilitate the initiation of dendritic Ca²⁺ spike and evoke BAC firing. These findings indicate that the proposed model is capable of reproducing in vitro experimental observations. By determining spike initiating dynamics, we have provided a fundamental link between dendritic Ca²⁺ spike and output APs, which could contribute to mechanically interpreting how dendritic Ca²⁺ activity participates in the simple computations of pyramidal neuron.

Power Spectral Density Analysis of Purkinje Cell Tonic and Burst Firing Patterns From a Rat Model of Ataxia and Riluzole Treated

Introduction:

Purkinje Cell (PC) output displays a complex firing pattern consisting of high frequency sodium spikes and low frequency calcium spikes, and disruption in this firing behavior may contribute to cerebellar ataxia. Riluzole, neuroprotective agent, has been demonstrated to have neuroprotective effects in cerebellar ataxia. Here, the spectral analysis of PCs firing in control, 3-acetylpyridine (3-AP), neurotoxin agent, treated alone and riluzole plus 3-AP treated were investigated to determine changes in the firing properties. Difference in the power spectra of tonic and burst firing was assessed. Furthermore, the role of calcium-activated potassium channels in the power spectra was evaluated.

Methods:

Analysis was performed using Matlab. Power spectral density (PSD) of PCs output were obtained. Peak frequencies were extracted from the spectrum and statistical comparisons were done. In addition, a multi-compartment computational model of a Purkinje cell was used. This computational stimulation allowed us to study the changes in the power spectral density of the PC output as a result of alteration in ion channels.

Results:

Spectral analysis showed that in the spectrum of tonic and burst firing pattern only high sodium frequency and low calcium frequency was seen, respectively. In addition, there was a significant difference between the frequency components of PCs firing obtained from normal, ataxia and riluzole treated rats. Results indicated that sodium firing frequency of normal, ataxic and treated PCs occurred in approximate frequency of 22.53±5.49, 6.46±0.23, and 31.34±4.07 Hz, respectively; and calcium frequency occurred in frequency of 4.22±2.02, 1.52±1.19, and 3.88±1.37 Hz, respectively. The simulation results demonstrated that blockade of calcium-activated potassium channels in the PC model changed the PSD of the PC model firing activity. This change was similar to PSD changes in ataxia condition.

Conclusion:

These alterations in the spectrum of PC output may be a basis for developing possible new treatment strategies to improve cerebellar ataxia.

Automating NEURON Simulation Deployment in Cloud Resources

Simulations in neuroscience are performed on local servers or High Performance Computing (HPC) facilities. Recently, cloud computing has emerged as a potential computational platform for neuroscience simulation. In this paper we compare and contrast HPC and cloud resources for scientific computation, then report how we deployed NEURON, a widely used simulator of neuronal activity, in three clouds: Chameleon Cloud, a hybrid private academic cloud for cloud technology research based on the OpenStack software; Rackspace, a public commercial cloud, also based on OpenStack; and Amazon Elastic Cloud Computing, based on Amazon's proprietary software. We describe the manual procedures and how to automate cloud operations. We describe extending our simulation automation software called NeuroManager (Stockton and Santamaria, Frontiers in Neuroinformatics, 2015), so that the user is capable of recruiting private cloud, public cloud, HPC, and local servers simultaneously with a simple common interface. We conclude by performing several studies in which we examine speedup, efficiency, total session time, and cost for sets of simulations of a published NEURON model.

