## Introduction

Synchronized movements to periodic events, such as dancing to music or clapping hands, are common in daily life. Such movements require information processing in the sensorimotor areas in the cerebral cortex, the basal ganglia, and the cerebellum1,2,3,4. In particular, when the cerebellum is damaged, rhythmic movements cannot be performed with a precision of tens to hundreds of milliseconds5,6. This is not surprising since the cerebellum is involved in regulating the timing of contraction of multiple muscles during motor execution, thereby enabling coordinated movements7,8. The cerebellum has also been shown to be important in the initiation of movement, especially its timing9,10,11. In theories of motor control, the essential role of the cerebellum is to generate internal models that allow for predictive motor control12. The internal models are formed by experience and updated through learning to maintain accurate movement13. This adaptive learning mechanism by the cerebellum optimizes not only the magnitude and direction of movement14,15,16 but also its timing17,18.

In addition to motor control, when synchronizing with external rhythms, it is also necessary to detect the periodicity of events and make sensory predictions accordingly. The importance of the basal ganglia in such processes has long been recognized. Evidence from clinical cases19,20, functional imaging with non-motor rhythmic tasks21,22, and physiological experiments in primates23,24 all suggest a role for the basal ganglia in beat-based timing25. Recent studies have also shown that the cerebellum is involved in rhythm processing. For example, studies using magnetoencephalography26 and experimental animals27,28 have demonstrated that rhythm perception without movement involves periodic entrainment of neuronal activity in the cerebellum and its connected brain regions. Such activities might be related to temporal attention29,30. Furthermore, the cerebellum has been shown to be involved in predicting the trajectory of visual stimuli in the absence of movement, indicating that the cerebellum also generates internal forward models for purely sensory events31,32,33. Thus, the cerebellum is likely to be responsible for information processing necessary for synchronized movements, including sensory prediction of periodic events, predictive motor control based on these internal representations, and error monitoring for updating the forward models. Since the cerebellum forms loop circuits with various areas in the cerebral cortex and therefore contains multiple functional modules34,35,36, it is possible that these functions are processed in parallel through the relevant networks. However, it remains largely unknown how the cerebellum represents and processes information for synchronized movements.

The present study aimed to investigate this in monkeys performing a series of predictive saccades to targets that alternately appeared on the left and right at regular intervals. Although predictive movements synchronized with external rhythms were previously thought to be behaviours specific to vocal learning species such as humans, songbirds, and dolphins37, it has recently been shown that monkeys also perform synchronized movements when reinforced with immediate rewards38,39. Human functional imaging studies using a similar task have reported increased regional blood flow in the cerebellar crus I during predictive (synchronized) saccades compared to reactive saccades40. We examined single neuron activity during synchronized saccades from the posterior portion of the cerebellar dentate nucleus, which receives input from the lateral lobules of the cerebellum and is involved in self-initiated eye movements41,42. We found that many neurons exhibited activity correlated with saccade timing and temporal errors, and some responded to eye movements in both directions with enhanced activity during predictive synchronized saccades. The results of electrical stimulation applied to the recording sites indicated that these signals were causally related to the timing of synchronized movements. The posterior portion of the cerebellar dentate nucleus may be part of multiple functional modules involved in the acquisition of internal rhythms, predictive motor control, and error detection.

## Results

### Classification of neurons and recording sites

To characterize the properties of the three types of neurons, we firstly calculated the directional index (DI, Methods) for the data of 550 ms SOA. Unilateral neurons showed strong directional selectivity (−0.82 ± 0.13, s.d. and 0.73 ± 0.19 for neurons with contralateral and ipsilateral preferences, respectively), while Bilateral neurons (−0.08 ± 0.31) and Postsaccade neurons (−0.16 ± 0.46) showed only weak directional modulation (Supplementary Fig. 3a). The distribution of the DI did not differ between the Bilateral and Postsaccade neurons (Tukey test, P = 0.78), while Unilateral neurons showed a different distribution from the others (P < 0.01). When the mutual information (MI) was computed to quantify the directional selectivity for each neuron (Supplementary Methods), the MI and DI correlated well (r = 0.94), and similar results were obtained (Supplementary Fig. 3d).

