Abstract
Without the cerebellum, organisms are challenged in the learning and execution of accurate and coordinated actions. It has a central position in the nervous system receiving and projecting to the spinal cord, midbrain, and cerebral cortex, implying convergence of sensory and motor streams. Its highly conserved neuroarchitecture would imply it is very good at what it does and that what it does is very general. Here we review theoretical, modeling, and computational work that has attempted to capture the dynamics and/or function of the cerebellum.
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Keywords
- Adaptive filter model
- Bottom-up models
- Cerebellar nucleus
- Coupled oscillators
- Dynamic models
- Golgi gating
- Granule cell models
- Reverberating loops model
- Spatiotemporal patterns in cerebellum
- Tidal wave
- Echo-state machine
- Forward and inverse models
- Functional models
- Golgi gating
- Granule cell models
- Inferior olivary models
- Coupled oscillators
- Echo-state machine
- Phase resets
- Synchronous groups
- Marr-Albus type models
- Adaptive filter model and distributed synaptic plasticity
- Information encoding and channel capacity
- Purkinje neuron single cell modeling
- Successes and failures
- Pellionisz and Llinas’s model
- Purkinje neuron single cell modeling
- Reverberating loops model
- Synchronous groups
- Tidal wave hypothesis
Introduction
Without the cerebellum, organisms are challenged in the learning and execution of accurate and coordinated actions. It has a central position in the nervous system, and it both receives and projects to the spinal cord and midbrain, implying convergence of sensory and motor streams, and in mammals also sends and receives from the cerebral cortex. Its highly conserved neuroarchitecture would imply it is very good at what it does and that what it does is very general.
A clue to its basal function is readily available from comparative neuroanatomy. The cerebellum first appears in gnathostomata fish, the jawed vertebrates, which underlie 99% of all vertebrates. In its most primitive instantiations, the cerebellum arises with the horizontal canal of the vestibular system, and thus the compensatory vestibular reflexes are among the first functions performed by the early cerebellum. Vestibular reflexes are implemented essentially by a feedback system, where visual and vestibular information are transformed into compensatory motor action. Further down the evolutionary road, the cerebellum retains that functionality (flocculus) and expands on it, by applying this compensatory function to a vast class of problems, exemplified in paradigms that study cerebellar function such as learning of timed reflexes as in eye blink conditioning (Brinke et al. 2015; Mauk et al. 1986; Ohyama et al. 2003; Rasmussen et al. 2013), visuomotor feedback and adaptation (Optican and Robinson 1980; Thier et al. 2000), force-field adaptation (Donchin et al. 2012), sequence learning (Spencer and Ivry 2009), postural corrections (Angelaki et al. 2009; Clark 1939), and rhythmic finger tapping (Del Olmo et al. 2007) to name a few. In addition to these, the cerebellum of mammals seems to be concerned with the learning and acquisition of novel motor behaviors, in association with the basal ganglia, thalamus, and the cerebral cortex. In humans, the cerebellum has been implicated in cognitive function (Schmahmann and Caplan 2006).
The million-dollar question in cerebellar modeling is how does the cerebellar structure enable these functions.
The cerebellum is one component of a tripartite system involving the inferior olive (IO) and the cerebellar nucleus (figure elsewhere). While early models have focused on the most conspicuous elements of the cerebellar circuitry, particularly the interaction between parallel fibers and Purkinje cells, the field has broadened its scope to include various models of the inferior olive and cerebellar nucleus. Given how tightly these three systems interact, all these models are included in the chapter.
There have been multiple forays at modeling the cerebellum, attempting to capture the abstract nature of cerebellar transformations (Albus 1971; Braitenberg 1987; Braitenberg et al.1997; Marr 1969). David Marr recognized in the large arbor of the Purkinje neuron the potential for pattern recognition, in analogy to the McCullogh-Pitts neuron (McCulloch and Pitts 1943), and Rosenblatt’s perceptron learning rule (Rosenblatt 1958). Eccles suggested that the cerebellum exhibited spatiotemporal transformations without being explicit about what they would be. Valentino Braitenberg saw in the orthogonal arrangement between parallel fibers and Purkinje cells temporal summation and coincidence detection (Braitenberg et al. 1997). Llinás saw in the oscillations in the inferior olive a form of binding different actions together through synchrony (Kazantsev et al. 2003). All these traditions have seen data that corroborates or questions their assumptions.
In a first approximation, computational models of the brain can be categorized as functional models and dynamical models, depending on assumptions and emphasis. Functional models , also called top-down, assume a set of functions and attempt to interpret the components of the architecture as realizing that function (such as finding the weights of a network that performs a certain transformation). Bottom-up models focus on the dynamical implications of selected aspects of anatomy and physiology. In one case, data on anatomy and physiology takes precedence, on the other, the focus is on enabling the assumed function with neural components with varying degrees of plausibility (Houk and Fagg 2014; Medina 2010; Medina and Lisberger 2008).
This chapter represents the multiplicity of the literature on cerebellar modeling. We survey multiple proposals and their perspectives, from those emphasizing physiological plausibility, through those that interpret the circuits in terms of their mathematical/physical properties (e.g., summation along delay lines – tidal waves/decorrelation – sparsity/transients – echo states/granular layer resonances), to those that predict physiological properties (Golgi network oscillations/resonation with cerebellar nucleus) to those that have been embodied in robots, as well as conceptual-mathematical models (forward and inverse models). The inferior olive has also received modeling attention, and we add a section devoted to it, as it is an essential element to the cerebellar system.
The chapter begins with introducing the concepts of forward and inverse models, which appear relevant to multiple aspects of the subsequent discussion. After that, we look into more detail into models in the literature, in a gradient from functional (top-down) to dynamic-physiological-anatomical models (bottom-up). Single cell models are briefly introduced in the proper context.
A Plethora of Models
Forward and Inverse Models
Without any knowledge about how the brain works, it is possible to perform a correct movement – a movement where the plan is identical to the execution. The brain is faced with the inverse problem of finding a sequence of muscle activations, to a given desired state. That is, knowing how to move is the same as knowing how to transform the desire to reach a state into the sequence that will bring it about. This is the definition of an inverse model.
How do we know that the movement is correct? When by the end of the movement, all looks as expected. For instance, the visual outcome of the action should match the desired/predicted movement. Expectations are not exclusively visual and can have any sensory feedback. And thus, to know whether a motor action was correct, we also have to know what we expect in terms of sensory feedback. This knowledge of how the outcome of an action should feel like is a “forward model.”
Forward and inverse models dissociate two aspects of motor function, planning and execution. This separation between forward and inverse is effectively a linearization of the tasks of the motor system. Wolpert and Kawato (1998) have proposed that motor control is in effect a collection of such forward and inverse models in “modules.” They propose a method to learn multiple paired forward-inverse models, and a context switcher – a responsibility estimator. The search for the counterparts of forward-inverse modules in the brain is a current field of research. As the theory is somewhat reticent about the actual neural implementation of said models, the identification of the cortical substrates is not obvious.
Nevertheless, modern discussions of cerebellar function have often attributed the role of forward model to the cerebellum (Wolpert and Kawato 1998), calculating sensorimotor expectations of a given action, a fact which has found confirmation in related brain regions (Han et al. 2000). The critical question in designing an inverse model is the encoding of “goal” or “end-state” as well as the form of the “sensory prediction” that inverse models need for training. For example, in a paradigm such as saccadic adaptation, in what form comes the information about the vector error (saccadic mismatch)? If we presume, as many do, that complex spikes encode error, they are burdened with a very specific signal to encode in very few spikes. Given the anatomy and the physiology of the Inferior Olive, specific encoding such as required by feedback theories seems contentious.