A unified model for two modes of bursting in GnRH neurons

Gonadotropin-releasing hormone (GnRH) neurons exhibit at least two intrinsic modes of action potential burst firing, referred to as parabolic and irregular bursting. Parabolic bursting is characterized by a slow wave in membrane potential that can underlie periodic clusters of action potentials with increased interspike interval at the beginning and at the end of each cluster. Irregular bursting is characterized by clusters of action potentials that are separated by varying durations of interburst intervals and a relatively stable baseline potential. Based on recent studies of isolated ionic currents, a stochastic Hodgkin-Huxley (HH)-like model for the GnRH neuron is developed to reproduce each mode of burst firing with an appropriate set of conductances. Model outcomes for bursting are in agreement with the experimental recordings in terms of interburst interval, interspike interval, active phase duration, and other quantitative properties specific to each mode of bursting. The model also shows similar outcomes in membrane potential to those seen experimentally when tetrodotoxin (TTX) is used to block action potentials during bursting, and when estradiol transitions cells exhibiting slow oscillations to irregular bursting mode in vitro. Based on the parameter values used to reproduce each mode of bursting, the model suggests that GnRH neurons can switch between the two through changes in the maximum conductance of certain ionic currents, notably the slow inward Ca(2+) current I s, and the Ca(2+) -activated K(+) current I KCa. Bifurcation analysis of the model shows that both modes of bursting are similar from a dynamical systems perspective despite differences in burst characteristics.

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.

Bioelectric memory: modeling resting potential bistability in amphibian embryos and mammalian cells

Background

Bioelectric gradients among all cells, not just within excitable nerve and muscle, play instructive roles in developmental and regenerative pattern formation. Plasma membrane resting potential gradients regulate cell behaviors by regulating downstream transcriptional and epigenetic events. Unlike neurons, which fire rapidly and typically return to the same polarized state, developmental bioelectric signaling involves many cell types stably maintaining various levels of resting potential during morphogenetic events. It is important to begin to quantitatively model the stability of bioelectric states in cells, to understand computation and pattern maintenance during regeneration and remodeling.

Method

To facilitate the analysis of endogenous bioelectric signaling and the exploitation of voltage-based cellular controls in synthetic bioengineering applications, we sought to understand the conditions under which somatic cells can stably maintain distinct resting potential values (a type of state memory). Using the Channelpedia ion channel database, we generated an array of amphibian oocyte and mammalian membrane models for voltage evolution. These models were analyzed and searched, by simulation, for a simple dynamical property, multistability, which forms a type of voltage memory.

Results

We find that typical mammalian models and amphibian oocyte models exhibit bistability when expressing different ion channel subsets, with either persistent sodium or inward-rectifying potassium, respectively, playing a facilitative role in bistable memory formation. We illustrate this difference using fast sodium channel dynamics for which a comprehensive theory exists, where the same model exhibits bistability under mammalian conditions but not amphibian conditions. In amphibians, potassium channels from the Kv1.x and Kv2.x families tend to disrupt this bistable memory formation. We also identify some common principles under which physiological memory emerges, which suggest specific strategies for implementing memories in bioengineering contexts.

Conclusion

Our results reveal conditions under which cells can stably maintain one of several resting voltage potential values. These models suggest testable predictions for experiments in developmental bioelectricity, and illustrate how cells can be used as versatile physiological memory elements in synthetic biology, and unconventional computation contexts.

Electronic supplementary material

The online version of this article (doi:10.1186/s12976-015-0019-9) contains supplementary material, which is available to authorized users.

NeuroManager: a workflow analysis based simulation management engine for computational neuroscience

We developed NeuroManager, an object-oriented simulation management software engine for computational neuroscience. NeuroManager automates the workflow of simulation job submissions when using heterogeneous computational resources, simulators, and simulation tasks. The object-oriented approach (1) provides flexibility to adapt to a variety of neuroscience simulators, (2) simplifies the use of heterogeneous computational resources, from desktops to super computer clusters, and (3) improves tracking of simulator/simulation evolution. We implemented NeuroManager in MATLAB, a widely used engineering and scientific language, for its signal and image processing tools, prevalence in electrophysiology analysis, and increasing use in college Biology education. To design and develop NeuroManager we analyzed the workflow of simulation submission for a variety of simulators, operating systems, and computational resources, including the handling of input parameters, data, models, results, and analyses. This resulted in 22 stages of simulation submission workflow. The software incorporates progress notification, automatic organization, labeling, and time-stamping of data and results, and integrated access to MATLAB's analysis and visualization tools. NeuroManager provides users with the tools to automate daily tasks, and assists principal investigators in tracking and recreating the evolution of research projects performed by multiple people. Overall, NeuroManager provides the infrastructure needed to improve workflow, manage multiple simultaneous simulations, and maintain provenance of the potentially large amounts of data produced during the course of a research project.

