Post-publication activity

Figure 1: Drawing of the brain at five levels of evolution. Note the increased size of the cerebellum (red).

The cerebellum derives its name as a diminutive of the word "cerebrum". This is particularly explicit in German, where the cerebellum is called Kleinhirn ("small brain"). This structure, present in all vertebrates, occupies a position immediately behind the tectal plate and straddles the midline as a bridge over the fourth ventricle. In addition, it is the only region of the nervous system to span the midline without interruption. Technically, the output of the cerebellum is exclusively inhibitory through the Purkinje neurons onto the cerebellar nuclei, but the cerebellar nucleus exerts both excitatory and inhibitory influences, on the thalamus and on the Inferior Olive, respectively (Ruigrok and Voogd, 1995).

The cerebellum has undergone enormous elaboration throughout phylogeny, in fact, more so than any other region of the central nervous system (CNS), but has maintained its initial neuronal structure, almost invariant. Thus, its size but not its wiring has changed with CNS evolution. As an example, the cerebellar cortex in a frog has an area approximately 12 mm² that is, 4 mm wide (in the mediolateral direction) and 3 mm long (in the rostrocaudal direction). In humans, the cerebellar cortex is a single continuous sheet with an area of 500 cm² (1,000 mm long and 50 mm wide) . This is 4 x \(10^3\) times more extensive than that of a frog (Braitenberg & Atwood, 1958). The increase in cortical extent has resulted in folding into very deep folia (Figure 2), allowing this enormous surface to be packed into a volume of 6 cm x 5 cm x 10 cm. Because the cerebellar cortex extends mainly rostro-caudally, most of the foldings occur in that direction.

Figure 2: Comparison of the surfaces of unfolded mammalian cerebella, from (Sultan and Braitenberg, 1993). Outlines of the shapes of the cerebellar cortices. These were obtained by connecting the ends of the most prominent folia. The scale is the same in the latera-lateral and anterior-posterior direction. The numbers indicate I: mouse: 2: bat; 3: flying fox: 4: guinea pig; 5: rabbit: 6: pigeon; 7: hare; 8: chinchilla; 9: squirrel: 10: dog; I I: cat; 12: macaque; 13: sheep: 14: human; 15: bovine. In the upper part of the figure the smaller cerebella are shown at a higher magnification.

Function

Fundamentally the cerebellar system (including the inferior olive and the cerebellar nuclei) acquires and implements sensorimotor “tactics” (how to) that contextualize the motor strategies (what, where and when) generated by the forebrain. Such tactics relate mostly to movement execution, timing, compensatory, and multi-limb coordination.

The basic transactions are implemented at the cerebellar nuclear level the output of which blends strategy and tactics by combining incoming ascending spinal cord information with descending motor commands initiated by intrinsic CNS activity and sensory input. Thus cerebellar function must be considered within the context of the rest of the nervous system, since it is not a primary way station for sensory or motor function, but combines both, during learning and execution. Though its destruction does not produce sensory deficits or paralysis, cerebellar lesion produces devastating inability for movement coordination and execution, and in some accounts, deficits in executive function (see Cerebellar Involvement in Cognition). Common behavioral functions subservient on an intact cerebellum include many oculomotor models (saccadic adaptation, vestibulo-ocular reflex, optokinetic reflex) as well as the acquisition of well timed conditioned responses (eye blink conditioning), as well as motoric compensations (force-field adaptation, motor remapping), or rhythmic perception (finger-tapping tasks).

Cerebellar Involvement in Cognition

The anatomical projections to and from the lateral hemispheres of the cerebellum of higher mammals would seem to imply a constant and substantial exchange between cerebellar and cortical networks beyond the sensory and motor cortices, involving multiple parietal and prefrontal regions (Sultan et al., 2012). In primates, the largest expansion of the cerebellar system was in the lateral cerebellum, dentate nucleus and principal olive (Voogd, 2010), a region associated with tool use (Imamizu et al., 2000) and higher level cognitive functions (Schmahmann, 1996; Stoodley & Schmahmann, 2009a; 2009b; Stoodley, Valera, & Schmahmann, 2010). Lesions and other degenerative conditions in the cerebellum of humans lead to impairments beyond the realm of motor function. The effects are felt in executive function, spatial cognition (Stoodley & Schmahmann, 2009a), verbal working memory (Cattaneo et al., 2011). These functions are consistent with the anatomical projections of the medial aspect of the lateral and posterior cerebellum (Sultan et al., 2012). Nevertheless, the dissociation of motor and cognitive components of behavior has proven challenging, for example, because of contamination of the signal through eye movements, which has led to debate about the actual contributions of the cerebellum to cognition (Glickstein, Sultan, & Voogd, 2011). Though still controversial, there maybe scientific value in studying the potential contributions of the cerebellum for cognition. One argument derives from Sherrington's observation that the origin of agency is subservient on the emergence of movement, and hence, cognition and motor function are inextricably linked as 'thought is the internalization of movement' (Sherrington, 1941).

Structure

Overview

The cerebellum is part of a system that comprises four main components:

The largest component is the cerebellar cortex, a tightly folded layer of neural tissue attached to the dorsal side of the pons, with white matter underneath.
A set of cerebellar nuclei, lying within the white matter, underneath the cortex.
The inferior olivary nucleus, a convoluted structure lying within the medulla oblongata, and sending projections known as climbing fibers to the cerebellar cortex and cerebellar nuclei.
A large collection of inputs known as mossy fibers, impinging on the cerebellum from various brain regions and sending projections to the cerebellar cortex and cerebellar nuclei.

This system structure is common to all vertebrates except some types of fish, which lack a discrete set of cerebellar nuclei but have cells with similar connectivity distributed within the deep layers of the cerebellar cortex.

