Chapter 2 Modeling the cerebellum: From adaptation to coordination

Motor Control and Sensory Motor Integration: Issues and Directions D.J. Glencross and J.P. Piek (Editors) 9 1995 Elsevier Science B.V. All rights reserved. Chapter 2

M O D E L I N G T H E C E R E B E L L U M : F R O M ADAPTATION TO COORDINATION

Michael A. Arbib and Nicolas Schweighofer

Center for Neural Engineering, University of Southern California Los Angeles, CA 90089-2520 W.T. Thach Washington University School of Medicine, Department of Anatomy and Neurobiology, St. Louis MO 63110-1031

We review data showing that the cerebellum is required for adaptation both of saccadic eye movements to consistent shifts in target position and of throwing when the subject wears a wedge prism. We then model the saccade adaptation in terms of plasticity of synapses from parallel fibers to Purkinje cells in cerebellar cortex, stressing the integration of cerebellar cortex and nuclei in microzones as the units for correction of motor pattern generators. The model uses a "window of eligibility" to ensure that error signals that elicit a corrective movement are used to adjust the original movement, not the secondary movement. We also find that correction involves not only adjustment of the original motor pattern generator (MPG) but modulated deployment of other MPGs to yield a successful overall movement. Finally, we extend this model to account for adaptation of throwing.

1. THE ROLE OF C E R E B E L L U M IN ADAPTATION The cerebellum has been implicated in adaptation of the metrics of movement to changing circumstances. In this section, we review two examples of adaptation - - for saccades and throwing - - and briefly note evidence for the role of the cerebellum. In later sections, we will develop a model for this role, arguing that the cerebellum "works"

by modulating and coordinating multiple Motor Pattern Generators (MPGs).

12

M.A. Arbib, N. Schweighofer & W.T. Thach

1.1 Saccade Adaptation Saccades are very fast eye movements of very short duration.

As pointed out by

Robinson (1986), the visual feedback delays are longer (about 40-80ms) than the movement itself (on the order of 50ms); and so saccades cannot be controlled by a normal feedback controller for accurately locating a visual target in the fovea. In a target perturbation experiment, a non-trained monkey (Goldberg et al., 1993) or a human subject (Albano and King, 1989)

has to make a saccadic eye movement

towards a target. During the saccade, the target is shifted to a new position but this shift is not perceived by the subject during the m o v e m e n t -

we speak of "saccadic

suppression". As the first saccade does not end at the new target position, it appears incorrect, and a second, corrective, saccade is generated with a latency comparable to the latency of the first (Albano and King, 1989). [In fact, small errors which are nunified by a single following corrective saccade appear to be a part of the normal human or monkey strategy (Optican, 1982). The reason is that the brainstem saccade generator uses a "noisy integrator", so completion of the saccade does not guarantee that the eye is on target.] However, over a few hundred trials, the amplitude and direction of the initial saccade changes and the amplitude of the corrective saccade decreases until the trained animal can saccade directly to the displaced target.

The gain changes gradually and

recovers gradually, and the gain for similar directions and amplitudes is also changed (Goldberg et al., 1993). The learning curve shows an exponential time course for the adaptation, with recovery apparently faster than learning (Figure la). The influence of the cerebellum and its associated structures on the execution of saccades can be observed in cerebellar patients and monkeys. Ritchie (1976)

made

symmetrical lesions of lobes VI and VII and found that large saccades made toward the primary position (centripetal) were grossly hypermetric, while those made away from primary position (centrifugal) were hypometric. Goldberg et al., (1993) studied monkeys with interpositus and fastigial nuclei lesions. Figure l b shows the lack of learning for the target perturbation experiment for monkeys after lesions of the interpositus and fastigial nuclei. The "learning" curve is a straight line (apart from noise). Moreover, due to the loss of the modulation supplied by the cerebellum, the saccades have a greater amplitude than those before the adaptation runs in the normal monkey and the performance, taken as the variance around the mean curve, is poorer after than before the lesion.

Thus, this results suggest that the

adaptation occurs in a system which includes the cerebellum and that the performance is somewhat degraded by cerebellar lesions. As noted by Ito (1984), the contribution of the vermis may be to modify central command signals executing a saccade, and Noda et al., (1990) showed that the cerebellum is indeed not the primary domain of the signal processing.

Cerebellar

impulses are projected downstream to saccade-programming circuits where visual

13

Modeling the Cerebellum: From Adaptation to Coordination

information has already been converted into motor-commanding signals. The cerebeUar eye movement map does not provide the total saccade command for a given frontocollicular eye movement command, but rather the correction that "modulates" the command issued by the superior colliculus and other regions in response to the retinal input.

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Figure 1. Effect of eerebellar lesion on saecadic adaptation. Each dot is a single trial. Continuous line is a ten trial running average of saccade amplitude plotted against the trial number of the middle of the epoch. (a) Normal monkey. Dashed line is at end of adaptation runs, after which the target is no longer displaced. Co) Monkey with lesion of interpositus and fastigial nuclei. There is no adaptation and performance is degraded. (From Goldberg et al. 1993). This supports the hypothesis that the cerebellum adjusts an MPG rather than being the MPG. Below, we shall further argue that coordination of MPGs is also required for successful saccade adaptation. But first we turn to a second data set, that on adaptation of throwing, to provide a broader challenge for our cerebellar modeling.

