Cerebellar cortex: its simulation and the relevance of Marr's theory.

Marr's theory of the cerebellar cortex as an associative learning device is one of the best examples of a theory that directly relates the function of a neural system to its neural structure. However, although he assigned a precise function to each of the identified cell types of the cerebellar cortex, many of the crucial aspects of the implementation of his theory remained unspecified. We attempted to resolve these difficulties by constructing a computer simulation which contained a direct representation of the 13,000 mossy fibres and the 200,000 granule cells associated with a single Purkinje cell of the cerebellar cortex, together with the supporting Golgi, basket and stellate cells. In this paper we present a detailed explanation of Marr's theory based upon an analogy between Marr's cerebellar model and an abstract model called the associative net. Although some of Marr's assumptions contravene neuroanatomical findings, we found that in general terms his conclusion that each Purkinje cell can learn to respond to a large number of different patterns of activity in the mossy fibres is substantially correct. However, we found that this system has a lower capacity and acts more stochastically than he envisaged. The biologically realistic simulated structure that we designed can be used to assess the computational capabilities of other network theories of the cerebellum.     

Optimal information storage and the distribution of synaptic weights: perceptron versus Purkinje cell

It is widely believed that synaptic modifications underlie learning and memory. However, few studies have examined what can be deduced about the learning process from the distribution of synaptic weights. We analyze the perceptron, a prototypical feedforward neural network, and obtain the optimal synaptic weight distribution for a perceptron with excitatory synapses. It contains more than 50% silent synapses, and this fraction increases with storage reliability: silent synapses are therefore a necessary byproduct of optimizing learning and reliability. Exploiting the classical analogy between the perceptron and the cerebellar Purkinje cell, we fitted the optimal weight distribution to that measured for granule cell-Purkinje cell synapses. The two distributions agreed well, suggesting that the Purkinje cell can learn up to 5 kilobytes of information, in the form of 40,000 input-output associations.     

Modulation of the dynamics of cerebellar Purkinje cells through the interaction of excitatory and inhibitory feedforward pathways

The dynamics of cerebellar neuronal networks is controlled by the underlying building blocks of neurons and synapses between them. For which, the computation of Purkinje cells (PCs), the only output cells of the cerebellar cortex, is implemented through various types of neural pathways interactively routing excitation and inhibition converged to PCs. Such tuning of excitation and inhibition, coming from the gating of specific pathways as well as short-term plasticity (STP) of the synapses, plays a dominant role in controlling the PC dynamics in terms of firing rate and spike timing. PCs receive cascade feedforward inputs from two major neural pathways: the first one is the feedforward excitatory pathway from granule cells (GCs) to PCs; the second one is the feedforward inhibition pathway from GCs, via molecular layer interneurons (MLIs), to PCs. The GC-PC pathway, together with short-term dynamics of excitatory synapses, has been a focus over past decades, whereas recent experimental evidence shows that MLIs also greatly contribute to controlling PC activity. Therefore, it is expected that the diversity of excitation gated by STP of GC-PC synapses, modulated by strong inhibition from MLI-PC synapses, can promote the computation performed by PCs. However, it remains unclear how these two neural pathways are interacted to modulate PC dynamics. Here using a computational model of PC network installed with these two neural pathways, we addressed this question to investigate the change of PC firing dynamics at the level of single cell and network. We show that the nonlinear characteristics of excitatory STP dynamics can significantly modulate PC spiking dynamics mediated by inhibition. The changes in PC firing rate, firing phase, and temporal spike pattern, are strongly modulated by these two factors in different ways. MLIs mainly contribute to variable delays in the postsynaptic action potentials of PCs while modulated by excitation STP. Notably, the diversity of synchronization and pause response in the PC network is governed not only by the balance of excitation and inhibition, but also by the synaptic STP, depending on input burst patterns. Especially, the pause response shown in the PC network can only emerge with the interaction of both pathways. Together with other recent findings, our results show that the interaction of feedforward pathways of excitation and inhibition, incorporated with synaptic short-term dynamics, can dramatically regulate the PC activities that consequently change the network dynamics of the cerebellar circuit.     

