THE DETECTION AND GENERATION OF SEQUENCES AS A KEY TO CEREBELLAR FUNCTION. EXPERIMENTS AND THEORY

Keywords

Abstract

We will present here some experiments and observations aimed at an explanation of the peculiarities of cerebellar structure. Of course such an explanation will have recourse to general ideas about the function of the cerebellum in the global sensory-motor control of the organism, and may in turn contribute to a more precise definition of that function. Our main point, however, is that there is information in structure and that neuroanatomy may profitably be consulted when other kinds of evidence fail to provide decisive clues.

Since some computational schemes which were proposed as models of cerebellar function were embarrassingly close to similar models proposed for the cerebral cortex, we shall start with a comparison of the two cortices, with special emphasis on the differences.

Size and number of neurons

Compared to the "large brain", the "small brain" is small only in volume, not in the surface of its cortex and even less in its linear extension. The antero-posterior length of the flattened cerebellar cortex of man exceeds two meters, that of the cow measures three meters and even that of the mouse 4 cm. Compare this to the diameter of the (roughly circular) flattened cerebral cortex of one hemisphere, which is 0.3 m in man and 1 cm in the mouse. The surface of the cerebellar cortex is about the size of one hemisphere of the telencephalic cortex.

The number of neurons is of the same order of magnitude (1010) in both cortices.Since both, the surface and the number of neurons can be taken as a rough estimate of the information handling capacity of a cortical structure, we may conclude that the complexity of the task of the cerebellar cortex matches that of the telencephalic cortex.

Isotropic vs anisotropic connections

A peculiarity of the cerebellar cortex, known since the early Golgi studies (Ramn y Cajal, 1911) is the lattice-like arrangement of its elements in the cortical neuropil. The great majority of axons and dendrites in the molecular layer, if their course is projected onto the plane of the cortex, run either in the anteroposterior or in the laterolateral direction. The distinction of two orthogonal directions in the cerebellar cortex is especially impressive if the physiological characteristics of the axons are considered, since the only axons running in the transversal direction,the parallel fibers, are excitatory, while the axons running at right angles to them belong exclusively to inhibitory interneurons.

The only notable exception is the (inhibitory) Golgi cell. Its dendritic and axonal ramifications occupy roughly circular territories in the cortical plane. While the other neuronal elements of the cerebellar cortex either shift signals laterolaterally or inhibit activity to the front and to the back of them, Golgi cells are local operators. They seem to put a brake, then and there, onto the activity of other cerebellar neurons and of afferent fibers when it exceeds a critical level.

In contrast to the cerebellum, the weave of the cerebral cortex is isotropic. There are no preferred directions (barring exceptional situations at the borders between some areas) and there is no geometrical separation of excitatory and inhibitory fibers.

Lack of excitatory loops

There are no neuronal connections within the cerebellar cortex that could sustain positive feedback. The only intrinsic excitatory neurons, the granule cells, only contact inhibitory neurons (intrinsic: stellate and basket cells, efferent: Purkinje cells) and never other neurons of their own kind. Thus a necessary prerequisite for an explosive build up of excitatory activity is missing in the cerebellar cortex. Nor can external loops be excitatory, for the only output of the cerebellar cortex, provided by Purkinje cell axons, is inhibitory (unless one postulates a complicated scheme involving serial inhibition and spontaneously active elements).

This is in striking contrast to the situation in the telencephalic cortex. There, the kind of synapse prevailing statistically over all the other kinds is one connecting excitatory pyramidal cells to other excitatory pyramidal cells (Braitenberg and Schz, 1991). There is a sheer infinite number of excitatory loops preformed in the neuropil of the cerebral cortex, and this may well be the key to an explanation of that structure. (A side effect of this is, of course, the proneness of the cerebral cortex to lapse into epileptic fits.)

Local operation

Intracortical fibers of the cerebellum run a few millimeters in the laterolateral direction, somewhat less in the anteroposterior direction. There are no corticocortical fibers leaving the cerebellar cortex through the white substance in order to re-enter it in another place. Therefore the operation which the cerebellar cortex performs on its input must be strictly local, limited to a region a few millimeters in size. Or, to put it differently, the output of the cerebellar cortex at one point can only be influenced by the input a few millimeters away.

Again, this is in contrast with one of the most striking structural properties of the cerebral cortex. The anatomy of the telencephalic hemispheres strongly suggests a global operation, since from any point of the cortex there is a wealth of connections to nearby and distant parts of the cortex, including connections to the opposite hemisphere. This, by the way, is one of the reasons why the large brain is larger than the small brain: much of its bulk is due to the large number of cortico-cortical fibers in the white substance. The slim white substance of the cerebellar hemispheres contains only afferent and efferent fibers.

Lack of separation of the hemispheres

The cerebellar cortex is one of the very few parts of the nervous system which shows no discontinuity at the midline. The cortical neuropil is continued without interruption between the hemispheres, the interaction between two points on either side of the midline presumably being exactly the same as between two points on one side.

In contrast, the telencephalic cortex consists of two separate hemispheres, joined by the fibers of the Corpus callosum, but not by continuous neuropil.

Orientation of the folds in one direction only

The anisotropy of the cerebellar cortex manifests itself even macroscopically. The folds which become necessary when a large surface is to fit into a small cranium, in the cerebellum have a tendency to run in one direction only. The consequence of this is that distances between the histological elements are not distorted by the folding along the direction of the folds, in the direction of parallel fibers.

The situation is quite different in the telencephalic cortex, where the folds, in larger brains, seem to run in all directions indiscriminately, and distances between neighbouring neurons are thereby freely stretched or compressed.

Morphological commentary

If anatomy is viewed in the spirit of "computer architecture", we may rephrase the above statements in computational terms and thereby obtain first hints at a theory of cerebellar function.

The cerebellar cortex is essentially "feed forward". Patterns of activity in the input are transformed into patterns of active output fibers through different sets of internal neurons which do not involve intracortical excitatory recurrent loops. The most numerous kind of intereneurons, the granular cells simply shift signals through their axonal branches (the parallel fibers) from their origin in the input in opposite directions along the laterolateral coordinate of the cortex at a low and fairly constant speed. Thus output neurons will be relaying input signals which arrived at different times in the past in different places, the farther back in time, the farther away their origin.

This being presumably the most important aspect of input-output transformation in the cerebellar cortex, we may assign a merely ancillary role to the fibers oriented in the other, the anteroposterior direction. These fibers are inhibitory and simply suppress activity on either side of a particularly active "beam of parallel fibers".

This view implies that the one-dimensional beam is the computational unit in the cerebellar cortex. The neuronal equipment of a laterolateral strip of cortex, consisting of parallel fibers each contacting a row of Purkinje cells (and conversely, Purkinje cells collecting signals from granular cell situated at various distances) suggests that one-dimensional spatio-temporal patterns are the adequate input to the cerebellar cortex.

If the laterolateral beam is responsible for the cerebellar response to one spatio-temporal pattern, the anteroposterior extension of the cerebellar cortex must be related to the number of such beams, or to the number of patterns that can be distinguished.

Comparative study of the flattened cerebellar cortex. (Sultan, 1994a; Sultan and Braitenberg, 1993)

In the light of this the macroscopical shape of the (unfolded) cerebellar cortex becomes important. The width of the flattened cortex should be related to the length of the sequences which are elaborated there, whereas the antero-posterior extension may have something to do with the variety of patterns.

Sultan (1994a) compared the cerebella of fourteen species of mammals and of one bird. For each species he reconstructed the outline of the cerebellar cortex by counting the number of folia, measuring the length of the folia and their anteroposterior extension. Histological analysis was used when there were doubts about the continuity of the cortex, e.g between its vermal and hemispheric portions.

The results are illustrated in Fig. 1 and 2. Fig. 1 shows the folial pattern, for selected species (macaque, sheep, dog, bat). Each horizontal line represents the length of one folium; neighbouring lines represent neighbouring folia; interrupted lines represent two folia which are not continuous. In Fig. 2 the folial patterns of three large cerebella (a: bovine, b: human, c: ovine) and three small cerebella (not to scale, d: squirrel, e: rabbit, f: mouse) are transformed into outline drawings in which length and width are shown in the correct proportions.