Cerebellar cortex anatomy

The cerebellar cortex is organized in three layers (Figure 3):

The granular layer (GL)
Where Mossy Fiber afferents arrive and contact granule cells and Golgi cells (and in the flocculus, the unipolar brush cells). Bellow the granular layer is the white matter, formed by the input and output nerve-fiber systems of this cortex.
The Purkinje cell layer (PC)
A flat sheet, containing the Purkinje cell somata, lying above the granule cell layer.
The Molecular Layer (ML)
The level peripheral to the Purkinje cell layer, containing the molecular layer interneurons (Baskets and Stellate cells), purkinje cell dendritic arbors and parallel fibers.

The cerebellar cortex is one of the least variable CNS structure with respect to its neuronal elements (Raymon y Cajal, 1904; Palay & Chan-Palay, 1974). The basic neuronal connectivity, present in all vertebrates, is composed of the Purkinje cell, the single output system of the cortex, and two inputs:

a monosynaptic input to the Purkinje cell, the climbing fiber, and
a disynaptic input, the mossy fiber granule cell-Purkinje cell system.

Concerning the cerebellar Purkinje cells (the largest neurons in the brain) they are the sole link between the cerebellar cortex and the cerebellar nuclei and its output is entirely inhibitory (Ito et al., 1964).

Figure 3: Light microscopy image of a young kitten (silver stain) showing the Purkinje cell layer, PC (a single row), the molecular layer, ML and the granule cell layer (GL). The dark layer bellow is the white matter containing efferent and afferent fibers to the cerebellar cortex. (S. Ramon y Cajal, unpublished image).

Neuronal types

Five types of neurons inhabit the cerebellar cortex, with a sixth type being found primarily in the floccular lobe and the vermis (the unipolar brush cell). Four are inhibitory (Purkinje, basket, stellate and Golgi cells) and two are excitatory (granule cells, unipolar brush cells) .

Granule cells: are the most numerous neurons in the vertebrate CNS (about 50 billion in humans). They have small somata, in humans around 5-8µm in diameter, with dendritic trees smaller than 39µm, composing about one third of the cerebellar mouse. They send an ascending axon through the cerebellar cortex which bifurcates at the molecular layer, giving rise to the parallel fibers.
Purkinje cell: are the largest neuron in the vertebrate CNS (Figure 4, PC). Their dendrites are flat (isoplanar) and stack in the cortex like pressed leaves. They receive, in humans, as many as two hundred thousand synapses. Purkinje cell axons provide the only output of the cerebellar cortex. Each cell has a large and extensive dendritic arborization, a single primary dendrite, a sphere-like soma (20-40 µm) and a long, slender axon that is myelinated when it leaves the granular layer. As the main Purkinje cell axon leaves the cortex, it gives off recurrent collaterals that ascend back through the granular layer above and below the Purkinje somata and ultimately form synapses with Golgi and basket cells.
Basket and Stellate cells: Are interneurons present in the molecular layer (Figure 4, ML). They are both inhibitory (GABAergic) on to Purkinje cells. Their axons run in the same direction as the dendrites of the Purkinje cells (see Figure 4, PC) are electrically coupled and receive both climbing fiber collaterals as well as parallel fibers originating in the granule layer.
1. Basket cells (Figure 4, BC) are found in the lower molecular layer. Their axons extend along the Purkinje cell layer at right angles to the direction of the parallel fibers. They may spread over a distance equal to 20 Purkinje cell widths and 6 deep and may contact as many as 150 Purkinje cell bodies. During its course, the horizontal segment of a basket cell axon sends off groups of collaterals that descend and embrace the Purkinje cell soma and initial segment. As many as 50 different basket cells are thought to wrap their axon terminals around each Purkinje cell soma, forming a basket-like meshwork resembling that on an old Chianti bottle (Hamori & Szentagothai, 1966). Basket cell axons also ascend to contact the Purkinje cell dendritic tree. There are about six times as many basket cells as Purkinje cells.
2. Stellate cells (Figure 4, SC) are generally found in the outer two-thirds of the molecular layer. The smallest stellate cells, in the most superficial regions of the molecular layer, have 5 to 9 µm-diameter somata, a few radial dendrites, and a short axon (, SC). Deeper stellate cells are larger, have more elaborate dendritic arborizations that radiate in all directions, and have varicose axons that can extend parallel to the Purkinje cell dendritic plane as far as 450 pm. There are about 16 times as many small stellate cells as there are Purkinje cells.
Golgi cells (Figure 4, GC) There are two sizes of Golgi cells: (I) large ones (somata 9-16 µm in diameter), which are found mainly in the upper part of the granular cell layer, and (2) smaller ones (somata 6-l l µm in diameter), which are found in the lower half of the granular layer. They have extensive radial dendritic trees that extend through all layers of the cortex (Figure 3). They receive input from the parallel fibers in the molecular layer and from climbing and mossy fiber collaterals in the granular layer. Their axons branch repeatedly in the granular layer, where they terminate on granule cell dendrites in the cerebellar glomeruli. There are approximately as many Golgi cells as Purkinje cells.
Unipolar brush cells are found in profusion in the flocculonodular lobe, and are commonly associated with vestibular signals. Their name derives from the brush like dendritic arbor, which is spiny (like the mossy fibers), short and tuftly branched, establishing synapses in the glomerular structures within which they contact granule cells. UBC's are often thought to multiply and distribute the mossy fiber signals carrying vestibular information. Interestingly, the UBCs are also find in the vermis, linked to postural movements, and outside the cerebellum, in the dorsal cochlear nucleus.