Detection of sequences in the cerebellar cortex: numerical estimate of the possible number of tidal-wave inducing sequences represented

The two major cortices of the brain--the cerebral and cerebellar cortex--are massively connected through intercalated nuclei (pontine, cerebellar and thalamic nuclei). We suggest that the two cortices co-operate by generating precise temporal patterns in the cerebral cortex that are detected in the cerebellar cortex as temporal patterns assembled spatially in the mossy fibers. We will begin by showing that the tidal-wave mechanism works in the cerebellar cortex as a read-out mechanism for such spatio-temporal patterns due to the synchronous activity they generate in the parallel fiber system which drives the Purkinje cells--the output neurons of the cerebellar cortex--to fire action potentials. We will review the anatomy of the mossy fibers and show that within a "beam", or "row" of cerebellar cortex the mossy fibers in principle could embed a vast number of tidal-wave generating sequences. Based on anatomical data we will argue that the cerebellar mossy fiber-granule cell-Purkinje cell system can potentially detect and--through learning--select from an enormous number of spatio-temporal patterns.     

BK Channels Control Cerebellar Purkinje and Golgi Cell Rhythmicity In Vivo

Calcium signaling plays a central role in normal CNS functioning and dysfunction. As cerebellar Purkinje cells express the major regulatory elements of calcium control and represent the sole integrative output of the cerebellar cortex, changes in neural activity- and calcium-mediated membrane properties of these cells are expected to provide important insights into both intrinsic and network physiology of the cerebellum. We studied the electrophysiological behavior of Purkinje cells in genetically engineered alert mice that do not express BK calcium-activated potassium channels and in wild-type mice with pharmacological BK inactivation. We confirmed BK expression in Purkinje cells and also demonstrated it in Golgi cells. We demonstrated that either genetic or pharmacological BK inactivation leads to ataxia and to the emergence of a beta oscillatory field potential in the cerebellar cortex. This oscillation is correlated with enhanced rhythmicity and synchronicity of both Purkinje and Golgi cells. We hypothesize that the temporal coding modification of the spike firing of both Purkinje and Golgi cells leads to the pharmacologically or genetically induced ataxia.     

Signal processing in cerebellar Purkinje cells

Mechanisms and functional implications of signal processing in cerebellar Purkinje cells have been the subject of recent extensive investigations. Complex patterns of their planar dendritic arbor are analysed with computer-aided reconstructions and also topological analyses. Local computation may occur in Purkinje cell dendrites, but its extent is not clear at present. Synaptic transmission and electrical and ionic activity of Purkinje cell membrane have been revealed in detail, and related biochemical processes are being uncovered. A special type of synaptic plasticity is present in Purkinje cell dendrites; long-term depression (LTD) occurs in parallel fiber-Purkinje cell transmission when the parallel fibers are activated with a climbing fiber innervating that Purkinje cell. Evidence indicates that synaptic plasticity in Purkinje cells is due to sustained desensitization of Purkinje dendritic receptors to glutamate, which is a putative neurotransmitter of parallel fibers, and that conjunctive activation of a climbing fiber and parallel fibers leads to desensitization through enhanced intradendritic calcium concentration. A microzone of the cerebellar cortex is connected to an extracerebellar neural system through the inhibitory projection of Purkinje cells to a cerebellar or vestibular nuclear cell group. Climbing fiber afferents convey signals representing control errors in the performance of a neural system, and evoke complex spikes in Purkinje cells of the microzone connected to the neural system. Complex spikes would modify the performance of the microzone by producing LTD in parallel fiber-Purkinje cell synapses, and consequently would improve the overall performance of the neural system. The primary function of the cerebellum thus appears to be endowing adaptability to numerous neural control systems in the brain and spinal cord through error-triggered reorganization of the cerebellar cortical circuitry.     