Functional Models
What is the function of the cerebellum? Attributing a specific function to a brain region is a notorious fallacy (Edelman and Gally 2001). Functional models are not bogged down by this remark, as the function is assumed – based on clinical observations (intention tremor in ataxias and cerebellar lesions), comparative neuroanatomy (compensatory movements), lesions, and a variety of experiments. The task for functional models is to enable the assumed function with selected neural components. For instance, one of the earliest functional models of the cerebellum, due to Marr (1969) and Albus (1971), imagines the Purkinje cell as analogous to a perceptron (Rosenblatt 1958), which is a mathematical function that linearly sums weighted inputs and applies a threshold, performing linear pattern separation. Though Purkinje cell physiology is not so simple, models of the cerebellum based on simple perceptrons have been able to perform a variety of functions attributed to the cerebellum (Casellato et al. 2014; Houk and Fagg 2014; Medina 2010), and some assumptions of early models remain valid today, if with interesting caveats.
Marr-Albus Type Models
It is illustrative to trace the evolution of Marr-Albus models to their historical assumptions and how they were updated over time.
In Marr-Albus type models, cerebellar function is essentially pattern recognition performed by the Purkinje neuron, in analogy with a perceptron rule. Inputs are weighted by parallel fiber synapses and the Purkinje neuron performs a hard threshold function. The Purkinje neuron is trained to recognize spatial patterns of parallel fibers carrying sensorimotor and other brain signals through a teaching signal, provided by the climbing fiber, which provides a “supervision” signal (Doya 2002). In Marr’s original suggestion, this would increase the efficiency of the synapses encoding the input pattern. Albus later suggested a decrease, which has been experimentally corroborated by Ito, as long-term depression (LTD) of the parallel fiber synapse (Ito 1989; Ito and Kano 1982).
According to Marr, a cerebellum made of pattern recognizers is able to perform two functions, learned movements and learned conditional reflexes. In the original formulation, the cerebrum produces a motor plan that activates inferior olivary cells in a sequence such as to produce the desired movement. As the desired movement is performed stepwise by the brain, the sensorimotor context (muscle and sensor activations converging into Purkinje cells) is learned in the parallel fiber to Purkinje cell synapses, upon the IO teaching signal. IO cells representing elemental movement would fire in the sequence determined by the cerebral cortex. In this theory, the cerebrum outsources movements to the cerebellum by having the Purkinje neuron recognizing (through parallel fiber synapses) the context of motion. After learning, the Purkinje neuron would be able to recognize input from parallel fibers, thereafter performing automated sequences on recognizing a sensorimotor context.
In this theory, the only role of the climbing fibers, and indeed, of the IO cells, is to copy the cerebral signals into single Purkinje neurons, carrying the sequence of activated motor outputs. Essentially, this model links the output of the Purkinje neuron to a motor command that activates the IO cell innervating the Purkinje neuron.
Marr and Albus conducted a very meticulous analysis of all the cerebellar components in association with the primary role of the neurons. Marr has calculated combinatorial properties of recognizable patterns, effectively computing the channel capacity and information encoding properties of the assumed pattern separation. More recently, there have been elaborations on those quantifications that incorporated new phenomena. Particularly, on the basis of the in vitro finding that a majority of parallel fiber Purkinke neuron synapses appear to be silent (in vitro only 1% generate synaptic current) (Barbour 1993), the information capacity of the Purkinje cell has been quantified (Barbour et al. 2007). Anatomical observations in the granular layer have also led to quantifications of pattern separation and have led to the statement that the granule layer would perform lossless encoding (Billings et al. 2014), a statement already present in Marr’s seminal paper.
Relationship with Reality: Are Purkinje Neurons Perceptrons?
The Purkinje neuron is a constantly active neuron, with tonic firing rates of “simple” spikes in the range between 20 and 200 Hz, providing a fairly graded output signal. As a teaching signal, the climbing fibers provide at most an imperfect feedback signal. The complex spike rate is low and highly variable, has very broad receptive fields, and unreliable spikes in a variety of event-triggered measurements (Frens et al. 2001; Herzfeld et al. 2015; Hoogland et al. 2015; Sauerbrei et al. 2015; Soetedjo et al. 2008). Because the complex spikes respond to a large array of stimuli, the mutual information for specific stimuli has a low upper bound. Moreover, the climbing fiber signal does more than change the recently active synapses (Medina and Lisberger 2008), as it is often accompanied by a postcomplex pause (Bloedel and Roberts 1971; Eccles et al. 1972), caused by a large calcium influx and the activation of calcium activated potassium channels (De Schutter and Steuber 2009). Reflecting on the graded nature of a calcium influx, the alterations of synapses can also be a graded phenomenon (Coesmans et al. 2004). In fact, systematic changes of Purkinje cell responses to complex spike with different spike counts have been recorded (Rasmussen et al. 2013). Modeling has shown that the number of spikelets in the olivary spike is related to the calcium influx in the olivary cell (De Gruijl 2012), though the extra spikes in the climbing fiber do not directly reflect on the number of spikelets in the Purkinje neuron (Mathy et al. 2009). Amidst these complications, and albeit the successes of MAI model, the physiology and dynamics of the cerebellum have cast doubt on its validity.
Extensions of MAI Model: Adaptive Filter Model and Distributed Synaptic Plasticity
One of the most conspicuous mismatches between prediction and reality in Marr’s model has been the strict assumption that the only locus of plasticity in the cerebellum is the Purkinje cell parallel fiber synapse. Recent research has shown that virtually every cerebellar synapse that has been tested for plasticity has shown long-term changes upon paired stimulations (Gao et al. 2012b).
In order to salvage the original model, many sites of plasticity have been grafted onto the original trunk of the MAI (Clopath et al. 2014; Hansel et al. 2001; Houk and Fagg 2014; Porrill and Dean 2007). These extensions have been able to reproduce some learning phenomena in simplified paradigms of cerebellar function, and particularly, in some cases reproduced learning rates observed in experiments (Clopath et al. 2014).
Information Encoding and Channel Capacity
If the Purkinje neuron is viewed as a binary pattern recognition, i.e., a McCullogh-Pitts unit, one may ask what is its information capacity , in bits. This question has been addressed by Barbour and colleagues on the basis of the observation that most parallel fiber-Purkinje synapses are silent (Barbour et al. 2007). It has also been addressed by Clopath under the assumption of correlated inputs (Clopath et al. 2012). Within a MAI formalism, the capacity of channels can be computed for mostly all elements of the circuit. Silver has recently suggested that the convergence ration of mossy fibers onto granule cell synapse (4–6) promotes lossless encoding of the mossy fiber input (Billings et al. 2014) (see also Heck and Sultan 2002). Marr himself has made a similar prediction on the basis of the ability of Purkinje neurons to decode “codons.” Though an interesting exercise, the meaning of the quantification of information channels is not immediately evident in motor control.
Successes and Failures of the MAI Model
MAI type models still face substantial criticism, reflected in the ongoing debates about the interpretation of the climbing fiber signal.
When the first models of the cerebellum were proposed, most of the information available was anatomical, which is reflected on the emphasis of early models from Marr, or Braitenberg. Alongside with developments of experimental science, models systematically incorporated more plausible physiological assumptions. Now there is a rich set of electrophysiological and molecular data that have complexified the picture and sharply contrasts the attractive simplicity of initial models. Some of the physiological phenomena could have broad implications for cerebellar function and modeling. Long-term potentiation of parallel fiber synapses is behaviorally quite important (Schonewille et al. 2010), climbing fibers do more than modify the synapses, as in calcium related pauses in Purkinje neurons (De Schutter and Steuber 2009), complex spike spikelets (De Gruijl et al. 2012; Jirenhed et al. 2007; Mathy et al. 2009; Rasmussen et al. 2013), and the suggestion that they can be graded error signals (De Gruijl 2012). Zebrin bands, i.e., distinct subdivisions of the cerebellum, which are related to different IO nuclei, have substantial physiological differences (Zhou et al. 2014). Anatomy has also shown that there are inhibitory feedbacks to the cerebellar cortex (Uusisaari and Knöpfel 2012; Wulff et al. 2009). Electrophysiology has shown the importance of inhibition in behavioral learning under the form of sharp ephaptic inhibition of Purkinje neurons by basket cells (Blot and Barbour 2013, 2014). The relationship between parallel fiber input and Purkinje cell output has become unclear, with increasing evidence that Purkinje cell spiking is largely driven by intrinsic mechanisms (Shin et al. 2007) and a possible pronounced role of the ascending branch of the parallel fiber (Bower 2010). It is an interesting question whether the functional simplicity of early models is compatible with these physiological observations.