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The simple inspection of these shapes does not lead to any intuitive interpretation. There is a common characteristics, valid for all mammals (not for the bird), namely a distinction of an anterior part in which the folial pattern is continuous between right and left, and a posterior region in which the cerebellar cortex is split in three parts, a median part (the posterior vermis) and two tailed extensions of the hemispheres. It is of course tempting to associate the anterior unsplit section with such computations as involve the whole body bilaterally, and the posterior paired extensions with paired and relatively independent parts of the body such as the extremities. The median posterior extension would then be associated with axial parts of the body which act in relative isolation from the limbs.

The comparison of the overall antero-posterior and latero-lateral extension of the cerebellar sheet in different animals is puzzling. If the width represents the duration of the longest sequences which the cerebellum can digest, the human cerebellum beats all the others in this respect (we have no cetaceans in our collection, which according to Riley, (1928) have particularly wide cerebella). The comparison of the human and bovine cerebellum indicates that the length of motor programs is not so much related to the bulk of the body but rather to their sophistication. But if the overall length of the cerebellum represents variety of motor programs, we cannot explain why the bovine cerebellum is the longest.

A quantitative analysis of the comparative data (Sultan and Braitenberg, 1993; Sultan, 1994a) with the methods of allometry (Jerison, 1973) does not yield much more insight. Like the weight of the brain as a whole, the surface of the cerebellar cortex, and (because of its fairly uniform thickness) its weight is roughly proportionate to the 2/3 power of the weight of the body (Fig. 3), confirming the parallel evolution of large and small brain already mentioned. The increase in surface of the cerebellar cortex with body size is mainly due to an increase in width for the smaller species, while in the larger species the length increases disproportionately (Fig. 4). In many of the allometric relationships investigated, the human cerebellum appears as a stray point, mainly because of its great width.

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One should like to know more about topographic and other maps on the surface of the cerebellum in order to make more sense of the comparative anatomy. This is a lacuna which makes itself felt in several contexts. The main obstacle to comprehensive mapping studies is the complicated folding of the cerebellar cortex, with only a small proportion of it available for investigation from the surface.

The basic unit of computation in the cerebellar cortex

Since we have already seen that the interaction of signals within the cerebellar cortex is limited to a region measuring only a few millimeters in either direction, it should be possible to give a complete description of the effects which the input signals have on each output neuron within that small region. The global operation could then be taken simply as the sum of these local effects. A complete description of course implies sufficient knowledge of the physiological properties of neurons and synapses, and it will be our next task to review the available information on this point.

Purkinje cells are the output elements of the cerebellar cortex. Their number in the human cerebellum is given as 15,000,000 (Braitenberg and Atwood, 1958; Mayhew, 1991), or 30,000,000 (Andersen et al. 1992), in the rat as about 300000 (Harvey and Napper, 1988; Mayhew, 1991). Their axons are inhibitory (Ito and Yohida, 1964; Ito et al. 1964; Ito and Yoshida, 1966; Ito et al. 1970) and terminate for the most part in the cerebellar nuclei and in the vestibular nuclei (Eccles et al. 1967a; Brodal, 1981), depending on their location in the cerebellum. Besides their terminations outside the cerebellar cortex, Purkinje cell axons have collaterals which terminate in neighbouring regions of the cerebellar cortex itself mainly on the perikarya of other Purkinje cells (Ramn y Cajal, 1911; Palay and Chan-Palay, 1974), but also on basket cells (Lemkey-Johnston and Larramendi, 1968; Leranth and Hmori, 1981).

The dendritic tree of the Purkinje cell is the seat of close to 200,000 synapses most of which (160,000 in the rat, Napper and Harvey, 1988) are localized on dendritic spines. The majority of these synapses are classified as excitatory both on eletronmicroscopic (Gray, 1961) and electrophysiological (Eccles et al., 1966) evidence. The vast majority of the excitatory synapses on the dendritic tree of Purkinje cells is provided by the axons of granular cells. The others stem from climbing fibers.

There are also inhibitory synapses on Purkinje cells, both on the dendritic tree and on the soma. Most of them come from two kinds of inhibitory interneurons, the stellate cells and the basket cells.

The very large number of synapses on the dendritic trees of Purkinje cells should be compared to the number of synapses on cortical pyramidal cells. Even the largest pyramidal cells don't quite come up to one tenth of the number of synapses on Purkinje cells. This is a feature which has never been explained in a satisfactory way. It is partly responsible for the ideas of those (Hounsgaard, 1989; Hounsgaard and Midtgaard, 1989; Ross et al. 1990; Cohen and Wu, 1990) who like to see the dendritic tree of Purkinje cells as a complex machine, itself capable of logical operations of the kind that are usually attributed to neuronal networks, not to single neurons.

A physiological peculiarity of Purkinje cells is their high firing rate under resting conditions, or rather, under conditions where no controlled stimulus is applied (Thach, 1970; Thach, 1972). Recording in monkeys during spontaneous movements, an average firing rate of about 40 spikes per second is seen to vary both upward and downward in relation to certain phases of the movement (Harvey et al. 1977; Fortier et al. 1989).

Another special feature is the "complex spike", a multiphasic event which involves the membrane potential of Purkinje cells in a standardized fashion. These complex spikes are clearly distinct from the ordinary, so-called simple spikes and occur with a much lower frequency of about 1 per second (Thach, 1972; Harvey et al. 1977).

Besides the spinal motoneuron, there is hardly a cell in the vertebrate nervous system that has been studied as extensively as the Purkinje cells from a biophysical angle (e.g. Llins and Sugimori, 1980; Hounsgaard and Midtgaard, 1988; Ghwiler and Llano, 1989; Hockberger et al. 1989; Konnerth et al. 1990; Regan, 1991; Llano et al. 1991; Torres-Aleman et al. 1992; Midtgaard et al. 1993). We shall not review these results here, for, as important as they undoubtedly are, we believe they are not essential to the argument we want to present.

Influences of other neurons onto Purkinje cells

It seems certain that complex spikes in Purkinje cells are the exclusive result of presynaptic climbing fiber activation (Colin et al. 1980; Ekerot and Oscarsson, 1981; Campbell et al. 1983). The special neuronal response may be due to the especially intimate anatomical relation between each Purkinje cell and the associated climbing fiber, but may also indicate a particular selection of membrane channels which are under the influence of the climbing fiber (Knpfel et al. 1991).

The other excitatory influence onto the Purkinje cell comes through the hundreds of thousands of contacts provided for each cell by the parallel fibers. There has been some discussion about the relative merits of the synapses which the ascending granular cell axons make with the Purkinje cells, and those made by the two branches, the parallel fibers proper (Llins, 1982; Bower and Woolston, 1983). Numerically the synapses on the parallel fibers prevail (Napper and Harvey, 1988; Sultan and Rotter, 1994). Moreover, in the case of parallel fibers that lie low in the molecular layer, the ascending branches connecting them with their parent granular cells cannot contact any Purkinje cells at all.

There is some anatomical evidence suggesting that parallel fibers situated low in the molecular layer are thicker than the ones above, and this corresponds also to some electrophysiological measurements (Nicholson and Llins, 1971; Heck, 1993, 1995c) showing different conduction velocities at different depths in the molecular layer. This may be important for a refinement of the theory which we will develop in this paper, but will be ignored at the present stage. This is justified by the observation that the global begavior of the mass of parallel fibers in the molecular layer is as if they all conducted spikes at the same velocity.

It is well known that granular cells in turn receive their excitatory input (mainly, if not exclusively) from the mossy fibers (Desclin and Colin, 1980; Brodal, 1981; Mason and Gregory, 1984). Thus Purkinje cell activity reflects activity in the two great systems of input fibers to the cerebellum, monosynaptically in the case of climbing fibers, bisynaptically through the mossy fiber- granular cell channel. Both systems are present everywhere throughout the entire extent of the cerebellar cortex, presumably with equal density.

There is an important difference in the way excitation reaches Purkinje cells through the two channels. In the case of the mossy fiber-granular cell-parallel fiber input the points of arrival on the dendritic membrane are the dendritic spines, while no such preference is apparent in the climbing fibers. In cerebral cortical as well as hippocampus physiology the opinion is widespread that synapses on spines may be plastic, i.e. subject to modification by learning (for a recent review see Horner, 1993). If this were confirmed, we would have a strong indication of the parallel fiber - Purkinje cell connection as the seat of memory in the cerebellum. Climbing fibers would then be assigned a more direct, preset function, in tune with the standardized response climbing fibers elicit in Purkinje cells.