Habituation rules for a theory of the cerebellar cortex

A quantitative model of cerebellar cortical function is described with a complete formalization based on (i) the topology of cerebellar cortical neuronal network, (ii) some particular synaptic properties of cell classes in cerebellum cortex, and (iii) the dynamics of excitation in this network. For (i), a construction of functional classes around one Purkinje cell is given and their existence is discussed. For (ii), as in Marr-Albus model, the modifiability of synapses between parallel fibres and Purkinje cell is assumed. But the formalization permits to consider the consequences of such a property at the level of glomerulus (with granule cells) which is known as a complex transformation system. For (iii) habituation rules are assumed. It is shown that this method leads to some interesting properties in the functioning of cerebellar cortex. Particularly, emitting frequency along a Purkinje cell axon results from a discrimination by the system between transformed input signals and an external noise due to all other contexts, and learning could be considered as the result of a conflict between a set of patterns and the transformed input signals. This model could be a basis for future numerical simulations.     

Delayed reverberation through time windows as a key to cerebellar function

We present a functional model of the cerebellum comprising cerebellar cortex, inferior olive, deep cerebellar nuclei, and brain stem nuclei. The discerning feature of the model being time coding, we consistently describe the system in terms of postsynaptic potentials, synchronous action potentials, and propagation delays. We show by means of detailed single-neuron modeling that (i) Golgi cells can fulfill a gating task in that they form short and well-defined time windows within which granule cells can reach firing threshold, thus organizing neuronal activity in discrete `time slices', and that (ii) rebound firing in cerebellar nuclei cells is a robust mechanism leading to a delayed reverberation of Purkinje cell activity through cerebellar-reticular projections back to the cerebellar cortex. Computer simulations of the whole cerebellar network consisting of several thousand neurons reveal that reverberation in conjunction with long-term plasticity at the parallel fiber-Purkinje cell synapses enables the system to learn, store, and recall spatio-temporal patterns of neuronal activity. Climbing fiber spikes act both as a synchronization and as a teacher signal, not as an error signal. They are due to intrinsic oscillatory properties of inferior olivary neurons and to delayed reverberation within the network. In addition to clear experimental predictions the present theory sheds new light on a number of experimental observation such as the synchronicity of climbing fiber spikes and provides a novel explanation of how the cerebellum solves timing tasks on a time scale of several hundreds of milliseconds. Received: 23 July 1999 / Accepted in revised form: 31 August 1999     

On the role of anatomy in learning by the cerebellar cortex

The properties due to the location of neurons, synapses, and possibly even synaptic channels, in neuron networks are still unknown. Our preliminary results suggest that not only the interconnections but also the relative positions of the different elements in the network are of importance in the learning process in the cerebellar cortex. We have used neural field equations to investigate the mechanisms of learning in the hierarchical neural network. The numerical resolution of these equations reveals two important properties: (i) The hierarchical structure of this network has the expected effect on learning because the flow of information at the neuronal level is controlled by the heterosynaptic effect through the synaptic density-connectivity function, i.e. the action potential field variable is controlled by the synaptic efficacy field variable at different points of the neuron. (ii) The geometry of the system involves different velocities of propagation along different fibers, i.e. different delays between cells, and thus has a stabilizing effect on the dynamics, allowing the Purkinje output to reach a given value. The field model proposed should be useful in the study of the spatial properties of hierarchical biological systems.     

A real-time spiking cerebellum model for learning robot control

We describe a neural network model of the cerebellum based on integrate-and-fire spiking neurons with conductance-based synapses. The neuron characteristics are derived from our earlier detailed models of the different cerebellar neurons. We tested the cerebellum model in a real-time control application with a robotic platform. Delays were introduced in the different sensorimotor pathways according to the biological system. The main plasticity in the cerebellar model is a spike-timing dependent plasticity (STDP) at the parallel fiber to Purkinje cell connections. This STDP is driven by the inferior olive (IO) activity, which encodes an error signal using a novel probabilistic low frequency model. We demonstrate the cerebellar model in a robot control system using a target-reaching task. We test whether the system learns to reach different target positions in a non-destructive way, therefore abstracting a general dynamics model. To test the system's ability to self-adapt to different dynamical situations, we present results obtained after changing the dynamics of the robotic platform significantly (its friction and load). The experimental results show that the cerebellar-based system is able to adapt dynamically to different contexts.     