1.2 Prism Adaptation of Throwing Martin et al. (in press), in Thach's laboratory, have studied the adjustment of eye, head (gaze), ann and hand in humans throwing a dart or ball at a target while wearing wedge prism spectacles. In throwing, the eyes (and head) fixate the target, and serve as reference aim for the arm. If wedge prism spectacles are placed over the eyes with the base to the fight, the optic path is bent to the subject's fight, and the eyes (and head)

14

M.A. Arbib, N. Schweighofer & W.T. Thach

move to the left in order to see the target so that the arm, calibrated to the line of sight, will throw to the left of the target. But with repeated throws, the calibration will change, and the arm will throw closer to and finally on-target. When, after adaptation, the prisms are first removed, the eyes are now on-target, but the eye-head-arm calibration for the previously left-bent gaze persists: the arm throws to the right of target by an amount almost equal to the original leftward error. With repeated throws, eye-head position and ann synergy are recalibrated: each throw moves closer to and finally on targeL Figure 2 shows the distance of the hit location to left or right of a target before donning the prisms (before the first dashed line), while wearing the prisms (between the dashed lines), and after removing the prisms (after the second dashed line). The initial points are all relatively close to the center of the target; the middle points (wearing prisms) start to the left of center, while the latter points (after prisms) start equally far to the fight. The failure to hit the target center after removal of the lenses comes as a surprise to all subjects in this study, and so it- and, we infer, the original adaptation - is unlikely to be due to a voluntary strategy. adaptation.

Operationally we attribute it to some subconscious

Nevertheless, conscious strategic corrections - "cheating" - are possible.

One subject, after donning the prisms, noted that the first throw hit approximately 40 cm to the left of center but his next throw was fight on target!

After being told "Quit

cheating! Throw to where you see the target, not to where you think the target actually is", his performance then followed the adaptation curve. Another subject had seen from the performance of prior subjects the error which the prisms introduced upon the first throw. She decided she would compensate for this by aiming at a point on the opposite side of the target. But unbeknownst to her, she was given a set of prisms with base to the left rather than to the right. Upon donning them, she threw with a doubled error. Is the adaptation visual and global, or motor and specific for the trained body parts, or somewhere in between?

To address this question, Martin et al. (in press) asked

subjects to make both fight hand and left hand throws, with and without prisms, to see if there was carry-over of adaptation on the one task to the other task. They found that prism adaptation occurred in the throwing arm, did not affect or abate with throws by the other arm, and readapted only during throws by the first arm. Therefore the adaptation could not be considered to be generally of vision, but instead to be to some extent specific for the trained body parts. Given this result, how specific is the adaptation to the task? Does the adaptation of the trained body parts carry over to their use in other tasks, or is it specific for the use of those body parts during the one task only? To address this question, we asked subjects to use the same ann to make underhand and overhand throws, with and without prism adaptation.

Modeling the Cerebellum: From Adaptation to Coordination

15

cm

Throw Figure 2. Performance of a human throwing a ball at a target with and without prisms. The vertical axis of the graph shows the distance of the hit location to left or fight of a target in a series of trials before donning the prisms (up to first dashed line), while wearing the prisms (up to second dashed line), and after removing the prisms. On wearing prisms, all subjects adapted the overhand throw.

For two subjects, the

subsequent underhand throw showed absolutely no effect of prior overhand adaptation.

16

M.A. Arbib, N. Schweighofer & W.T. Thach

In these subjects, prior overhand adaptation persisted in subsequent overhand throws despite intervening underhand throws, and readapted with repeated overhand throws. In 4 subjects, the results were similar, except for an apparent carry-over of the overhand prism adaptation to the first subsequent underhand throw. Nevertheless, this disappeared with the second throw, and it is therefore unclear to what extent this represented an adaptive change.

In these as in the first 2 subjects, the prior overhand adaptation

survived the intervening underhand throws, persisting undiminished in subsequent overhand throws, and readapting only after repeated overhand throws. Two subjects showed persistent carryover from prior overhand adaptation to underhand throws, but only one showed carryover of overhand adaptation to the underhand throws which then readapted without any apparent adaptation left in the final overhand throw. To test if one can learn to store more than the one gaze-throw calibration simultaneously, Martin et al. asked subjects to make 200 throws while wearing the prisms and 250 without each day, 4 days per week for 7 weeks. They measured the progress on the 5th day of each week .with 25 throws before, 100 throws during, and 75 throws after wearing the prisms. This made a total of 900 throws with prisms and 1100 throws without prisms each week for 7 weeks. Over time and practice, the first throw with the prisms landed closer to the target, and the first throw without the prisms (aftereffect) also landed closer to the target. By 7 weeks, throws were on-target for the first trial wearing and the first trial after removing the "known" prisms. This suggests that 2 adaptations (no-prisms and known-prisms) may be stored' simultaneously and separately. A subject who had adapted to one prism behaved as if naive when presented with a novel prism. Both non-prism and "known" prism calibrations were affected; both had to be readapted independently. Prism adaptation in macaques is abolished by cerebeUar lesion (Baizer & Glickstein, 1974). Weiner et al. (1983) gave more detailed results in patients with cerebeUar disease, and showed that adaptation was not impaired by disease of corticospinal or basal ganglia systems. Martin et al. (in press) also applied their paradigm to patients with cerebellar disease.

In a patient who had multiple sclerosis, with tremor and ataxia (no other

deficits), no adaptation was seen after donning and doffing the prisms. With a patient with fight cerebellar hemisphere infarct, tremor and ataxia, little adaptation is seen after donning and doff'rag the prisms. Martin et al. also found that two patients with MRIdocumented inferior olive hypertrophy (a degenerative disease of the inferior olive, which is the exclusive source of the cerebellar climbing fibers) could not adapt, despite otherwise normal performance. Both patients had ataxia of gait (damage of the inferior olive leads immediately to malfunction and ultimately to atrophy of the cerebellum [Strata, 1987; Murphy and O'Leary, 1971] ), but the upper extremity movements were relatively normal. This suggests that the adaptation mechanism could be dissociated at least in degree from those of coordination and performance.

