Behavioral correlates of Purkinje cell ensemble covariance

Abstract

Purkinje cells (PCs) mediate adaptive cerebellar output through inhibitory control of the cerebellar nuclei. Because many PCs converge, their influence on target neurons likely depends on population activity. However, despite a large literature on individual PC relationships with behavior, little is known about how multiple PCs collaborate to influence movement. In simultaneously recorded PCs during mouse forelimb reach behavior, we found that different kinematic variables correlated either with PC firing rates or covariance: reach velocity and deceleration correlated with firing rates, while the coordination of these kinematic variables, which influences reach amplitudes, was related to covariance. In a model, physiological levels of PC covariance elevated nuclear firing rates, which could promote faster reach deceleration than the same conditions with low covariance. These findings suggest a role for coherent PC population activity to coordinate movement.

Little is known about how cerebellar population activity controls movements in ways distinct from individual neuron correlations. Here the authors show that Purkinje neuron population covariance uniquely influences motor coordination in mice.

Introduction

Cerebellar output coordinates movement and modulates cerebral cortical dynamics^1–5. This output is shaped by cerebellar Purkinje cells (PCs) activity. GABAergic PCs fire at high spontaneous rates, tonically inhibiting pacemaker neurons in the cerebellar output nuclei^6–8. While PC firing suppression is one mechanism to disinhibit nuclear neurons, individual nuclear neurons receive convergent input from many PCs and excitatory mossy fibers, raising the importance of population activity in controlling cerebellar output^9–12. Several studies support this population coding framework, where activity from multiple PCs improves linear decoding of kinematic variables such as eye, limb, and whisker velocity^11,13–15. Beyond firing rates alone, the temporal structure of population activity also influences nuclear integration. A logical extension of the rate coding hypothesis of PC population activity predicts that broader-timescale covariance of convergent PCs will influence synaptic integration and nuclear output, since coordinated firing rate changes will exert more coherent control over nuclear neurons without interference from unmodulated convergent PCs. Moreover, synchronized inhibition across PCs can facilitate nuclear firing by transiently reducing summated tonic inhibition^16–19, forming an additional mechanism of nuclear disinhibition^14,15. It is not known whether coordination of PC population activity across different timescales correlates with cerebellar control of movement, and if so, whether such coding modalities preferentially structure specific kinematic variables.

Cerebellar output is organized into discrete modules that exert directional control over specific effectors, such as the eyes and limbs, through firing rate codes^12,20,21. For limb control, electrophysiological and optogenetic studies in mice have identified directional output modules within the anterior interposed nucleus (IntA) that exert either an inward pull or outward push on the limb^20,22. In many IntA neurons, firing rates scale linearly with outward deceleration (or inward acceleration)^20,23. Within the deceleration module, PC rates inversely scale with deceleration, consistent with a disinhibitory relationship with nuclear coding^14,20. Because PC activity is also correlated with peak velocity¹³—and faster reaches both accelerate and decelerate more steeply—endpoints remain relatively stable despite kinematic variability. These coordinated relationships between multiple phases of the reach, and their linear coding with PCs and IntA neurons, have led to the view that these regions regulate motor coordination by translating inputs to the cerebellum that convey corollary discharge and goal information into well-regulated PC pauses that precisely guide the limb to the target^10,24. However, in some movements, kinematic scaling between peak velocity and deceleration is higher or lower than average, which can influence endpoints, leading to under- or overshooting targets. It is not known if cerebellar coding influences such kinematic scaling mismatches.

Thus, because both rates and covariance may influence nuclear firing rates which then influence kinematics, we asked whether PC covariance is associated with kinematic variables in addition to firing rate^15–18,25. Here, we analyzed simultaneously recorded PCs during mouse reaching movements to examine both the presence and behavioral relevance of PC covariance during a reach-to-grasp task in mice. We used hierarchical clustering to identify groups of PCs with distinct reach-related firing patterns. We then performed pair-wise analyses of these PCs, within and between clusters, and found that when reach deceleration was faster than expected for a given peak velocity there was a pronounced increase in PC covariance, spanning 1 to 100 s of milliseconds. This observation suggested a role for varying PC covariance in adjusting the gain between kinematic variables within a movement. Computational modeling and dynamic clamp results support a model in which increased PC covariance enhances nuclear disinhibition to levels sufficient to account for the behavioral variables. These correlative data suggest that PC covariance may be a coding dimension that influences motor coordination, reinforcing a principle of PC population coding that is not evident from single PC firing rates alone.

Results

Purkinje cell activity patterns during reach in mice

We analyzed previously published¹³ recorded PCs with high density Neuropixels 1.0 probes in forelimb-associated lobules 4/5 and Simplex during skilled reaching movements in head-fixed mice (25 recording sessions from 7 mice). We identified PCs and putative PCs (together called PCs) based on firing patterns, spike waveform, and/or presence of complex spike-associated pauses¹³ (Fig. 1a; See Methods). During reach-to-grasp movements, mouse PC populations exhibit diverse simple spike firing patterns. We previously categorized these PC activity patterns as dominated by either firing rate suppression or facilitation¹³. However, this classification overlooked more complex time-varying firing dynamics that may shape distinct phases of reaching movements. Therefore, to gain a more detailed view of PC codes during reach, we performed hierarchical clustering on peri-event time histograms (PETHs) of reach-aligned PCs (Fig. 1b–c). Most reach-modulated PCs fell into one of four clusters (Supplementary Fig. 1), characterized by reach aligned-suppression (cluster 1); early-facilitation (cluster 2); early-suppression (cluster 3); or reach aligned-facilitation (cluster 4) (Fig. 1b–c). We observed PCs from multiple clusters in all 25 sessions (Fig. 1d) but with varying ratios across 7 mice (Fig. 1d, colored dots). Kinematics did not differ significantly across sessions, arguing against the idea that behavioral variability accounts for response types (p = 0.13 comparing endpoints; p = 0.84 comparing peak velocity on sessions with >50% cluster 1 vs > 50% cluster 4).