Cerebellar LTD and pattern recognition by Purkinje cells

Many theories of cerebellar function assume that long-term depression (LTD) of parallel fiber (PF) synapses enables Purkinje cells to learn to recognize PF activity patterns. We have studied the LTD-based recognition of PF patterns in a biophysically realistic Purkinje-cell model. With simple-spike firing as observed in vivo, the presentation of a pattern resulted in a burst of spikes followed by a pause. Surprisingly, the best criterion to distinguish learned patterns was the duration of this pause. Moreover, our simulations predicted that learned patterns elicited shorter pauses, thus increasing Purkinje-cell output. We tested this prediction in Purkinje-cell recordings both in vitro and in vivo. In vitro, we found a shortening of pauses when decreasing the number of active PFs or after inducing LTD. In vivo, we observed longer pauses in LTD-deficient mice. Our results suggest a novel form of neural coding in the cerebellar cortex.     

Enhancement of both long-term depression induction and optokinetic response adaptation in mice lacking delphilin

In the cerebellum, Delphilin is expressed selectively in Purkinje cells (PCs) and is localized exclusively at parallel fiber (PF) synapses, where it interacts with glutamate receptor (GluR) delta2 that is essential for long-term depression (LTD), motor learning and cerebellar wiring. Delphilin ablation exerted little effect on the synaptic localization of GluRdelta2. There were no detectable abnormalities in cerebellar histology, PC cytology and PC synapse formation in contrast to GluRdelta2 mutant mice. However, LTD induction was facilitated at PF-PC synapses in Delphilin mutant mice. Intracellular Ca(2+) required for the induction of LTD appeared to be reduced in the mutant mice, while Ca(2+) influx through voltage-gated Ca(2+) channels and metabotropic GluR1-mediated slow synaptic response were similar between wild-type and mutant mice. We further showed that the gain-increase adaptation of the optokinetic response (OKR) was enhanced in the mutant mice. These findings are compatible with the idea that LTD induction at PF-PC synapses is a crucial rate-limiting step in OKR gain-increase adaptation, a simple form of motor learning. As exemplified in this study, enhancing synaptic plasticity at a specific synaptic site of a neural network is a useful approach to understanding the roles of multiple plasticity mechanisms at various cerebellar synapses in motor control and learning.     

Adaptive Robotic Control Driven by a Versatile Spiking Cerebellar Network

The cerebellum is involved in a large number of different neural processes, especially in associative learning and in fine motor control. To develop a comprehensive theory of sensorimotor learning and control, it is crucial to determine the neural basis of coding and plasticity embedded into the cerebellar neural circuit and how they are translated into behavioral outcomes in learning paradigms. Learning has to be inferred from the interaction of an embodied system with its real environment, and the same cerebellar principles derived from cell physiology have to be able to drive a variety of tasks of different nature, calling for complex timing and movement patterns. We have coupled a realistic cerebellar spiking neural network (SNN) with a real robot and challenged it in multiple diverse sensorimotor tasks. Encoding and decoding strategies based on neuronal firing rates were applied. Adaptive motor control protocols with acquisition and extinction phases have been designed and tested, including an associative Pavlovian task (Eye blinking classical conditioning), a vestibulo-ocular task and a perturbed arm reaching task operating in closed-loop. The SNN processed in real-time mossy fiber inputs as arbitrary contextual signals, irrespective of whether they conveyed a tone, a vestibular stimulus or the position of a limb. A bidirectional long-term plasticity rule implemented at parallel fibers-Purkinje cell synapses modulated the output activity in the deep cerebellar nuclei. In all tasks, the neurorobot learned to adjust timing and gain of the motor responses by tuning its output discharge. It succeeded in reproducing how human biological systems acquire, extinguish and express knowledge of a noisy and changing world. By varying stimuli and perturbations patterns, real-time control robustness and generalizability were validated. The implicit spiking dynamics of the cerebellar model fulfill timing, prediction and learning functions.     