Purkinje Neuron Single Cell Modeling
The Purkinje neuron is one of the neurons most frequently modeled in physiological detail, with compartmental models dating back to 1977, with a 62 compartmental model based on the Hodgkin Huxley formalism, with three ionic sorts, sodium, potassium, and a leak current (Pellionisz et al. 1977).
Later models increased the level of morphological detail, along with differential distributions of more ion channels over the dendrite and simulation of calcium dynamics (De Schutter and Bower 1994a, b) (Fig. 1b). This model has predicted parallel fiber mediated calcium influx (De Schutter and Bower 1994c) and that LTD of parallel fiber synapses causes decreased calcium influx, leading to shorter pauses in simple spike firing (Steuber et al. 2007). This is a counterintuitive result, as this would mean that Purkinje cell would increase its firing rate after LTD. This prediction has been confirmed in vitro, but not yet in vivo under physiological conditions. There has been little progress in modeling the complete Purkinje cell since, but extensive voltage clamp measurements of channels in isolated Purkinje cell somata have led to a detailed model of somatic spike initiation (Khaliq and Raman 2006) that is used extensively (de Solages et al. 2008; Ostojic et al. 2015; Phoka et al. 2010).
Increasing the level of magnification further, Purkinje cell models also focused on single dendritic branches and even on complex biochemical expression networks in single dendritic spines (Anwar et al. 2012, 2013). Using reaction diffusion formalisms, it has been shown that the stochastic gating of calcium-activated K+ channels causes the large variability of dendritic calcium spikes that may have large implications for plasticity mechanisms, which depend on the calcium concentration in the spine probabilistically (Antunes and De Schutter 2012).
Dynamical Models
The Cerebellum Implements Spatiotemporal Transformations
As early as 1958, Braitenberg has proposed that the lattice structure of the cerebellum causes transformations of spatial patterns into temporal patterns and vice versa (Braitenberg and Atwood 1958). A similar suggestion appears in “Cerebellum as a Computer” from Eccles et al. (1967), where the authors mention spatiotemporal patterns but do not elaborate past that suggestion. As spatiotemporal patterns are very general, the burden is to be explicit about the sensorimotor patterns and what are their entailments.
Spatiotemporal patterns in the cerebellum may have a great variety of origins – in the delay line properties of parallel fibers, in the oscillatory resonances of the Golgi-granule cell network (Maex and De Schutter 1998; Solinas et al. 2010; Vervaeke et al. 2012), in the coupled oscillations of the inferior olive (Llinas and Yarom 1981), in the longitudinal distribution of climbing fiber afferents, in the organization of cerebellar Zebrin stripes (Marshall and Lang 2004; Shinoda et al. 2000; Sugihara et al. 2001), in the loops between the cerebellar modules in the cortex, deep cerebellar nuclei (DCN), and IO (reviewed from a functional perspective in Glickstein et al. 2011).
The models above, in one way or another, imply spatiotemporal transformations although it is not immediately obvious how the spatiotemporal patterns would map into behavior. The interpretation of cerebellar output is a responsibility of its targets, and it is not clear what is the general principle that unifies all cerebellar output.
Mathematical Properties of the Circuit (Tidal Waves)
One of the earliest proposals interpreting the cerebellum in terms of its anatomical features was the timing hypothesis, which observed that granule cell “activity will reach different Purkinje cell arbors at different times” (Braitenberg and Atwood 1958). The next instantiation of this idea specified that parallel fiber propagation times would directly reflect the order of muscle activations (Braitenberg 1965). The proponent himself later deemed this unlikely, given that even if the parallel fibers are very long axons with very slow conduction velocities (~0.5 m/s), the propagation is of the order of 10 ms, much shorter than the 50 plus ms necessary for even the shortest of movements. This led to the “Tidal wave ” hypothesis (Braitenberg 1983, 1987), which shifted the emphasis from the propagation delays of individual parallel fibers into temporal summation along the parallel fiber bundles in the folia (Fig. 2).
Against empirical tests, the proposal met with conflicting results. While some did observe sequential activity between Purkinje neurons arranged along a parallel fiber (Ebner and Bloedel 1981; Eccles et al. 1966), others did not encounter them, unless inhibitory interneuron influence on Purkinje neurons was blocked. A parsimonious conclusion is that temporal summation along parallel fiber bundles indeed exists, albeit gated by inhibitory interneuron intervention (Bower 2010).
Reverberating Loops and Golgi Gating
Due to anatomically closed loops, components of the extended olivocerebellar circuit influence themselves, with delays (Apps and Hawkes 2009; Ekerot et al. 1987; Jörntell et al. 2010; Voogd et al. 2003, 2010). For example, a pause in Purkinje cell spiking, whether from an inhibitory interneuron or from the complex spike, feeds back on itself through two routes (mesodiencephalic junction as well as the cerebellar nuclei) which re-converge onto the same inferior olivary nucleus, which ultimately projects back to Purkinje neurons in the same microzone (Apps and Hawkes 2009). The activity of the cerebellar cortex thus influences itself in a time-delayed fashion. This observation has been spelled out with attention to detail in Kistler’s “reverberating loops” model (De Zeeuw et al. 2000; Kistler and van Hemmen 1999), which identifies two reverberating loops within the cerebellar system, one due to rebound firing of the cerebellar nuclear cells (B. D. Armstrong and Harvey 1966; De Gruijl et al. 2013; Ruigrok and Voogd 1995) creating delayed responses on the IO and the other due to Golgi cell oscillatory gating of granule cell activity.
Golgi cell’s low frequency of oscillation, broad receptive fields (Prsa et al. 2009; Vos et al. 1999), gap junctioned network (Dieudonne 1998; Dugué et al. 2009), and relatively limited dynamic range in response to input have led many to propose its role as an oscillatory gating of the granule cells (Kistler and van Hemmen 1999; Maex and De Schutter 1998; Solinas et al. 2010; Vervaeke et al. 2012). Kistler has further proposed that this oscillatory gating would generate reverberatory loops in the cerebellar system within the 100 ms scale, and this discretization step would be essential to production of motor sequences. An abrupt signal into the granule layer would lead Golgi cells to transiently and synchronously oscillate, gating their own input by inhibiting surrounding granule cells. Glutamate spillover depression of inhibition most likely promotes a winner takes all scenario that is compatible with all the theories mentioned here (Hull and Regehr 2012; Mitchell and Silver 2000).
Though sought for, the predicted ability of Golgi cells to create gated oscillations has not been conclusively confirmed. Nevertheless, some corroboration for the assumption may be derived from the frequency spectrum of the local field potential of the granular layer, which has been observed to correlate with the predicted bands.
The spatial properties of the putative Golgi gating spatial extent has been investigated in the model by Solinas (Solinas et al. 2007, 2010), which produces “center-surround” inhibition properties. It is worth noting that the shape of the center surround in this model is determined by the range and shape of the Golgi cell axonal arbor, which is a crucial assumption of the model. This overlooks, for instance, the fact that the axonal arbor is not round or cylindrical, rather flat and constrained by zebrin bands (Hawkes et al. 2008).