Finally, they studied

Modeling the Cerebellum: From Adaptation to Coordination

17

patients with lesions presumed to involve mossy fibers (see the next section for a description of cerebellar input pathways) of the middle cerebellar peduncle, who also show impaired adaptation (cases of ataxic hemiparesis, with contralateral lesions of the basis pontis involving leg corticospinal and ann pontocerebellar fibers, according to Fisher, 1978). This is not to say that the cerebellar cortex, the inferior olive, and the mossy fibers are equivalent or equipotential in their control of learning, but only that they are all necessary.

Their roles are quite different, but the differences are only to be

revealed by integrated modeling and experiments that asks questions about each. 2. M I C R O C O M P L E X E S AND THE MODULATORY R O L E OF THE CEREBELLUM In modeling the cerebellum, we stress that the cerebellar cortex and nuclei form an integrated system, and we view this system as divided into small structural and functional units inserted into various extracerebellar systems. Figure 3 reproduces one of these units, named a cerebellar corticonuclear microcomplex (Ito, 1984). A microcomplex is composed of a cerebellar microzone and a small number of nuclear cells and receives (to simplify) two kinds of input, mossy fibers and climbing fibers, the output being carried by the deep nuclear cells. Both mossy fibers and climbing fibers supply collaterals to the nuclear cell group as well as passing to the corresponding microzone of cerebellar cortex. The set of mossy fiber inputs are transformed by the granules cells whose axons form the parallel fibers. A typical Purkinje cell may receive input from on the order of 100,000

Microzone

(cerebellar cortex) pf

gc

PC

mossy fibers

input

output error

signal

Figure 3. A corticonudear microcomplex, the structural-functional unit of the cerebellum, involving a patch of cerebellar cortex and the patch of cerebellar nucleus to which its Purkinje cells project, cf, climbing fiber; PC, Purkinje cell; gc, granule cell; 10, inferior olive. (Adaptedfrom Ito, 1984).

18

M.A. Arbib, N. Schweighofer & W.T. Thach

parallel fibers yet will always receive input from only one climbing fiber (the axon of a cell in the inferior olive, IO).

The parallel fibers are long enough (Mugnaini ~'}83) to

provide synapses across many other microzones. The set of parallel fibers crossing a given microzone constitutes a general context for the present sensorimotor actions in the form of a large set of signals providing information about the state of activity of various structures, from the higher level to sensory ones.

The granule cells ensure that the

parallel fibers each carry some combination of activity on several mossy fibers, rather than simply relaying their activity. We now note several crucial facts and hypotheses: 1) The only output cells of a microzone are Purkinje cells (PCs).

As PCs have

inhibitory action upon nuclear cells (while collaterals of mossy fibers excite the nuclear cells), the signal flow from the nuclear ceils is modulated by the microzone action. 2) The climbing fibers are commonly considered as error detectors and evidence has been accumulated for this (see Ito (1990) for a review). Thus, climbing fibers convey signals encoding errors in the performance of the system in which a given microcomplex is installed. 3) Climbing fiber signals induce LTD (long-term depression of synaptic strength) in those parallel fiber~PC synapses which were coactivated with the climbing fibers (within a certain time window). In the next section we will see how this general mode of function can be applied to model the saccadic system. The model is due to Schweighofer et al. (in press), to whom we refer the reader for a more detailed review of the experimental data which ground the model. In Section 4 we will vary that model to account for the adaptation of throwing. 3. MODELING THE ROLE OF CEREBELLUM IN SACCADE ADAPTATION 3.1 The Structure of the Model

Figure 4a shows the overall structure of the model. Goldberg et al. (1993) found that stimulation of SC (the superior colliculus, known to provide a retinotopic control surface for saccades) produces saccades which are not adapted to target perturbations. This result suggests that the path concerned with target location and involving the cerebellum comes from "higher up" than the superior coMculus. It could be from the FEF (frontal eye fields of the cerebral cortex), the posterior parietal cortex, or even the visual cortex or the lateral geniculate nucleus. Schweighofer et al. assume that the cerebro-pontovermal side path for saccade adaptation starts from the FEF, going through a pontine nucleus, then through the lobules Vie and/or VII of the vermis to the FOR (fastigial oculomotor regions), ending up in the parapontine reticular formation (PPRF) where the brainstem saccade generators reside (at least for horizontal movements).

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Figure 4. System views of cerebeUar modulation of the saccadic system and in the adaption of the throw to deviating prisms: (a) The saccadic system must be adaptive. Note that "delayed feedback" (which is relayed via the [not shown] I0) is a form of visual input, but it is segregated from that which serves as input to the non-adaptive pathway. (b) Putative mechanisms of adjustment between eye position and synergy of the muscles in the trunk and arm involved in throwing. Afferent information on eye position arriving in intermediate zone lobulus simplex is carried over parallel fibers to purkinje cells which project to cells in the dentate nucleus which control eye, neck, arm and hand muscle synergies. As in the saccade model, the cerebellum provides a correction to the main pathway, but here the correction is further "upstream", via premotor cortex.