Fig. 1. Identifying PC response classes during reaching movements.

a In head-fixed mice, Neuropixels 1.0 recordings⁸⁶ targeted Lobules 4/5 and Simplex while mice performed skilled reaches to retrieve a food pellet. Right, top: overlay of 100 waveforms from a single pPC (average in dark green). Bottom: normalized simple spike waveforms of all PCs and putative PCs (n = 404 units); b Four PC activity clusters were identified using hierarchical clustering (see Methods). Top: All normalized PETHs, sorted by cluster (colored bars on left; units per cluster= 110, 93, 84 and 117 units classified in cluster 1 through 4, respectively), aligned to reach endpoint (maximal outward extent, black vertical line). Dots on the right denote CS-identified PCs (43.9%, 14.6%, 22%, 19.5% of the CS-identified PCs were classified as cluster 1 through 4, respectively; Chi-squared test; X² = 6.8; p = 0.03; degrees of freedom=3). Bottom: pairwise correlation coefficients for all units with dendrograms corresponding to the four identified clusters; c, Firing rates (mean ± s.e.m) for the different clusters aligned to movement onset (top) or movement endpoint (bottom). Black traces: average outward forelimb position; d, The fraction of simultaneously recorded PCs within each cluster per session. Each session is a row; each colored circle indicates an animal. Colors correspond to the different clusters, as in c; Source data are provided as a Source Data file.

Given that 1-ms PC synchrony influences nuclear integration, in vitro¹⁷, we analyzed whether simple spike synchrony, defined as 1-ms covariance exceeding chance levels (see Methods), occurred between simultaneously recorded pairs of PCs within or between clusters. We computed rate-corrected cross-correlograms (CCGs) for simultaneously recorded pairs in each cluster (Cluster 1: 465 pairs; cluster 2 pairs: 194 pairs; cluster 3 pairs: 206 pairs; cluster 4: 387 pairs). While all clusters had significant CCGs at 0 ms lag (p = 4.7 × 10⁻¹⁰, 9.1 × 10⁻⁴, 0.0024 and 0.004 for cluster 1 through 4, respectively), 1-ms synchrony was most prominent for cluster 1 and 2 PCs (Fig. 2a–b, blue and red) with only modest above-chance synchronization in clusters 3 and 4 (Fig. 2a–b, green and orange; CCG peaks mean ±s.e.m., cluster 1: 2.9 ± 0.5 sp/s; cluster 2: 1.6 ± 0.5 sp/s; cluster 3: 1.3 ± 0.4 sp/s; cluster 4: 0.7 ± 0.2 sp/s). These observations held in complex spike-identified PCs as well (Supplementary Fig. 2).

Fig. 2. Cluster-dependent synchronization.

a Rate-corrected cross-correlograms (CCG)⁵⁰ for each PC cluster. Traces show mean ± s.e.m; b 0-lag CCG values for all PC pairs (dots). Averages, circles. Clusters 3 and 4 CCGs were significantly lower than Cluster 1 (mean ± s.e.m. per 1-s for each cluster: cluster 1- 2.9 ± 0.5, n = 465 pairs; cluster 2- 1.6 ± 0.5, n = 194 pairs; cluster 3- 1.3 ± 0.4, n = 206 pairs; cluster 4- 0.7 ± 0.2, n = 387 pairs. All clusters had significant 0-lag CCG values (p = 4.7×10^-10, 9.1 × 10^-4, 0.0024, 0.004 and effect sizes are: 0.27, 0.21, 0.19 and 0.15 for cluster 1 through 4, respectively; two-sided one-sample t-test); Differences between clusters: cluster 1 vs. 2: p = 0.1; 1 vs. 3: p = 0.027; 1 vs. 4: p = 1 × 10^-4; 2 vs. 3: p = 0.59; 2 vs. 4: p = 0.06; 3 vs. 4: p = 0.21. Percentage of pairs with significant CCGs: Cluster 1: 30%, cluster 2: 27%, cluster 3: 25%, cluster 4: 20%. Significance levels were calculated with two-sided two-sample t-test, threshold for significance corrected for multiple comparisons is p = 0.013; c Average joint peri-event time histograms (jPETHs) for each cluster aligned to reach endpoint (white lines) binned at 1-ms. Warmer colors depict higher cofiring; d Mean diagonals of the jPETH matrices (dashed diagonals in c) for each cluster (baseline-corrected), aligned to endpoint ± s.e.m (shading). Top, mean outward forelimb position; e, Same as c, showing the jPETH for mixed cluster 1 x cluster 2 pairs. Black circle diagonal indicates time of cross-correlation values in g; f, Top, rate corrected CCG for cluster 1 x cluster 2 pairs (n = 477 pairs). Bottom, mean jPETH diagonal for mixed cluster 1 x cluster 2 pairs. Shadings correspond to s.e.m; g, Mean ± s.e.m. of the jPETH 0-lag cofiring measured 0.1 s before movement endpoint (black circle in e). Colors correspond to the different clusters (mean ± s.e.m.; cofiring at time 0, cluster 1: 34.2 ± 2.2 sp²/s²; cluster 2: 32.7 ± .4.5 sp²/s²; cluster 1-2: 39.8 ± 2.8 sp²/s²; cluster 1 vs. 2: p = 0.74; cluster 1 vs. 1-2: p = 0.12; cluster 2 vs. 1-2: p = 0.18; two-sided two-sample t-test) and circles to the pair-by-pair cofiring values. The percent of pairs with significant 0-lag peaks are indicated within the bars (cluster 1: 43 ± 2%; cluster 2: 17 ± 3%; cluster 1-2: 30 ± 2%; error is standard error of a binomial distribution- $counterror=\surd \frac{p*\left(1-p\right)}{n}$, where p is the percent of significant units and n is the total number of pairs; Source data are provided as a Source Data file.

Next, to analyze behavioral modulation of PC cofluctuation, we constructed joint peri-event time histograms- (jPETHs)^1,26,27 aligned to reach endpoint for simultaneously recorded cell pairs, binned at 1 ms (Fig. 2c,d) and averaged across pairs. Because jPETHs are sensitive to shared event-locked firing rate modulation, we applied a “shift-predictor” that removes correlations expected from overlapping PETHs—which are maximized by our clustering method -- leaving residual structure that reflects trial-by-trial cofluctuation and fine temporal coordination. Shift-corrected jPETHs for Cluster 1, 2, and 1-2 pairs revealed a marked increase in PC cofluctuation during reach, evident as a prominent diagonal band (Fig. 2c,d blue, red, Fig. 2e,fn = 477 pairs). In contrast, reach-associated cofluctuation within cluster 3 and 4 PCs and other between-cluster PCs was modest or absent (Fig. 2c,d green and orange; Supplementary Fig. 3).