The second reverberating loop was proposed by Kistler and involves the delayed reverberation from the cerebellar nucleus. In this idea, a complex spike on the Purkinje neuron lifts inhibition from the cerebellar nucleus, which produces a set of rebound spikes (D. M. Armstrong and Harvey 1968; Witter et al. 2013), which in turn reset oscillations on the IO, and 100 ms later, rebound IO spikes, subsequently translated to complex spikes in the cerebellum, potentially closing the loop. This is an attractive idea because it implies temporal binding of cerebellar actions on the motor system, within a plausible window for motor control. Several caveats should be mentioned. The complex spike pause is highly variable in duration and comprises only a small fraction of the pauses in Purkinje neuron spiking (Warnaar et al. 2015); these other pauses can also be synchronized among Purkinje neurons (Shin and De Schutter 2006). Moreover the presence of rebound spikes in DCN neurons is specific to certain classes of neurons (Najac and Raman 2015), though other forms of phase-locking between Purkinje neuron and DCN neuron activity have been described recently (Person and Raman 2012). The latter, together with recent optogenetic studies confirming the functioning of the Purkinje neuron–DCN–IO loop (Chaumont et al. 2013), suggests the possible presence of more complex forms of reverberating loops. The existence of multiple loops in the olivocerebellar system would seem essential, although it has not so far been tied with a necessary correlate.
Single Cell Models of the Granular Layer
The dynamical behavior of network models evidently depends on the dynamics of the component cells. The two main players in the granule cell layer have seen devoted modeling efforts. Network oscillations are often an emergent property of single cell oscillations. Golgi cell models have reproduced the single cell oscillations on the basis of a persistent sodium current and a slow potassium current (Solinas et al. 2007). Golgi cell model produces robust oscillations upon both phasic and tonic depolarizing input, which implies that synaptic input to Golgi cells inhibits those cells that excite it, effectively gating the granule cells to particular windows (Table 1).
Granule cell models have shown high reliability of mossy fiber signal transmission. A full-fledged compartmental model of the turtle granule cell with differential ion channel expressions shows dynamic models: a linear relationship between mossy fiber activity and granule cell firing (Gabbiani et al. 1994). A later model based on measurements from rodent cells emphasized resonant properties of this neuron (D’Angelo et al. 2001) and was extended by showing that the active sodium conductance sharpens the action potential being propagated in the ascending axon to an almost instantaneous signal transmission (<200 us) (Diwakar et al. 2009).
Inferior Olivary Models
Coupled Oscillators and Electrotonic Coupling
Weakly coupled oscillator s are able to maintain phase differences. Moreover, they can often be reset, which can be regarded as a short-term memory mechanism. The inferior olivary physiology supports these assumptions in spades, as IO cells have been shown to robustly oscillate (due to an interplay between calcium low threshold channels and calcium activated potassium channels) (Llinas and Yarom 1981; Manor et al. 1997). Crucially, these cells are robustly coupled electrotonically (De Zeeuw et al. 1997; Llinas et al. 1974), causing phase shifts in neighboring cells. Finally, both inhibitory and excitatory stimuli to the IO cells produces phase resets, according to phase of the oscillation (van der Giessen 2007). These three facts stand as compelling evidence that the IO acts as a network of coupled oscillators. Multiple models have examined these properties and their consequences in the network (Kazantsev et al. 2004; Latorre et al. 2013; Lefler et al. 2013; Schweighofer et al. 1999; Torben-Nielsen et al. 2012).
Not all olivary cells maintain robust oscillations, however (Bazzigaluppi and Bazzigali 2013; Bazzigaluppi et al. 2012). Intracellular recordings in vivo (but under anesthesia) have verified that around one third of cells robustly oscillate, with another third exhibiting transient oscillations, and a third of nonoscillating cells. This proportion varies considerably from subnucleus to subnucleus of the olive, indicating that robust oscillations may not be necessary for all olivary function. Interestingly, modeling has shown that even nonoscillating cells, when coupled, may engage a group of coupled cells in oscillations (Torben-Nielsen et al. 2012), suggesting that the combination of physiological properties of the cells in a group determines the group behavior.
Phase Resets and Synchronous Groups
Upon perturbations, olivary cells may produce a spike and reset, as a function of perturbation phase, sign, and intensity. This endows a group of coupled oscillators with the ability of maintaining phase differences, as a function of their correlated input (Jacobson et al. 2009; Kazantsev et al. 2004; Latorre et al. 2013; Torben-Nielsen et al. 2012). It is natural to interpret this property as underlying the temporal organization of muscle synergies. Synchronous groups have indeed been observed experimentally (Lang et al. 2006; Welsh et al. 1995), as and correlate with different phases of the activation of muscle ensembles. This could imply that simultaneous groups of Purkinje neurons would be activated in functional sequences, rather than exclusively synchronously.
Early oscillatory models of cerebellar function attribute to the complex spike a direct effect on the outcome of movements, rather than a teaching/error signal. This seems consistent with the outcome of experiments, which observe stark motor deficits from pharmacological, lesion, and genetic model experiments (Gao et al. 2012a; Llinas et al. 1975; Schonewille et al. 2011). Nevertheless, the complex spike has been shown to modify the response properties of Purkinje neuron simple spikes (Medina and Lisberger 2008), and so, it does not seem improbable that the two functions may coexist. And indeed, even features of the complex spike shape have also been shown to correlate with particular alterations of the Purkinje neuron simple spike responses (Rasmussen et al. 2013; Warnaar et al. 2015), such as the number of spikelets. The number of spikelets has been studied in a small network setting (De Gruijl 2012), from which the authors concluded that phase relationships and synchronicity between cells could explain the number of spikelets.
Echo State Machines
Still in the topic of dynamical models of the cerebellum, it has been suggested that the architecture of the cerebellum supports the encoding of transients, as in an echo-state machine (Jaeger 2003). Interestingly, this model has suggested that the cerebellar architecture may be interpreted as encoding the passage of time (Yamazaki and Tanaka 2007).
Cerebellar Nucleus
Most models that feature a cerebellar nucleus often assume a very simplified version of its fairly intricate architecture and physiology. Until recently, the cerebellar nuclei did not receive extensive attention (Uusisaari and De Schutter 2011). Detailed anatomical and physiological work has shown that the cerebellar nucleus has a nonnegligible set of cell types, about five, and that these cells differ in their innervation schemes from the cerebellar cortex (some receiving somatic inhibition from Purkinje neurons, some distal dendritic input), anatomical properties (cell sizes), response properties (some with fast responses and rebounds), while some have much broader responses. A further complication of the overall picture of the olivocerebellar system is that some of the cells send feedback projection to the cerebellar cortex (Ankri et al. 2015), effectively dispelling the assumption that the cerebellum is exclusively a feed-forward system.
From the perspective of cerebellar nuclei output, two cells are of particular interest. The nucleo-olivary gabaergic projection cells and the glutamatergic projection cells have substantially different projection schemes and physiological responses (Houck and Person 2014; Kalmbach et al. 2011; Najac and Raman 2015; Uusisaari and Knöpfel 2012), the latter being fast firing and producing rebounds and the former being a slowly modulating cell. Most models of the cerebellar nucleus tend to include only the former, although it was often assumed that the output was of the fast/rebounding type. And as there are differences in the projection cells, it is likely that circuit dynamics of the DCN still reserve discoveries in the interactions of the poorly described interneurons (Table 2).
Conclusion
Consilience: Meeting in the Middle
Our exposition focuses on cerebellar modeling from two perspectives, functional and dynamical, which should be ultimately complementary. Nevertheless, the interfaces between functional models and dynamical necessities of the circuit are not always evident. Though many layers of biophysical phenomena appear exquisitely adapted, the functional relevance of biological mechanisms is not always apparent. Physiologically accurate modeling does not assume we know what the cerebellum does, but it does assume that things exist for a reason, unless there are compelling reasons to think otherwise. Functional modeling produces meaningful motor behavior, at the expense of biological accuracy. Although all models are fated to exclude something, both types of models are effective testbeds for assumptions both biological and functional.