Neuronal morphology generates high-frequency firing resonance

The attenuation of neuronal voltage responses to high-frequency current inputs by the membrane capacitance is believed to limit single-cell bandwidth. However, neuronal populations subject to stochastic fluctuations can follow inputs beyond this limit. We investigated this apparent paradox theoretically and experimentally using Purkinje cells in the cerebellum, a motor structure that benefits from rapid information transfer. We analyzed the modulation of firing in response to the somatic injection of sinusoidal currents. Computational modeling suggested that, instead of decreasing with frequency, modulation amplitude can increase up to high frequencies because of cellular morphology. Electrophysiological measurements in adult rat slices confirmed this prediction and displayed a marked resonance at 200 Hz. We elucidated the underlying mechanism, showing that the two-compartment morphology of the Purkinje cell, interacting with a simple spiking mechanism and dendritic fluctuations, is sufficient to create high-frequency signal amplification. This mechanism, which we term morphology-induced resonance, is selective for somatic inputs, which in the Purkinje cell are exclusively inhibitory. The resonance sensitizes Purkinje cells in the frequency range of population oscillations observed in vivo.

Simulation of alcohol action upon a detailed Purkinje neuron model and a simpler surrogate model that runs >400 times faster

BACKGROUND

An approach to investigate brain function/dysfunction is to simulate neuron circuits on a computer. A problem, however, is that detailed neuron descriptions are computationally expensive and this handicaps the pursuit of realistic network investigations, where many neurons need to be simulated.

RESULTS

We confront this issue; we employ a novel reduction algorithm to produce a 2 compartment model of the cerebellar Purkinje neuron from a previously published, 1089 compartment model. It runs more than 400 times faster and retains the electrical behavior of the full model. So, it is more suitable for inclusion in large network models, where computational power is a limiting issue. We show the utility of this reduced model by demonstrating that it can replicate the full model's response to alcohol, which can in turn reproduce experimental recordings from Purkinje neurons following alcohol application.

CONCLUSIONS

We show that alcohol may modulate Purkinje neuron firing by an inhibition of their sodium-potassium pumps. We suggest that this action, upon cerebellar Purkinje neurons, is how alcohol ingestion can corrupt motor co-ordination. In this way, we relate events on the molecular scale to the level of behavior.

Computational neurobiology is a useful tool in translational neurology: the example of ataxia

Hereditary ataxia, or motor incoordination, affects approximately 150,000 Americans and hundreds of thousands of individuals worldwide with onset from as early as mid-childhood. Affected individuals exhibit dysarthria, dysmetria, action tremor, and diadochokinesia. In this review, we consider an array of computational studies derived from experimental observations relevant to human neuropathology. A survey of related studies illustrates the impact of integrating clinical evidence with data from mouse models and computational simulations. Results from these studies may help explain findings in mice, and after extensive laboratory study, may ultimately be translated to ataxic individuals. This inquiry lays a foundation for using computation to understand neurobiochemical and electrophysiological pathophysiology of spinocerebellar ataxias and may contribute to development of therapeutics. The interdisciplinary analysis suggests that computational neurobiology can be an important tool for translational neurology.

Integration of modeling with experimental and clinical findings synthesizes and refines the central role of inositol 1,4,5-trisphosphate receptor 1 in spinocerebellar ataxia

A suite of models was developed to study the role of inositol 1,4,5-trisphosphate receptor 1 (IP3R1) in spinocerebellar ataxias (SCAs). Several SCAs are linked to reduced abundance of IP3R1 or to supranormal sensitivity of the receptor to activation by its ligand inositol 1,4,5-trisphosphate (IP3). Detailed multidimensional models have been created to simulate biochemical calcium signaling and membrane electrophysiology in cerebellar Purkinje neurons. In these models, IP3R1-mediated calcium release is allowed to interact with ion channel response on the cell membrane. Experimental findings in mice and clinical observations in humans provide data input for the models. The SCA modeling suite helps interpret experimental results and provides suggestions to guide experiments. The models predict IP3R1 supersensitivity in SCA1 and compensatory mechanisms in SCA1, SCA2, and SCA3. Simulations explain the impact of calcium buffer proteins. Results show that IP3R1-mediated calcium release activates voltage-gated calcium-activated potassium channels in the plasma membrane. The SCA modeling suite unifies observations from experiments in a number of SCAs. The cadre of simulations demonstrates the central role of IP3R1.