The inhibitory influences onto Purkinje cells have been observed both in intracellular (Crepel, 1974; Konnerth et al. 1990; Midtgaard, 1992) and extracellular (Eccles et al. 1967b) recording. They are mediated mainly by two classes of interneurons which are distinguished by their point of arrival on the somatodendritic membrane of the Purkinje cell. Basket cells surround with their terminations the perikaryon and the initial segment of the Purkinje cell axon, while stellate cells tend to terminate on the dendritic tree. They are otherwise very similar in the length and geometrical distribution of their axons in the cortical plane (Sultan, unpublished data), in the shape of their own dendritic trees and in their synaptic relations (Rakic, 1972; Palay and Chan-Palay, 1974). Both kinds of inhibitory interneurons are fed excitation through the same mossy fiber - granular cell - parallel fiber channel that also serves the Purkinje cell, with the difference, however, that parallel fibers contact the dendrites of i

The Golgi cell, the third type of inhibitory interneuron has more complex synaptic relations. As we have mentioned already, their role seems to be essentially to check excess local activity. They are therefore not directly involved in the spatio-temporal transformations which are at the center of our attention. They are a striking exception to the anisotropy of the network formed by Purkinje, stellate, basket and granular cells.

Geometry of the arrangement of input elements around Purkinje cells

In no other part of the nervous system does the shape of neuronal elements suggest their functional relations as clearly as in the cerebellar cortex. The dendritic tree of Purkinje cells in all species studied (excepting perhaps the most rudimentary cerebella such as those of Cyclostomes, Johnston, 1902a; Johnston, 1902b; Arins Kappers et al. 1936) extends much farther in the anteroposterior direction than in the direction at right angles to it. In the human cerebellum, for example, the dendritic tree of individual Purkinje cells is contained in a rectangular block measuring about 350 m in the vertical direction z (corresponding to the thickness of the molecular layer), about as much in the anteroposterior direction y (the direction pependicular to the folia) but only about 35 m in the laterolateral direction x (the direction along the folia).

(For reasons of word economy we will use the x,y,z frame from here on. Another reason for using the intrinsic coordinates of the cerebellar cortex rather than the body coordinates right-left, forward-backward, up-down is that the two frames of reference do not coincide in large parts of the cerebellum. In the posterior vermis, which in many species is tucked under the body of the cerebellum, the y coordinate is actually inverted with respect to the "forward" coordinate of the body, and the z coordinate, the "vertical" in the histological jargon, almost nowhere corresponds to the vertical of the body, due to the folding of the cerebellar cortex.)

The rectangular boxes containing the dendritic trees of individual Purkinje cells are quite separate from each other, the overlap between the territories of one cell and its neighbours being very small or non-existent both in the x and y direction. This again is a rare exception among the various kinds of neuropil which have been studied with Golgi and similar methods. Most everywhere, very notably in the cerebral cortex, dendritic trees are intermingled, each overlapping the territories of thousands of other dendritic trees. In the cerebellum they seem to keep out of each other's way. In the cerebral cortex the "relative dendritic density" (Braitenberg and Schz, 1991) of pyramidal cells, i.e. the ratio of the dendritic length of one neuron to the total dendritic length present in the territory of its ramification is about 1: 1000. In the cerebellum the same ratio, for Purkinje cells as well as stellate and basket cells is close to 1: 1.

The neatly separated flat dendritic trees of Purkinje cells (and stellate cells) acquire a special meaning in connection with the peculiar geometry of the axonal fiber felt in which they are embedded. The bulk of the axonal fiber population in the molecular layer is provided by the parallel fibers which all run in the x direction, i.e. perpendicular to the y,z-plane in which the dendrites of their target neurons ramify. The whole arrangement seems to imply that for each dendritic tree there is a predetermined place along the length of the parallel fibers at which it is to receive its synaptic input. The idea that what really matters is the time at which the synapses become active, is old but still enticing.

Rise and fall of the timing hypothesis

Although the original proposal (Braitenberg and Atwood, 1958) was phrased in rather general terms ("transformation of spatial into temporal patterns and vice versa", "activity...will reach different Purkinje cell trees after different time intervals, depending on their distance", "arrival of a front of activity at any one Purkinje cell implies activity at different loci at different times in the past"), the version that met with most favour (Braitenberg, 1961; Braitenberg and Onesto, 1962; Braitenberg, 1965; Kornhuber, 1974) was that which sees parallel fibers as generators of time delays between the activation of different muscles involved in one movement. And this was also the most vulnerable form of the timing hypothesis, since with the known length of parallel fibers of a few millimeters and the measured (Eccles et al. 1967a) velocity of conduction of about 0.5 m/s the delays generated could hardly be more than 10 ms. This is not enough for the timing of movements which typically take about 200 or 300 ms

Moreover, the vague hope that some internal circuitry could perhaps relay signals from one delay line to the next in order to generate the longer delays was thwarted by the finding (Eccles et al. 1967a) that, except for the granular cells/ parallel fibers themselves, all interneurons in the cerebellar cortex are inhibitory.

Tidal waves

An alternative scheme (Braitenberg, 1983; Braitenberg, 1987) was therefore proposed which makes use of the special geometry of the cerebellar cortex without incurring in the difficulties which made the timing hypothesis obsolete in its original form. The emphasis is on conduction velocity rather than on time intervals. The idea is the following.

We stylize the molecular layer of the cerebellar cortex as a tissue which conducts signals along the direction +x and -x at the velocity (vo), the velocity of conduction in parallel fibers (dashed lines, Fig. 5a). A small patch of excitation at one point (asterisk) produces two waves running in opposite directions, quickly exhausted because of the limited length of parallel fibers. The density of excitation is nowhere very great, since the parallel fibers which are fed by the input at any point are only a small fraction of all the parallel fibers present there, due to their staggered arrangement. In fact, at a location x,y in the molecular layer all the parallel fibers converge which have their parent granular cells located within a range of x-l, y and x+l, y, where l is the length of each branch of a parallel fiber. Today's estimates for 2l vary between 4.7 and 6 mm (Brand et al. 1976; Mugnaini, 1976; Schild, 1980; Mugnaini, 1983; Harvey and Napper, 1988; Pichitpornchai et al. 1994).

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The situation is different for moving input (fig 5b). Suppose a patch of input excitation moves in the direction x at the velocity (vo) (asterisks). Then each new input will add excitation to that already travelling in the parallel fibers, and the density of excitation in the travelling wave, or tidal wave, will increase up to a maximum when all the available parallel fibers participate in the wave. This maximum is reached after the input has travelled for a distance l, and the density of excitation stays constant thereafter if the movement of the input continues in the same direction at the same speed.

The maximum density of excitation in the travelling wave corresponds to an activation of one half of all the synapses provided by parallel fibers, since only the branches of the parallel fibers running in the direction in which the wave moves contribute excitation to the wave. The signals running in the opposite direction are dispersed and stay behind the wave. Activation of all the parallel fiber synapses in the molecular layer would be possible only if moving input converges from opposite directions creating two waves that cross each other.

The supposition that tidal waves arise in the cerebellar cortex is a simple consequence of anatomy and can hardly be otherwise, given the basic facts of physiology. Moreover, it is an inescapable conclusion that the local density of excitation A, the height of the tidal wave so to say, depends critically on the velocity at which the input "moves" along the cerebellar cortex. If E is the total amount of excitation introduced (say, the number of spikes elicited by the input in the granular cells), (x is the distance travelled by the input, v the velocity at which the input moves, vo the velocity of conduction in parallel fibers, d the "wavelength" of the spikes on the parallel fibers (i.e. the duration of the spike multiplied by the velocity of conduction), then the local density of excitation

                               E

             A = --------------------------------

                     ((x / v) |v - vo| + d.

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The question is, what sense can we make of the idea of the cerebellum

critically tuned to the speed at which certain sequences occur in its input. Before turning to this theoretical problem, we want to assure ourselves that the physiology of the cerebellum does indeed behave in the way we expect from anatomy.

Experimental verification of tidal waves

The test experiments where performed in vitro by D. Heck (1993; 1995a, 1995b, 1995c) on acute slices of rat cerebellar cortex. The problem was approached in two stages. In a first set of experiments the existence of velocity dependent waves in the molecular layer was demonstrated by extracellular recording. In subsequent experiments the effectiveness of these waves in exciting postsynaptic elements was shown by intracellular recording of Purkinje cells.