We also found that the time course of preparatory activity differed between groups, with the peak timing of Unilateral neurons (−140 ± 76 ms, s.d.) being preceded by saccades to a greater degree than that of Bilateral neurons (−58 ± 87 ms, unpaired t-test, t63 = 4.06, P = 0.0001, Supplementary Fig. 3b). To further assess the time course of preparatory activity, the timing and magnitude of peak activity and the slope of ramping activity were compared across SOAs (Supplementary Fig. 4). Although many individual neurons showed significant changes for different SOAs, the time course of the population activity of Bilateral neurons exhibited a significant change only in the slope of ramping activity and that of Unilateral neurons in the slope and peak timing. When the preparatory activity was temporally scaled, the time course of activity for different SOAs appeared to be very similar for both types of neurons (Supplementary Fig. 4c). The temporal scaling of preparatory activity was consistent with the previous studies in the medial frontal cortex44,45,46 and striatum46,47, as well as those in the cerebellum in the range of hundreds of milliseconds42.

The recording sites of these groups of neurons differed in the dorsoventral direction (one-way ANOVA, F3,94 = 6.14, P = 0.0008, Supplementary Fig. 1b) but not in the anteroposterior (F3,94 = 1.22, P = 0.31) and mediolateral directions (F3,94 = 1.96, P = 0.13). Postsaccade neurons and Unilateral neurons with a contralateral directional preference were predominantly distributed dorsally in the dentate nucleus, while Unilateral neurons with an ipsilateral preference and Bilateral neurons were widely distributed. This suggests that these neurons may send information to different areas in the brain and are involved in different aspects of behavioural control.

### Enhancement during motor synchronization

Previous functional imaging studies have shown that cerebellar activity increases during synchronized eye movements40. We also found that some neurons in the cerebellar nuclei exhibited differential activity between the synchronized (predictive) and reactive saccades. In the example Bilateral neuron shown in Fig. 2b, the activity was greater during synchronized than reactive saccades. In contrast, in the case of the Unilateral neuron shown in Fig. 2c, the activity before ipsilateral saccades was comparable between the conditions. To compare the activity of individual neurons between conditions, the prediction index (PI) was calculated for each neuron during saccades in the preferred direction (550 ms SOA, Methods). The PI is positive if the firing modulation for synchronized saccades is greater than that for reactive saccades, and negative in the opposite case. The PI averaged 0.10 ± 0.16 (s.d.), 0.01 ± 0.09, and 0.03 ± 0.11 for Bilateral, Unilateral, and Postsaccade neurons, respectively, and only the PIs for Bilateral neurons were significantly greater than zero (one sample t-test, t30 = 3.36, P = 0.002). We obtained similar results for different SOAs, with the PIs for only Bilateral neurons being greater than zero (P < 0.02, Supplementary Fig. 5b). These results suggest that the elevated activity during synchronized saccades found in previous studies might be related to the increased activity of Bilateral neurons (Fig. 2a, top panel). We also found a significant negative correlation between the DI and PI for all neurons (r = −0.31, P = 0.003, Supplementary Fig. 3c), indicating that neurons with weak directionality tended to have greater activity during synchronized saccades than during reactive saccades.

### Neural correlations with the next saccade timing and the current temporal error

We also performed decoding analysis to see if information about saccade timing and task condition can be extracted from each neuron population (Supplementary Methods), and found that all types of neurons carried significant information of the ISI, temporal error and other task conditions at different timing (Supplementary Fig. 9).

### Separation of sensory and motor responses

Recent studies have shown that the cerebellum is also involved in sensory prediction without movement execution27,31. Is the activity of the different groups of neurons associated with periodic sensory prediction necessary for synchronized saccades, or solely associated with motor execution? Since synchronized saccades are made at the predicted timing of target presentation, it is difficult to separate them. To address this, we focused on neuronal activity during the transition from reactive to synchronized saccades at the beginning of each trial (Fig. 1c), where sensory prediction may precede saccade execution.