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.

Dendritic diameters affect the spatial variability of intracellular calcium dynamics in computer models

There is growing interest in understanding calcium dynamics in dendrites, both experimentally and computationally. Many processes influence these dynamics, but in dendrites there is a strong contribution of morphology because the peak calcium levels are strongly determined by the surface to volume ratio (SVR) of each branch, which is inversely related to branch diameter. In this study we explore the predicted variance of dendritic calcium concentrations due to local changes in dendrite diameter and how this is affected by the modeling approach used. We investigate this in a model of dendritic calcium spiking in different reconstructions of cerebellar Purkinje cells and in morphological analysis of neocortical and hippocampal pyramidal neurons. We report that many published models neglect diameter-dependent effects on calcium concentration and show how to implement this correctly in the NEURON simulator, both for phenomenological pool based models and for implementations using radial 1D diffusion. More detailed modeling requires simulation of 3D diffusion and we demonstrate that this does not dissipate the local concentration variance due to changes of dendritic diameter. In many cases 1D diffusion of models of calcium buffering give a good approximation provided an increased morphological resolution is implemented.

Digital expression profiling of the compartmentalized translatome of Purkinje neurons

Underlying the complexity of the mammalian brain is its network of neuronal connections, but also the molecular networks of signaling pathways, protein interactions, and regulated gene expression within each individual neuron. The diversity and complexity of the spatially intermingled neurons pose a serious challenge to the identification and quantification of single neuron components. To address this challenge, we present a novel approach for the study of the ribosome-associated transcriptome-the translatome-from selected subcellular domains of specific neurons, and apply it to the Purkinje cells (PCs) in the rat cerebellum. We combined microdissection, translating ribosome affinity purification (TRAP) in nontransgenic animals, and quantitative nanoCAGE sequencing to obtain a snapshot of RNAs bound to cytoplasmic or rough endoplasmic reticulum (rER)-associated ribosomes in the PC and its dendrites. This allowed us to discover novel markers of PCs, to determine structural aspects of genes, to find hitherto uncharacterized transcripts, and to quantify biophysically relevant genes of membrane proteins controlling ion homeostasis and neuronal electrical activities.

Realistic modeling of neurons and networks: towards brain simulation

Realistic modeling is a new advanced methodology for investigating brain functions. Realistic modeling is based on a detailed biophysical description of neurons and synapses, which can be integrated into microcircuits. The latter can, in turn, be further integrated to form large-scale brain networks and eventually to reconstruct complex brain systems. Here we provide a review of the realistic simulation strategy and use the cerebellar network as an example. This network has been carefully investigated at molecular and cellular level and has been the object of intense theoretical investigation. The cerebellum is thought to lie at the core of the forward controller operations of the brain and to implement timing and sensory prediction functions. The cerebellum is well described and provides a challenging field in which one of the most advanced realistic microcircuit models has been generated. We illustrate how these models can be elaborated and embedded into robotic control systems to gain insight into how the cellular properties of cerebellar neurons emerge in integrated behaviors. Realistic network modeling opens up new perspectives for the investigation of brain pathologies and for the neurorobotic field.

Using Strahler's analysis to reduce up to 200-fold the run time of realistic neuron models

The cellular mechanisms underlying higher brain functions/dysfunctions are extremely difficult to investigate experimentally, and detailed neuron models have proven to be a very useful tool to help these kind of investigations. However, realistic neuronal networks of sizes appropriate to study brain functions present the major problem of requiring a prohibitively high computational resources. Here, building on our previous work, we present a general reduction method based on Strahler's analysis of neuron morphologies. We show that, without any fitting or tuning procedures, it is possible to map any morphologically and biophysically accurate neuron model into an equivalent reduced version. Using this method for Purkinje cells, we demonstrate how run times can be reduced up to 200–fold, while accurately taking into account the effects of arbitrarily located and activated synaptic inputs.