Slices 0.4 mm thick were excised from the vermis of 3 to 4 week old rats by means of a vibrotome. The cuts were oriented parallel to the long axis of the folia and usually contained several long sections of the cerebellar cortex cut at different angles with respect to the pial surface. An example is shown in Fig. 7. Also shown on the figure are the positions of the tips of 11 stimulating electrodes (white arrow heads) arranged in a row within the granular layer of one of the folia, one cut as nearly as possible perpendicular to the cortical surface. This made it possible to record from the molecular layer overlying the stimulated region of the granular layer and hence presumably containing the parallel fibers belonging to the stimulated granular cells. For details of the slice preparation and of the electrophysiological technique the original publications (Heck, 1993, 1995b, 1995c) should be consulted. The stimulating electrodes were epoxy insulated tungsten wires spaced at distances of about 130 m. Negative

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The recording electrode was placed in the molecular layer at a position well beyond the end of the row of stimulating electrodes in the granular layer, at distances ranging from 200 to 1100 m from the last electrode in the row.

With this arrangement the amplitude and time course of the signal induced in the molecular layer by different sequences of stimuli in the granular layer could be observed. The signal appeared in the form of a biphasic deflection riding on the large stimulus artefact (Fig. 8). The most striking difference was between the response to "movement" of the stimulus toward the recording electrode and "movement" in the opposite direction, away from the recording electrode. In the first case a sizable signal was observed (fig 8a), and none in the second (fig 8b). Movement was simulated by sequential activation of the stimulating electrodes starting at one end of the row, with delays between one electrode and the next which were variable but constant for each run.

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There are two good reasons for believing that the observed signal is indeed due to conduction in parallel fibers. The first reason is pharmacological. The signal is abolished by addition to the bathing medium of the sodium channel blocker tetrodotoxin (TTX), which shows that it is no passive stimulus artifact. The signal persists after application of synaptic blockers such as 6-cyano-7-nitro-quinoxaline -2,3-dione (CNQX), excluding the possibility of some transsynaptic effect.

The second reason for believing that the signal reveals summation in the system of parallel fibers is its amplitude, which after sequential activation is much larger than

after activation of individual stimulating electrodes. The signal due to "movement" toward the recording electrode far exceeds the signal obtained with stimulation of even the stimulating sites closest to the recording electrode. This clearly shows that also the activation of more distant stimulating sites contributes to the phenomenon, most likely through conduction in parallel fibers.

The case becomes even stronger when the dependance of the signal amplitude on the velocity of the "movement" ( i.e. on different delays in the stimulating sequence) is taken into consideration. Fig. 9 shows this dependence for two experiments on slice preparations from two different animals. In both cases, when "movement toward" was tested, the signal fell below the noise level for velocities smaller than about 0.15- 0.2 m/s, or larger than about 0.5- 0.6 m/s. The maximum response was obtained for velocities around 0.3 m/s in one case and 0.4 m/s in the other. This would imply that in the acute slice of the rat cerebellum parallel fibers conduct at a velocity of 0.3 to 0.4 m/s, well comparable to, even if slightly less than the velocity measured in vivo (Dow, 1949; Eccles et al. 1967a; Garwicz and Andersson, 1992).

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Excitatory synaptic influence of parallel fibers on Purkinje cells

Several authors (Shambes et al. 1978b; Llins, 1982; Bower and Woolston, 1983) expressed doubts on the effectiveness of the excitatory synapses between parallel fibers and Purkinje cells. Although these synapses are plentiful on electronmicrographs, their excitatory action is not apparent in experiments in which the granular layer of the cerebellar cortex is stimulated locally. The activity of Purkinje cells is only increased in a narrow region immediately overlying the site of stimulation (Shambes et al. 1978b; Bower and Woolston, 1983). This was taken as an indication of the insignificant weight of the synapses on the horizontal branches (=the parallel fibers) of the granular cell axon, as compared to the synapses on the ascending stem of the same axon which, though much fewer, were supposed to act much more strongly.

If this were the whole truth, the "tidal waves" in the molecular layer which we had postulated and even observed electrophysiologically would be functionally quite meaningless, since their cumulative strength develops in the horizontal parallel fibers and would then fail to produce postsynaptic effects. Unless, of course, the point of the whole arrangement was precisely this, to amplify the input when it is moving and to suppress static input. If this were the intended function, individually weak parallel fibers together with the provision for summation of the effects of very many of them on individual Purkinje cell would be ideal. In this case experiments with "moving" input could produce sizeable effects where static input fails completely.

To test this possibility, Heck (1995a, 1995c) recorded intracellularly from Purkinje cell bodies in acute slices of the (guinea pig) cerebellum, prepared in the fashion already described. The electrodes were KCl filled micropipettes with resistance greater than 70 MW. Again the stimuli were delivered through an array of 11 electrodes, in various successions imitating movement at various speeds and directions (Fig. 10). The intracellular records show increased simple spike activity in Purkinje cells precisely in the situations where in the previous experiment (Fig. 9) a large tidal wave was observed with extracellular recording. Since the extracellular wave almost certainly reflects mass action in the parallel fiber system, we may conclude, in disagreement with the above authors, that parallel-fiber to Purkinje-cell - synapses are very effective when the input is presented to them in the right way.

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Application of the tidal wave concept to a theory of cerebellar function.

Having thus convinced ourselves that in the cerebellar cortex the main link between input and output (the one involving the greatest number of synapses and neurons) is gated by certain spatio-temporal patterns in the input, we turn to the question of how this affects our view of the function of the cerebellum in general.

The main obstacle to a comprehensive view of the role of the cerebellum in its interplay with the rest of the nervous system is our insufficient knowledge of afferent and especially efferent connections. Several regions of the brain and spinal cord are known to project onto the cerebellum directly or indirectly, via mossy or climbing fibers (Brodal, 1981), but the exact pattern of the projection is still a matter of controversy (Oscarsson, 1976; Leicht and Schmidt, 1977; Leicht et al. 1977; Oscarsson, 1979; Ekerot and Larson, 1980; Woolston et al. 1981; Kassel et al. 1984; Gonzalez et al. 1993). More obscure still are the ways in which the ouput of the cerebellum is inserted into the main stream of motor control, to which it supposedly contributes. These are open questions that limit the poignancy of any functional scheme we may propose. To be on the safe side, we shall start our speculations from the facts we consider certain, namely from the structure of the cerebellar cortex interpreted in the way we have

The concept of a beam of parallel fibers, first introduced by Eccles et al. (1967a) is central to our scheme. We may define it as a latero-lateral strip of cerebellar cortex reaching from the right to the left margin of the cerebellum (in the anterior and central part of the cerebellar map, Fig. 1 and 2) or from the lateral margin to the medial margin in the paired tail-like extensions, or from the right to the left margin of the posterior vermis. Note that in most cases, in larger animals at least, a beam of parallel fibers is longer (in the x direction) than the individual parallel fibers. The width of the beam could be defined as the width of the dendritic ramification of individual Purkinje cells (in the y direction), for that is the width that includes all the parallel fibers which may converge onto one and the same neuron. Another way of defining the extension of the beam in the y direction is based on the geometry of basket and stellate cell axons. These inhibit Purkinje cells in the front and in the

It should be noted that individual beams are not separated from each other by unmovable anatomical barriers. There are no discontinuities in the layout of neuronal elements along the y direction (nor in the x direction, for that matter). Rather, we should think of the beam as a physiological entity which owes its existence to a particular situation in the input. The situation is that of input excitation reaching the beam in a certain temporal sequence.

The tidal wave set up by the input will affect not only Purkinje cells, as shown in Heck's experiments, but stellate and basket cells as well, since these also (and exclusively) receive their excitatory input from parallel fibers. There is an important consequence to this. Due to the geometry of the inhibitory (basket and stellate) interneurons, if a certain beam is excited by just the right input sequence, it will strongly inhibit neighbouring beams in the +y and -y direction. "Lateral inhibition" as described by Ratliff (1965) is embodied in this, even if the term "lateral" is misleading in the present situation, where the inhibition hits neighbouring beams in front and in the back of the beam in question. In any case, the principle of "winner takes all" will be valid among beams, with the beam most precisely matched to the input sequence suppressing activity in neighbouring beams, which may be also excited by the input, but less strongly so.