Type-1 metabotropic glutamate receptor in cerebellar Purkinje cells: a key molecule responsible for long-term depression, endocannabinoid signalling and synapse elimination

The cerebellum is a brain structure involved in the coordination, control and learning of movements, and elucidation of its function is an important issue. Japanese scholars have made seminal contributions in this field of neuroscience. Electrophysiological studies of the cerebellum have a long history in Japan since the pioneering works by Ito and Sasaki. Elucidation of the basic circuit diagram of the cerebellum in the 1960s was followed by the construction of cerebellar network theories and finding of their neural correlates in the 1970s. A theoretically predicted synaptic plasticity, long-term depression (LTD) at parallel fibre to Purkinje cell synapse, was demonstrated experimentally in 1982 by Ito and co-workers. Since then, Japanese neuroscientists from various disciplines participated in this field and have made major contributions to elucidate molecular mechanisms underlying LTD. An important pathway for LTD induction is type-1 metabotropic glutamate receptor (mGluR1) and its downstream signal transduction in Purkinje cells. Sugiyama and co-workers demonstrated the presence of mGluRs and Nakanishi and his pupils identified the molecular structures and functions of the mGluR family. Moreover, the authors contributed to the discovery and elucidation of several novel functions of mGluR1 in cerebellar Purkinje cells. mGluR1 turned out to be crucial for the release of endocannabinoid from Purkinje cells and the resultant retrograde suppression of transmitter release. It was also found that mGluR1 and its downstream signal transduction in Purkinje cells are indispensable for the elimination of redundant synapses during post-natal cerebellar development. This article overviews the seminal works by Japanese neuroscientists, focusing on mGluR1 signalling in cerebellar Purkinje cells.     

Suppression of inhibitory synaptic potentiation by presynaptic activity through postsynaptic GABA(B) receptors in a Purkinje neuron

At inhibitory synapses on a cerebellar Purkinje neuron, the depolarization caused by heterosynaptic climbing fiber activation induces long-lasting potentiation accompanied by an increase in GABA(A) receptor responsiveness. Here we show that activation of a presynaptic inhibitory interneuron during the conditioning postsynaptic depolarization suppresses the potentiation. The suppression is due to postsynaptic GABA(B) receptor activation by GABA released from presynaptic terminals. The results suggest that GABA(B) receptor activation decreases the activity of cAMP-dependent protein kinase through the G(i)/G(o) proteins. The presynaptic activity-dependent suppression of synaptic plasticity is a novel regulatory mechanism of synaptic efficacy at individual synapses and may contribute to the learning and computational ability of the cerebellar cortex.     

Kainate receptors differentially regulate release at two parallel fiber synapses

Presynaptic kainate receptors (KARs) facilitate or depress transmitter release at several synapses in the CNS. Here, we report that synaptically activated KARs presynaptically facilitate and depress transmission at parallel fiber synapses in the cerebellar cortex. Low-frequency stimulation of parallel fibers facilitates synapses onto both stellate cells and Purkinje cells, whereas high-frequency stimulation depresses stellate cell synapses but continues to facilitate Purkinje cell synapses. These effects are mimicked by exogenous KAR agonists and eliminated by blocking KARs. This differential frequency-dependent sensitivity of these two synapses regulates the balance of excitatory and inhibitory input to Purkinje cells and therefore their excitability.     

Probabilistic secretion of quanta in the central nervous system: granule cell synaptic control of pattern separation and activity regulation.