Mathematical model of bursting in dissociated purkinje neurons

In vitro, Purkinje cell behaviour is sometimes studied in a dissociated soma preparation in which the dendritic projection has been cleaved. A fraction of these dissociated somas spontaneously burst. The mechanism of this bursting is incompletely understood. We have constructed a biophysical Purkinje soma model, guided and constrained by experimental reports in the literature, that can replicate the somatically driven bursting pattern and which hypothesises Persistent Na⁺ current (I_NaP) to be its burst initiator and SK K⁺ current (I_SK) to be its burst terminator.

The sodium-potassium pump controls the intrinsic firing of the cerebellar Purkinje neuron

In vitro, cerebellar Purkinje cells can intrinsically fire action potentials in a repeating trimodal or bimodal pattern. The trimodal pattern consists of tonic spiking, bursting, and quiescence. The bimodal pattern consists of tonic spiking and quiescence. It is unclear how these firing patterns are generated and what determines which firing pattern is selected. We have constructed a realistic biophysical Purkinje cell model that can replicate these patterns. In this model, Na(+)/K(+) pump activity sets the Purkinje cell's operating mode. From rat cerebellar slices we present Purkinje whole cell recordings in the presence of ouabain, which irreversibly blocks the Na(+)/K(+) pump. The model can replicate these recordings. We propose that Na(+)/K(+) pump activity controls the intrinsic firing mode of cerbellar Purkinje cells.

Computational analysis of calcium signaling and membrane electrophysiology in cerebellar Purkinje neurons associated with ataxia

Background

Mutations in the smooth endoplasmic reticulum (sER) calcium channel Inositol Trisphosphate Receptor type 1 (IP3R1) in humans with the motor function coordination disorders Spinocerebellar Ataxia Types 15 and 16 (SCA15/16) and in a corresponding mouse model, the IP3R1^{delta18/delta18} mice, lead to reduced IP3R1 levels. We posit that increasing IP3R1 sensitivity to IP3 in ataxias with reduced IP3R1 could restore normal calcium response. On the other hand, in mouse models of the human polyglutamine (polyQ) ataxias, SCA2, and SCA3, the primary finding appears to be hyperactive IP3R1-mediated calcium release. It has been suggested that the polyQ SCA1 mice may also show hyperactive IP3R1. Yet, SCA1 mice show downregulated gene expression of IP3R1, Homer, metabotropic glutamate receptor (mGluR), smooth endoplasmic reticulum Ca-ATP-ase (SERCA), calbindin, parvalbumin, and other calcium signaling proteins.

Results

We create a computational model of pathological alterations in calcium signaling in cerebellar Purkinje neurons to investigate several forms of spinocerebellar ataxia associated with changes in the abundance, sensitivity, or activity of the calcium channel IP3R1. We find that increasing IP3R1 sensitivity to IP3 in computational models of SCA15/16 can restore normal calcium response if IP3R1 abundance is not too low. The studied range in IP3R1 levels reflects variability found in human and mouse ataxic models. Further, the required fold increases in sensitivity are within experimental ranges from experiments that use IP3R1 phosphorylation status to adjust its sensitivity to IP3. Results from our simulations of polyglutamine SCAs suggest that downregulation of some calcium signaling proteins may be partially compensatory. However, the downregulation of calcium buffer proteins observed in the SCA1 mice may contribute to pathology. Finally, our model suggests that the calcium-activated voltage-gated potassium channels may provide an important link between calcium metabolism and membrane potential in Purkinje cell function.