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20

M.A. Arbib, N. Schweighofer & W.T. Thach

From Goldberg et al.'s experiments, we infer the existence of one or several neural maps where adaptation occurs at specific spatial positions, based on the positions of the first target: Indeed, the adaptation is selective to a set of saccades with similar amplitude and direction. To account for the crude "correction map" found in the cerebellar cortex, we will keep the major thesis of the D&A model (Dominey and Arbib, 1992) that a functional topography that preserves saccade direction and amplitude is maintained through multiple projections between brain regions until it is finally transformed into a temporal pattern of activity that drives and holds the eyes onto the target. The preserved topography is a map coding for amplitude and direction of an eye movement vector that, when combined with the current eye location, will center the eye on the target. The Schweighofer et al. model adds an adaptive component, postulating (in line with the experimental data) that saccadic gain change for a particular region of space (around the targe0 will be accomplished within the functional topography of the granule cell layer and the Purkinje cell layer. We now present the essential features of the model, but refer the reader to the original paper for the equations and parameter settings which constitute the formal description of the model A retinotopic map is sent via the pontine nucleus to a set of mossy fibers we call mfret. To simplify, we assume that the pontine nucleus cells (from which the retinotopic mossy fibers arise) are mere relays in which the precise target map is somewhat lost by divergence-convergence. The spread of activity is modeled by a gaussian distribution of weights to form a "blurry ~ topographic connection from the motor layer of the FEF. This "coarse coding" speeds up adaptation (Albus, 1981), allowing one to update a group of cells which are "close" to the selected cell so that learning is thus extended to saccades of similar amplitude and direction, as seen in Goldberg et al.'s experiments. In the model, another set of mossy fibers carries eye position. The position coding cells receive proprioception signals from the oculomotor muscles.

Another factor to be

taken into account is the correlation observed between FOR firing and the saccade duration as well as the anatomical connections from the brainstem SGs (saccade generators) back to the vermis (Yamada and Noda, 1987). These fibers carry a temporal signal, such that the firing of the FOR neurons (as well as PCs) will be somewhat synchronized with the SG's activities. The model represents three types of cerebellar neurons - granule cells, Purkinje cells and FOR (cerebellar nucleus) cells - but no cerebellar interneurons are taken into consideration.

The granule cells generate a statistical distribution of combinations of

mossy fibers carrying retinotopic signals, two position signals and a temporal signal. Each Purkinje cell receives inputs from all the parallel fibers. The parallel fiber - Purkinje cell weights are modifiable.

[For the present study we make the following simplifying

Modeling the Cerebellum: From Adaptation to Coordination

21

assumption: As the climbing fibers, on their way to the cerebellar cortex, send collaterals to the deep nuclei, the excitation of the fastigial neurons by these collaterals will nullify the strong inhibition caused by the complex spikes, which are the response of the PCs to the climbing fiber f'u'ing. The present model thus omits the "real time", as distinct from the "training", role of the climbing fibers.] The PC axons then converge to the FOR, whose cells they inhibit. Collaterals of the mossy fibers also converge to nuclear cells, and give an excitatory projection. It is the output of the nuclear cells that provides the correction signal to the saccadic generators. We must demonstrate that it can be adapted in the fight direction (i.e., corresponding to the corrective saccade) so long as the initial saccade requires correction. We assume that the post-saccadic information available to correct an erroneous motor command is an encoding of the motor activity (or corollary discharge) for a visually guided corrective saccade and thus posit that the IO receives both sensory and motor information: A pre-IO neuron receives visual and motor inputs and has the role of an "error detector". An "error" will be detected if a) a target is on the retina but b) not on fovea, and c) a saccade has just been completed. If these three conditions are fulfilled, the output of this neuron "ungates" the saccadic IO cells. The error detector system comprises three neurons: a "goal" neuron, a "memory" cell and a "pre IO" neuron. The motivation for having a "goal" neuron is the need for a signal that continues until the target acquisition is complete. Thus, this cell starts fuing as soon as the target is not on the fovea and fires until the eye acquires the target: It encodes the goal of the saccadic system. Moreover, when a saccade is generated, some SG signals are sent to a "movement memory" cell. These two neurons project to the "pre IO" neuron. When active, this neuron "ungates" the path from the sensory inputs to the IO. Therefore the corrective saccade will send an error signal to the appropriate microcomplex (direction of the error) with some amplitude information (the IO f'u~.s with a higher probability for a large saccade).

3.2 Adaptation: Problems and Solutions The error information is delayed relative to the efferent signal because the error can only be assessed after the movement has been completed. If, after the first saccade, the absence of a target on the fovea and the presence of a target "nearby" signals an error, then adaptation changes the "gain" of the agonist-antagonist pair of muscles responsible for the first (erroneous) saccade. This implies the need for a short term memory system capable of retaining the appropriate parameters of the first saccade (position on the retina and corresponding eye position). Moreover, the climbing fiber error signal corresponding to the erroneous first saccade reaches the cerebellar cortex after the signal carried by the

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M.A. Arbib, N. Schweighofer & W.T. Thach

parallel fibers corresponding to the second saccade (see below). The problem is to associate the learning not with the second saccade but with the first one. To address these temporal problems, we will assume synapse eligibility

(Klopf,

1982; Sutton and Barto, 1981; Houk et al., 1990). The form postulated by Houk et al. is that activation of a dendritic spine by a parallel fiber leads to the release of a chemical called a second messenger in the spine, where its concentration acts as a short term memory. We say the synapse is eligible if the concentration is above some threshold. If an error signal is provided to the whole cell by the climbing fiber, the resulting increase in Ca ++ in the cell will, it is posited, only affect the eligible synapses, and therefore only their efficacies are changed.

If a parallel fiber-PC synapse participates in synaptic

transmission, it becomes eligible to be weakened by LTD if a climbing fiber signal is sent somewhat later. However, we now add to this view of eligibility the requirement that, when the error signal arrives, the concentration of the messenger should tend to be largest for synapses involved in the initial saccade. We therefore introduce the concept of a "time window of eligibility". The example we chose in our model is a concentration over time having the response of a second order system for the concentration [2nd] of the messenger (although, alternatively, we can imagine a second messenger following a first order equation but with a significant delay). Ideally, the concentration matches in time the occurrence of the error signal and the concentration decays relatively fast to ensure a minimum of interference with the next saccade. In the model there are two 1000 elements weight vectors wltd, one for each PC, as each parallel fiber makes a synaptic contact with each PC. With the assumption that the rise in calcium concentration is rapid compared to the second messenger dynamics, the weight update rule is, at each time step, for the ith synapse and for each PC, d_d__wltdi = - a I 0 12ndli dt Here, a is the learning coefficient, I 0 the binary climbing fiber error signal, and wmax > wltdi 30. But if this equation alone were operative, all the weights would tend to zero.