Although the shift predictor controls for shared event-locked firing rate changes, several additional confounds could in principle account for the observed jPETHs, including slow rate cofluctuations and shared response latency. To address these possibilities, we performed a set of shuffle-based analyses that control for known statistical confounds and provide structured baselines against which task-related changes in cofluctuation can be quantified. Each analysis controlled for firing rate fluctuations in a distinct manner, and all revealed that cofluctuation in Clusters 1 and 2 exceeded chance levels. We describe these controls in turn below.

Trial shuffling, which disrupts trial-by-trial coupling while preserving peri-event firing rate profiles, abolished signatures of cofluctuation in Cluster 1 and 2 jPETHs and between them (Supplementary Fig. 4a–d). Cofiring was significantly higher in simultaneously recorded pairs compared to trial-shuffled pairs (Supplementary Fig. 4a–d; mean ± s.e.m.; Difference between shuffle and measured: Cluster 1: −31.7 ± 1.9 sp²/s², paired t-test; p = 1.7 × 10⁻⁴⁷; Cluster 2: −36.2 ± 7.1 sp²/s², p = 8.9 × 10⁻⁷; Cluster 1-2: −33.3 ± 2.3 sp²/s², p = 6.7 × 10⁻⁴⁰; 0-lag cofiring peak: Cluster 1: −36.3 ± 3.1 sp²/s², p = 1.4 × 10⁻²⁷; Cluster 2: −39.6 ± 6.9 sp²/s², p = 4.3 × 10⁻⁸; Cluster 1-2: −47.7 ± 3.8 sp²/s², p = 2.8 × 10⁻³¹).

To distinguish slow rate cofluctuations from finer temporal coordination, we next analyzed cofiring after randomly jittering spike times within a 75 ms window, which preserves broader covariance structure. Under this manipulation, observed 0-lag cofiring peaks in Clusters 1 and 2 remained significantly higher than jittered controls, such that above-chance 1 ms synchrony accounted for >50% of the total cofiring measured in the jPETH (Supplementary Fig. 4e–h; difference, jitter–measured: g: Cluster 1: −10.7 ± 0.8 sp²/s², p = 6.3 × 10⁻³⁵; Cluster 2: −11.5 ± 2.4 sp²/s², p = 3.1 × 10⁻⁶; Cluster 1-2: −11.2 ± 1 sp²/s², p = 1.3 × 10⁻²⁴; h: Cluster 1: −17.8 ± 2.6 sp²/s², p = 2.3 × 10⁻¹¹; Cluster 2: −19.5 ± 5.9 sp²/s², p = 0.001; Cluster 1-2: −27.1 ± 3.3 sp²/s², p = 2×10⁻¹⁵). Finally, using a simpler jittering method where each spike is moved by a random shift between 1 and 75 ms, also accounted for >50% of the total cofiring (Supplementary Fig. 4i–l; difference, jitter–measured: i: Cluster 1: −14.2 ± 1 sp²/s², p = 3×10⁻³⁸; Cluster 2: −13.6 ± 2.5 sp²/s², p = 2.1 × 10⁻⁷; Cluster 1-2: −14.3 ± 1.2 sp²/s², p = 7.3 × 10⁻²⁷; l: Cluster 1: −17.7 ± 2.5 sp²/s², p = 1.1 × 10⁻¹¹; Cluster 2: −18.8 ± 5.8 sp²/s², p = 0.002; Cluster 1-2: −27.5 ± 3.3 sp²/s², p = 6.4 × 10⁻¹⁶). This provides an estimate of millisecond-scale synchrony and suggested not all cofiring was a consequence of 1-ms synchrony; however, jitter-based approaches cannot fully dissociate precise synchrony from shared trial variability, and we return to longer-timescale covariance later.

We next computed jPETHs for randomly selected units that were not necessarily recorded simultaneously, a manipulation that preserves behavior-related firing rate fluctuations while disrupting spike timing relationships (Supplementary Fig. 5). Plotting excess cofiring (the difference between observed cofiring in simple spikes, in units sp²/s², and that expected by chance) against firing rate revealed no monotonic relationship between these metrics, particularly across clusters. Notably, cofiring within Cluster 1 tended to be inversely correlated with firing rate, whereas within Cluster 2 these relationships ran in the opposite direction. The negative correlation between cofiring and firing rate in Cluster 1 contrasted with null expectations based on shuffled data (Supplementary Fig. 5d), in which increased firing rates lead to greater apparent synchrony.

Finally, the observed increase in cofiring was not attributable to latency covariation. Cofiring survived controls that (1) explicitly shifted spike times within each trial to behavioral onset (Supplementary Fig. 6a–c)^28,29 and (2) replaced each trial’s activity with that from another trial matched for behavioral onset time (Supplementary Fig. 6d–f). Taken together, these analyses indicate that PC cofluctuation is structured, time-varying, and selectively expressed across PC clusters, consistent with shared inputs imparting covariance during reach.

To quantify reach-associated cofiring, we extracted the 0-ms lag values from jPETHs 100 ms before reach endpoint (black circle, Fig. 2e,g). A pair of PCs was classified as exhibiting significant cofiring if its 0-ms lag value exceeded the 95th percentile of values along the downward (anti-)diagonal, which captures lead–lag structure of cofiring around this shared reference time. Using this criterion, 43% of Cluster 1 pairs and 30% of Cluster 1 × 2 pairs exhibited significant cofiring 100 ms before endpoint (black circles in Fig. 2e and Fig. 2g; cluster 1: 34.2 ± 2.2 sp²/s²; cluster 2: 32.7 ± 4.5 sp²/s²; cluster 1-2: 39.8 ± 2.8 sp²/s²; p > 0.05 for all; exact values in legends).

Together, these results point to a robust, behaviorally linked increase in cofiring during reach in specific subgroups of PCs.

