Barto, A. G.; Fagg, A. H.; Sitkoff, N., and Houk, J. C. A cerebellar model of timing and prediction in the control of reaching. Neural Computation. 1999; 11:565-94.
Abstract:  A simplified model of the cerebellum was developed to explore its potential for adaptive, predictive control based on delayed feedback information. An abstract representation of a single Purkinje cell with multistable properties was interfaced, using a formalized premotor network, with a simulated single degree-of-freedom limb. The limb actuator was a nonlinear spring-mass system based on the nonlinear velocity dependence of the stretch reflex. By including realistic mossy fiber signals, as well as realistic conduction delays in afferent and efferent pathways, the model allowed the investigation of timing and predictive processes relevant to cerebellar involvement in the control of movement. The model regulates movement by learning to react in an anticipatory fashion to sensory feedback. Learning depends on training information generated from corrective movements and uses a temporally asymmetric form of plasticity for the parallel fiber synapses on Purkinje cells.

Braitenberg, V. The cerebellar network: attempt at a formalization of its structure. Network. 1993; 411-17.
Abstract:  The well defined histology of the cerebellar cortex makes it possible to translate its structure directly into statements about the transformation of cerebellar input into output.  In the system of 'parallel fibres' the input activity is shifted in space and time in an anisotropic way.  One consequence of this is that input patterns moving across the cerebellar cortex at a velocity corresponding to that of conduction in parallel fibres should elicit a much larger response that static input.

Braitenberg, V.. The cerebellum revisited. J.Theoret.Neurobiol. 1983; 2237-241.
 

Braitenberg, V.; Heck, D., and Sultan, F. The detection and generation of sequences as a key to cerebellar function. Experiments and theory. Behav.Brain Sci. 1997; 20(2): 229-45
Abstract:  Starting from macroscopic and microscopic facts of cerebellar histology, we propose a new functional interpretation that may elucidate the role of the cerebellum in movement control. The idea is that the cerebellum is a large collection of individual lines (Eccles's "beams": Eccles et al. 1967a) that respond specifically to certain sequences of events in the input and in turn produce sequences of signals in the output. We believe that the sequence-in/sequence-out mode of operation is as typical for the cerebellar cortex as the transformation of sets into sets of active neurons is typical for the cerebral cortex, and that both the histological differences between the two and their reciprocal functional interactions become understandable in the light of this dichotomy. The response of Purkinje cells to sequences of stimuli in the mossy fiber system was shown experimentally by Heck on surviving slices of rat and guinea pig cerebellum. Sequential activation of a row of eleven stimulating electrodes in the granular layer, imitating a "movement" of the stimuli along the folium, produces a powerful volley in the parallel fibers that strongly excites Purkinje cells, as evidenced by intracellular recording. The volley, or "tidal wave," has maximal amplitude when the stimulus moves toward the recording site at the speed of conduction in parallel fibers, and much smaller amplitudes for lower or higher "velocities." The succession of stimuli has no effect when they "move" in the opposite direction. Synchronous activation of the stimulus electrodes also had hardly any effect. We believe that the sequences of mossy fiber activation that normally produce this effect in the intact cerebellum are a combination of motor planning relayed to the cerebellum by the cerebral cortex, and information about ongoing movement, reaching the cerebellum from the spinal cord. The output elicited by the specific sequence to which a "beam" is tuned may well be a succession of well timed inhibitory volleys "sculpting" the motor sequences so as to adapt them to the complicated requirements of the physics of a multijointed system.
 

Bullock, D.; Fiala, J. C., and Grossberg, S. A neural model of timed response learning in the cerebellum. Neural Networks. 1994; 7(6/7):1101-1114.
 

Buonomano, D. V. and Mauk, M. D. Neural network model of the cerebellum: temporal discrimination and the timing of motor responses. Neural Comput. 1994; 6:38-55.
Abstract:  Substantial evidence has established that the cerebellum plays an important role in the generation of movements. An important aspect of motor output is its timing in relation to external stimuli or to other components of a movement. Previous studies suggest that the cerebellum plays a role in the timing of movements. Here we describe a neural network model based on the synaptic organization of the cerebellum that can generate timed responses in the range of tens of milliseconds to seconds. In contrast to previous models, temporal coding emerges from the dynamics of the cerebellar circuitry and depends neither on conduction delays, arrays of elements with different time constants, nor populations of elements oscillating at different frequencies. Instead, time is extracted from the instantaneous granule cell population vector. The subset of active granule cells is time-varying due to the granule--Golgi-- granule cell negative feedback. We demonstrate that the population vector of simulated granule cell activity exhibits dynamic, nonperiodic trajectories in response to a periodic input. With time encoded in this manner, the output of the network at a particular interval following the onset of a stimulus can be altered selectively by changing the strength of granule -> Purkinje cell connections for those granule cells that are active during the target time window. The memory of the reinforcement at that interval is subsequently expressed as a change in Purkinje cell activity that is appropriately timed with respect to stimulus onset. Thus, the present model demonstrates that a network based on cerebellar circuitry can learn appropriately timed responses by encoding time as the population vector of granule cell activity.

Pavlasek, J. Timing of neural commands: a model study with neuronal networks. Biological Cybernetics. 1997 Nov; 77(5):359-65.
Abstract:  Networks constructed of biologically realistic model neurons (neuroids) were used to study how in a neural assembly using pulse (interval)-coded information slow rhythmical oscillations with possible mode transitions might occur and how the efferent commands might be structured and their phase-shifts created. The simulations show that slow oscillations (in the hertz range) can be derived from reverberatory spiking in relatively short closed loops (fewer than ten neuroids) with the inputs protected against disturbing afferent signals and the outputs coupled by convergence on a common neuroid. Slow oscillations can be modified by a tonic activity entering the network; this activity changes the transmission time in the coupled loops involved. The structuring of the regulatory commands (in the millisecond range) was achieved by simulation of sequential activity propagation in a non-ring neuronal assembly supervised by a tonic activity in a set of inputs. The tonic activity acted as an instructive signal influencing the pattern of the functional connectivity in such a way that a particular efferent command was generated by the instructed network.