Conclusion

Thus, we have established an initial platform for computational evaluation and prediction of ataxia pathophysiology. Specifically, the model has been used to investigate SCA15/16, SCA1, SCA2, and SCA3. Results suggest that experimental studies treating mouse models of any of these ataxias with appropriately chosen peptides resembling the C-terminal of IP3R1 could adjust receptor sensitivity, and thereby modulate calcium release and normalize IP3 response. In addition, the model supports the hypothesis of IP3R1 supersensitivity in SCA1.

Neural circuit and its functional roles in cerebellar cortex

OBJECTIVE

To investigate the spike activities of cerebellar cortical cells in a computational network model constructed based on the anatomical structure of cerebellar cortex.

METHODS AND RESULTS

The multicompartment model of neuron and NEURON software were used to study the external influences on cerebellar cortical cells. Various potential spike patterns in these cells were obtained. By analyzing the impacts of different incoming stimuli on the potential spike of Purkinje cell, temporal focusing caused by the granule cell-golgi cell feedback inhibitory loop to Purkinje cell and spatial focusing caused by the parallel fiber-basket/stellate cell local inhibitory loop to Purkinje cell were discussed. Finally, the motor learning process of rabbit eye blink conditioned reflex was demonstrated in this model. The simulation results showed that when the afferent from climbing fiber existed, rabbit adaptation to eye blinking gradually became stable under the Spike Timing-Dependent Plasticity (STDP) learning rule.

CONCLUSION

The constructed cerebellar cortex network is a reliable and feasible model. The model simulation results confirmed the output signal stability of cerebellar cortex after STDP learning and the network can execute the function of spatial and temporal focusing.

Potassium currents of olfactory bulb juxtaglomerular cells: characterization, simulation, and implications for plateau potential firing

Odor identity is encoded by the activity of olfactory bulb glomeruli, which receive primary sensory input and transfer it to projection neurons. Juxtaglomerular cells (JGCs) may influence glomerular processing via firing of long lasting plateau potentials. Though inward currents have been investigated, little is known regarding potassium current contribution to JGC plateau potentials. We pursued study of these currents, with the overarching goal of creating components for a computational model of JGC plateau potential firing. In conditions minimizing calcium-activated potassium current (I(K(Ca))), we used whole cell voltage clamp and in vitro slice preparations to characterize three potassium currents in rat JGCs. The prominent component I(kt1) displayed rapid kinetics (τ(10%-90% rise), 0.6-2 ms; τ(inactivation), 5-10 ms) and was blocked by high concentration 4-aminopyridine (4-AP) (5 mM) and tetramethylammonium (TEA) (40 mM). It had half maximal activation at -10 mV (V(½)max) and little inactivation at rest. I(kt2), with slower kinetics (τ(10%-90% rise), 11-15 ms; τ(inactivation), 100-300 ms), was blocked by low concentration 4-AP (0.5 mM) and TEA (5 mM). The V(½)max was 0 mV and inactivation was also minimal at rest. Sustained current I(kt3) showed sensitivity to low concentration 4-AP and TEA, and had V(½)max of +10 mV. Further experiments, in conditions of physiologic calcium buffering, suggested that I(K(Ca)) contributed to I(kt3) with minimal effect on plateau potential evolution. We transformed these characterizations into Hodgkin-Huxley models that robustly mimicked experimental data. Further simulation demonstrated that I(kt1) would be most efficiently activated by plateau potential waveforms, predicting a critical role in shaping JGC firing. These studies demonstrated that JGCs possess a unique potassium current profile, with delayed rectifier (I(kt3)), atypical A-current (I(kt1)), and D-current (I(kt2)) in accordance with known expression patterns in olfactory bulb (OB) glomeruli. Our simulations also provide an initial framework for more integrative models of JGC plateau potential firing.