Distinguishing kinematic correlates of PC covariance from PC rates

If covariance plays a role in Purkinje cell population coding, it should have measurable behavioral consequences. Given the interest in 1-ms synchrony as a potential mechanism for modulating nuclear activity, we first focus on millisecond-scale cofiring, noting that cofiring values derived from jPETHs include both trial-by-trial cofluctuation and precise spike-timing. We return to these factors separately later. Previous studies set up strong predictions on how PC simple spike synchrony in this region might influence limb kinematics. First, in vitro studies show that increased synchrony enhances nuclear firing^17,18. Second, higher firing rates in IntA during reach are associated with faster limb deceleration on outreach²⁰. Combining these observations, we predicted that increased cofiring during the reach would facilitate more nuclear firing, strengthening the inward pull on the limb, resulting in faster deceleration and hypometric endpoints. Previous studies have shown that facilitating and suppressing PCs encode mouse reach kinematics better when considered as a population¹³, thus we focused on rates and synchrony observed across Cluster 1 + 2 PCs (Fig. 3), with alternative groupings in Supplemental Figs. 7−8.

Fig. 3. Synchrony between cluster 1 and 2 PCs is strongly related to kinematics.

a Schematic outlining how reaches were sorted by endpoint; b, Average jPETHs between simultaneously recorded cluster 1 x cluster 2 PCs for the quarter of trials with the largest overshoot (left) or undershoot (right); c Average cofiring from diagonals of the matrices in b, for farthest (green) and nearest (magenta) endpoint reaches. Black trace shows outward velocity and gray the outward position (shaded area mean ± s.e.m.; undershoot: 185.2 ± 13.8 sp²/s², overshoot: 86.1 ± 7.7 sp²/s²; two-sided paired t-test; p = 4.5 × 10⁻¹⁷; Cohen’s d = 0.4; n = 477 pairs); d Firing rates during reaches with the farthest (green) or nearest (magenta) endpoints (firing rate change from baseline in time before −0.25 s relative to endpoint, mean± s.e.m.- overshoot: 2.03 ± 1.5 sp/s, undershoot: 6.5 ± 1 sp/s; two-sided paired t-test; p = 0.0003; Cohen’s d = 0.16; n = 477 pairs); e Schematic outlining how reaches were sorted by deceleration; f Average jPETHs for slowest deceleration quartile (left) and fastest deceleration quartile (right), aligned to time of peak velocity; g Average cofiring during reaches with the fastest or slowest deceleration (mean ± s.e.m.; fast: 104.6 ± 8.3 sp²/s², slow: 92.7 ± 9.8 sp²/s²; two-sided paired t-test; p = 0.2; Cohen’s d = 0.06; n = 477 pairs); h Average firing rates during reaches with the fastest or slowest deceleration (mean± s.e.m.; fast: -10.5 ± 1.4 sp/s, slow: 0.6 ± 1.6 sp/s; two-sided paired t-test; p = 1.5 × 10^-33; Cohen’s d = 0.34; n = 477 pairs); i Schematic outlining how reaches were sorted by peak velocity; j Average jPETHs for reaches within the lowest peak velocity quartile (left) or highest peak velocity quartile (right); k Average cofiring during reaches within the fastest peak velocity quartile (magenta) or slowest peak velocity quartile (green) (mean ± s.e.m.; fast: 88 ± 9.6 sp²/s², slow: 73.8 ± 10.3 sp²/s²; two-sided paired t-test; p = 0.26; Cohen’s d = 0.07; n = 477 pairs); l Average firing rates during reaches within the fastest quartile (magenta) or slowest quartile (green) reaches (mean± s.e.m.; fast: 11.7 ± 1.4 sp/s, slow: 22.4 ± 1.3 sp/s; two-sided paired t-test; p = 1.5×10^-12; Cohen’s d = 0.35; n = 477 pairs); m Schematic outlining how reaches were sorted by deceleration error; n Average jPETHs for reaches with the largest negative deceleration error (left) or highest positive deceleration error (right), associated with low and high-gain reaches; o Average cofiring during reaches with high and low deceleration error (mean± s.e.m.; high: 123.5 ± 10 sp²/s², low: 60.7 ± 10.7 sp²/s²; two-sided paired t-test; p = 2 × 10^-9; Cohen’s d = 0.28; n = 477 pairs); p Average firing rates during reaches with high and low deceleration error (mean± s.e.m.; high: 6.8 ± 1.1 sp/s, low: 5.7 ± 1.2 sp/s; two-sided paired t-test; p = 0.22; Cohen’s d = 0.04; n = 477 pairs); Source data are provided as a Source Data file.

To test these predictions, we first sorted reaches by limb displacement at endpoint, defined as the position of maximum outward extent (Fig. 3a–c). We then compared the jPETHs and cofiring between the top and bottom quartiles of the kinematic distributions. Consistent with our prediction, trials with hypometric endpoints showed higher cofiring and broader covariance across Cluster 1 + 2 PCs before the endpoint (Fig. 3a–c, shaded area mean ± s.e.m; undershoot: 185.2 ± 13.8 sp²/s², overshoot: 86.1 ± 7.7 sp²/s²; p = 4.5 × 10⁻¹⁷), with only minor firing rate differences (firing rate change- overshoot: 2.03 ± 1.5 sp/s, undershoot: 6.5 ± 1 sp/s; p = 0.0003). Contrary to our prediction, however, when we performed the same analyses but stratified by deceleration rate, we observed little difference in cofiring in reaches from the top and bottom deceleration quartiles (Fig. 3e–g; 104.6 ± 8.3 sp²/s², slow: 92.7 ± 9.8 sp²/s²; p = 0.2). Similarly, cofiring was comparable between low and high peak velocity reaches (Fig. 3i–k; fast: 88 ± 9.6 sp²/s², slow: 73.8 ± 10.3 sp²/s²; p = 0.26). By contrast, firing rates scaled with these variables, with a larger rate suppression associated with faster deceleration (Fig. 3h; fast: −10.5 ± 1.4 sp/s; slow: 0.6 ± 1.6 sp/s; p = 1.5 × 10⁻³³) and higher peak velocity (Fig. 3l, Peak velocity: fast- 11.7 ± 1.4 sp/s; slow: 22.4 ± 1.3 sp/s; p = 1.5 × 10⁻¹²). Similar relationships between cofiring and kinematics were also observed for Cluster 1 PCs analyzed as a segregated cluster but not Cluster 2 alone (Supplementary Figs. 7–8).