The Unilateral neuron shown in Fig. 5a displayed strong activity before leftward saccades, with the peak of the activity gradually preceding the target as the saccade reaction time shortened for the first, third, and fifth leftward targets (vertical red lines). In the right panel, the activity aligned with the first, second, third, fourth, and subsequent targets and saccades show that this neuron consistently exhibited increased activity before leftward saccades (the second column with orange shading). In contrast, the Bilateral neuron in Fig. 5b exhibited a transient decrease in activity after the first target onset, followed by an increase in activity that peaked at the time of the second target (the first vertical blue line). Subsequently, the activity peaked around the time of the target presentation with some directional selectivity. When the data were aligned with either target onset or saccades (right panel), the neuronal activity was more consistent with the target appearance than with saccade initiation.

To assess whether the activity of each neuron was better aligned with sensory prediction or saccade, we calculated the sensorimotor index (SMI, Methods) for the second saccades. The SMI was negative if the neuronal activity was associated with saccades and positive if it was associated with the visual stimulus, and the index was calculated to be −0.90 and 0.27 for the neuron shown in Fig. 5a and b, respectively. As shown in Fig. 5c, the SMIs of most Unilateral and Postsaccade neurons were negative, with means (s.d.) of −0.45 ± 0.34 and −0.33 ± 0.40, respectively. In Bilateral neurons, about half showed a positive SMI, with a mean of −0.04 ± 0.41, which was significantly greater than that of the other types of neurons (Tukey test, P < 0.02). The SMI calculated for saccades in the non-preferred direction also differed between Postsaccade and Bilateral neurons (unpaired t-test, t46 = 3.52, P = 0.001, Supplementary Fig. 10b). These results suggest that Bilateral neurons may play a role in sensory prediction whereas Unilateral and Postsaccade neurons are involved in motor control.

To further confirm these findings, we performed three additional analyses. First, the similarity of the time course (rather than the magnitude) of neuronal activity was examined by calculating a correlation coefficient between the data for the second reactive saccades and later synchronized saccades (>5th). When the data were aligned with saccades, Unilateral neurons showed more consistent response than Bilateral neurons (Supplementary Fig. 10c, left panel). However, when the data were aligned with the target onset, Bilateral neurons exhibited more consistent response than Unilateral neurons (right panel).

Second, we applied the time-warping analysis that was previously used to dissociate sensory from motor responses during synchronized tapping49 (Supplementary Methods). Briefly, this analysis quantifies the likelihood of spike occurrence under the assumption of sensory or motor alignments. Although all neurons showed a preference for motor alignment during synchronized saccades (because movement occurred at the predicted rather than actual sensory timing), only Bilateral neurons significantly changed their preference toward sensory alignment during the second saccades (paired t-test, two-sided, t32 = 3.19, P = 0.003, Supplementary Fig. 11a).

Finally, we also examined the change in the time course of target-aligned neuronal activity with the order of target presentation (Supplementary Fig. 11b). When the difference in the time course of activity for Unilateral neurons between the first and the later cycles was quantified using the ROC analysis (Supplementary Methods), the area under the curve (AUC) averaged 0.71 ± 0.15 (s.d.) and was significantly different from 0.5 (one sample t-test, two-sided, t32 = 8.26, P < 10−8). In contrast, for Bilateral neurons, the activity lasted longer in the first cycle, but the centre of the activity was almost the same as that during synchronized saccades (AUC, 0.51 ± 0.15, P = 0.66). These results further suggest that Unilateral neurons are involved in saccade generation, while Bilateral neurons may reflect the temporal prediction of target appearance both in the initial cycles and during synchronization.