References
Albus JS (1971) A theory of cerebellar function. Math Biosci 10(1/2):25–61. doi:10.1016/j.gaitpost.2012.10.015
Angelaki DE, Klier EM, Snyder LH (2009) A vestibular sensation: probabilistic approaches to spatial perception. Neuron 64(4):448–461. https://doi.org/10.1016/j.neuron.2009.11.010
Ankri L, Husson Z, Pietrajtis K, Proville R, Léna C, Yarom Y et al (2015) A novel inhibitory nucleo-cortical circuit controls cerebellar Golgi cell activity. eLife 4. doi:10.7554/eLife.06262
Antunes G, De Schutter E (2012) A stochastic signaling network mediates the probabilistic induction of cerebellar long-term depression. J Neurosci 32(27):9288–9300. https://doi.org/10.1523/JNEUROSCI.5976-11.2012
Anwar H, Hong S, De Schutter E (2012) Controlling Ca2+-activated K+ channels with models of Ca2+ buffering in Purkinje cells. Cerebellum (London, UK) 11(3):681–693. https://doi.org/10.1007/s12311-010-0224-3
Anwar H, Hepburn I, Nedelescu H, Chen W, De Schutter E (2013) Stochastic calcium mechanisms cause dendritic calcium spike variability. J Neurosci 33(40):15848–15867. https://doi.org/10.1523/JNEUROSCI.1722-13.2013
Apps R, Hawkes R (2009) Cerebellar cortical organization: a one-map hypothesis. Nat Rev Neurosci 10(9):670–681. https://doi.org/10.1038/nrn2698
Armstrong BD, Harvey RJ (1966) Responses in the inferior olive to stimulation of the cerebellar and cerebral cortices in the cat. J Physiol 187(3):553–574
Armstrong DM, Harvey RJ (1968) Responses to a spino-olivo-cerebellar pathway in the cat. J Physiol 194(1):147–168
Barbour B (1993) Synaptic currents evoked in Purkinje cells by stimulating individual granule cells. Neuron 11(4):759–769
Barbour B, Brunel N, Hakim V, Nadal J-P (2007) What can we learn from synaptic weight distributions? Trends Neurosci 30(12):622–629. https://doi.org/10.1016/j.tins.2007.09.005
Bazzigaluppi P, Bazzigali (2013) The inferior olive: coupling, oscillations and bursting activity. Erasmus MC, Rotterdam
Bazzigaluppi P, Ruigrok TJ, Saisan P, de Jeu MTG, De Zeeuw CI (2012) Properties of the nucleo-olivary pathway: an in vivo whole-cell patch clamp study. PLoS One 7(9):e46360. https://doi.org/10.1371/journal.pone.0046360.t003
Billings G, Piasini E, Lorincz A, Nusser Z, Silver RA (2014) Network structure within the cerebellar input layer enables lossless sparse encoding. Neuron 83(4):960–974. https://doi.org/10.1016/j.neuron.2014.07.020
Bloedel JR, Roberts WJ (1971) Action of climbing fibers in cerebellar cortex of the cat. J Neurophysiol 34(1):17–31
Blot A, Barbour B (2013) Analysis of the study of the cerebellar pinceau by Korn and Axelrad. Biorxiv preprint server http://doi.org/10.1101/001123
Blot A, Barbour B (2014) Ultra-rapid axon-axon ephaptic inhibition of cerebellar Purkinje cells by the pinceau. Nat Neurosci 17(2):289–295. https://doi.org/10.1038/nn.3624
Bower J (2010) Model-founded explorations of the roles of molecular layer inhibition in regulating Purkinje cell responses in cerebellar cortex: more trouble for the beam hypothesis. Front Neurosci 4:1–37
Braitenberg V (1965) A note on the control of voluntary movements. In: Proceedings from cybernetics of neural processes, Napoli, pp 1–8
Braitenberg V (1967) Is the cerebellar cortex a biological clock in the millisecond range? Prog Brain Res, 1–15
Braitenberg V (1983) The cerebellum revisited. J Theoret Neurobiol 2:237–241
Braitenberg V (1987) The cerebellum and the physics of movement: some speculations. In: Cerebellum and neuronal plasticity. Plenum Press, New York, pp 193–208
Braitenberg V, Atwood R (1958) Morphological observations on the cerebellar cortex. J Comp Neurol 109(1):1
Braitenberg V, Heck D, Sultan F (1997) The detection and generation of sequences as a key to cerebellar function: experiments and theory. Behav Brain Sci 20(2):229–245; discussion 245–277
Casellato C, Antonietti A, Garrido JA, Carrillo RR, Luque NR, Ros E et al (2014) Adaptive robotic control driven by a versatile spiking cerebellar network. PLoS One 9(11):e112265. https://doi.org/10.1371/journal.pone.0112265
Chaumont J, Guyon N, Valera AM, Dugué GP, Popa D, Marcaggi P et al (2013) Clusters of cerebellar Purkinje cells control their afferent climbing fiber discharge. Proc Natl Acad Sci 110(40):16223–16228. https://doi.org/10.1073/pnas.1302310110
Clark SL (1939) Responses following electrical stimulation of the cerebellar cortex in the normal cat. J Neurosci 2(1):19–35
Clopath C, Nadal J-P, Brunel N (2012) Storage of correlated patterns in standard and bistable Purkinje cell models. PLoS Comput Biol 8(4):e1002448. https://doi.org/10.1371/journal.pcbi.1002448.g005
Clopath C, Badura A, De Zeeuw CI, Brunel N (2014) A cerebellar learning model of vestibulo-ocular reflex adaptation in wild-type and mutant mice. J Neurosci 34(21):7203–7215. https://doi.org/10.1523/JNEUROSCI.2791-13.2014
Coesmans M, Weber JT, De Zeeuw CI, Hansel C (2004) Bidirectional parallel fiber plasticity in the cerebellum under climbing fiber control. Neuron 44(4):691–700. https://doi.org/10.1016/j.neuron.2004.10.031
D’Angelo E, Nieus T, Maffei A, Armano S, Rossi P, Taglietti V et al (2001) Theta-frequency bursting and resonance in cerebellar granule cells: experimental evidence and modeling of a slow k+-dependent mechanism. J Neurosci 21(3):759–770
De Gruijl JR (2012) Timing and graded signals in the inferior olive. Erasmus MC, Rotterdam
De Gruijl JR, Bazzigaluppi P, de Jeu MTG, De Zeeuw CI (2012) Climbing fiber burst size and olivary sub-threshold oscillations in a network setting. PLoS Comput Biol 8(12):e1002814. https://doi.org/10.1371/journal.pcbi.1002814
De Gruijl JR, Bosman LWJ, De Zeeuw CI, de Jeu MTG (2013) Inferior olive: all ins and outs. In: Handbook of the cerebellum and cerebellar disorders. Springer Netherlands, Dordrecht, pp 1013–1058. https://doi.org/10.1007/978-94-007-1333-8_43
De Schutter E, Bower JM (1994a) An active membrane model of the cerebellar Purkinje cell II. Simulation of synaptic responses. J Neurophysiol 71(1):401–419
De Schutter E, Bower JM (1994b) An active membrane model of the cerebellar Purkinje cell. I. Simulation of current clamps in slice. J Neurophysiol 71(1):375–400
De Schutter E, Bower JM (1994c) Simulated responses of cerebellar Purkinje cells are independent of the dendritic location of granule cell synaptic inputs. Proc Natl Acad Sci U S A 91(11):4736–4740
De Schutter E, Steuber V (2009) Patterns and pauses in Purkinje cell simple spike trains: experiments, modeling and theory. Neuroscience 162(3):816–826. https://doi.org/10.1016/j.neuroscience.2009.02.040
de Solages C, Szapiro G, Brunel N, Hakim V, Isope P, Buisseret P et al (2008) High-frequency organization and synchrony of activity in the Purkinje cell layer of the cerebellum. Neuron 58(5):775–788. https://doi.org/10.1016/j.neuron.2008.05.008