This mismatch—where cofiring was highest in reaches with hypometric endpoints despite normal deceleration and peak velocity—led us to explore alternative kinematic factors contributing to hypometric endpoints. A hallmark of coordinated reaches is the linear scaling of deceleration with peak velocity²⁴. Despite this relationship, the gain between these variables fluctuates reach to reach. We reasoned that hypometric endpoints could arise if deceleration was disproportionately high for a given peak velocity, even if absolute deceleration was not higher than average. We quantified this ‘deceleration error’ as the residual of deceleration relative to the session-wide least-squares relationship between peak velocity and deceleration (Fig. 3m–o). Reaches with positive deceleration error have a high gain relating peak velocity to deceleration and vice versa. Note that this metric does not imply an erroneous reach. Interestingly, we found much higher cofiring in Cluster 1 + 2 PCs during reaches with disproportionately fast deceleration relative to peak velocity, and vice versa (Fig. 3m–o; high gain (positive deceleration error): 123.5 ± 10 sp²/s², low gain (negative deceleration error): 60.7 ± 10.7 sp²/s²; p = 2 × 10⁻⁹). Indeed cofiring was linearly correlated with both endpoint position and deceleration error for Cluster 1 and Cluster 1 + 2 PCs, emerging 60 ms before movement endpoint (Supplementary Fig. 9). Importantly, cluster 1 + 2 PC firing rates were not significantly different across these conditions (Fig. 3p: high: 6.8 ± 1.1 sp/s; low: 5.7 ± 1.2 sp/s; p = 0.22). Similar relationships were seen for Cluster 1 neurons analyzed alone (Supplementary Figs. 7) but not Clusters 2 (Supplementary Figs. 8), 3 or 4 (not shown). Thus, cofiring is associated with faster deceleration for a given velocity, accounting for hypometric reach endpoints.

Kinematic associated PC covariance across timescales

Given the ambiguity in the drivers of jPETH-based cofiring, which reflect a combination of trial-by-trial rate cofluctuation and millisecond-scale synchrony, we next assessed population coordination using complementary metrics that isolate these components. With the goal to eventually model the role of PC covariance on CbN, we created a pseudo-population of PCs built from simultaneously recorded pairs whose cofiring fell within the ranges observed during high- or low-deceleration error trials, then jittered spikes within increasingly broad windows (Fig. 4a-d). Up to 100 ms jitter, this visualization preserved differences in jPETH structure between high- and low-gain conditions, suggesting that cofluctuating firing rates partially drive these differences. To define this contribution, we computed correlations between peri-event time histograms (PETHs) across conditions. PETHs from Cluster 1 and Cluster 1–2 PC pairs were significantly more correlated during high-gain compared to low-gain reaches, consistent with the broad temporal structure observed in the jPETHs (Fig. 4b; high gain: 0.18 ± 0.01; low-gain: 0.12 ± 0.1; p = 4.8 × 10⁻⁵). In contrast, differences in PETH correlations were modest for reaches stratified by deceleration or peak velocity (high deceleration: 0.19 ± 0.012, low deceleration: 0.22 ± 0.014, p = 0.03; fast velocity: 0.21 ± 0.013, slow velocity: 0.18 ± 0.014; p = 0.03).

Fig. 4. PC ensemble covariance across timescales is selectively enhanced during high-deceleration gain reaches.

a jPETHs of PC pseudo-populations built from simultaneously recorded pairs whose cofiring fell within the ranges observed during high- (top row) or low- (bottom row) deceleration error reaches. Each column shows jPETHS following spike jittering noted on top. White lines correspond to time of movement endpoint; b Time-resolved correlations of PC PETHs cluster 1 + 2 pairs (n = 477 pairs) across conditions sorted by deceleration, peak velocity and deceleration error. Colors follow convention in Fig. 3. Data presented as mean (line) ± s.e.m. (shaded area); c Rate corrected CCG for cluster 1 + 2 pairs for the empirical data from high-(purple) and low-gain (green) reaches (0-lag CCG values; mean± s.e.m.; high error: 4.8±0.8 sp/s; low error: 0.8±1 sp/s; two-sided paired t-test, p = 0.001; n = 477 pairs); d Rate-corrected CCGs for PC pseudopopulations. Magenta shows the CCG for high deceleration error and green for low deceleration error, with different maximal jitter window. Pearson’s correlations between jittered and actual CCG- high deceleration error, non-jittered vs. 1-ms jitter: r = 0.82. p = 5×10^-51; non-jittered vs. 20-ms: r = 0.4, p = 3.4 × 10^-9; difference: p = 5×10^-31; Low deceleration error, actual vs. 1-ms jitter: r = 0.83. p = 4×10^-52; actual vs. 20-ms: r = 0.41, p = 9.8×10^-10; difference: p = 1.3 × 10^-20. Insets show the rate corrected CCG in a window between –5 and 5 ms, for clarity, vertical scale bars = 10 sp/s in left panel and 5 sp/s in middle and right panels. Data presented as mean (line) ± s.e.m. (shaded area); e Relationship between the rate corrected CCG peaks at 0-lags and the 1-ms cofiring values calculated from the jPETHs diagonals, shown for the whole data as seen in Fig. 2f (black circles and line correspond to mean ± s.e.m.; Linear model fit: Slope=0.12, R² = 0.74, p = 0.0004), for the high deceleration error (magenta; slope=0.18, R² = 0.64, p = 0.04) or low deceleration error (green line; slope=0.02, R² = 0.02, p = 0.8). Formula on top corresponds to the relation between CCG values and 1-ms cofiring as depicted by the black line; Source data are provided as a Source Data file.