Perrett, S. P.; Ruiz, B. P., and Mauk, M. D. Cerebellar cortex lesions disrupt learning-dependent timing of conditioned eyelid responses. J.Neurosci. 1993; 131708-1718.
Abstract:  Among the many issues surrounding the involvement of the cerebellum in motor learning, the relative roles of the cerebellar cortex and cerebellar nuclei in Pavlovian conditioning have been particularly difficult to assess. While previous studies have investigated the effects of cerebellar cortex lesions on the acquisition and retention of conditioned movements, we have examined the effects of these lesions on the timing of Pavlovian eyelid responses. The rationale for this approach arises from previous studies indicating that this timing is a component of Pavlovian eyelid responses that is learned and that involves temporal discrimination. To permit within-animal comparisons, rabbits were trained to produce differently timed responses to high- and low-frequency auditory conditioned stimuli (CSs). Before the lesion the conditioned responses to both CSs were appropriately timed--each peaked near the time at which the unconditioned stimulus was presented for that CS.  However, after the lesion both CSs could elicit similarly timed conditioned responses that peaked inappropriately at very short latencies. The changes in responses timing were sensitive to the size of the lesion, particularly its rostral-caudal extent. Similar results were obtained in animals trained with one CS, indicating that the disruption of response timing is not related to impaired auditory discrimination. Because response timing is learned and therefore requires synaptic plasticity, these data suggest that there are at least two sites of plasticity involved in the motor expression of Pavlovian eyelid responses. Plasticity at one site is necessary for the learned timing of conditioned responses, while plasticity at another site is revealed by the inappropriately timed responses observed following removal of the cerebellar cortex. This lesion-induced dissociation of the expression of motor responses and their learned timing supports a synthesis of competing views by suggesting that motor learning involves both the cerebellar cortex and cerebellar nuclei. We hypothesize that motor learning involves a decrease in strength of the granule cell-Purkinje cell synapses (e.g., Ito and Kano, 1982) in the cerebellar cortex and an increase in strength of the mossy fiber-cerebellar nuclei synapses (e.g., Racine et al., 1986). Finally, these data suggest that the cerebellar cortex may mediate the temporal discriminations that are necessary for the learned timing of conditioned responses.

Sugihara, I.; Lang, E. J., and Llinás, R. Uniform olivocerebellar conduction time underlies Purkinje cell complex spike synchronicity in the rat cerebellum. J.Physiol.(Lond.). 1993; 470243-271.
Abstract:  1. The issue of isochronicity of olivocerebellar fibre conduction time as a basis for synchronizing complex spike activity in cerebellar Purkinje cells has been addressed by latency measurement, multiple-electrode recording and Phaseolus vulgaris leucoagglutinin (PHA-L) tracing of climbing fibres in the adult rat. 2. The conduction time of the olivocerebellar fibres was measured by recording Purkinje cell complex spike (CS) responses from various areas of the cerebellum. The CSs were evoked by stimulating the olivocerebellar fibres near the inferior olive. In spite of a difference in length, as determined directly by light microscopy, the conduction times of different climbing fibres were quite uniform, 3.98 +/- 0.36 ms (mean +/- S.D., n = 660). 3.  Multiple-electrode recording of spontaneous Purkinje cell CS activity was employed to study the spatial extent of CS synchronicity in the cerebellar cortex. Recordings of CS were obtained from Purkinje cells located on the surface and along the walls of lobule crus 2a. The rostrocaudal band-like distribution of simultaneous (within 1 ms) CS activity in Purkinje cells extended down the sides of the cerebellar folia to the deepest areas recorded (1.6-2.6 mm deep). As shown in previous experiments, the distribution of simultaneous CS activity did not extend significantly (500 microns) in the mediolateral axis of the cerebellar cortex. 4. In two animals a detailed determination of the length of the olivocerebellar fibre bundles was performed by staining the fibres with PHA-L injected into the contralateral inferior olive. This measurement included fibre bundles terminating in twenty-six different areas, ranging from the tops of the various folia to the bottoms of the fissures in both the hemisphere and the vermis. There was a 47.5% difference between the length of the longest measured fibre bundle (15.8 mm, terminating in lobule 6b, zone A) and the length of the shortest measured fibre bundle (8.3 mm, terminating in the cortex at the base of the primary fissure, zone D), after correction for tissue shrinkage. To attain an isochronous conduction time the conduction velocities for these two fibre bundles were calculated to be 4.22 m/s and 2.37 m/s, respectively. 5. By interpolating between measured points a simple formula was derived to estimate the average length of olivocerebellar fibres terminating in any given area of the cerebellar cortex, excluding the paraflocculus, the flocculus and the most lateral regions of the hemisphere. 6. We investigated the most likely mechanisms by which conduction velocity variations with length could result in global isochronicity.  We found that longer branches tended to have thicker diameters than shorter branches, indicating more rapid conduction velocities; however, other parameters such as internodal distance could not be unambiguously determined.  7.  We conclude that the isochronicity of the climbing fibre activation of Purkinje cells is the result of differential conduction velocity in the climbing fibre system such that longer fibres conduct faster than shorter ones.  Thus while the cerebellar cortex is a deeply folded structure, the conduction time for the climbing fibre system is tuned such that the cortex functions, in the time domain, as an unfolded surface.