Figure 4: Drawing of the two cerebellar afferent circuits and intrinsic neurons. The climbing fiber-Purkinje cell circuit. A fine branch of an axon from the inferior olivary nucleus (CF) climbs over the extensive arborization of the Purkinje Cell (PC) dendritic tree. The Purkinje cell is viewed in profile here since it is drawn from a coronal section of the cerebellar cortex. In the glomeruli, activity in the mossy fibers (MF) excites granule cells (GrC), whose axons project toward the surface of the cortex where they bifurcate to form parallel fibers (PF); these in turn pass through many Purkinje cell dendrites with which they form excitatory synapses. Two main types of intrinsic neurons are depicted: the Golgi cells (GC), with cell bodies just below the Purkinje cell layer; and the basket cells (BC), with cell bodies in the molecular layer. (Modified from (Ramon y Cajal, 1904).)

Cerebellar zones and microzones

In the 1960's converging lines of research discovered that in mammals the cerebellum is organized in longitudinal zones (reviewed in Voogd, 2010). These zones became designated Zebrin stripes due to the appearance of the cerebellum when stained with aldolase C (a glycolytic enzyme). The zonal pattern emerges early in development, before the migration of Purkinje neurons through the granular layer. This zonal pattern exists both for the efferent connections (mossy fiber system and olivary climbing fibers) and the afferent connections (cerebellocortico-nuclear projections, Apps & Hawkes, 2009; Voogd, Pardoe, Ruigrok, & Apps, 2003). Essentially, all the cerebellar systems, including the cerebellar nucleus and the inferior olive, seem to be organized according to the longitudinal zones of the cerebellar cortex. Furthermore, these projections seem to be congruent, i.e., mossy fiber and climbing fiber projections overlap (Voogd et al., 2003), further supporting the idea that these zones may be functionally distinct, especially considering that neurons seem to respect zonal boundaries (Golgi cells, Sillitoe et al., 2008), and that Purkinje cells from different zones have been observed to possess distinct baseline firing rates (Zhou et al., 2014). Other anatomical correlates of zones exist, as white matter boundaries and axonal calibers. Knowledge about zones advances our understanding of the interactions between cerebellum and cerebral cortex, as reviewed in (Glickstein et al., 2011).

Figure 5: Cerebellar longitudinal zones, and striped patterns as revealed by Zebrin (aldolase C) staining. (from: Voogd, 2003)

Figure 6: Cerebellar longitudinal zones, and striped patterns as revealed by Zebrin (aldolase C) staining. (from: Glickstein et al., 2009)

Receptive Fields

Fractured somatotopy of sensory projections

The response properties of Purkinjes to sensory stimuli have been mapped, and the picture that emerges is that of a mosaic. Along a folia, clusters of purkinje neurons will respond to similar stimuli, where there is no apparent topographical contiguity with the skin. There is no obvious sensory or motor homunculus, like in the cerebral cortex. This has been called "fractured somatotopy" (Shambes, et. al 1978). A coarse representation of the body seems to be symmetrical about the sagittal plane, with the head of the homunculus in the center and the lower limbs pointing outwards (see and ) a one to one relation directly to Purkinje Cells (PC, Figure 4 and ) and the mossy fibers afferents (MF, Figure 4 and ) a one to many system, represent opposite extremes of connectivity among afferents in the CNS.]]

Mossy Fiber-Parallel Fiber Pathway'

The second cerebellar afferents, the mossy fibers, originate from many CNS regions, to include the vestibular nerve and nuclei, spinal cord, reticular formation, and basilar pontine nuclei as part of the cortico-ponto-cerebellar pathway one of the most massive in the brain. Mossy fibers enter through all three cerebellar peduncles (inferior, middle, and superior) and send collaterals to the cerebellar nuclei before branching in the white matter and synapsing on the granule cells (Shinoda et al., 1992). Thus, unlike the climbing fibers, mossy fibers do not synapse directly on Purkinje cells but rather on the small granule cells lying directly below them (Figure 4, GrC). The synapses between mossy fibers and granule cells occur as the fine branches of the mossy fibers twine through the granular layer. The contacts are made as the mossy fiber enlarges and generates tight knottings along its length. These portions of contact are called mossy fiber rosettes. One mossy fiber may have 20-30 rosettes (see Figure 4). The mossy fibers go on to activate Purkinje cells via the parallel fibers described above.

Output: Cerebellar nuclei

There are three cerebellar nuclei (CN) on each side of the midline; each receives input via Purkinje cell axons (, PC) from the region of cortex directly above it and projects to specific brain regions. The most medial nucleus, the fastigial, receives input from the midline region of the cerebellar cortex, the vermis. It projects caudally to the pons, medulla, vestibular nuclei, and spinal cord and rostrally to the ventral thalamic nuclei. Lateral to the vermis are the newer parts of the cerebellar cortex, the paravermis, which projects to the interpositus nucleus (which itself is divided into anterior and posterior divisions), and the hemispheres, which project to the dentate nucleus. The latter two cerebellar nuclei project rostrally to the red nucleus and ventral thalamic nuclei and caudally to the pons, medulla, cervical spinal cord, and reticular formation. There is a pattern of innervation of the cerebellar nuclei within this broad radial organization whereby the rostrocaudal and mediolateral groups of Purkinje cell axons parcel each cerebellar nucleus into well-defined territories (Voogd et al., 1980), see also Figure .

Figure 7: Convergence of Purkinje neuron axons to cerebellar nucleus (courtesy of Hermina Nedelescu and Bernd Kuhn. GFP mouse provided by Sascha Du Lac).