Our previous control analyses indicated that rate differences could not account for all synchrony, so to isolate the rate-corrected synchrony across these conditions, we computed rate-corrected CCGs. CCG peaks were also significantly higher during high-gain than low-gain reaches, indicating an increase in excess millisecond-scale synchrony (Fig. 4c–d). 0-lag rate-corrected CCGs averaged 4.8 ± 0.8 sp/s in high-gain conditions but 0.8 ± 1 sp/s in low-gain conditions (p = 0.001) (Fig. 4c). Jittering spike timing by 1 or 100 ms in the pseudo-populations in 4a selectively attenuated 0-lag peaks (Fig. 4d; (0-lag CCG: control: 22.6 ± 0.32 sp/s; 9 ± 0.37 sp/s with 1-ms jitter; p = 2 × 10⁻¹³⁶).

Since both rate cofluctuations and rate-corrected CCGs were distinct across conditions, we next tested whether these values covary as well. We therefore plotted cofiring measured with jPETH diagonals against the 0-lag CCG peak values. There was a strong linear relationship between the two metrics (R² = 0.64; p = 0.04; Fig. 4e), for high-gain reaches, but a weaker relationship and shallower slope for low-gain reaches (R² = 0.02; p = 0.8). Together, these findings indicate that both fine-timescale synchrony and broader firing rate coordination are enhanced during high-gain reaches, consistent with a shared population-level mechanism of coherence that may influence motor output.

PC covariance modulates CbN rates associated with reach kinematics in a model

We hypothesize that PC covariance differences in hypo and hypermetric reaches account for firing rate changes in the cerebellar nuclei that directly influence deceleration error (Fig. 5a–c). To formalize this idea, we next used a simple model³⁰ to examine how physiological cofluctuations among 40 convergent PCs influenced postsynaptic neurons (Fig. 5a). Spontaneous CbN model neuron intrinsic firing was 90 sp/s without inhibition through a tonic excitatory conductance³¹; inhibitory synaptic conductances matched physiological measurements^17,31,32, including 2.5 ms decay (τ), 5 nS inhibitory conductance and −75 mV GABA_A receptor reversal potential (see Table 1). We also estimated the CbN firing rates expected to account for excess deceleration (‘deceleration error’) based on the linear relationship between IntA firing rates and deceleration established in vivo (Fig. 5d; see Methods). This regression was then used to estimate the difference in deceleration error attributable to covariance (Fig. 5g).

Fig. 5. Wider covariance dynamics enhance CbN disinhibition.

a Schematic illustrating relationship of deceleration gain to endpoint variance. When deceleration is higher than expected for a given peak velocity (purple, top), endpoints become hypometric, and vice versa; b Summary of PC ensemble firing patterns for high- and low-deceleration-gain reaches; c Deceleration error (excess deceleration expected relative to peak velocity, see text) corresponding to hypo- (magenta) or hyper- (green) metric reaches (Deceleration error; mean±s.e.m., hypometric: 39.8 ± 9.6 cm/s²; hypermetric: −14.3 ± 8.3 cm/s²; p = 0.0035; two-sided paired t-test); d Deceleration error as a function of estimated CbN firing rate. CbN firing was estimated based on recordings in IntA during reach²⁰. Scatter plot shows an example session, where each circle corresponds to the predicted CbN FR (y-axis) and empirical deceleration error (x-axis) for a single reach (23 reaches). Black line indicates the regression line for the session, red, average regression across sessions (25 sessions). Top, formula for the red regression line; e Schematic diagram of the model where cerebellar nuclear cells (CbN) received input from 20 PC pairs- chosen based on high or low deceleration error related jPETHs. Because these were recorded firing patterns, they preserved physiological covariance across multiple timescales; f CbN model firing rate as a function of average PC ensemble cofiring. Linear fit: slope = +0.52 CbN sp/s per 1 PC sp²/s², p = 4.6 × 10⁻⁵. R² = 0.91. PC firing rates remained stable throughout the increasing cofiring values (mean ± sd: 71.4 ± 6.5 sp/s; r = 0.17, p = 0.67, Pearson’s correlation). Colored circles represent the rate-corrected CCG peak at each cofiring value (Pearson’s correlation; R² = 0.66; p = 0.001), colorbar shown on top; g, To identify the contribution of PC covariance across timescales to CbN firing rates and associated deceleration error, we plot the estimated deceleration error (left y-axis) and corresponding simulated CbN firing (right y-axis) as a function of the maximal applied jitter, for the high and low deceleration error-related PC activity. Dashed rectangle depicts range shown on the right. Here, we redisplay the estimated change in CbN firing as a function of the jitter magnitude, normalizing the estimated firing rate for high- and low-gain reaches. Note the strong dependence of high-gain trials on fine-timescale covariance with much less sensitivity for low-gain, reflecting the presence of covariance in the inputs. The black line indicates the percentage of the total firing rate difference between high- and low-gain firing rates estimated in f (left) that is a consequence of PC covariance within that window. Decay constant for high gain=3.13 ± 0.54. Shadings depict 95% confidence interval. Data presented as mean (lines) ± s.e.m. (shaded areas); Source data are provided as a Source Data file.

To test how PC covariance affects nuclear activity in the model, we applied PC spike trains from the pseudo-population of PCs built from simultaneously recorded pairs representing high- and low-gain reaches (Fig. 5e). As expected from previous studies, nuclear firing increased linearly with average PC 1 ms cofiring within the sampled population (Fig. 5f; slope=0.52 CbN spike per 1 sp²/s² PC cofiring). To isolate the contribution of 1-ms spike-timing from broader covariance, we jittered spikes within increasingly broad windows. This approach also allowed us to assess how variance at broader timescales, inherently embedded in the recorded ensembles, influences model output. PC patterns with high-gain covariance drove higher firing rates in CbN compared to low-gain conditions (Fig. 5g, right axis), which mapped onto distinctly different estimated deceleration error (Fig. 5g, left axis). Jittering by just 1 ms reduced firing rates in high gain conditions by ~12 sp/s (−11.9 ± 0.3 sp/s; Fig. 5g right inset), but notably the drop in spike output continued as longer jitter windows were imposed (Fig. 5f, pink), indicating a strong effect of the broader PC covariance on CbN firing as well. PCs with covariance seen in low gain conditions, drove less nuclear firing overall and was less sensitive to jittering (Fig. 5f, green, −2.2 ± 0.2 sp/s with 1-ms jitter). Ultimately, this analysis revealed that PC covariance under 20 ms accounted for up to 25% of the firing rate differences seen across high- and low-gain reaches (with a mere 1-ms jitter accounting for 9 ± 1% of the difference), and that covariance under 80 ms accounted for positive deceleration error. These simulations suggest that PC coordination across timescales may contribute to nuclear output that structures reach kinematics.