Thus, we implemented a weight normalization which can be thought of as providing nonspecific LTP (long-term potentiation of synaptic strength), in order to keep the sum of synaptic weights for each PC constant. Cl~e normalization adopted is a subtractive normalization. In simulations we also tried a multiplicative normalization; however, this gave a learning curve somewhat different from the data.) additional increment: ~ i Wltdi n

Each weight receives the

Modeling the Cerebellum: From Adaptation to Coordination

23

where the sum is over all the weights of one PC, and n = 1000, the total number of synapses per PC. This potentiation has no direct functional role as far as the behavior is concerned, except that it somewhat degrades the performance of the whole system. This effect is reduced by the large number of synapses. In the non-adaptive pathway, a light in the upper half of the retina does not elicit a downward movement since the corresponding signals are not transmitted to the "downward" generator. However, to correct for errors, this constrained access is not possible for the adaptive system. For instance, suppose that the first target fails on the upper sector of the map and the second on the lower half. In this case, the first saccade amplitude should be decreased as it is hypermetric.

To compensate for this error, a

decrease of the agonist innervation pulse and an increase of the antagonist pulse is needed. Therefore, the adaptive saccadic system requires more than simple gain control, but adaptation of coordination between the different saccade generators. Consequently, and as seen in the microstimulation experiments, each microcomplex should be able to influence the SGs of the antagonist and agonist muscle.

In other words, for each

direction for the first saccade there are two degrees of freedom: Either adaptation occurs in the same direction or in the opposite direction. Adaptation requires coordination. We refer the reader to Schweighofer et al. (in press) for further specification of this coordination, and for the results given by this model. 4. MODELING THE ROLE OF CEREBELLUM IN ADAPTATION OF DART THROWING

As already exemplified in our model of saccade adaptation, our theory of cerebeUar function posits that the cerebellar output affects motor repertoires resident in movement generators located elsewhere, and that this effect is not only modulatory, controlling the gain of these MPGs, but also combinatorial - mixing motor elements within and across generators such as to adapt old and develop new synergies of multiple body parts. Coordination is effected by the parallel fibers which via Purkinje cell beams spans the width of up to two different body representations within the deep cerebellar nuclei. We concur with the evidence that the parallel fiber-Purkinje cell synapse is adjustable, under the influence of the climbing fiber, and that this is the probable mechanism for changing synergic combinations. In our model of the adaptation of throwing to wearing prisms, the essential adjustment is proposed to be between eye position and synergy of the muscles in the trunk and arm involved in throwing. The target is seen, and the eye foveates and fLxates the center of the target. Afferent information on eye position (not necessarily excluding visual) arrives in the visual and "face" tactile receiving areas in intermediate zone lobulus

24

M.A. Arbib, N. Schweighofer & W.T. Thach

simplex (cf. Snider and Stowell, 1944; Snider and Eldred, 1952). Information is carried over parallel fibers to Purkinje cells located more laterally in the hemisphere which project to cells in the dentate nucleus that control eye, neck, arm and hand muscle synergies. With repeated throws, adjustments are made in the strength of the parallel fiber input to Purkinje cells (and possibly cortical intemeurons - basket, stellate and Golgi cells - but they are not included in the present model) such that the changing output produced from the Purkinje cells in response to the eye position (and visual) input modulates the throw sufficiently for it to hit the target. 4.1. Notes on Neural Coding

We consider the movement to be decomposed into two parts: 1) aiming and then 2) throwing. Only horizontal shift of gaze by the prisms will be considered, and we thus assume that aiming is realized solely by orientation of the shoulder joint in the horizontal plane. Trunk, elbow and wrist rotation are not taken into account in our model. Only the desired position of the shoulder after aiming (but before throwing) is coded, i.e., we do not model the movement itself, but only the horizontal adaptation of the endpoint of aiming. This is supported by the study of Flanders et al. (1992) who showed that the pointing movement to a target is controlled independently in the elevation and horizontal directions. This parcellation may facilitate comparing a target location signal with signals of the limb position so as to yield a motor error signal. If arm position and target location are represented in a common coordinate system (centered at the shoulder), a simple combination between these internal representations may suffice to compute the initial part of the movement. Note that this parcellation representation in arm reaching is earlier, closer to the sensory side of the nervous system than in the saccadic system. Population studies in the premotor and motor cortices show that cells code for all the possible direction of arm movements in 3D space; no separate coding for elevation or azimuth coding by two sub-populations have been found, which might suggest that the two different processing mechanisms would be coded by the same population of neurons. The segregation would therefore be based on functional assemblies and not spatial ones (Bumod and Caminiti, 1992). We model one such assembly, coding for horizontal desired position. We model the arm (Figure 5) with a reference frame centered at the shoulder, with x pointing towards the target, y to the left, and z upwards. In order to reproduce the data, the minimum model of the arm will have three degrees of freedom: vertical and horizontal shoulder rotations (around y and z, respectively) and elbow rotation. We do not model the third degree of freedom at the shoulder joint of the real ann. The rest position is with the ann along - z with the elbow joint maximally extended, while the endpoint of