Cell types

Classically, the cells of the cerebellar nuclei have been ascribed to two distinct categories, Excitatory (Glutamatergic) and inhibitory (Gabaergic) with a ratio of about one to one. Recently, a finer categorization has been proposed, which includes local neurons (Uusisaari and Knöpfel, 2012), and Glycinergic neurons (putatively providing feedback to the cerebellar cortex). Glutamatergic projection neurons reach different regions of the neuraxis (, ECN), while GABAergic neurons provide feedback to the inferior olive exclusively (, ICN). The cerebellar nuclei are not simply “throughput” stations; rather, the synaptic integration that takes place here is a fulcrum for cerebellum function. Indeed, it is here that information from the cerebellar cortex is integrated with direct input from the mossy and climbing fibers. The fact that half of the cerebellar nuclear neurons are GABAergic and project to the inferior olive, as their only target, underlines the importance of decoupling at the inferior olive as a central control system for motricity (Llinas, 2009).

Convergence from Purkinje neurons

Estimates of convergence from Purkinje cells to DCN cells have ranged between 400:1 (Cat, Palay and Chan-Palay, 74), to 40:1 (in Mice, estimated in Person and Raman, 2012). This convergence ratio possibly includes both Glutamatergic and Gabaergic projection cells. It is worthy of mention that Glycinergic cells have been observed in the nuclei, whose projections seem to ascend back to the cerebellar cortex, although the targets have not yet been determined.

Dynamics and Electrophysiological circuit properties

There are three main neuronal circuits in the cerebellum: two in the cortex, which relate to the two afferent systems as described above, and two main circuits involving the cerebellar nuclei. These latter, as stated above, are excitatory, relating to connectivity with the rest of the brain. The other, a re-entrant inhibitory cerebellar system, terminates in the inferior olive exclusively.

Cerebellar input circuits

Mossy fiber circuit

The sequence of events that follows the stimulation of mossy fibers was first suggested by Janos Szentagothai at the Semmelweis University School of Medicine in Budapest: the stimulation of a small number of mossy fibers activates, through the granule cells and their parallel fibers, an extensive array of Purkinje cells and all three types of inhibitory interneurons (Eccles, Llinas, & Sasaki, 1966a). Subsequent interactions of the neurons tend to limit the extent and duration of the response. The activation of Purkinje cells through the parallel fibers is soon inhibited by the basket cells and the stellate cells, which are activated by the same parallel fibers (see Figure 4). Because the axons of the basket and stellate cells run at right angles to the parallel fibers, the inhibition is not confined to the activated Purkinje cells; those on each side of the beam or column of stimulated Purkinje cells are also subject to strong inhibition. The effect of the inhibitory neurons is therefore to sharpen the boundary and increase the contrast between those cells that have been activated and those that have not. At the same time, the parallel fibers and the mossy fibers activate the Golgi cells in the granular layer. The Golgi cells exert their inhibitory effect on the granule cells and thereby quench any further activity in the parallel fibers. This mechanism is one of negative feedback: through the Golgi cells, the parallel fiber extinguishes its own stimulus. The net result of these interactions is the brief firing of a relatively large but sharply defined population of Purkinje cells.

Climbing fiber circuit

In the normal adult cerebellum, a one-to-one relationship exists between a climbing fiber and a given Purkinje cell (i.e. each Purkinje cell receives one climbing fiber); however, each olivary axon branches to provide climbing fibers to approximately 10 Purkinje cells (Eccles, Llinas, & Sasaki, 1966b). The branching patterns of olivocerebellar axons are not random, but rather the branches of an individual axon predominantly remain within a relatively narrow plane that is aligned to the rostrocaudal axis (Sugihara et al., 2001) (see Figure 8). Moreover, neurons from the same region of the inferior olive tend to project to the same rostrocaudally oriented strip of cerebellar cortex (Voogd et al., 2003). Thus, the projection pattern of the olivocerebellar pathway divides the cerebellar cortex into a series of parasagittally-oriented zones. Interestingly, the projection pattern of the olivocerebellar pathway is largely in register with corticonuclear (Purkinje cell axons to cerebellar nuclei) and cerebellar nucleo-olivary projections, such that a series of reentrant loops are formed. For example, climbing fibers from the principal nucleus of the inferior olive project to the lateral part of the cerebellar hemisphere and also send collaterals to the dentate nucleus. In turn, the dentate is targeted by Purkinje cells of the lateral part of the hemisphere, and its GABAergic cells project back to the principal olivary nucleus. Although climbing fibers have Purkinje cells as their primary targets, they also activate other neurons of the cerebellar cortex. For example, they activate Golgi cells, which will inhibit the input through the mossy fibers (see Figure 4). Thus, when climbing fibers fire, their Purkinje cells are dominated by this input. The climbing fiber input to basket and stellate cells sharpens the area of activated Purkinje cells. An additional feature of the anatomy of the olivocerebellar system is of particular note with regard to its action on the cerebellum: olivary neurons are electrotonically coupled by gap junctions (Llinas et al., 1974; Sotelo et al., 1974; Llinas & Yarom, 1981). In fact, immunoflouresccnce and mRNA studies indicate that the inferior olive has one of the highest densities of connexin 36 (Belluardo, 2000 #14198; Condorelli, 1998 #14191), the protein from which neuronal gap junctions are usually formed (Rash et al., 2000). This electrotonic coupling is thought to allow olivary neurons to synchronize their activity. Interestingly, most of these gap junctions occur between dendritic spines that are part of complex synaptic arrangements known as glomeruli. Olivary glomeruli, in addition to the gap-junction-coupled dendritic spines, contain both excitatory and inhibitory presynaptic terminals, whose function is thought to be to control the efficacy of the electrotonic coupling (De Zeeuw,1990).

Figure 8: Climbing fiber synapses with Purkinje neurons end in the parasaggital plane.