When the relationship between the SMI and the absolute value of the DI was examined for all neurons, there was a significant negative correlation between them (r = −0.55, P = 1.5 × 10−8, Fig. 5d). There was also a significant correlation between the SMI and PI, with neurons that increased their activity during synchronized movements reflecting the sensory component more strongly (r = 0.38, P = 1.8 × 10−4, Fig. 5e). The relationships between these three indices are summarized in Supplementary Fig. 11c.

### Causal role in synchronized saccades

To determine the causal relationship between these neuronal activities and behavioural performance, electrical microstimulation was applied to the recording sites during synchronized saccades (550 ms SOA). A train of stimulation pulses (100 ms in duration at 333 Hz, 100 µA) was delivered 100, 200, 300 or 400 ms after the saccade, and the effects on the next saccade timing (or ISI) were examined (Fig. 6a). The box-whisker plot in Fig. 6b summarizes the effect sizes for all stimulation conditions for the 40 sites that showed significant effects in any of the conditions (Dunnett test, P < 0.05, n = 32 and 8 for monkeys I and J, respectively). The stimulation affected only the timing of the next saccades, and monkeys adjusted after the next ISI to compensate temporal errors induced by the stimulation (Supplementary Fig. 12).

The effects of electrical stimulation varied greatly from site to site, and significant facilitatory and suppressive effects on saccades were found in various proportions, even under the same conditions (Fig. 6b). To examine the differences by stimulation site in detail, the data were clustered into three categories based on the effect size in each condition (Fig. 6c and d). In the first group, electrical stimulation promoted saccades in both directions (vertical cyan bar on the right in Fig. 6c). In the second group, stimulation facilitated contralateral but delayed ipsilateral saccades, and the effect was greater when stimulation was delivered immediately after the previous saccade (magenta). In the third group, only contralateral saccades were delayed regardless of stimulation timing (orange).

We assumed that the changes in saccade timing for each site could be explained by a certain combination of the effects of stimulation to the three types of neurons (Methods). We estimated the relative impact of electrical stimulation to each group of neurons by calculating simple Pearson correlations between neuronal activity at specific 100 ms periods and the ISI (Supplementary Fig. 13a–c). Our model well explained the stimulation effects such that the coefficients of determination (CDs) were significantly greater than those obtained from the permutation data (1000 iterations, Wilcoxon’s rank-sum test, P < 0.01, Supplementary Fig. 13d) and the CDs for 72.5% of the stimulation sites were greater than the median of the permutation data. The triangular plot in Fig. 6e summarizes the relative contribution of each neuronal population to the changes in saccade timing derived from the fitted data. The data showed that the first group (cyan) was mainly accounted for by the stimulation effects of Bilateral and Unilateral neurons, and that the second group (magenta) was due to the Postsaccade neurons. In addition, Bilateral neurons and Unilateral neurons were stimulated in various relative proportions, while Bilateral neurons and Postsaccade neurons were rarely stimulated simultaneously. This observation was consistent with the results of our neuronal recordings, which showed that Bilateral neurons and Postsaccade neurons were distributed separately, mainly in the ventral and dorsal portions of the dentate nucleus, respectively (Supplementary Fig. 1). In fact, the stimulation sites for the second group tended to be distributed more dorsally than the other groups (one-way ANOVA, F2,39 = 3.37, P = 0.045, Supplementary Fig. 13e). Consistent with these findings, the effects of stimulation at dorsal and ventral sites showed similar properties to the second and first groups, respectively (Supplementary Fig. 13f).

## Discussion

During synchronization, the brain needs to maintain an internal rhythm to make a series of predictive movements50. Since our monkeys had been trained on the synchronized saccade task for many months, it is likely that they were able to quickly generate an internal rhythm in each trial. In fact, we have previously shown that the animals can continue to make saccades for many cycles even after removing the temporal error by presenting a target at the time of eye movements (error-clamp condition)38. We found in this study that Unilateral and Postsaccade neurons consistently increased their activity in relation to saccades, but many Bilateral neurons were active at the time of target onset during the transition from reactive to synchronized saccades in each trial (Fig. 5). These results suggest that Bilateral neurons may represent the expected timing of target onset, rather than promoting a specific motor output.