The prior simulations focused on the consequences of removing physiologically realistic PC cofluctuations to model output; here, we shift focus to the inverse manipulation, introducing synchrony to PC spike trains that follow average firing rate changes. Model details were the same as above, except that rather than using recorded PC spike trains, we simulated PC activity using a Poisson process replicating the time-varying firing rates of 40 PCs randomly chosen from clusters 1 and 2 PCs in our recordings (Supplementary Fig. 10a). We applied physiological or stochastic levels of synchrony to these populations by altering the cofiring probability per millisecond and compared postsynaptic responses across the 2 s simulated epoch (Supplementary Fig. 10b, top). As expected^16,17, PC firing rate suppression promoted nuclear firing through disinhibition (Supplementary Fig. 10b, bottom), with more firing during physiological PC synchrony than stochastic. Synchrony promoted longer inter-arrival-times for IPSCs (“ISIs”, Supplementary Fig. 10c; mean ± std; ISIs, stochastic: 1.07 ± 0.01 ms; physiological: 1.1 ± 0.02 ms), which in turn facilitated a higher average membrane potential in the postsynaptic cell (stochastic: −46.1 ± 2.5 mV; phys: −45.6 ± 2.5 mV) and consequently higher firing rates (Supplementary Fig. 10d; stochastic: 25.3 ± 11.2 sp/s; phys: 28.5 ± 11.4 sp/s). CbN firing rose linearly with CCG measurements in the physiological range (Supplementary Fig. 10e), with a rate-corrected CCG of 0.02 (similar to values seen in highly-synchronized or in CS-identified PCs, which might indicate a within-microzone PCs¹⁰) resulting in an excess of 10 sp/s in nuclear firing (Pearson’s correlation; r = 0.75; slope: 0.4 nuclear sp/s per 0.001 PC mean pairwise CCG increase). Broader synchrony windows of 5 ms, facilitated nuclear firing even more strongly¹⁸, suggesting that 1-ms synchrony reflects a lower bound of effects of PC covariance on nuclear disinhibition. We return to this effect more thoroughly below.

Finally, we tested CbN responses to PC rates and synchrony using dynamic clamp in acute adult mouse brain slices (Supplementary Fig. 10f). We assayed nuclear neuron responses to pools of 40 PCs, matching net firing rate patterns recorded in vivo. Inhibitory postsynaptic conductances were modeled with physiological time constants and amplitude¹⁷, specifically 2.5 ms IPSC decay times and 5 nS steady-state unitary conductance (See Methods). With no above-chance population synchrony, these naturalistic PC trains resulted in an almost complete suppression of spontaneous nuclear neuron firing, reducing the high spontaneous firing rate of 53.2 ± 6.6 sp/s at baseline to 2.34 ± 1 sp/s within the first second of the train, with no DC current injection. The suppression of PC net firing rates that occurred during reach partially disinhibited nuclear neurons from these depressed rates, promoting firing up to an average of 11.5 ± 2.3 sp/s at their peak during the maximal disinhibitory phase (Supplementary Fig. 10g,h). Adding physiological levels of 1 ms-synchrony to PC rate suppression increased nuclear firing rates 72.6%, to 20.4 sp/s (Supplementary Fig. 10i; no sync: 11.83 ± 3.04 sp/s. sync: 20.42 ± 3.41 sp/s from 80-100% velocity quintile; paired t-test: p = 1.28 × 10⁻⁴; n = 13 cells). These data suggested that both rate and synchrony promote disinhibition of nuclear firing. Finally, we explored the interaction between PC rates and synchrony in driving nuclear firing, varying both rate and synchrony across a broad range of values. Both PC firing rate and PC synchrony influenced nuclear firing, with the strongest disinhibition occurring with low PC rates and high PC synchrony, and the strongest inhibition with the converse pattern, high PC rates and low PC synchrony (Supplementary Fig. 10j–l; n = 17 cells; Two-ways ANOVA; F₆(PC firing)=21.16, p = 1.24 × 10⁻¹⁹; F₁(Covariance) = 24.06, p = 1.79 × 10⁻⁶). Together, these data show that synchrony enhances the efficacy of PC firing rate suppression on CbN neuron firing, such that rate changes will more powerfully disinhibit CbN firing in the presence of synchrony¹⁸.

Discussion

We found a relationship between Purkinje cell ensemble covariance and behavior. Using hierarchical clustering, we identified four classes of PCs in mouse intermediate cortical lobules based on firing rate patterns during reaching. Both rate and covariance were correlated with kinematic variables in mouse forelimb reaches but had interesting dissociations. While PC rate suppression strongly correlated with limb deceleration, covariance, including 1-ms synchrony, was more strongly related to deceleration gain – defined as disproportionate deceleration for a given peak velocity. High deceleration gain accounted for hypometric reach endpoints observed in conditions of high covariance. Finally, we linked high covariance and high deceleration gain using a computational model, finding that covariance across timescales spanning 1–200 ms can increase nuclear firing rates aligning with their roles in mediating deceleration. These results suggest that PC covariance structure in the intermediate cerebellum influences downstream cerebellar nuclear firing and subsequent kinematics^14,15,33.

Across species, reaches show relatively symmetric early- and late-phase acceleration and deceleration phases, characteristic of their well-known bell-shaped velocity profiles, mediated by timed alternation of extensor and flexor muscles^24,34–39. In uncoordinated reaches, or those that have pronounced online corrections, the early accelerative phase becomes mismatched with deceleration, resulting in asymmetric velocity profiles and inaccurate or variable endpoint positions⁴⁰. The organization of these kinematic relationships is sensitive to cerebellar manipulations^41–43, motivating mechanistic studies investigating the role of cerebellar computations in coordinating acceleration with deceleration. When we analzyed PC firing rates in relation to limb peak velocity, deceleration, and deceleration gain (Fig. 3), we found that PC rates correlated with peak velocity and deceleration rate but not deceleration gain. PC rate suppression occurred near peak velocity, preceding deceleration but after acceleration, arguing that PCs could contribute directly to the decelerative phase by responding to speed-related information entering the cerebellum via mossy fibers.