De Zeeuw CI, Koekkoek SK, Wylie DR, Simpson JI (1997) Association between dendritic lamellar bodies and complex spike synchrony in the olivocerebellar system. J Neurophysiol 77(4):1747–1758
De Zeeuw CI, van Hemmen L, Kistler WM (2000) Time window control: a model for cerebellar function based on synchronization, reverberation, and time slicing. Prog Brain Res 124:275–297
Del Olmo MF, Cheeran B, Koch G, Rothwell JC (2007) Role of the cerebellum in externally paced rhythmic finger movements. J Neurophysiol 98(1):145–152. https://doi.org/10.1152/jn.01088.2006
Dieudonne S (1998) Submillisecond kinetics and low efficacy of parallel fibre-Golgi cell synaptic currents in the rat cerebellum. J Physiol 510(Pt 3):845–866
Diwakar S, Magistretti J, Goldfarb M, Naldi G, D’Angelo E (2009) Axonal Na+ channels ensure fast spike activation and back-propagation in cerebellar granule cells. J Neurophysiol 101(2):519–532. https://doi.org/10.1152/jn.90382.2008
Donchin O, Rabe K, Diedrichsen J, Lally N, Schoch B, Gizewski ER, Timmann D (2012) Cerebellar regions involved in adaptation to force field and visuomotor perturbation. J Neurophysiol 107(1):134–147. https://doi.org/10.1152/jn.00007.2011
Doya K (2002) Metalearning and neuromodulation. Neural Netw 15(4–6):495–506
Dugué GP, Brunel N, Hakim V, Schwartz E, Chat M, Lévesque M et al (2009) Electrical coupling mediates tunable low-frequency oscillations and resonance in the cerebellar Golgi cell network. Neuron 61(1):126–139. https://doi.org/10.1016/j.neuron.2008.11.028
Ebner TJ, Bloedel JR (1981) Temporal patterning in simple spike discharge of Purkinje cells and its relationship to climbing fiber activity. J Neurophysiol 45(5):933–947
Eccles JC, Llinas RR, Sasaki K (1966) Parallel fibre stimulation and the responses induced thereby in the Purkinje cells of the cerebellum. Exp Brain Res (Experimentelle Hirnforschung Expérimentation Cérébrale) 1(1):17–39. https://doi.org/10.1007/BF00235207
Eccles JC, Ito M, Szentágothai J (1967) The cerebellum as a neuronal machine. Springer, Berlin
Eccles JC, Sabah NH, Schmidt RF, Táboríková H (1972) Cutaneous mechanoreceptors influencing impulse discharges in cerebellar cortex. II. In Purkynĕ cells by mossy fiber input. Experimental Brain Research Experimentelle Hirnforschung Expérimentation Cérébrale 15(3):261–277
Edelman GM, Gally JA (2001) Degeneracy and complexity in biological systems. Proc Natl Acad Sci U S A 98(24):13763–13768. https://doi.org/10.1073/pnas.231499798
Ekerot CF, Gustavsson P, Oscarsson O, Schouenborg J (1987) Climbing fibres projecting to cat cerebellar anterior lobe activated by cutaneous A and C fibres. J Physiol 386:529–538
Fagg AH, Sitko N, Barto AG, Houk J (1997) A computational model of cerebellar learning for limb control. Mh. In: Neural Computation Meeting proceedings, NCM 1997
Frens MA, Mathoera AL, van der Steen J (2001) Floccular complex spike response to transparent retinal slip. Neuron 30(3):795–801
Gabbiani F, Midtgaard J, Knöpfel T (1994) Synaptic integration in a model of cerebellar granule cells. J Neurophysiol 72(2):999–1009
Gao Z, Todorov B, Barrett CF, van Dorp S, Ferrari MD, van den Maagdenberg AMJM et al (2012a) Cerebellar ataxia by enhanced CaV2.1 currents is alleviated by Ca2+-dependent K+-channel activators in Cacna1aS218L mutant mice. J Neurosci 32(44):15533–15546. https://doi.org/10.1523/JNEUROSCI.2454-12.2012
Gao Z, van Beugen BJ, De Zeeuw CI (2012b) Distributed synergistic plasticity and cerebellar learning. Nat Neurosci 13:619–635. doi:10.1038/nrn3312
Glickstein M, Sultan F, Voogd J (2011) Functional localization in the cerebellum. Cortex 47(1):59–80. https://doi.org/10.1016/j.cortex.2009.09.001
Han VZ, Grant K, Bell CC (2000) Reversible associative depression and nonassociative potentiation at a parallel fiber synapse. Neuron 27(3):611–622
Hansel C, Linden DJ, D’angelo E (2001) Beyond parallel fiber LTD: the diversity of synaptic and non-synaptic plasticity in the cerebellum. Nat Neurosci 4(5):467–475. https://doi.org/10.1038/87419
Hawkes R, Sillitoe RV, Chung SH, Fritschy JM, Hoy M (2008) Golgi cell dendrites are restricted by Purkinje cell stripe boundaries in the adult mouse cerebellar cortex. J Neurosci 28(11):2820–2826. https://doi.org/10.1523/JNEUROSCI.4145-07.2008
Heck D, Sultan F (2002) Cerebellar structure and function: making sense of parallel fibers. Hum Mov Sci 21(3):411–421
Herzfeld DJ, Kojima Y, Soetedjo R, Shadmehr R (2015) Encoding of action by the Purkinje cells of the cerebellum. Nature 526(7573):439–442. https://doi.org/10.1038/nature15693
Hoogland TM, De Gruijl JR, Witter L, Canto CB, De Zeeuw CI (2015) Role of synchronous activation of cerebellar Purkinje cell ensembles in multi-joint movement control. Curr Biol 25(9):1–10. https://doi.org/10.1016/j.cub.2015.03.009
Houck BD, Person AL (2014) Cerebellar loops: a review of the nucleocortical pathway. Cerebellum 13(3):378–385. https://doi.org/10.1007/s12311-013-0543-2
Houk J, Fagg A (2014) A computational model for cerebellar learning for limb control. 1–16
Hull C, Regehr WG (2012) Identification of an inhibitory circuit that regulates cerebellar Golgi cell activity. Neuron 73(1):149–158. https://doi.org/10.1016/j.neuron.2011.10.030
Ito M (1989) Long-term depression. Annu Rev Neurosci 12:85–102
Ito M, Kano M (1982) Long-lasting depression of parallel fiber-purkinje cell transmission induced by conjunctive stimulationo of parallel fibers and climbing fibers in the cerebellar cortex. Neurosci Lett 33:253–258
Jacobson GA, Lev I, Yarom Y, Cohen D (2009) Invariant phase structure of olivo-cerebellar oscillations and its putative role in temporal pattern generation. Proc Natl Acad Sci 106(9):3579–3584. https://doi.org/10.1073/pnas.0806661106
Jaeger D (2003) No parallel fiber volleys in the cerebellar cortex: evidence from cross-correlation analysis between Purkinje cells in a computer model and in recordings from anesthetized rats. J Comput Neurosci 14(3):311–327
Jirenhed DA, Hesslow G, Bengtsson F (2007) Acquisition, extinction, and reacquisition of a cerebellar cortical memory trace. J Neurosci 27(10):2493–2502. https://doi.org/10.1523/JNEUROSCI.4202-06.2007
Jörntell H, Bengtsson F, Schonewille M, De Zeeuw CI (2010) Cerebellar molecular layer interneurons – computational properties and roles in learning. Trends Neurosci 33(11):524–532. https://doi.org/10.1016/j.tins.2010.08.004
Kalmbach BE, Voicu H, Ohyama T, Mauk MD (2011) A subtraction mechanism of temporal coding in cerebellar cortex. J Neurosci 31(6):2025–2034. doi:10.1523/JNEUROSCI.4212-10.2011
Kazantsev VB, Nekorkin VI, Makarenko VI, Llinas RR (2003) Olivo-cerebellar cluster-based universal control system. Proc Natl Acad Sci U S A 100(22):13064–13068. https://doi.org/10.1073/pnas.1635110100