The implications of probabilistic secretion of quanta for the functioning of neural networks in the central nervous system have been explored. A model of stochastic secretion at synapses in simple networks, consisting of large numbers of granule cells and a relatively small number of inhibitory interneurons, has been analysed. Such networks occur in the input to the cerebellum Purkinje cells as well as to hippocampal CA3 pyramidal cells and to pyramidal cells in the visual cortex. In this model the input axons terminate on granule cells as well as on an inhibitory interneuron that projects to the granule cells. Stochastic secretion at these synapses involves both temporal variability in secretion at single synapses in the network as well as spatial variability in the secretion at different synapses. The role of this stochastic variability in controlling the size of the granule cell output to a level independent of the size of the input and in separating overlapping inputs has been determined analytically as well as by simulation. The regulation of granule-cell output activity to a reasonably constant value for different size inputs does not occur in the absence of an inhibitory interneuron when both spatial and temporal stochastic variability occurs at the remaining synapses; it is still very poor in the presence of such an interneuron but in the absence of stochastic variability. However, quite good regulation is achieved when the inhibitory interneuron is present with spatial and temporal stochastic variability of secretion at synapses in the network. Excellent regulation is achieved if, in addition, allowance is made for the nonlinear behaviour of the input-output characteristics of inhibitory interneurons. The capacity of granule-cell networks to separate overlapping patterns of activity on their inputs is adequate, with spatial variability in the secretion at synapses, but is improved if there is also temporal variability in the stochastic secretion at individual synapses, although this is at the expense of reliability in the network. Other factors which improve pattern separation are control of the output to very low activity levels, and a restriction on the cumulative size of the excitatory input terminals of each granule cell. Application of the theory to the input neural networks of the cerebellum and the hippocampus shows the role of stochastic variability in quantal transmission in determining the capacity of these networks for pattern separation and activity regulation.     

A computational mechanism for unified gain and timing control in the cerebellum

Precise gain and timing control is the goal of cerebellar motor learning. Because the basic neural circuitry of the cerebellum is homogeneous throughout the cerebellar cortex, a single computational mechanism may be used for simultaneous gain and timing control. Although many computational models of the cerebellum have been proposed for either gain or timing control, few models have aimed to unify them. In this paper, we hypothesize that gain and timing control can be unified by learning of the complete waveform of the desired movement profile instructed by climbing fiber signals. To justify our hypothesis, we adopted a large-scale spiking network model of the cerebellum, which was originally developed for cerebellar timing mechanisms to explain the experimental data of Pavlovian delay eyeblink conditioning, to the gain adaptation of optokinetic response (OKR) eye movements. By conducting large-scale computer simulations, we could reproduce some features of OKR adaptation, such as the learning-related change of simple spike firing of model Purkinje cells and vestibular nuclear neurons, simulated gain increase, and frequency-dependent gain increase. These results suggest that the cerebellum may use a single computational mechanism to control gain and timing simultaneously.     

Interrelated modification of excitatory and inhibitory synapses in three-layer olivary-cerebellar neural network

The model of three-layer olivary-cerebellar neural network with modifiable excitatory and inhibitory connections between diverse elements is suggested. The same Hebbian modification rules are proposed for Purkinje cells, granule (input) cells, and deep cerebellar nuclei (output) cells. The inverse calcium-dependent modification rules for these cells and hippocampal/neocortical neurones or Golgi cells are conceivably the result of the involvement of cGMP and cAMP in postsynaptic processes. The sign of simultaneous modification of excitatory and inhibitory inputs to a cell is opposite and determined by the variations in pre- and/or postsynaptic cell activity. Modification of excitatory transmission between parallel fibers and Purkinje cells, mossy fibers and granule cells, and mossy fibers and deep cerebellar nuclei cells essentially depends on inhibition effected by stellate/basket cells, Golgi cells and Purkinje cells, respectively. The character of interrelated modifications of diverse synapses in all three layers of the network is influenced by olivary cell activity. In the absence (presence) of a signal from inferior olive, the long-term potentiation (depression) in the efficacy of a synapse between input mossy fiber and output cell can be induced. The results of the suggested model are in accordance with known experimental data.