Virtual NEURON: a strategy for merged biochemical and electrophysiological modeling

Because of its highly branched dendrite, the Purkinje neuron requires significant computational resources if coupled electrical and biochemical activity are to be simulated. To address this challenge, we developed a scheme for reducing the geometric complexity; while preserving the essential features of activity in both the soma and a remote dendritic spine. We merged our previously published biochemical model of calcium dynamics and lipid signaling in the Purkinje neuron, developed in the Virtual Cell modeling and simulation environment, with an electrophysiological model based on a Purkinje neuron model available in NEURON. A novel reduction method was applied to the Purkinje neuron geometry to obtain a model with fewer compartments that is tractable in Virtual Cell. Most of the dendritic tree was subject to reduction, but we retained the neuron's explicit electrical and geometric features along a specified path from spine to soma. Further, unlike previous simplification methods, the dendrites that branch off along the preserved explicit path are retained as reduced branches. We conserved axial resistivity and adjusted passive properties and active channel conductances for the reduction in surface area, and cytosolic calcium for the reduction in volume. Rallpacks are used to validate the reduction algorithm and show that it can be generalized to other complex neuronal geometries. For the Purkinje cell, we found that current injections at the soma were able to produce similar trains of action potentials and membrane potential propagation in the full and reduced models in NEURON; the reduced model produces identical spiking patterns in NEURON and Virtual Cell. Importantly, our reduced model can simulate communication between the soma and a distal spine; an alpha function applied at the spine to represent synaptic stimulation gave similar results in the full and reduced models for potential changes associated with both the spine and the soma. Finally, we combined phosphoinositol signaling and electrophysiology in the reduced model in Virtual Cell. Thus, a strategy has been developed to combine electrophysiology and biochemistry as a step toward merging neuronal and systems biology modeling.

Model of very fast (> 75 Hz) network oscillations generated by electrical coupling between the proximal axons of cerebellar Purkinje cells

Very fast oscillations (VFO; > 75 Hz) occur transiently in vivo, in the cerebellum of mice genetically modified to model Angelman syndrome, and in a mouse model of fetal alcohol syndrome. We recently reported VFO in slices of mouse cerebellar cortex (Crus I and II of ansiform and paramedian lobules), either in association with gamma oscillations (approximately 40 Hz, evoked by nicotine) or in isolation [evoked by nicotine in combination with gamma-aminobutyric acid (GABA)(A) receptor blockade]. The experimental data suggest a role for electrical coupling between Purkinje cells (blockade of VFO by drugs reducing gap junction conductance and spikelets in some Purkinje cells); and the data suggest the specific involvement of Purkinje cell axons (because of field oscillation maxima in the granular layer). We show here that a detailed network model (1000 multicompartment Purkinje cells) replicates the experimental data when gap junctions are located on the proximal axons of Purkinje cells, provided sufficient spontaneous firing is present. Unlike other VFO models, most somatic spikelets do not correspond to axonal spikes in the parent axon, but reflect spikes in electrically coupled axons. The model predicts gating of VFO frequency by g(Na) inactivation, and experiments prolonging this inactivation time constant, with beta-pompilidotoxin, are consistent with this prediction. The model also predicts that cerebellar VFO can be explained as an electrically coupled system of axons that are not intrinsic oscillators: the electrically uncoupled cells do not individually oscillate (in the model) and axonal firing rates are much lower in the uncoupled state than in the coupled state.

New dynamics in cerebellar Purkinje cells: torus canards

We describe a transition from bursting to rapid spiking in a reduced mathematical model of a cerebellar Purkinje cell. We perform a slow-fast analysis of the system and find that-after a saddle node bifurcation of limit cycles-the full model dynamics temporarily follow a repelling branch of limit cycles. We propose that the system exhibits a dynamical phenomenon new to realistic, biophysical applications: torus canards.

Fully implicit parallel simulation of single neurons

When a multi-compartment neuron is divided into subtrees such that no subtree has more than two connection points to other subtrees, the subtrees can be on different processors and the entire system remains amenable to direct Gaussian elimination with only a modest increase in complexity. Accuracy is the same as with standard Gaussian elimination on a single processor. It is often feasible to divide a 3-D reconstructed neuron model onto a dozen or so processors and experience almost linear speedup. We have also used the method for purposes of load balance in network simulations when some cells are so large that their individual computation time is much longer than the average processor computation time or when there are many more processors than cells. The method is available in the standard distribution of the NEURON simulation program.