Kazantsev VB, Nekorkin VI, Makarenko VI, Llinas RR (2004) Self-referential phase reset based on inferior olive oscillator dynamics. Proc Natl Acad Sci U S A 101(52):18183–18188. https://doi.org/10.1073/pnas.0407900101
Khaliq ZM, Raman IM (2006) Relative contributions of axonal and somatic Na channels to action potential initiation in cerebellar Purkinje neurons. J Neurosci 26(7):1935–1944. https://doi.org/10.1523/JNEUROSCI.4664-05.2006
Khaliq ZM, Gouwens NW, Raman IM (2003) The contribution of resurgent sodium current to high-frequency firing in Purkinje neurons: an experimental and modeling study. J Neurosci 23(12):4899–4912
Kistler WM, van Hemmen L (1999) Delayed reverberation through time windows as a key to cerebellar function. Biol Cybern 81(5–6):373–380
Lang EJ, Llinas RR, Sugihara I (2006) Isochrony in the olivocerebellar system underlies complex spike synchrony. J Physiol 573(Pt 1):277–9– author reply 281–2. https://doi.org/10.1113/jphysiol.2006.571101
Latorre R, Aguirre C, Rabinowitch M, Varona P (2013) Transient dynamics and rhythm coordination of inferior olive spatio-temporal patterns. Front Neural Circuits 1–18. https://doi.org/10.3389/fncir.2013.00138/abstract
Lefler Y, Torben-Nielsen B, Yarom Y (2013) Oscillatory activity, phase differences, and phase resetting in the inferior olivary nucleus. Front Syst Neurosci 1–9. https://doi.org/10.3389/fnsys.2013.00022/abstract
Llinas RR, Yarom Y (1981) Properties and distribution of ionic conductances generating electroresponsiveness of mammalian inferior olivary neurones in vitro. J Physiol 315:569–584
Llinas RR, BAKER R, Sotelo C (1974) Electrotonic coupling between neurons in cat inferior olive. J Neurophysiol 37(3):560–571
Llinas RR, Walton K, Hillman DE, Sotelo C (1975) Inferior olive: its role in motor learing. Science (New York, NY) 190(4220):1230–1231. https://doi.org/10.1126/science.128123
Maex R, De Schutter E (1998) Synchronization of golgi and granule cell firing in a detailed network model of the cerebellar granule cell layer. J Neurophysiol 80(5):2521–2537
Manor Y, Rinzel J, Segev I (1997) Low-amplitude oscillations in the inferior olive: a model based on electrical coupling of neurons with heterogeneous channel densities. J Neurophysiol 77:2736
Marr D (1969) A theory of cerebellar cortex. J Physiol 202(2):437–470
Marshall SP, Lang EJ (2004) Inferior olive oscillations gate transmission of motor cortical activity to the cerebellum. J Neurosci 24(50):11356–11367. https://doi.org/10.1523/JNEUROSCI.3907-04.2004
Mathy A, Ho SSN, Davie JT, Duguid IC, Clark BA, Häusser M (2009) Encoding of oscillations by axonal bursts in inferior olive neurons. Neuron 62(3):388–399. https://doi.org/10.1016/j.neuron.2009.03.023
Mauk MD, Steinmetz JE, Thompson RF (1986) Classical conditioning using stimulation of the inferior olive as the unconditioned stimulus. Proc Natl Acad Sci U S A 83(14):5349–5353
McCulloch WS, Pitts W (1943) A logical calculus of the ideas immanent in nervous activity. Bull Math Biophys
Medina JF (2010) A recipe for bidirectional motor learning: using inhibition to cook plasticity in the vestibular nuclei. Neuron 68(4):607–609. https://doi.org/10.1016/j.neuron.2010.11.011
Medina JF, Lisberger SG (2008) Links from complex spikes to local plasticity and motor learning in the cerebellum of awake-behaving monkeys. Nat Neurosci 11(10):1185–1192. https://doi.org/10.1038/nn.2197
Mitchell SJ, Silver RA (2000) Glutamate spillover suppresses inhibition by activating presynaptic mGluRs. Nature 404(6777):498–502. https://doi.org/10.1038/35006649
Najac M, Raman IM (2015) Integration of Purkinje cell inhibition by cerebellar nucleo-olivary neurons. J Neurosci 35(2):544–549. https://doi.org/10.1523/JNEUROSCI.3583-14.2015
Neymotin SA, Lee H, Park E, Fenton AA, Lytton WW (2011) Emergence of physiological oscillation frequencies in a computer model of neocortex. Front Comput Neurosci 5:19. https://doi.org/10.3389/fncom.2011.00019
Ohyama T, Nores WL, Murphy M, Mauk MD (2003) What the cerebellum computes. Trends Neurosci 26(4):222–227
Optican LM, Robinson DA (1980) Cerebellar-dependent adaptive control of primate saccadic system. J Neurophysiol 44(6):1058–1076
Ostojic S, Szapiro G, Schwartz E, Barbour B, Brunel N, Hakim V (2015) Neuronal morphology generates high-frequency firing resonance. J Neurosci 35(18):7056–7068. https://doi.org/10.1523/JNEUROSCI.3924-14.2015
Pellionisz A, Llinas RR (1977) A computer model of cerebellar Purkinje cells. Neuroscience 2(1):37–48
Pellionisz A, Llinas RR, Perkel DH (1977) A computer model of the cerebellar cortex of the frog. Neuroscience 2(1):19–35
Person AL, Raman IM (2012) Purkinje neuron synchrony elicits time-locked spiking in the cerebellar nuclei. Nature 481(7382):502–505. https://doi.org/10.1038/nature10732
Phoka E, Cuntz H, Roth A, Häusser M (2010) A new approach for determining phase response curves reveals that Purkinje cells can act as perfect integrators. PLoS Comput Biol 6(4):e1000768. https://doi.org/10.1371/journal.pcbi.1000768
Porrill J, Dean P (2007) Cerebellar motor learning: when is cortical plasticity not enough? PLoS Comput Biol 3(10):e197. https://doi.org/10.1371/journal.pcbi.0030197
Prsa M, Dash S, Catz N, Dicke PW, Thier P (2009) Characteristics of responses of Golgi cells and mossy fibers to eye saccades and saccadic adaptation recorded from the posterior vermis of the cerebellum. J Neurosci 29(1):250–262. https://doi.org/10.1523/JNEUROSCI.4791-08.2009
Rasmussen A, Jirenhed DA, Zucca R, Johansson F, Svensson P, Hesslow G (2013) Number of spikes in climbing fibers determines the direction of cerebellar learning. J Neurosci 33(33):13436–13440. https://doi.org/10.1523/JNEUROSCI.1527-13.2013
Rosenblatt F (1957) The perceptron – a perceiving and recognizing automaton. Report 85-460-1. Cornell Aeronautical Laboratory
Rosenblatt F (1958) The Perceptron--a perceiving and recognizing automaton. Cornell Aeronautical Laboratory, Report 85-460-1
Ruigrok TJ, Voogd J (1995) Cerebellar influence on olivary excitability in the cat. Eur J Neurosci 7(4):679–693
Sauerbrei BA, Lubenov EV, Siapas AG (2015) Structured variability in Purkinje cell activity during locomotion. Neuron 87(4):840–852. https://doi.org/10.1016/j.neuron.2015.08.003
Schmahmann J, Caplan D (2006) Cognition, emotion and the cerebellum. Brain 129(Pt 2):290–292. https://doi.org/10.1093/brain/awh729
Schonewille M, Hoebeek FE, Belmeguenai A, Koekkoek SK, Houtman SH, Boele HJ et al (2010) Purkinje cell-specific knockout of the protein phosphatase PP2B impairs potentiation and cerebellar motor learning. Neuron 67(4):618–628. https://doi.org/10.1016/j.neuron.2010.07.009