In defining the beam as the anatomical module in which a tidal wave may develop, one important question remains unanswered: What is the length of the beam? Is it really the entire length of the folium, which in some parts of the cerebellum may span the entire width of the organ, or should we think of the beam as limited to the length of individual parallel fibers? There is a rationale for both interpretations. As we have seen, "moving input" recrutes parallel fibers into a tidal wave of increasing amplitude, up to a maximum which is reached when the input has travelled the length of one parallel fiber (a few millimeters). So, in a sense, the phenomenon is limited to that length. On the other hand, if the input keeps moving in the same direction, the tidal wave will continue undiminished after it has reached its maximum, moving along with the input and strongly exciting the folium in its entire length.

Since eventually we want to apply the tidal wave mechanism to movement physiology, we prefer to think of beams as having the length of folia. Some of the folia of larger cerebella are at least ten centimeters long. The time it takes for a wave to travel this length at the intrinsic velocity vo is about 200 ms. This is the duration of a rapid limb movement in a large mammal, or the duration of a syllable in human speech. It is not impossible to think of the length of a beam as providing temporal order for such motor performances. It should be kept in mind, however, that the majority of folia, even in larger cerebella, are shorter than ten centimeters and could only be held responsible for shorter sequences.

In order to make this idea more concrete, we must try to establish what kind of input reaches the cerebellar cortex, and in what geometrical order it is presented there.

Somatotopic and other maps on the cerebellar cortex

Some of the rules that govern the projection of parts of the body onto the cerebellar surface were known to clinicians and to physiologists working with macroelectrodes (Snider and Stowell, 1942; Adrian, 1943; Combs, 1954), long before the details of cerebellar physiology were elucidated by Eccles and his school. It was known e.g. that the cerebellum had prevalently uncrossed relations with the motor periphery, and correspondingly, crossed relations with the motor cortex. Also, the homunculus was known to be represented at least twice on the cerebellar cortex, once in the anterior lobe, with its head pointing backward, and again in the posterior lobes (corresponding to the posterior paired extension of our scheme fig 1 and 2), where it is split in two separate halves, heads pointing medially on the flattened cerebellar map.

Later the projections were studied with more refined methods, separately for the climbing fiber and mossy fiber input (Oscarsson, 1976; Leicht and Schmidt, 1977; Leicht et al. 1977; Oscarsson, 1979; Ekerot and Larson, 1980; Woolston et al. 1981; Robertson et al. 1982; Kassel et al. 1984; Robertson, 1984; Robertson, 1985; Gonzalez et al. 1993). The general rules were confirmed on the whole, but the details were surprising. For one thing, the Swedish group headed by Oscarsson, concentrating on the anterior lobe of the cat cerebellum found distinct zones, or "sagittal strips" each characterized by its special kind of climbing fiber input. These strips, about 1 mm wide and running through the entire extent of the anterior lobe in a sagittal (y in our terminology) direction confirmed earlier anatomical descriptions (Jansen and Brodal, 1940; Voogd, 1969) of fibers in the white substance of the cerebellum arranged in a similar manner. While most of the strips on both sides of the midline received input from the sam

This organization in a succession of sagittal strips is said to permeate the entire cerebellar system even outside the cerebellar cortex proper. Individual strips of the cortex are served by special portions of the afferent inferior olivary nucleus which provides the climbing fibers, and similarly on the efferent side each strip projects on its private region of the cerebellar nuclei (Voogd and Bigar, 1980; Trott and Armstrong, 1987a; Trott and Armstrong, 1987b; Andersson et al. 1987).

Although the detailed mapping of the anterior lobe by Oscarsson and his group relied on climbing fiber input after electrical stimulation of individual peripheral nerves, the same group mantains that if mossy fiber input is mapped instead, a map is obtained which is not identical but very similar to the other (Oscarsson, 1976; Ekerot and Larson, 1980).

For the purpose of our theory the Swedish findings provide ample food for thought. Since parallel fibers span a few millimeters on both sides of their point of origin, they obviously link several sagittal zones. Whatever function we attribute to the parallel fibers, for instance that of building up tidal waves with sequential input, the function must involve more than one sagittal zone. Take as an example the sagittal zone which receives input from the animal's paw, together with the neighbouring zone which represents more proximal input. A tidal wave will arise in the parallel fibers linking the two zones if the proximal input follows (or precedes) the distal input after a time interval corresponding to the conduction delay in the fibers. The whole arrangement makes a good detector for something travelling through the limb in one direction or the other at just the right speed.

The Swedish map is detailed enough so that we can make an appraisal of the speed at which something must travel through the animal in order to elicit a strong stimulus, i.e. a tidal wave in the cerebellar cortex. For this we must know the "magnification factor", to use a term from visual physiology, i.e. the length of the beam which represents a certain length of the body. The magnification factor varies a great deal in the different representations of the cat's somatotopy on the anterior lobe of the cerebellum. If we accept that the length of the cat is represented on 1 mm cortex, as in the so-called b-zone of Oscarsson's map, then a succession of stimuli must travel through the cat's body at a speed of several hundred m/s in order to make the input move through the cortex at the velocity of 0.5 m/s, the intrinsic velocity of conduction in parallel fibers which makes tidal waves of maximal amplitude (taking the length of the cat as 400 mm, a signal must travel at a speed of 200 m/s for it to take 2 ms from

The magnification factor is very different in the global mapping of the cat's body on the entire anterior lobe, the mapping that was known already to the old electrophysiologists and was found later to be superimposed on the more detailed maps discovered by the Swedish group. Here a folium several centimeters in length reaches from one margin of the cortex to the other and represents, e.g., the left forelimb on the left, the trunk in the middle and the right forelimb on the right. On such a map the intrinsic speed vo of conduction in parallel fibers corresponds to a sequence of events traversing the body at a velocity of about 10 m/s.

Not only the magnification factor varies in the different somatotopic maps, but also the orientation of the body representation with respect to the intrinsic coordinates x and y of the cerebellar cortex. If we ask in what direction the image of the cat on the anterior lobe is traversed by the parallel fibers, the answer is different in the various representations. In the compressed image of the length of the cat in the "b-zone" the parallel fibers correspond to the long axis of the animal. If we take the part of the global representation near the midline, the parallel fibers run through the image transversally from right to left. And finally around the "d-zone", as we have already seen, the direction of parallel fibers correspond to the direction distal-proximal in the representation, i.e. vertical in the standing animal. It is as if the multiple somatotopic representations had the purpose of letting the cerebellum monitor sequences of events, or movements, in whatever direction they may run through the anim

(Place Fig. 11 about here)
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Fractured maps

Even greater surprises came from some studies (Shambes et al. 1978a; Shambes et al. 1978b; Woolston et al. 1981; Welker, 1987) of sensory projections onto the granular layer of the cerebellar cortex. Extracellular recording with micoelectrodes, very likely of mossy fiber afferents (or of their synaptic partners, the granular cells), revealed hardly any trace of an orderly representation of parts of the body on connected regions of the cerebellum. When the authors scanned several folia of the lateral parts of the hemisphere in various species of animals, neighbouring positions in the granular layer as a rule seemed to correspond to "receptive fields" quite far apart in the animal's anatomy. Vice versa, most places of stimulation on the animal's surface would produce responses in more than one place on the cerebellar cortex. If there was any orderly mapping to be seen, it remained confined to single patches of the mosaic-like projection, i.e. to a region no larger and usually much smaller than 1 mm across. Fro

This was in contrast with the very concept of a somatotopic map which had inspired some previous descriptions, both clinical and experimental, of the relation between parts of the body and parts of the cerebellum. Whereas the physiological maps produced by Oscarsson and his group had added some detail and some local exceptions to the traditional picture, without contradicting it in principle, the findings of Welker and his group were difficult to reconcile with the estblished prejudices.

It seems that no serious attempt was ever made to arrive at a fruitful synthesis of the contrasting opinions. Quite likely both schools highlight separate aspects of the truth, with the incompatible statements only arising where undue generalisations are made. The idea of sagittal stripes with uniform input throughout their length, as it is implicit in Oscarsson's map (e.g. in the "b-zone"), undoubtedly was based on intra- and extrapolation from more limited observations, and deserves further experimental verification. Similarly, Welker's fractured maps were obtained by recording from the granular layer underlying the summit of the folia. This method necessarily disregards the part of the cerebellar cortex which is burried in the sulci, possibly obscuring a continuity between apparently disparate "patches". Moreover, the granular layer beneath the convex parts of the cortex is very thick and contains both granular cells and afferents belonging to a much wider region than just the summit of the folium. In the

All in all, the two sets of data are not really comparable and it may be more correct to say that they complement, rather than contradict each other.