Cerebellar Output

Cerebellar cortex-cerebellar nuclei circuit

Electrical activation of mossy fiber inputs to the cerebellar system generates an early excitation in the cerebellar nuclei because the collaterals terminate directly on the cerebellar nuclear cells (see Figure 4). The same information then proceeds to the cerebellar cortex, which in turn produces an early excitation of Purkinje cells to be translated into inhibition at the cerebellar nucleus. This inhibition is followed by a prolonged increase in excitability of the cerebellar nuclear cells. The increased excitability is the result of two actions: (1) disinhibition due to reduced Purkinje cell activity, which in turn results from the inhibitory action of basket and stellate cells after the initial activation of Purkinje cells, and (2) cerebellar nuclear cell intrinsic properties (see later). The Purkinje cell inhibition is also due indirectly to the inhibitory action of the Golgi interneuron, which, by preventing the mossy fiber input from reaching the molecular layer, reduces the excitatory drive to Purkinje cells. The cerebellar nuclear projection neurons themselves send axon collaterals to cortical inhibitory interneurons including basket cells, which thus provide recurrent inhibition of the cerebellar nuclear neurons, as seen in spinal motoneurons

As stated above the climbing fiber and the mossy fiber-granule cell-parallel fiber pathways are the two main types of afferents to the cerebellum, as a whole, and to the Purkinje cells in particular. These afferent systems differ dramatically in their interactions with the Purkinje cells. Thus the Purkinje cell and its climbing fiber afferent have a one-to-one relationship, whereas the relationship between the Purkinje cell and the mossy fiber-parallel fiber system can be characterized as many-to-many. Moreover, the directionality of the parallel fibers imparts a mediolateral orientation to Purkinje cell activation by the mossy fiber-parallel fiber system, whereas the climbing fiber system, as we shall see, is organized to produce synchronous activation of specific groupings of Purkinje cells, groupings that often have a rostrocaudal orientation. Their electrophysiological and anatomical differences lead to distinct functional roles for these two systems, which we discuss later.

Concerning the climbing fiber system, as a result of the electrotonic coupling between inferior olivary neurons and the topography of the olivocerebellar projection, this system generates synchronous (on a millisecond time scale) complex spike activity in rostrocaudal bands of Purkinje cells (). These bands are normally only about 250 µm wide in the mediolateral direction but can be several millimeters long in the rostrocaudal direction and may extend down the walls of the cerebellar folia and across several lobules . Thus, instead of providing the primary drive for activity in the olivocerebellar system, the main role of olivary afferents is to determine the pattern of “effective” electronic coupling between olivary neurons and thereby the distribution of synchronous complex spike activity across the cerebellar cortex. This idea is supported by results showing that spontaneous climbing spike activity persists following the block of glutamatergic and GABAergic input to the inferior olive (Lang, 2001, 2002).

The activity of the cerebellar nuclei thus is regulated in three ways;

by excitatory input from collaterals of the cerebellar afferent systems,
by inhibitory inputs from Purkinje cells activated over the mossy fiber pathways, and
by inputs from Purkinje cells activated by the climbing fiber system.

The effect of these inputs on cerebellar nuclear cells is shown by the intracellular recording of the response of these neurons to white matter stimulation (, A to E) at increasing levels of stimulating intensity. The stimulus activates a variety of axons that are running through the white matter, and as a result the response of these cells has five parts as shown in the . An initial EPSP due to antidromic activation of the mossy fiber collaterals to the nuclear cell followed by an IPSP (pink arrow), which results from direct excitation of Purkinje axons projecting to the nuclear cell and a second EPSP-IPSP sequence (red arrow) with a latency of 3-4.5 msec. These latter EPSP IPSP sequence results from climbing fiber collateral activation directly the cerebellar nuclear cells, and the IPSP is generated as a result of climbing fiber activation of Purkinje cells that in turn project onto the cerebellar nucleus. Finally, the second IPSP is terminated by a rebound response, which is due to the intrinsic membrane properties of the cerebellar nuclear cells themselves and a smaller IPSP (, red circle). Thus, the response in Fig. 4 is a combination of the properties of the synaptic circuit and the intrinsic properties of the Purkinje and cerebellar nuclear cells.

Thus the effect of synchronous olivocerebellar activity on the output of the cerebellar nuclei generates a punctate and rather powerful synaptic EPSP-IPSP sequences are often followed by a rebound depolarization followed by a return to hyperpolarization, as seen in (red dot). Thus if a sufficient number of inferior olivary neurons, having a common rhythmicity, are activated synchronously, a large and equally synchronous activation of Purkinje cells will occur. This in turn produces a large IPSP followed by a rebound burst response in the cerebellar nuclear cells. (, last set of superimposed recordings).

In contrast to the punctate nature of cerebellar activation by the olivocerebellar system, the mossy fiber-parallel fiber system provides a continuous and very delicate regulation of the excitability of the cerebellar nuclei, brought about by the tonic activation of simple spikes in Purkinje cells, which ultimately generates the fine control of movement known as motor coordination. The fact that the mossy fibers inform the cerebellar cortex of both ascending and descending messages to and from the motor centers in the spinal cord and brainstem gives us an idea of the ultimate role of the mossy fiber system: it informs the cortex of the place and rate of movement of limbs and puts the motor intentions generated by the brain into the context of the status of the body at the time the movement is to be executed. By contrast the very abrupt and power activation of Purkinje cells by the climbing fibers and the fact that the inferior olive has an intrinsic rhythmicity at 5 to 10 Hz suggest the climbing fiber input is more concerned with motor timing (Llinas, 2009, 2011).