Previous studies have shown that internal rhythms emerge in predicting the appearance of periodic stimuli even in the absence of movement51, and that the cerebellum is involved in this process26. In particular, recent studies in primates have demonstrated that neurons in the posterior portion of the dentate nucleus exhibit predictive activity that peaks at the timing of periodic visual stimuli27,52. Furthermore, evidence shows that the cerebellum also plays a role in generating internal forward models for purely sensory aperiodic events, such as trajectory of slowly moving objects31,32,33,53. The activity of Bilateral neurons described so far could be regarded as an internal model of periodically alternating visual stimulus, which guides synchronized saccades.

The three types of neurons were localized in a restricted region of the cerebellar dentate nucleus. While there was no difference in the anterior-posterior or medial-lateral coordinates of the recording sites between the groups, there was a clear difference in the depth of the recording sites (Supplementary Fig. 1). Since the stimulation effects on saccade timing can be explained as a combination of the effects of excitation of these groups of neurons, each group may be involved in multiple processes necessary for synchronized saccades in parallel, by sending information to different brain regions. Neurons in the posterior portion of the cerebellar dentate nucleus are known to project directly to the superior colliculus (SC)54 and indirectly via the thalamus to the frontal eye fields (FEF), supplementary eye field (SEF), and posterior parietal cortex (PPC)34. Recent studies using trans-synaptic tracers have reported that the FEF and PPC receive signals mainly from the ventral portion of the dentate nucleus34,55, whereas the trans-thalamic projections to the SEF originate from both the dorsal and ventral portions56.

Supplementary Fig. 15 shows a diagram of the hypothesis that the cerebellum controls synchronized saccades through multiple loops with oculomotor areas in the cerebral cortex. Bilateral neurons participate in the link with the SEF and represent the temporal prediction of the target sequence. Postsaccade neurons participate in the link with the PPC and SEF, which plays a role in monitoring errors and updating internal rhythms. Unilateral neurons participate in the link with the FEF and/or SC and regulate saccade timing. Although this multiple-loop hypothesis is highly speculative, it will provide a practical working hypothesis for future studies exploring signals in these cortical areas and the SC during synchronized saccades.

Two issues must be considered when interpreting the present results. First, the possible involvement of the medial part of the cerebellum in synchronized saccades cannot be ruled out. For oculomotor control, vermal lobules VI–VII regulate signals in the brainstem saccade generator through the fastigial nucleus57. This pathway is important for the adaptation of saccade amplitude and direction15, but its involvement in saccade timing might become apparent in future studies. For example, recent studies in rodents have shown that both the medial58 and lateral59 cerebellum are involved in self-timed somatic movements. The second issue is that in the present study, the animals alternated saccades in opposite directions, whereas many studies of synchronized movement employ tasks with a repetition of identical movements, such as tapping. However, everyday activities such as dancing and clapping involve adjusting the timing of multiple movements, and many similarities can be found with synchronized eye movements. The present results may be generalized by exploring neuronal activity in the medial frontal cortex and the striatum, which has been examined during tapping24,60, using the synchronized saccade task.

## Methods

### Animal preparation

Two adult male Japanese macaques (monkeys I and J, 7–9 years old, 8–9 kg) were used. These animals were previously used in a series of behavioural experiments38,61,62. All experimental protocols were evaluated and approved in advance by the Hokkaido University Animal Care and Use Committee and were in accordance with the Guidelines for Proper Conduct of Animal Experiments (Science Council of Japan, 2006). Animal health and well-being were carefully monitored by animal care staff and experimenters, and food intake, water supply, stool volume, and overall physical condition were checked and recorded daily. To motivate the animals to perform the tasks, their water intake was regulated during weekday training and experiments, but they had free access to water on weekends. There was no strict dietary restriction, and a variety of vegetables, fruits, nuts and grains were provided daily.