25

Modeling the Cerebellum: From Adaptation to Coordination

aiming depends on the strategy, i.e., overarm or underarm, as well as the target location (see Figure 5). In both underarm and overarm strategies, the elbow adopts a characteristic angle prior to the throw phase, and we assume it is the shoulder rotation around z that provides the horizontal component of aiming m whose adaptation we study. Arm Configurations

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~x z

Shoulder

Figure 5. Simplified model of the arm. Note the two degrees of freedom for the shoulder and one for the elbow. Adaptation in the present model is limited to the horizontal shoulder rotation. During adaptation, the cerebellum shifts the population coding of the desired shoulder position after aiming by an amount opposite to the initial deviation. If we follow the "Georgopoulos view" (Georgopoulos, Schwartz, and Kettner, 1986) that premotor and motor

cortex

neurons

code

a

"population vector"

S

which

codes

shoulder

direction, it is tempting to posit that the nuclear output is a signal corresponding to a vector

C

in world coordinates which is added to the gaze vector G

"adapted" shoulder planar vector

to yield the

S = G + C. Since the shoulder vector should point

toward the center of the target, adaptation ends when

cos(C, S) = - cos(G, S).

However, when the glasses are taken off, G points towards the target, and C does not

26

M.A. Arbib, N. Schweighofer & W.T. Thach

change until re-adaptation takes place. In the case of a 40 ~ prism, the shoulder will point at -40 ~ yet if S = G + C, the new S would be at - 20-. Therefore this first model cannot account for the data. The actual transformation needed is a vectorial rotation and not a sum. If the cerebellum learns how to rotate the desired shoulder position coding by 40 ~ removal of the lenses will give an error of 40 ~, as in the experiment. The most common neuron considered in neural network modeling forms (some function of) a linear combination of its inputs. However, a rotation is a bilinear combination of its inputs, as the following very simple example shows. Suppose that the neural and world representation of a vector in 2D is the same, i.e., that any vector can be decomposed on an orthonormal basis and that all vectors are unitary. In this case, S, G and C are neural and physical quantities: If c, g and s are the angles between the different vectors and x, the rotation: g ---> s = c + g is given by

cos(s) = cos(c) cos(g) - sin(c) sin(g)

and

sin(s) = sin(c) cos(g) + sin(g) cos(c)

which is a bilinear transformation. The mapping from the 4 inputs to the 2 outputs involve two matrices, of 4 weights each (in our particular case, 4 of these values are 0). Burnod et al., (1991) show that for any two vectorial representation a la Georgopoulos, there exist a bilinear transformation corresponding to a rotation. In the first studies on population coding

(Georgopoulos et al., 1986) it was first

assumed that neurons were coding the direction of the movement. However, it is more likely (Kalaska et al., 1992; Mussa-Ivaldi, 1988; Sanger, 1994) that the recorded cells are encoding arm movement variables (not only direction) related to the shoulder movement during reach, due to a stereotyped coupling of shoulder muscle activity and joint motions to handpath during normal reaching behavior. Consistent with this is the finding by Caminiti et al. (1991) that cortical cells' preferred directions change with initial shoulder angles. It is to be noted that the distribution of preferred direction across cell population is approximately uniform, suggesting that single cells are not confined to coding shoulder motion variables along one of the 3 cardinal degree of freedom of movement of the shoulder joint. Rather the code seems to involve all degrees of freedom but in different ratios for different cells. 4.2 The Structure of the Model

A subject throws where she looks. Before the throw, head and trunk are "towards the target". This ability, under normal circumstances, does not involve the cerebellum, as is shown by the "accurate" throwing (the mean is on target, though the variance is high) by the cerebellar patient when not wearing prisms. Yet, from the inability of cerebellar

Modeling the Cerebellum: From Adaptation to Coordination

27

patients to adapt their throw to prisms (Martin et al., in press), we infer that the shoulder area of the dentate - - which is part of the lateral cerebellar system B is necessary for the aiming adaptation, in accord with the putative role of the dentate in planning and preparation of the movement. Moreover, the cerebellum is involved in a side path projecting to the premotor cortex, area 6, via the ventral thalamus (Shinoda et al., 1992). As shown by the underarm vs. overarm experiments, the adaptation is not a recalibration of the visual coding of space, i.e., the transformation from eye-head coordinates to body-centered coordinates (which would be task neutral) but acts on a direct transformation of coordinates specific to the task.

Each motor schema is a

controller with its own more or less private coordinate system. Figure 4b gives an overview of the system:

The premotor cortex "prepares"

information on desired horizontal shoulder position for the motor cortex.

Since the

lateral cerebellum projects (via the dentate nucleus) to premotor cortex, we postulate that it is this cerebellar signal that adjusts the premotor cortex appropriately to changing circumstances (e.g., prisms), and we here model how the cerebellar circuitry can adapt on the basis of a delayed error signal provided by vision of where the dart lands relative to the target. The system (human + dart) is an open loop system since the error in dart throwing is available only after the movement and so the error cannot be used to correct the given movement; however, over trials, the correct match between gaze and throw is learned. Before throwing, the subject foveates the target and therefore the internal representation of gaze, and hence of the desired shoulder position (before it undergoes adaptation), is changed. This information is coded in a distributed manner, providing robustness to lesion and noise: A signal distributed over many noisy nonlinear channels may be summed to yield an accurate signal. This system has been modeled using leaky integrator neurons in our NSL simulation environment. To reproduce all the experiments, the mossy fiber inputs that we consider are (Figure 6):

(~)

(b)

1 Figure 6. This row of cells shows the mossy fiber inputs. Three different moralities are encoded, each carded by a distinct sub-population of neurons. From right to left, the peaks of mossy fiber activity encode the horizontal shoulder position (derived from the horizontal gaze direction) in the case (a) of the 25~ prism on and (b) for a case without prisms; the vertical shoulder position which is in overhand position; and the cognitive inputs which corresponds to "mental set" (e.g., the knowledge that one is wearing prisms). There are 180 neurons, each f'Lringin the 0 - 60 spike/s range.