Synaptic Plasticity

The most celebrated example of synaptic plasticity in the cerebellum is long term depression (LTD), which can be induced by conjunctive activation of parallel fibers and climbing fibers (Ito & Kano, 2012), a feature which figures centrally in many models of cerebellar function ever since David Marr theorized that this would be the only changeable synapse in the cerebellum (Marr, 1969; Albus, 1917). Marr suggested that plasticity between Purkinje and parallel fibers occurs through potentiation, guided by climbing fiber stimulus. Masao Ito (Ito & Kano, 1982) later shown the opposite: that it is conjunctive activation of parallel fibers and climbing fibers that lead to synaptic depression of parallel fibers, effectively lending support to James Albus's proposal (Albus, 1971), that depression was the effective mechanism of cerebellar plasticity.

Since then, diverse forms of plasticity have been observed between virtually all classes of cerebellar neurons (reviewed in Gao et al. 2012), under a variety of induction protocols, showing that the cerebellar circuitry is vividly modifiable. Lesions and knockouts of many plasticity mechanisms lead to motor deficits and motor learning impairments, in different degrees of severity (see for instance, Vinueza Veloz, 2014). In the case of LTD, experiments on an LTD deficient mutant mice show a mild impairment in oculomotor reflexes (a common experimental model of cerebellar learning) and eye blink conditioning and it indicates that other forms of plasticity are also at play. It is likely as well, that essential plasticity happens at many levels of the motor circuits.

Figure 9: Diverse forms of plasticity are observed in the cerebellar cortex and cerebellar nuclei (from Gao et al., 2012)

Models of the Cerebellum

The functional significance and computational capabilities of the cerebellum have been focus of intensely creative theoretical attention, spanning a broad range of levels of analysis and of abstraction. Models range from physiological mechanisms to the control of motor behavior, from motor learning and adaptation, to information encoding and capacity, from cellular mechanisms to electrophysiological properties of field potentials.

Some models emphasize the temporal parameters of the cerebellar processing, e.g., regarding the parallel fibers as delay lines along which signals accumulate (Braitenberg, Heck, & Sultan, 1997), while some view the oscillatory activity of the inferior olivary nucleus as providing the essential timing signals for the coordination of sensorimotor loops (Porras & Llinas, 2014). Other models consider the cerebellar cortex as a metric space-time tensor (Pellionisz & Llinas, 1979). Somewhat orthogonally, other models relate cerebellar activity to metabolic and energy budgets (Howarth, Gleeson, & Attwell, 2012; Howarth, Peppiatt-Wildman, & Attwell, 2009).

Though many models agree on the high-level functional role of the cerebellum in implementing sensorimotor transformations (either as metric tensors, or as sensori-motor associations), the multitude of models proposed meanwhile reflects some dissonance among theories and by extension, represents the state of our understanding of cerebellar function. It is conceivable, however, that some synthesis between the existing models awaits, or that some functions attributed to the cerebellum by models will only exist in the broader context of brain networks.

Figure 10: Abstract representation of the cerebellar structure by Braitenberg (1961).

Historical Notes

Historically, two broad classes of abstract models for cerebellar function have been proposed, one hinging on the plasticity of the PF-PN synapse, and its assumed supervised learning (via climbing fibers), while the other emphasizes the spatio-temporal processing capabilities of the cerebellar system, including PF propagation velocities and oscillatory dynamics of the olivo-cerebellar loop; functionally, the first class of models emphasizes learning of sensorimotor associations, while the other emphasizes on-the-fly control of muscle ensembles/synergies. Though many cerebellar researchers will side with one or another of these views, it is conceivable that the dichotomy they represent may be resolvable, as these mechanisms do not appear to be mutually exclusive, given that they operate in somewhat different time scales, one on-the-fly control (posture/reaching/manipulating) while the other in learning compensatory behavior (e.g., VOR/OKR/Saccadic Adaptation/Force-Field adaptation).

The cerebellum implements spatio-temporal transformations

As early as 1958, Braitenberg has proposed that the lattice structure of the cerebellum evinces transformations of spatial patterns into temporal patterns -- and vice versa (Braitenberg & Atwood, 1958) (Figure 10). A similar suggestion appears in 'cerebellum as a computer', Eccles, Ito and Szentagothai, who have highlighted its abilities to create spatio-temporal patterns, though they have not spelled out how these spatio-temporal patterns may look like, their analysis remaining at the level of the role of the components of the circuit (Eccles, Ito, & Szentágothai, 1967). Spatiotemporal patterns in the cerebellum have a great variety of origins -- in the delay line properties of parallel fibers, in the oscillatory resonances of the golgi-granule cell network (Solinas, Nieus, & D'Angelo, 2010; Vervaeke, Lorincz, Nusser, & Silver, 2012), in the coupled oscillations of the inferior olive (Llinas & Yarom, 1981), in the longitudinal distribution of climbing fiber afferents, in the organization of cerebellar Zebrin stripes(Sugihara, Marshall, & Lang, 2007; Sugihara, Wu, & Shinoda, 2001).

Symmetries of the cerebellum

Prima facie, the cellular arrangement in the cerebellum implies two symmetries in the propagation of activity, a mirror symmetry (across the midline) and a translational symmetry (along the parallel fibers) (Braitenberg, 1993), relations which have been proposed as underlying timing and sequencing functions (Braitenberg et al. 1997). The compelling convergence in cerebellar geometry would seem to support these views, although it has been contested (Bower, 2001). For instance, the planar structure of the Purkinje neuron would have the function to maximize temporal concentration of the summations of post synaptic potentials, while the sagittal divergence of climbing fibers creates synchronous bands of excitation/inhibition.

Despite the multitude of spatio-temporal patterns available to the cerebellar anatomy/physiology, few models currently existing that place emphasis on these large scale spatio-temporal interactions, in the context of known cerebellar function.

Figure 11: Topological unfoldings of the cerebellum.