Some uncontroversial statements about cerebellar maps

Tentatively, we gather the following picture from the fragmentary information in the literature. Somatosensory input reaches a large part of the cerebellar surface both through climbing and mossy fibers. The two systems of afferents seem to project body maps on the cerebellum which are roughly in register. There seems to be a tendency for sagittal strips of cerebellar cortex to receive similar or perhaps even identical (via branched fibers) input. In some places it may be patches, rather than stripes of similar input. Be it patches or stripes, there are breaks in the representation, fragments of body maps located next to each other with no apparent common coordinates and with abrupt transitions between them. There are local maps superimposed on global maps, and in some places apparently local disorder superimposed on an otherwise fairly regular representation.

Contrary to what a previous generation of cerebellar physiologists liked to think, ("the head ganglion of the proprioceptive system") the cerebellum is fed at least as much information from the surface of the body as from the deep (muscle, tendon and joint) senses (Thach, 1967; Gellman et al. 1983; Robertson, 1984; Gellman et al. 1985; Gibson and Gellman, 1987; Welker, 1987; Andersson et al. 1987).

Besides input from the somatosensory periphery, relayed through the inferior olive (climbing fibers) and the pontine nuclei (mossy fibers), the cerebellum receives at least as much from the telencephalic cortex. The cortico-ponto-cerebellar fiber tract is massive, as Glickstein (1987) never tired to point out, and provides mossy fibers all over the cerebellar surface. But the cortex finds its way into the cerebellum also via climbing fibers, apparently through direct cortico-olivary connections (Sasaki et al. 1977; Andersson and Eriksson, 1981; Andersson and Nyquist, 1983). The cerebro-cerebellar projections provide good additional evidence for the reality of topographic maps on the cerebellar cortex. Where the somatotopy is well defined on the cerebral cortex, such as in the primary motor area, the projection onto the cerebellum is point to point and respects the overall somatotopy there. Other regions of the cerebral cortex, where somatotopy plays a minor part, such as the frontal and parietal areas, proje

There is also the celebrated vestibular input to the cerebellum, perhaps the most ancient one, probably at the root of the development of the cerebellum in early vertebrate evolution. It is mainly localized in the most posterior portions of the flattened cerebellar map. At the present stage we do not intend to incorporate the vestibular system in our thinking on the function of the cerebellum in general. Whatever idea comes to mind to explain the role of the cerebellar machinery in the vestibular context, it does not seem to apply in the control of limb movements, and vice versa.

Input sequences

We are now in a position to define more precisely the sequences of events which may result in tidal waves in the molecular layer of the cerebellar cortex. If we compound everything we have learned about mossy fiber input both from the cerebral cortex and from the periphery, proceeding along a folium we may meet a succession of input points arranged in a regular somatotopic way, or fragments of somatotopic progressions abruptly set next to each other, perhaps with different scales of the representation, or a succession of input points of the cerebro-ponto-cerebellar pathway, with or without geometrical order, or a mixture of input points from ascending (spino-cerebellar) and descending (cortico-cerebellar) pathways. The question is, what such a sequence of input points represents when they are activated in succession at the speed of conduction vo in parallel fibers. We have already seen that in the simplest case it may represent movement of something through the body in a certain direction at a certain veloc

One more point can be made if the doctrine of the sagittal zones is seen in conjunction with the macroscopic cerebellar maps of Fig. 1 and 2. The anterior lobe of the cerebellum is roughly triangular in outline. The sagittal zones which run in an antero-posterior direction through the whole extent of the anterior lobe must be converging anteriorly. If it is true that they receive the same or very similar input throughout, we should conclude that the same (or similar) sequences are represented on successive folia with different magnification factors. This means that different folia respond to the same sequence presented at different speeds.

All of this makes sense in the light of some speculations on the cerebro-cerebellar interplay which we will now propose.

Control of movement by the cerebral cortex

While there is evidence (Georgopoulos et al. 1986; Schwartz et al. 1988; Georgopoulos et al. 1988) for sets of neurons in the cortex whose activity correlates with the direction of movement, rather than with the length or force of the muscle, it is still true that the output from the motor cortex ultimately controls directly the spinal motoneurons of the ( and ( kind. The activity of corticospinal neurons must produce activity in the motoneurons and hence tension in individual muscles, and movement ultimately results from changes in the distribution of their activity. Rather than being programmed as such by the cortex, movement would then be a passive consequence of a changed pattern of muscular tensions. This corresponds in essence to an idea propounded by Feldmann (1966; 1974a; 1974b; 1986) and by Bizzi (1976; 1982a; 1982b; 1984; 1986), the equilibrium point hypothesis.

We like to subscribe to that theory wihout going into the arguments that have been advanced in its favour or against it. What is attractive about the idea is the conceptual simplicity which it provides. We may think of the telencephalon as of a machine which knows (= has incorporated information about) the state of the world, including the animal's body and its position in the world. It also knows the rules that govern the transitions from one state of the world to the next, and it knows the set of values associated with the states and the transitions. It will occasionally take the initiative and force a transition from one state to a more advantegeous one by "rethinking"the animals conformation or position in the world. The transition to the new state envisioned by the telencephalon automatically implies a change in the signals that reach the motor system (cortico-spinal neurons are part of the thinking machinery), and such changes entail actual movements.

What we don't expect the telencephalic cortex to do, is to consider all the mechanical consequences which come with the transition from one conformation of the body to the next. Quite apart from friction and viscosity there is both inertia and elasticity in the structures involved, and the two together must by necessity result in oscillation. There are centrifugal and Coriolis forces which grossly affect movements of a limb with several joints. Such parasitc forces are by no means negligible: with the elbow set at a right angle, the centrifugal force acting on the forearm exceeds gravity already with a rotation in the shoulder joint as slow as 1 cycle per second (corresponding to a conductor beating adagio). And finally there are forces transmitted from any moving joint to all the other joints of the body by sheer mechanical coupling. The situation is so complicated that traditional methods of mathematical physics fall short of a complete analysis. For the mechanics of arm movements see Hollerbach and Flash

It is our contention that the cerebellum takes care of the physics which is implied but not explicitly contained in the simple motor commands emanating from the cerebral cortex. The symptoms which are observed after cerebellar lesions, including clumsiness, overshoot of movements and oscillations are not unlike what you see in a puppet when the puppeteer controls the positions of the limbs only in a global way, leaving the execution of the movement to the passive mechanical properties of the system. We claim that the intact cerebellum corrects for the parasitic forces which accompany every movement, and thus brings the actual execution closer to the smooth transition between two positions intended by the cerebral cortex.

This separation of tasks between the two great cortices, one essentially cognitive and the other mechanical must be related to the spectacular differences between the two structures which served as a starting point for our essay. We have no difficulty in interpreting the telencephalic cortex as an associative memory in which a replica of the causal structure of the outside world is built into the synaptic relations between neurons (Braitenberg and Schz, 1991). The question is now, what the very different structure of the cerebellar cortex could have possibly to do with the physics of movement.

The sequence in - sequence out mode of operation

We propose the following scheme. The functional unit is the beam of parallel fibers with the attached postsynaptic elements. It is excited by a sequence of events spelled out by the order of mossy fibers along its length. The individual events are signals from the motor cortex interspersed with signals from the periphery. The whole sequence therefore means a movement to be executed under given conditions of the motor system. The neurons of the motor cortex fire sufficiently in advance of the actual onset of movement (up to 100 ms, Evarts, 1972), so that the information reaches the cerebellum early enough for it to make its own contribution to the motor command.

The output is triggered by the tidal wave which travels through the excited beam. We imagine that some of the Purkinje cells along the beam are more strongly excited by the parallel fibers than the others. What the beam passes on to the cerebellar nuclei, is a sequence of signals produced by selected Purkinje cells at times specified by the moving wave of excitation.

It should be remembered that the beam is not limited to the length of parallel fiber branches. Both the input sequence to which it is tuned and consequently also the sequence of output signals it emits may have durations considerably exceeding the 10 ms limit.

The excited beam inhibits neighbouring beams so as to prevent possible superposition of output signals from Purkinje cells.