28

M.A. Arbib, N. Schweighofer & W.T. Thach

1) Desired arm configuration at the end of aiming. This position is calculated from eye position muscles or from a corollary discharge for the control of gaze. aiming, proprioceptive inputs of the end-point are not available.

Before

Instead, a desired

vertical shoulder position should be available to the cerebellum via mossy fibers (this is coherent with fact that there are no direct inputs from the periphery to the lateral cerebellum). This desired position is well learned and cerebellar patients can throw well if no adaptation is required. 2) Desired vertical shoulder position (to distinguish underarm from overarm throws). 3) Cortical projections for some form of "mental set".

This does not have to

explicitly code knowledge for the present purpose, but must differ depending on whether the subject is or is not wearing prisms, and whether the prisms are known or unknown. This input is necessary to explain the ability of highly practiced subjects to immediately switch "gain" when donning or removing known prisms.

The cortical input to the

cerebellar cortex is known to be large. Indeed, the cortico-pontine fibers form a very large group which arises from the whole cerebral cortex. The data shows that "prism knowledge" has a large influence on the response, and this knowledge has an orthogonal representation at the parallel fiber level. Orthogonality of the inputs is a important issue in this model, as will be discussed in conclusion. We have earlier seen that the adaptation should act by coding a rotation of the shoulder vector, rather than providing a correction vector which is to be added on. The model provided here combines the gaze vector G with a rotation signal coming from the ventral thalamus using neurons (postulated to be in premotor cortex) which perform not only additions but also some kind of multiplication. Such neurons have been proposed on theoretical grounds (the sigma pi neurons of Rumelhart et al., 1986); and highly nonlinear neurons such as the large pyramidal cells or Purkinje cells are supposed to have dendritic spikes which allow them to function as event detectors (Andersen et al., 1987; Burnod et al., 1991; Gluck et al., in press). It is to be noted that the transformation need not be too accurate and that learning and modulation can compensate for non-perfectly realized mathematical requirements. The algorithm we use to perform this rotation was developed by Hoff (1987, unpublished work) and uses sigrna-pi neurons. Burnod et al., (1991) reached a similar result using cortical columns, whose output are a combination of sums and product of the inputs. We have found that similar computations using Gaussian representations of the variables yields comparable results, and that even spatial "humps" of activities yield satisfying results. We next turn to the error detection system. With Flanders et al. (1992), we assume that the target location and arm position share the same coordinate system, with error "in

Modeling the Cerebellum: From Adaptation to Coordination

29

register" with the shoulder position. A leftward error activates a "leftward" group of IO cells. These cells receive a weighted retinotopic projection from the retina so that a large error will give rise to several spikes. Therefore, the climbing fibers fire to give the direction and amplitude of the error given by a visual projection to the IO which retains some retinotopy. A large error will activate (with a certain probability) different PCs than will a smaller error. However, there is a gradient of cf firing activities within each microzone. In a more complete model, vertical shoulder position should also be able to undergo adaptation (to vertical prisms for instance, or muscle lesions, aging etc.). The same parallel fiber set would therefore overlap four microzones instead of the two in the present model and the IO would encode horizontal as well as vertical errors.

4.3. Adaptation The purely feedforward nature of the movement (the error is not corrected during the movement as the error is not known before the dart hits the wall holding the target) and the delay between motion generation and error detection require again the concept of eligibility Q especially since the throw is made in between the aiming and the receipt of the error signal. In the present case, we use a variant of the eligibility model developed for saccadic adaptation. With wltd the vector of adjustable weights for parallel fiber-PC synapses, the synaptic adjustment rule we consider is: dwltd

=

{- c.lO + d.(1 - 10) }.[2nd]

dt with wmax>wltd > 0, c>>d>0, 10 the binary climbing fiber error signal, c the learning coefficient (LTD) and d the "forgetting" coefficient. Some kind of LTP in the learning rule is necessary: If only LTD were occurring, all the weights would tend to zero. The difference from the saccade rule is using the d term to achieve this. As the climbing fiber fh-es only if an error occurs, the weights increase most of the time, but very slowly since c>>d. As a consequence, if the system is tuned at a certain moment, the increase of the weights will sooner or later induce an error. One (or a few) corrective saccades will then be generated, and because c>>d, the weights soon regain the correct values. Each variant of the eligibility-based learning rule has certain advantages, and it is a topic of current research both to explore their properties formally and to define new experiments which can better determine the rule that best describes plasticity of these synapses in the real cerebellum.

r~

Figure 7. These three simulations show the spatial behavior of the cerebellar neurons in the course of adaptation to 25 ~ prisms and underhand throwing with a known prism: Each row graphs the activity of a layer of cells just prior to the throw. First row: The mossy fiber input. Second row: The fETingrates of 15 PCs (vertical scale 0-110 spikes/s, as in the following groups of neurons). Third row: The nuclear cells firing rates. The nuclear cells are inhibited by the PCs, and are driven by a high background rate. Fourth row: The response of the 20 thalamic neurons. Last row: The 40 premotor neuron activities. The shoulder position is derived from the premotor layer activity by the population vector transformation. (a) Situation before the first throw with prism on (corresponding to (a) in figure 6). The apparent uniform background rate of the PCs is due to random connections from the parallel fibers with initial random weights. The background firing rate in the nucleus provides facilitation to the premotor activity which is deviated from the middle due to the gaze input. The premotor activity corresponds to a 25 ~ angle between the shoulder and the forward direction. (Note that the activity peak doesn't represent the shoulder direction, as the latter is given by the "center of mass"). Co) End of adaptation to a prism. The depressed PC activities (on the left) release activity in the nuclear cells and in the thalamic neurons. The premotor distributed activity is pushed back in the middle (the corresponding shoulder position is 0~ (c) After re-adaptation to the non-prism situation (corresponding to (b) in figure 6). The PC layer shows another depression on the right side: It is not forgetting but re-learning that occurs. Also note the quasi-constant thalamic total activity (during adaptation) due to the negative feedback achieved by the reticular thalamic complex, which results in contrast enhancement and consequently in a better shift of the premotor "hump". The shoulder angle is 0 ~ again.