Schonewille M, Gao Z, Boele H-J, Vinueza Veloz MF, Amerika WE, Šimek AAM et al (2011) Reevaluating the role of LTD in cerebellar motor learning. Neuron 70(1):43–50. https://doi.org/10.1016/j.neuron.2011.02.044
Schweighofer N, Doya K, Kawato M (1999) Electrophysiological properties of inferior olive neurons: a compartmental model. J Neurophysiol 82(2):804–817
Schweighofer N, Doya K, Fukai H, Chiron JV, Furukawa T, Kawato M (2004) Chaos may enhance information transmission in the inferior olive. Proc Natl Acad Sci U S A 101(13):4655–4660. https://doi.org/10.1073/pnas.0305966101
Shin SL, De Schutter E (2006) Dynamic synchronization of Purkinje cell simple spikes. J Neurophysiol 96(6):3485–3491. https://doi.org/10.1152/jn.00570.2006
Shin SL, Hoebeek FE, Schonewille M, De Zeeuw CI, Aertsen A, De Schutter E (2007) Regular patterns in cerebellar Purkinje cell simple spike trains. PLoS One 2(5):e485
Shinoda Y, Sugihara I, Wu HS, Sugiuchi Y (2000) The entire trajectory of single climbing and mossy fibers in the cerebellar nuclei and cortex. Prog Brain Res 124:173–186. https://doi.org/10.1016/S0079-6123(00)24015-6. Elsevier
Soetedjo R, Kojima Y, Fuchs AF (2008) Complex spike activity in the oculomotor vermis of the cerebellum: a vectorial error signal for saccade motor learning? J Neurophysiol 100(4):1949–1966. https://doi.org/10.1152/jn.90526.2008
Solinas S, Forti L, Cesana E, Mapelli J, De Schutter E, D’Angelo E (2007) Computational reconstruction of pacemaking and intrinsic electroresponsiveness in cerebellar Golgi cells. Front Cell Neurosci 1:2. https://doi.org/10.3389/neuro.03.002.2007
Solinas S, Nieus T, D’Angelo E (2010) A realistic large-scale model of the cerebellum granular layer predicts circuit spatio-temporal filtering properties. Front Cell Neurosci 4:12. https://doi.org/10.3389/fncel.2010.00012
Spencer RMC, Ivry RB (2009) Sequence learning is preserved in individuals with cerebellar degeneration when the movements are directly cued. J Cogn Neurosci 21(7):1302–1310. https://doi.org/10.1162/jocn.2009.21102
Steuber V, Mittmann W, Hoebeek FE, Silver RA, De Zeeuw CI, Häusser M, de Schutter E (2007) Cerebellar LTD and pattern recognition by Purkinje cells. Neuron 54(1):121–136. https://doi.org/10.1016/j.neuron.2007.03.015
Steuber V, Schultheiss NW, Silver RA, Schutter E, Jaeger D (2010) Determinants of synaptic integration and heterogeneity in rebound firing explored with data-driven models of deep cerebellar nucleus cells. J Comput Neurosci 30(3):633–658. https://doi.org/10.1007/s10827-010-0282-z
Sugihara I, Wu HS, Shinoda Y (2001) The entire trajectories of single olivocerebellar axons in the cerebellar cortex and their contribution to cerebellar compartmentalization. J Neurosci 21(19):7715–7723
Ten Brinke MM, Boele H-J, Spanke JK, Potters JW, Kornysheva K, Wulff P et al (2015) Evolving models of Pavlovian conditioning: cerebellar cortical dynamics in awake behaving mice. CellReports 13(9):1977–1988
Thier P, Dicke PW, Haas R, Barash S (2000) Encoding of movement time by populations of cerebellar Purkinje cells. Nature 405(6782):72–76. https://doi.org/10.1038/35011062
Torben-Nielsen B, Segev I, Yarom Y (2012) The generation of phase differences in a network model of the inferior olive subthreshold oscillations. PLoS Comput Biol 1–10. https://doi.org/10.1371/journal.pcbi.1002580
Uusisaari M, De Schutter E (2011) The mysterious microcircuitry of the cerebellar nuclei. J Physiol 589(14):3441–3457. https://doi.org/10.1113/jphysiol.2010.201582
Uusisaari MY, Knöpfel T (2012) Diversity of neuronal elements and circuitry in the cerebellar nuclei. Cerebellum (London, England) 11(2):420–421. https://doi.org/10.1007/s12311-011-0350-6
van der Giessen RS (2007) Role of electrotonic coupling in the olivocerebelar system. Erasmus MC, Rotterdam
Velarde MG, Nekorkin VI, Kazantsev VB, Makarenko VI, Llinas RR (2002) Modeling inferior olive neuron dynamics. Neural Netw 15(1):5–10
Vervaeke K, Nusser Z, Silver RA (2010) Rapid desynchronization of an electrically coupled interneuron network with sparse excitatory synaptic input. Neuron 67(3):435–451. https://doi.org/10.1016/j.neuron.2010.06.028
Vervaeke K, Lorincz A, Nusser Z, Silver RA (2012) Gap junctions compensate for sublinear dendritic integration in an inhibitory network. Science (New York, NY) 335(6076):1624–1628. https://doi.org/10.1126/science.1215101
Voogd J, Pardoe J, Ruigrok TJ, Apps R (2003) The distribution of climbing and mossy fiber collateral branches from the copula pyramidis and the paramedian lobule: congruence of climbing fiber cortical zones and the pattern of zebrin banding within the rat cerebellum. J Neurosci 23(11):4645–4656
Voogd J, De Zeeuw CI, Schraa-Tam CKL, Geest JN (2010) Visuomotor cerebellum in human and nonhuman primates. Cerebellum (London, England) 11(2):392–410. https://doi.org/10.1007/s12311-010-0204-7
Vos BP, Volny-Luraghi A, De Schutter E (1999) Cerebellar Golgi cells in the rat: receptive fields and timing of responses to facial stimulation. Eur J Neurosci 11(8):2621–2634
Warnaar P, Couto J, Negrello M, Junker M, Smilgin A, Ignashchenkova A et al (2015) Duration of Purkinje cell complex spikes increases with their firing frequency. Front Cell Neurosci 9:1–30. https://doi.org/10.3389/fncel.2015.00122
Welsh JP, Lang EJ, Suglhara I, Llinas RR (1995) Dynamic organization of motor control within the olivocerebellar system. Nature 374(6521):453–457. https://doi.org/10.1038/374453a0
Witter L, Canto CB, Hoogland TM, De Gruijl JR, De Zeeuw CI (2013) Strength and timing of motor responses mediated by rebound firing in the cerebellar nuclei after Purkinje cell activation. Front Neural Circuits 7:133. https://doi.org/10.3389/fncir.2013.00133
Wolpert DM, Kawato M (1998) Multiple paired forward and inverse models for motor control. Neural Netw 11(7–8):1317–1329
Wulff P, Schonewille M, Renzi M, Viltono L, Sassoè-Pognetto M, Badura A et al (2009) Synaptic inhibition of Purkinje cells mediates consolidation of vestibulo-cerebellar motor learning. Nat Neurosci 12(8):1042–1049. https://doi.org/10.1038/nn.2348
Yamazaki T, Tanaka S (2007) The cerebellum as a liquid state machine. Neural Netw 20(3):290–297. https://doi.org/10.1016/j.neunet.2007.04.004
Zhou H, Lin Z, Voges K, Ju C, Gao Z, Bosman LW et al (2014) Cerebellar modules operate at different frequencies. eLife 3:e02536. https://doi.org/10.7554/eLife.02536
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Negrello, M., De Schutter, E. (2022). Models of the Cortico-cerebellar System. In: Pfaff, D.W., Volkow, N.D., Rubenstein, J.L. (eds) Neuroscience in the 21st Century. Springer, Cham. https://doi.org/10.1007/978-3-030-88832-9_171
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