The procedures for animal surgery and recording experiments were identical to those described previously63. Briefly, in separate surgeries under general isoflurane and nitrous oxide anaesthesia, a pair of plastic head holders was installed to the skull using titanium screws and dental acrylic, and a scleral search coil was implanted under the conjunctiva. Analgesics were administered during each surgery and for the following few days. After recovery from surgery, the monkeys were trained in the synchronized saccade task38 for several months. During the training and the subsequent experimental sessions, the monkey’s head was secured to the primate chair in a darkened booth, and horizontal and vertical eye position were recorded using the search coil technique (MEL-25, Enzanshi Kogyo). After training on eye movement tasks, a third surgery was performed to place a recording chamber for vertical electrode penetration aimed at the deep cerebellar nuclei. The location of the chamber was verified postsurgically using MRI. Daily recording sessions began after full recovery from the surgery. Topical or systemic antibiotics were administered as necessary.

### Visual stimuli and behavioural task

Experiments were controlled by a Windows-based stimulus presentation and data acquisition system (TEMPO, Reflective Computing). Visual stimuli were presented on a 27-inch liquid crystal display monitor (XL2720Z, BenQ, refresh rate 120 or 144 Hz) that was located 40 cm away from the eyes and subtended 73° × 46° of visual angle. Throughout the experiment, two landmarks (white unfilled 1° squares) were presented ±7° horizontally (Fig. 1a) on a dark background, and all visual stimuli were presented within the landmark. Each trial started with the appearance of an initial fixation point (blue or purple square, 10.9 cd/m2) at either landmark location. After a random 1000–2000 ms period, the fixation point was extinguished and a saccade target (red or green square, 33.9 cd/m2) was presented at the other landmark location. The saccade target was alternately presented at the landmark locations with a stimulus onset asynchrony (SOA; 400, 550, or 700 ms) for 8000 or 12000 ms (11–30 target steps) and was visible until the appearance of the opposite target. The animals were trained to follow the alternating targets with their eyes. The trial was aborted immediately if the monkey made an anticipatory saccade to the first target in the sequence (reaction time < 100 ms), or if the inter-saccadic interval (ISI) was shorter than 25% SOA or longer than twice of the SOA, or if the eyes were deviated >3.5° vertically from the target locations.

### Recording procedures

Single neuron activity was recorded from three dentate nuclei of the two monkeys. Single tungsten microelectrodes (~1.0 MΩ at 1 kHz, Alpha Omega Engineering or FHC Inc.) were lowered through a 23-gauge stainless steel guide tube using a grid system (Crist Instruments). The electrodes were advanced remotely using a micromanipulator (MO-97S, Narishige) attached to the recording chamber. Signals obtained from the electrodes were amplified, bandpass filtered (300 Hz to 10 kHz), and monitored online using oscilloscopes and an audio device. Once a task-related neuron was encountered, waveforms of action potentials were isolated using software with real-time template-matching algorithms (ASD, Alpha Omega Engineering). The occurrence of each action potential was saved in files as a time stamp with the data of eye movements and visual stimuli during the experiments. We found task-related neurons in the caudal part of the cerebellar dentate nucleus (Fig. 1d). Most neurons were recorded from stereotaxic coordinates 8 mm posterior to the interaural line and 9 mm lateral to the midline in monkey I, and 8 mm posterior and 8 mm lateral in monkey J (Supplementary Fig. 1).

### Electrical microstimulation

To examine the causal role of neuronal activity, electrical stimulation was applied to the recording sites during the synchronized saccade task. A train of 0.2 ms biphasic pulses at 333 Hz for 100 ms was delivered as electrical stimulation. The current intensity was monitored by measuring the voltage across a 1 kΩ resistor placed in series with the electrode, and was adjusted to 100 μA. Electrical stimulation was applied at 100, 200, 300, or 400 ms following every four synchronized saccades (550 ms SOA). The stimulation timing was constant during each trial (~ four times per trial), but varied from trial to trial. We have never delivered stimulation during the reactive saccade trials.