Routes of cerebellar output

Again disregarding the small part of the cerebellum related to vestibular functions, the axons of Purkinje cells, the only fibers exiting the cerebellar cortex, terminate on cells of the cerebellar nuclei with inhibitory synapses (Ito et al. 1964; Ito and Yoshida, 1966; Ito et al. 1970). This would be dead output unless the cerebellar nuclei received excitation from other sources. We know that they receive collaterals from the very same fibers, mossy (Shinoda et al. 1992) and climbing (Dietrichs and Walberg, 1989; Van Der Want et al. 1989), that bring excitation into the cerebellar cortex. What comes to mind is the idea proposed by Eccles (Eccles, 1973) of the cerebellum sculpting the motor commands, i.e. taking away the part of the excitation that is superfluous or damaging in the execution of a movement. This is in line with the separation of tasks that we have envisaged, the cerebral cortex (together with the spinal afferents) specifying the motor commands in a global way, and the cerebellum contributing

The further destiny of cerebellar output is not very clear. There are two main routes. The cerebellar nuclei have strong connections to the cerebral cortex via the thalamus, with excitatory synapses at both levels. Through this route the cerebellum can influence the motor command at its origin. The nuclei also have excitatory terminations at the level of the red nucleus. This being a relay station for some of the cortico-spinal pathways, here we have an alternative, more direct route through which the cerebellum can insert its information in the flow of motor signals between brain and spinal cord. Note that there are no inhibitory links along either route, so that no further change of sign occurs beyond the inhibition of Purkinje cells on cerebellar nuclear neurons and the idea of "sculpting" remains valid.

Sculpting motor commands for optimal performance. A model

This is not the place to go into the physics of body movement with the thoroughness it deserves, nor are we competent to do so. We shall only construct a simple model to show the principle (Fig. 12).

Fig. 12a illustrates a system (the push-me-pull-you) which is characterized by an internal degree of freedom and by internal elasticity. The two equal masses M1 and M2 are connected by a spring and can be made to move without friction along a straight line by exernally applied forces. If a force is applied for a certain time to M1 from the left, the center of mass of the system will move toward the right at a constant velocity v. However, the momentum of the system will be exchanged periodically between M1 and M2, each of the two partial masses reachuing the velocity 2v when the other one is momentarily a rest (Fig. 12b). The internal oscillation of the system has a period T which depends on the masses and on the elasticity of the springs.

(Place Fig. 12 about here)
[FIGURES ARE ONLY AVAILABLE IN THE PAPER VERSION]

The question may be asked of how to stop the push-me-pull-you once it is set in motion. We should like not only the movement of its center of mass to stop, but also the internal oscillation. For this to happen we may deliver to M2 a pulse equal in magnitude and opposite in direction to that delivered before to M1. This will stop the movement of the center of mass in any case, and will also stop the oscillation if it is applied at a time when M1 is without motion and M2 carries all the momentum. This occurs at a time T/2 after the system has been set in motion, and then again at time T+T/2, 2T+T/2 etc.These times are also the ones where the spring has come back to its resting position, so that no energy, potential or kinetic is left in the system. At all other times it is not possible to stop the motion of the system by applying a single pulse without a residual oscillation.

Consider a slight modification of this model (fig 12c) which brings it closer to movement physiology. Again we have two masses connected by springs, but one (L) is contained in the other (C) as a small carriage within a larger one. Thus the forces can be applied only to one of the masses, C, by two motors Ag and Ant pulling in opposite directions. Here again, if we want to shift the carriage C by applying pairs of opposite pulses separated by a time interval, we are limited to certain intervals (in this case multiples of T) if we want to avoid residual oscillation.

This means, of course, that we can only make the carriage C travel multiples of an elementary distance corresponding to multiples of T at a given velocity. Otherwise the carriage will not stay put but will oscillate around the common mass center of L and C. This is a serious limitation of the principle of "bang-bang control" so dear to some movement physiologists.

There is a way around this difficulty. Suppose the carriage C is set in motion by an initial pulse delivered by Ag. It can be stopped at any time by a stronger pulse in Ant which not only counteracts the initial momentum but produces momentum in the opposite direction. This momentum is in turn annulled by a second pulse in Ag which followes the Ant-pulse after the critical time T so as to absorb the internal oscillation. With this strategy there is no limitation to the choice of delay between the initial Ag and the Ant-pulse, and therefore no restrictive condition on the distance travelled. The only condition for a complete stand-still of the system is the fixed delay between the Ant-pulse and the second Ag-pulse, which does not change for movements of different amplitude.

The relation of this mechanical model to real limb movement is the following. The carriage C stands for the part that is to be moved, e.g. the (stretched) arm around the shoulder. The contents of C, masses and springs, represent everything that is losely but elastically connected to the arm, including the forearm, the hand and all the muscles and other tissues of the arm itself. Ag ant Ant are antagonistic muscle groups of the shoulder joint which produce equal and opposite forces for holding, and differential forces for movement. If a multi-joint and multi-muscle system such as the entire arm seems poorely represented by our simple mechanical model, think of movement of one segment only: there are always considerable masses of soft tissue elastically fastened to the moving bone.

The so-called triphasic pattern of innervation, reminiscent of the three pulse sequence which we had postulated for optimal "bang-bang-bang" control in Fig. 12c, is generally observed in rapid limb (Wadman et al. 1979; Wadman et al. 1980) and even head (Hannaford and Stark, 1987) movements. As we would predict from the model, the interval between the initial activation of the agonist and that of the antagonist increases with the amplitude of the movement, while the later part of the sequence, involving the antagonist and second agonist pulse, remains constant (Wadman et al. 1979). That this programming is the business of the cerebellum is a feeling widespread among physiologists (for a review see Brooks and Thach, 1981) and clinicians (Dichgans and Diener, 1984; Hore et al. 1991). There are various ways in which the cerebrum and the cerebellum could cooperate. Both the agonist and the antagonist innervation may be diffusely programmed by the cerebral cortex, with the cerebellum sculpting the raw form by well

The role of the cerebellum in the mechanics of movement as sketched here, to our knowledge was first proposed by Nahvi (1985). His idea of the neuronal machinery involved, however, is different from ours.

Role of the inhibitory interneurons in the cerebellar cortex

According to our scheme, every tidal wave arising in a beam of the cerebellar cortex besides activating Purkinje cells excites also a large number of inhibitory interneurons (stellate and basket cells) which in turn strongly inhibit the Purkinje cells of neighbouring beams. Quite likely the sum of the inhibitory action off beam exceeds the excitation produced by the wave on the Purkinje cells along the beam. Globally, therefore, excitation of the beam could be expected to produce an increase rather than decrease of activity in the cerebellar nuclei, since more Purkinje cells are turned off than on, and consequently less inhibition reaches the nuclei. This could be at the origin of the observation that activity in nuclear neurons covaries with activity in Purkinje cells (Thach, 1972). If some Purkinje cells along an excited beam are observed together with nuclear cells which receive afferents from a larger pool of Purkinje cells, one could well get that impression.

The overall effect of excitatory input to the cerebellum could be an increase of the excitatory output from cerebellar nuclei, contrary to expectations. However, on a purely speculative level we prefer to think that what counts in the output are the signals emanating from the one most strongly excited beam, with the inhibitory interneurons serving only to isolate the beam from its neighbours. In this view the cerebellum remains essentially inhibitory. Of course our view also implies that the connections between the Purkinje cells and the motor periphery are highly specific and that they are able to operate in isolation, since otherwise the effect of Purkinje cells on the beam would be drowned out by the contrary action of Purkinje cells in neighbouring beams.

Our believe that the interesting input-output transformations occur between parallel fibers and Purkinje cells, with no participation of inhibitory interneurons except in their ancillary function of background suppression, rests on a theoretical prejudice. When two kinds of neurons are served by the same input fibers (e.g.parallel fibers), but one has spines on its dendrites and the other does not, we tend to think that the synapses on spines (e.g. on Purkinje cells) are subject to learning while the synapses on smooth dendrites (e.g. of basket cells) are unalterable by experience. For a review of the empirical basis of this prejudice see Horner (1993). We like to think that the cerebellar cortex just like the cerebral cortex stores information by modifying the strength of its synapses, and the parallel fiber synapses on the spines of Purkinje cells dendrites are the best candidates for that.

Two kinds of learning in the cerebellum

At this stage our theorizing will continue completely unbridled by empirical evidence, not for lack of research on plasticity in the cerebellum, but for lack of a consensus among the researchers involved.