Modeling the Cerebellum: From Adaptation to Coordination

31

4.4. Results

Simulated prism adaptation experiments are shown in Figures 7-9. Note that with a model performing a pure rotation, the off-prism initial error is exactly opposite to the initial with- prism error (Figure 7). Both forgetting and relearning are present in the system: forgetting is included in the learning rule and is due to normalization. If there is no climbing fiber activity at a particular site, the weights are very slowly increased. As the error signal carded by the climbing fibers decreased the weights, there is forgetting of the previously learned patterns. By contrast, re-adaptation to the non-prism situation after adaptation to prisms is due to learning rather than forgetting, as can be seen on Figure 7c. In the PC layer, there is a large depression after complete re-adaptation. Figure 8 shows the trial-by-trial behavior corresponding to the neural adaptation shown in Figure 7 for three of the trials. Experience with a -25 degree prism

25 ......

15 A

lO

l's lID

Q

| ~.~ O

-15

~

trials

Figure 8. Simulated 25~ prism adaptation experiments. This figure and the following show the adaptation of angle between the shoulder angle (derived from the premotor activity) and the "forward" direction after aiming over trials. The adaptation requires 20 trims, somewhat more than re-adaptation to the non-prism condition. Note that the irregularities in the curve are due to the probabilistic firing of the climbing fibers. Figure 9 show the Over/Underhand experiments. Note that in this case the vertical shoulder position is different for the two throwing strategies. The model reproduces the experimental data reported by Thach: There is some transfer from overhand to underhand in

M.A. Arbib, N. Schweighofer & W.T. Thach

32

the prism adaptation.., though in some subjects there is no transfer, while in others the transfer is total. We adopted a middle ground with some overlap in the mossy fiber inputs between the two positions and a not too large mossy fiber input for the vertical shoulder position. Overhand I Underhand

o 0 G S o -S

J

0

"= - I0 Q

,'o

I

i

I

30

40

,

20

SO

trials

Figure 9. Over/Underann experiment. In this case, the vertical shoulder position is different for the two throwing strategies. The fu'st part of the learning curve shows adaptation with an overhand strategy. After the prisms are removed, an underhand throw strategy is used. Then 0ast peak) throwing is made overhand again. The model reproduces a typical case reporteal by Martin et al: There is only partial transfer from overhand to underhand in the prism adaptation. 5. D I S C U S S I O N The proposed model embeds the cerebellum in a very general framework, applying to both saccade adaptation and dart throwing.

Even though the direct mapping from sensory to

motor output is somewhat plastic, adaptation to novel context does not occur reliably without the cerebellum. Our model uses the same cerebellar model for two different types of adaptation, with a similar coding of the error n

but in the dart throwing model the

cerebellum projects "upstream" to the premotor cortex instead of "downstream" to the brainstem. The microzone concept holds in both cases.

The unification of diverse

information - from sensory signals to cerebral codes for "mental set" - is made possible by the large number of granule cells, each of which forms a sample of diverse mossy fiber signals. Coordination of the modulation of different MPGs is made possible by the long length of the parallel fibers, overlapping different microzones. One sees a tendency for the

Modeling the Cerebellum: From Adaptation to Coordination

33

cerebellum to be less and less concerned with the actual movement when moving from vermis to the "new" lateral hemispheres. Finally, we note that the concept of eligibility addresses a key problem in the analysis of adaptive behavior: how can reinforcement or error signals to a network affect those cells which were active some time earlier? Extending the ideas of Klopf, Sutton and Barto, and Houk, we suggest that a short-term memory internal to individual synapses may provide a "window of eligibility" when the delay between activity and feedback is on the order of a few hundred milliseconds. Although it takes us beyond the reach of the present study, we note that a more explicit form of short-term memory seems required to link events more widely separated in time.

Important clues for future modeling, and for the design of

adaptive systems, may come from the phenomenon of trace conditioning. Here, an animal without cerebellum cannot be conditioned in a simple conditioned response; and an animal with cerebellum and without hippocampus can be conditioned only if the delay between unconditioned and conditioned stimulus is at most a few hundred milliseconds. The animal must have both cerebellum and hippocampus intact if it is to be conditioned when this delay is much longer (Moyer, DeYoe, and Disterhoft, 1990). The hypothesis is that the hippocampus holds a trace during the intervening period, bringing yet another neural network into play.

ACKNOWLEDGMENTS The research at USC was supported in part by Grant N00014-92-J-4026 from the Office of Naval Research for research on "Cerebellum and the Adaptive Coordination of Movement".

Appendix: List of Abbreviations cf Climbing fiber D&A

Dominey and Arbib model

FEF

Frontal eye fields

FOR

Fastigial oculomotor regions

LTD

Long term depression

LTP

Long term potentiation

SC

superior coUiculus

PPRF

Paramedian pontine reticular formation

IO

Inferior olive

SG

Saccade generator

PC

Purkinje cell

34

M.A. Arbib, N. Schweighofer & W.T. 1bach

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