### Histological procedures

After completion of the experiments in monkey I, several marking lesions were made by injecting direct current (10–20 µA, tip negative, ~1 mC) through electrodes placed at known coordinates. Several days after this procedure, the animal was sedated, administered analgesics, deeply anesthetized with pentobarbital, and transcardially perfused with phosphate-buffered saline followed by 3.5% paraformaldehyde. The brain was then removed, fixed, and equilibrated with 30% sucrose. Histological sections (100 μm, coronal) were cut on a freezing microtome (HM440E, Microm) and stained with cresyl violet. The recording sites were reconstructed according to stereotaxic coordinates, and the depth of the electrode tip relative to the dorsal border of the dentate nucleus, which was verified physiologically during the experiments (Fig. 1d and Supplementary Fig. 1).

### Behavioural data analysis

The eye position signals obtained directly from the eye coil device (MEL-25, Enzanshi Kogyo) were digitized at a 16-bit resolution, sampled at 1 kHz, saved to a file during the experiments, and analysed offline using Matlab (Mathworks). During the experiments, saccades were detected online when horizontal eye position crossed the centre of the screen. However, for the offline analysis, saccade onset was detected when the angular eye velocity exceeded 200°/s and an eye displacement was >7°. We measured two temporal parameters of saccades. The inter-saccadic interval (ISI) was the onset interval of successive targeting saccades in opposite directions. The temporal error was the time from the target onset to saccade initiation and was equivalent to the latency of reactive saccades. When we examined the time course of saccade latency in each trial (Fig. 1c), the data of reactive saccades for different SOAs were plotted every 550 ms.

### Calculation of indices that characterize neuronal activity

To characterize neuronal activity, we calculated the following modulation indices for individual neurons and compared them across neuronal groups. The directional index (DI) was defined as (IpsiContra)/(Ipsi + Contra), where Ipsi and Contra indicate the magnitude of neuronal activity measured for ipsilateral and contralateral saccades, respectively. Neuronal activity was measured from the spike density function during a 200 ms period starting from 250 ms before (Unilateral and Bilateral neurons) or immediately after (Postsaccade neurons) saccades.

The prediction index (PI) was calculated as (PredReact)/(Pred + React), where Pred and React represent the magnitude of firing modulation during predictive and reactive saccades in the preferred direction, respectively. For predictive saccades, the activity was measured for the sixth or later saccades in the sequence in trials with a 550 ms SOA. For reactive saccades, the data were aligned with saccades following a 550 ms fixation interval. To measure the firing modulation, we initially searched for the peak of the spike density function during ±275 ms of the saccades in the preferred direction. We then searched for the minimum value of the activity within ±275 ms of the peak. The magnitude of the firing modulation was measured as the difference between these values. Similarly, the PI for different SOAs was also calculated by measuring the firing modulation for saccades with the same length of measurement interval as the SOA. The modulation of neuronal activity for different task conditions and saccade direction were also evaluated by computing the mutual information (Supplementary Methods and Supplementary Fig. 3).

### Correlation between neuronal activity and saccade timing

The cerebellum is thought to be involved in the predictive control of movement and the detection of errors that are necessary for learning. To evaluate the information carried by each dentate nuclear neuron, the correlations between neuronal activity and the timing of saccades were examined. When investigating the relationship between neuronal activity and motor control, data were aligned with synchronized saccades, and partial correlations of trial-by-trial neuronal firing rate with the timing of the next saccade (ISI) were calculated every 200 ms (10 ms steps), controlling for temporal errors of saccades and the previous ISI (Fig. 3c–e). Since the goal of this analysis was to evaluate the signals regulating saccade initiation, the partial correlation was calculated for the data until the next saccade. Therefore, the number of trials contributing to the partial correlation decreased shortly before the next target onset. To examine the relationship between neuronal activity and timing errors, data were again aligned with synchronized eye movements, and partial correlations with temporal error (saccade latency) were calculated after controlling for the previous ISI (Fig. 4c–e). The partial correlation coefficients at 200–400 ms and 0–200 ms following synchronized saccades, where large correlations were found, were used as indices for evaluating the involvement of neuronal activity in motor control and error monitoring, respectively, and the groups of neurons were compared (Fig. 3f, 4f, Supplementary Figs. 6d and 8d).