Besides dendritic spines, "mossy fibers" such as found in various places in the nervous system (e.g. also in the hippocampus) are suggestive of learning processes. When the synapses of one long axonal segment, instead of being randomly distributed along its length (as on pyramidal cell axon collaterals of the cerebral cortex, Hellwig et al. (1994), or on parallel fibers in the cerebellum, Sultan (1994) are lumped together in a few places into gigantic synaptic conglomerations, the"mossy excrescences" or "rosettes", we may ask what determines the position of these special synapses and suspect that it is some sort of learning process.

Tentatively, we propose the following. There is an early imprinting of the cerebellar input which decides at what places the mossy fibers establish synapses with the granular cells. Mossy fibers branch profusely especially along the direction y, less so along x (Cajal, 1911; Scheibel, 1977; Sultan, 1994b), but their synapses seem to be distributed in a haphazard way in the granular layer. We imagine that early experience sees to it that every beam receives a different combination of mossy fiber input, perhaps one that is likely to be significant because it has occurred often in an early phase.

Later another learning process sets in which continues throughout life, the tuning of parallel fiber synapses on Purkinje cells guided by the search for the energetically most economic, mechanically simple and smooth movements. We do not know under what sensory guidance this learning takes place, but it is quite plausible that the somatosensory input to the cerebellum, which comes largely from the skin, plays a role there. If the inelegant movement of the beginner is distinguished from the elegant movement of the accomplished athlete by all the slinging and oscillating that goes on in a parasitic way in various parts of his body, the relevant input for learning may well come from receptors that signal local tension in the skin.

Together, the early imprinting and the later synaptic refinement could achieve what we take to be the role of the cerebellum in motor control. First a catalogue of sensory-motor patterns is set up in the succession of beams, then each item in the catalogue is associated with the appropriate temporal pattern of cerebellar output to optimise the motor performance. The occurrence of a particular sensory-motor pattern elicits a tidal wave in a particular beam which triggers the corresponding output sequence in real time.

Role of the climbing fibers

The climbing fibers have not played any role in our thinking about the cerebellum. Yet, elimination of climbing fibers by an early destruction of their source, the inferior olivary nucleus, is said to have deleterious consequences on cerebellar function (Llins et al. 1975; Soechting et al. 1976). The rate at which climbing fibers fire, not much more than once in a second, makes it unlikely that they are involved in the immediate business of fast motor control. They have been variously interpreted as tutors which supervise the learning process between parallel fibers and Purkinje cells (Marr, 1969; Albus, 1971; Ito et al. 1982), or as emergency lines which take over in exceptional situations, e.g. when unforseen obstacles get in the way of a planned movement (Gellman et al. 1985). The two ideas are not incompatible: when an acquired motor routine leads to undesirable effects, it is time to change the routine.

Epilogue

To present yet another theory of cerebellar function is to tax the patience of the specialists. If we recommend our ideas to their attention, although sketchy and likely to be changed in many ways, it is because they are anchored in the most indisputable set of data, the neuroanatomical ones. Cerebellar structure was known before, and there were many speculations about it, but only recently (Heck, 1993, 1995a, 1995c) did one of them turn into physiological reality. Our point is that input sequences are the key to cerebellar function, a strong point not likely to be taken back. Working from this central point out, we got onto unsafe ground in many ways. But we see the uncertainties now in a different perspective, and this may in itself be fruitful.

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Figure legends

FIG. 1. Folial patterns of the cerebella of the macaque, sheep, dog and bat. Each horizontal line represents the length of the folium in the latero-lateral direction. The number of lines represents the anterior-posterior direction (not the same scale as the latero-lateral direction, compare Fig. 2). Modified from Sultan and Braitenberg, 1993.

FIG. 2. The shape of the unfolded cerebellar cortex of three large animals (a: bovine; b: human; c: macaque) and three small animals (d: squirrel; e: rabbit; f: mouse). Note the different scale of the large and the small cerebella. The anterior-posterior extension of both the human and the bovine cerebellum exceeds two meters. Modified from Sultan and Braitenberg, 1993.

FIG. 3. The logarithm of the cerebellar surface (mm2) is plotted against the logarithm of body weight (gram). The regression line has a slope of 0.62 (without the human cerebellum), indicating a dependance of the cerebellar surface on the 2/3 power of body weight. Note that the human cerebellum appears as a stray point. (Sultan and Braitenberg, 1993)

FIG. 4. Maximal cerebellar width plotted against the cerebellar surface on a log-log scale. The dependance is steeper for the smaller cerebella than for the larger ones. Again the human cerebellum is a stray point. (Sultan and Braitenberg, 1993)

FIG. 5. a: The effect of a short local stimulus (asterisk) to the cerebellum in time and in space. The latero-lateral direction of the cerebellum is plotted horizontally and time in the direction up-down. The horizontal line represents a succession of points in time. The wave set up by the stimulus travels in both directions and dies out after it has moved the length l , corresponding to one branch of the parallel fiber.

FIG. 5. b: If a succession of stimuli (asterisks), imitating movement along the folium is presented summation of activity in the parallel fiber system occurs. A wave of maximal amplitude builds up when the stimuli follow each other at the velocity vo of conduction in the parallel fibers. After sessation of the stimuli the tidel wave setup continues for the length l with diminishing amplitude and then dies out.

FIG. 6. Maximal amplitude of the tidel wave (A) as a function of velocity. Vo is the velocity of conduction in the parallel fibers, d is the spread of the signal along the parallel fibers (the wave length of the signal), (x is the length of the stretch over which the stimulus was presented. In a. the "wave length" is 10( that in b. Modified from Braitenberg, 1993.

FIG. 7. Horizontal section through the rat cerebellum (Nissel-stain). The position of the tips of 11 stimulating electrodes in the granule layer are shown by white arrow heads. The position of the recording electrode is also indicated.

FIG. 8. Extracellular recording after sequential stimulation at sites indicated in Fig. 7. On the left side of the recording the stimulus artefacts (negative excursions), clipped by the recording instrument of the 11 stimuli are shown, which build up into the large negative wave (clipped). As the recorded potential goes back to its resting level, the wave set up by the parallel fiber activity induced by the stimulus (deflection between the two asterisks) is evident in A were the stimuli moved toward the electrode and absent in B were movement was in the oppoiste direction.

FIG. 9. The amplitude of the recorded parallel fiber activity (see Fig. 8) as a function of the "velocity" of the stimulus. A: When the "movement" is toward the electrode, the signal is maximal for velocities arround 0.3- 0.4 m/s. B: Movement away from the recording electrode does not produce any appreciable signal at any velocity, nor does (C) a random sequence of stimuli have any effect. Modified from Heck, 1993.

FIG. 10. Intracellular recording from Purkinje cell after stimulation through a comb of electrodes as described in Fig. 9. A- F: Supperposition of ten trials each. Again the stimulus artefact is shown on the left. With the stimulus moving away from the electrode (A, C, E) only some spontaneous activity is observed in the Purkinje cell. With the stimulus moving toward the recording site (B, D, F) the spike activity in the Purkinje cell is increased and the spikes tend to occure at about 5 ms after the sessation of the stimulus. (From Heck, 1995a). G,H shows the time cource of the EPSP averaged over 20 trials after stimulation moving away from the recorded Purkinje cell (G) and towards it (H). (From Heck, 1995c)

FIG. 11. (from Braitenberg, 1983) Simplified version of the somatotopy in the anterior lobe of the cat cerebellum as established by Oscarsson and his coworkers. The direction of the parallel fibers is horizontal on this map. It corresponds, in different somatotopic maps to the longitudinal (x), transversal (y) or vertical (z) coordinate of the cat's body, marked on the maps as H-T (head to tail), L-R (left-right) and D-P (distal-proximal). Data from Andersson and Eriksson, 1981; Ekerot and Larson, 1980; Oscarsson, 1969, 1980.

FIG. 12. A: The push-me-pull-you two masses connected by a spring. They can be made to oscillate toward and away from their common center of masses (B, left). If the system is set in motion by a push from the left, the center of mass moves with a constant velocity (oblique straight line, B right) while the masses oscillate against each other. The effect is that each of the masses in turn stops its movement when the other reaches maximal velocity. C: Two motors (Ag, Ant) pulling in opposite direction a carriage c containing a movable mass l suspended between springs. Mommentum imparted to the carriage c is transmitted to the load l which will tend to oscillate when the carriage is stopped by a mommentum in the opposite direction. The stop of the carriage without residual internal oscillation requires acertain sequence of applied force. See text.