Somewhat independently from the notion of spatio-temporal pattern transformations, later theories and associated models depart, rather than from the unique features of cerebellar anatomy, from the resemblance of purkinje neurons and the perceptron model of associative learning, as proposed by Marr (1969) or Albus (1971). Such models draw support from physiological findings, such as the plastic synapse between the PN and the PF, or the coupled oscillations of the inferior olive.

Top-Down and Bottom-Up models

Mauk and Medina have proposed the distinction between two broad classes of cerebellar models (Medina & Mauk, 2000). Bottom-up models, in which physiology takes the forefront, and top-down models, which focus on reproducing the presumed functions of the cerebellar system, while simplifying or abstracting away particularities of the known physiology and anatomy. Though this is a somewhat stiff classification system, it is acceptable as a guide for the discussion below.

Top-Down models will typically inspire themselves in linear systems theory, to explain motor control, and adaptive changes. Top-down models often are focused on learning of compensatory gains, as for example, in the Vestibulo-ocular Reflex, the optokinetic reflex, smooth pursuit or saccadic adaptation. Other commonly modeled paradigms are the eye-blink conditioning (Herreros & Verschure, 2013), or arm and joint control (Houk & Fagg, 2014; Neymotin, Lee, Park, Fenton, & Lytton, 2011).

Bottom-Up models depart from known anatomical and dynamical properties of the cerebellar networks and attempt to interpret the output in terms of functions they may execute. This category would include Braitenberg's suggestion that PNs flatness leads to maximization of temporal concentration, or that sequences of activated PFs may sum into a 'tidal wave', as well as many models of the inferior olive in oscillatory control.

Marr-Albus Models

In the majority of cerebellar models, the plastic synapse between the parallel fibers and Purkinje neurons figures centrally. The arrangement of PNs and PFs resembling a perceptron, led David Marr to propose that the role of the PN was to perform pattern separation. Modern members in this category are Adaptive Filter models (Dean & Porrill, 2011) and Forward Models (Wolpert & Kawato, 1998). The apparent similarity of the PN with a perceptron has also invited information theoretic approaches to quantify its encoding capacity (Clopath, Nadal, & Brunel, 2012; Barbour 2007). Learning models are commonly criticized on the emphasis on the PN-PF synapse and the neglect of the complex physiology and anatomy of the cerebellar system, including the oscillatory behavior of the inferior olive or rebound firing at the cerebellar nuclei. Notwithstanding, such models have however proved successful in robotic models of motor behavior (Caselato 2014; Houk 2014) .

Role of The Climbing Fiber

Climbing fiber as a teaching signal

One watershed of opinions that arises in cerebellar models lies on the interpretation of the climbing fiber signal. After Marr, Albus and Ito, it has become common to regard olivary spikes (complex spikes) as error signals, used for training the purkinje neuron parallel fiber synapse. On the other front, others regard climbing fibers as timing signals, related to the onset of movement. Both views have been systematically tested in experiments, and both views have shown fragility when confronted with results. Though LTD has been repeatedly observed, both in vivo and in vitro, the preponderance of PF-PN LTD for motor learning has been tested with LTD deficient mouse models, which have only mild learning impairments (Hansel, Linden, & D'angelo, 2001; Hoebeek et al., 2011). Other arguments against the functional preponderance of LTD include it's lack of temporal specificity of the presumed error in a system that is known to produce sophisticated, fast and precise motor timing. Critics also point out that all synapses in the cerebellum are highly modifiable (though they often fail to draw attention to the sheer numbers of PF-PN synapse, far outnumbering any other in the cerebellum). The perceptron analogy to the purkinje neuron, and hence the role of the climbing fiber as a teaching signal, has been criticized on the grounds that the perceptron it is mutatis mutandis applied to various other neuronal systems, such as the cerebral cortex, the hippocampus, or the basal ganglia, and is quite divorced from considerations of anatomy and physiology.

Climbing fiber as a timing signal

That the CF synapse could serve as a timing signal stands to reason, as this powerful synapse causes complex spikes which in turn cause synchronous pauses in multiple PNs (de Schutter & Steuber, 2009), preferentially in a parasagittal arrangement (parallel to the Purkinje Neuron arbor) (Lang, Llinas, & Sugihara, 2006; Lang, Sugihara, Welsh, & Llinas, 1999; Welsh, Lang, Sugihara, & Llinas, 1995). Pauses in PNs will lift inhibition and create a firing window in the CN, that under experimental conditions may also cause rebound firing in the CN (Steuber, Schultheiss, Silver, Schutter, & Jaeger, 2010; Witter, De Zeeuw, Canto, Hoogland, & De Gruijl, 2013). Though there are doubts whether the rebound firing is physiological or functional, it is clear that the release of the torrential inhibition during PNs pauses (post-complex spike or not) in the CN will create a well timed window for higher firing rates, also owing to the sharp profile of the GABAergic inhibitory synapse from Purkinje cell axons (Person & Raman, 2012). This disinhibition of the CN leads to inhibition in the inferior olive, which may create phase resets in populations of olivary cells (Kazantsev, Nekorkin, Makarenko, & Llinas, 2004; Lefler, Torben-Nielsen, & Yarom, 2013).

Particularly, the nucleo-olivary inhibition arrives on the glomeruli, where inferior olive cells have electrotonic synaptic transmission through gap junctions (De Zeeuw et al., 1998). Recently it has been shown that the inhibitory potentials from the nucleus can shunt these connections, potentially decoupling networks of olivary neurons (Lefler, Yarom, & Uusisaari, 2014).

Nevertheless, if the climbing fiber is crucial for timing of motor activities or well timed errors, peristimulus time histograms tend to show broad peaks (20-50ms) in response to either sensory stimulation or movement onset.