### Stimulation data analysis

The effects of electrical microstimulation were quantitatively analysed for 49 sites where stimulation was delivered for ≥20 synchronized saccades in each condition. The effect size (Cohen’s d) for each condition was calculated as (μstim – μcont)/sqrt[(σstim2 + σcont2)/2], where μ and σ indicate the mean and standard deviation of the ISI with (stim) or without (cont) stimulation, respectively. The Dunnett test was used to evaluate the effect of electrical stimulation at each site and the timing of saccades in either direction. If a significant effect was found in at least one stimulation condition, the data obtained from that site were included for further analysis (n = 32 and 8 sites for monkeys I and J, respectively, Fig. 6b, bottom). The data from 40 stimulation sites were classified into three groups using the hierarchical clustering method based on the effect sizes under eight different stimulation conditions (Fig. 6c and d). Since the stimulus effect at each site is expected to be determined by combinations of task-related neurons in the surrounding area, we attempted to model the stimulation effect by a certain combination of the effects of stimulation to the three groups of neurons. The stimulation effects on the different groups of neurons were estimated based on simple Pearson correlations between each neuron’s activity and the ISI calculated at the four electrical stimulation times and averaged for each group of neurons (Supplementary Fig. 13a–c). Because there were only nine Unilateral neurons with contralateral preference, the correlation coefficient for the population of Unilateral neurons was calculated by combining the values of the ipsilateral and contralateral neurons and flipping the same value (Supplementary Fig. 13c, green dotted lines). Since these values are considered to reflect the strength of the influence of each group’s activity on saccade timing, we assumed that the effect of electrical stimulation would be a linear sum of these values with appropriate weights. Thus, the effect sizes in eight different conditions for each stimulation site (d) will be described as,

$${{{{{\rm{d}}}}}}={{{{{{\rm{w}}}}}}}_{{{{{{\rm{ipsi}}}}}}}{{{{{{\rm{Uni}}}}}}}_{{{{{{\rm{ipsi}}}}}}}+{{{{{{\rm{w}}}}}}}_{{{{{{\rm{contra}}}}}}}{{{{{{\rm{Uni}}}}}}}_{{{{{{\rm{contra}}}}}}}+{{{{{{\rm{w}}}}}}}_{{{{{{\rm{bi}}}}}}}{{{{{\rm{Bi}}}}}}+{{{{{{\rm{w}}}}}}}_{{{{{{\rm{post}}}}}}}{{{{{\rm{Post}}}}}}$$
(1)

where w indicates the weight for each group of neurons and Uni, Bi, Post are the impacts of stimulation effect of each group of neurons on saccade timing calculated above as the Pearson correlation at eight different timing (Supplementary Fig. 13c). We constrained the weight (w) of each group to be positive because electrical stimulation increases the activity of nearby neurons. The values of w calculated by the least-squares method for each stimulation site were normalized and are plotted in Fig. 6e.

### Statistical analysis

The unpaired t-test or analysis of variance (ANOVA) was used to compare the indices, peak latencies, and recording sites among the neuronal groups, and the one sample t-test was used to compare the mean of each group with a constant value. To evaluate the effects of electrical stimulation, Dunnett’s test was performed between the ISI in each stimulation condition and in the control. All tests were performed using the Statistical toolbox for Matlab. To evaluate the partial correlation between neuronal activity and saccade timing for each neuron, the distribution of the r-values obtained from the permutated data (1,000 iterations) was compared with the actual data (Fig. 3f, 4f, Supplementary Figs. 6d and 8d). Other statistical methods are described in the relevant text or figure legends.

Further details of the analysis in the Supplementary Figures can be found in the Supplementary Methods and in the relevant figure legends.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.