It seems likely that a synthesis between the different roles of the complex spike is possible. Particularly in bottom-up models, which show how functions may emerge from the network dynamics, though much work remains to be done in order to bring the existing models to common ground.

Spatio-temporal Tensors / Feedforward Models

Over the years, multiple authors suggested that the cerebellum generates predictions on the basis of a space-time metric tensors representing sensorimotor feedbacks, which incrementally translate desired postural state (in joint coordinates) into appropriate motor actions. This view is similar to the more recent suggestion that the cerebellum embodies feedforward models (e.g., Wolpert and Kawato, 1998). The latter family of models commonly assumes that the CF is a teaching signal, role which is debated by the former, which considers it a timing signal. Naturally, the function to be performed by the cerebellar system, that is, motor prediction, is similar in both cases, though the means thereto vary.

Oscillatory control

Many authors note that fine control of movement smooth and precise movement relies on a tight coordination act between the oscillatory activity of agonist and antagonist muscles. Braitenberg provides an interesting deconstruction of the movements for different kinds of bowing in the violin, e.g., staccato, martelé, and legato, detaché, in terms of the oscillatory components that would produce them (Braitenberg, 1987).

The inferior olive produces a robust low frequency rhythm (4-10 Hz), which may sharply phase lock with muscle contractions. Remarkably, the fastest possible frequency of muscular contraction is directly correlated with the maximum frequency of olivary oscillation (10 Hz tetanic rhythm). These facts support the idea that cerebellar activity is chiefly concerned with the production of rhythms related to motor activity, which may underlie the generation of smooth movement and sophisticated motor patterns (Braitenberg, 1987). Some models based on the oscillatory contractions based on the inferior olivary rhythm have been proposed.

Similar control schemes based on integration of coupled oscillators have been proposed in the literature, to control muscle sets and even to drive an underwater vehicle (e.g., Rokni & Sompolinsky, 2011; Porras & Llinas, 2014).

However, It is not yet clear how the oscillations of the inferior olive are integrated into motor behavior through the output of the cerebellum through the climbing fiber circuit. Many models of the cerebellum tend to simplify away the oscillatory nature of the complex spike production, focusing instead on its role as a teaching signal. It is likely however, that synchronous complex spikes create pauses in simple spikes, which by relieving cerebellar nuclear cells from inhibition, create excitatory projections downstream to the reticular nucleus and spinal cord, and upward, to the cerebral cortex, according to olivary rhythms.

Frequently Modeled Circuits and Components

The selection of relevant features to simulate/emulate is the essence of modeling. In this section we review a few models and components and the roles attributed to them.

Most models assume one or more of the following components:

Parallel Fiber - Purkinje neuron synapse & climbing fiber
The high convergence of parallel fibers on Purkinje cell suggests that some kind of pattern recognition may be take place at this synapse. This synapse is known to undergo both long term depression (LTD), in the presence of climbing fiber signals and long term potentiation (LTP), in its absence.
The powerful synapse from a single axon from the inferior olivary nucleus delivers the complex spike. The co-occurrence of PF EPSCs and CFs is known to cause LTD of parallel fiber synapses, and conversely, no CF results in potentiation. Thus, the CF may control the direction of plasticity (Albus, 1971; Marr, 1969).
Large number of granule cells
The immense number of granule cells and the few connections of each granule cell often led to the suggestion that the granular layer plays the role of pattern separation, which has been buttressed by recent studies on the coding capacities of the granule layer(Billings, Piasini, Lorincz, Nusser, & Silver, 2014).
Golgi cell oscillatory gating of the granular layer
Some models suggest that the negative feedback to Golgi cells promote oscillatory gating of mossy fiber activity (Solinas et al., 2010; Vervaeke et al., 2012). In vitro studies have verified that resonant frequencies tend to exist, particularly in the beta band.
Conduction delays of parallel fibers
Parallel fibers are unusually long, thin, unmyelianated axons, with low conduction speeds (.1m/s - .3m/s). This has led Eccles and Braitenberg to suggest that PFs could act as delay lines and PNs, would respond as integrators of coincidence, leading to Braitenberg's 'tidal wave hypothesis', in which mossy fibers activated in a linear sequence along the parallel fiber axis contribute to a wavefront that would either accumulate to activate Purkinje cells and inhibitory interneurons.
Oscillations of the olivary nuclei
The climbing fiber signal originates in a nucleus known to produce oscillatory activity in vivo, as a function of the cells being intrinsic oscillators, and heavily gap junctioned (De Gruijl, De Zeeuw, Paolo, & de Jeu, 2012; Latorre, Aguirre, Rabinowitch, & Varona, 2013; Torben-Nielsen, Segev, & Yarom, 2012). The robust oscillatory frequencies of the olive, and the ability of sensory input to reset them, has been proposed to be the primary timing mechanism for the control of multiple muscle ensembles for coordinated motor behavior. Moreover, there is the possibility that inhibitory stimuli to glomeruli structures may decouple synchronous networks.
Reverberatory feedback with the Cerebellar Nuclei
It has been shown in vivo and in vitro that the disinhibition of the cerebellar nucleus caused by the climbing fiber pause may trigger rebound activity in the nucleus (Witter et al., 2013). This has been hypothesized to gate the output signal of the cerebellum (Kistler & van Hemmen, 1999).
Multiple firing patterns of Purkinje neurons
Purkinje neurons have a complex distribution of ion channels in its membrane, with differential distributions between soma, smooth dendrites and spiny dendrites. PN is known to produce distinct types of firing and to be modulated in a variety of ways depending on channel kinetics. Particularly, it has been proposed that pauses in purkinje cells may be the most readily learned output pattern, through LTD (Steuber et al, 2011).

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Cerebellum

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