We found prominent beyond-chance covariance in Clusters 1 and 2 PCs and between them. This covariance structure was temporally dynamic during reaching movements and occurred independent of increasing firing rates. Unlike firing rates, which were strongly correlated with velocity and deceleration, covariance was not. Rather, PC covariance was related to the gain of the relationship between peak velocity and deceleration, such that high covariance occurred when deceleration was fast relative to a given peak velocity. In this sense, covariance constitutes an additional axis of control over cerebellar output. Rate and covariance modulation may therefore allow the cerebellum to superimpose flexible gain with learned motor control signals which would be encoded in synaptic weights. In this framework, PC firing rates report net parallel fiber drive through learned synaptic weights encoding feedforward control signals, whereas covariance, through unknown drivers, regulates the efficacy of mossy fiber excitation in downstream cerebellar output neurons. These observations also reinforce the emerging consensus around population coding among PC populations, with motor potent control signals emerging from the difference in firing rates across competing modules^10,44. Notably, reflex gains that differ with sensory experience have also been suggested to be modulated by PC synchrony¹⁵. We observed excess 1-ms synchrony in conditions that also had broader high covariance, suggesting the possibility of shared underlying mechanisms, such as gap-junction mediated molecular layer interneurons (MLI) inhibition⁴⁵, thus our observations point to a potentially shared role for covariance outside firing rates in regulating cerebellar output relative to input.

These data lead us to advance the following hypothesis for how PC population covariance influences reach kinematics: During reach, mossy fibers fire proportional to reach velocity, driving both PC and nuclear firing rates through learned synaptic weights^46–48. This rate code would account for the linear relationship seen between velocity and deceleration, and eumetric endpoints, but would not account for variability of deceleration relative to peak velocity and variable endpoints. Under conditions of high PC covariance, the efficacy of velocity-tuned mossy fibers would increase in the nuclei, driving higher nuclear firing rates, boosting relative deceleration. This modulatory role of PC covariance on nuclear response gain could account for the excess deceleration we observed associated with high covariance. This conjecture should be tested when technologies improve to adjust population structure without influencing overall firing rates.

The presence of Purkinje neuron beyond-chance synchrony across different species^10,14,49—at least in some cerebellar regions⁵⁰—raises the important question of the circuit mechanisms that promote it. Many proposed mechanisms have received empirical support, including shared parallel fiber afferents⁴⁹, shared climbing fibers⁵¹, shared MLI inputs⁴⁵, as well as ephaptic and gap junctional coupling between cerebellar cortical cell types^52–54. The latter phenomena have been shown to synchronize MLIs which, through feedforward inhibition, regulate spike-timing and synchrony in PCs by shortening integration time windows for coincident excitation^53–59. The relationship between MLIs and synchrony is particularly intriguing given the enhancement of PC co-firing during firing rate suppression and longer-window covariance¹⁴. Given this collection of potential mechanistic drivers of synchrony, it is interesting to note that not all reach-related PCs synchronized. We saw little synchrony or broader covariance between Cluster 3 or 4 PCs, which may suggest fewer shared inputs or a more disbursed anatomical distribution. Neuromodulatory states may also influence PC synchrony or other population relationships. The cerebellum receives noradrenergic, cholinergic, dopaminergic and hormonal inputs which could influence input-output relations of cerebellar circuitry to enhance or suppress spike synchrony and/or influence the efficacy of PC synchrony on nuclear neurons^60–64. That said, PC synchrony may not be either engaged or effective in cerebellar regions with slower linear coding, such as the vestibular flocculus and vestibular nuclei, where IPSC kinetics are either slower or have not been tested^50,65,66. It will be interesting to investigate whether such circuit elements may be differentially organized in regions with or without regulated PC synchrony, or if mechanisms exist that explicitly degrade PC synchrony.

While these data suggest a role for PC synchrony in regulating reach kinematics in intermediate cerebellum, the generalizability of these observations is unclear. In the macaque flocculus, synchrony levels do not fluctuate with behavior and levels of synchrony are low, diminishing the likelihood that synchrony serves a role in structuring behavior there⁵⁰. Moreover, IPSC kinetics and response to synchrony within different cerebellar nuclei vary^65–67. Regional specializations across cerebellar cortical regions also exist, with distinct distributions of unipolar brush cells, zebrin striping, density of Golgi cells, and cerebello-olivary interactions^68–71. These circuit motif variations may lead to distinct computations supporting behaviors with diverse control system requirements. Smooth pursuit eye movements, which require the floccular complex, are non-ballistic, thus inertial non-linearities are minimal. By contrast, saccadic eye movements, which are supported by the oculomotor vermis, require accelerative and decelerative signals to overcome inertial forces. It is possible that these or other unique control requirements require distinct neural mechanisms across cerebellar regions. In line with this reasoning, the present data corroborate Heck et al.⁴⁹, finding evidence for behaviorally modulated synchrony during reaching movements, which also require marked control of accelerative and decelerative forces.

Overall, our study presents evidence indicating that PC covariance structure is strongly associated with kinematics and works with firing rate to regulate downstream neuron activity and control movement. It will be interesting for future studies to investigate similar coding principles in long-range inhibitory neurons in other brain regions⁷², such as striatal and pallidal regions of the basal ganglia, noting that both healthy and pathological states may be influenced by synchronization^73–75. Another important avenue of future investigation will be whether differential sensitivity to synchronous firing between large premotor projection neurons and small nucleo-olivary neurons mediates multiplexed coding in the cerebellum^67,76. Moreover, associating pathological synchrony with cerebellar disorders could be a fruitful avenue of investigation^77–79.

Methods

Experimental methods

All procedures followed National Institutes of Health guidelines and were approved by the Institutional Animal Care and Use Committee at the University of Colorado Anschutz Medical Campus. Animals were housed in an environmentally controlled room, kept on a 12 h light/dark cycle and had free access to food and water, except during behavioral training and testing times. Adult C57BL/6 (Charles River Laboratories) mice of either sex (5 males and 2 females) were used in the experiments, but sex was not considered in the study design. The present dataset comprises high-density recordings previously reported¹³. Accordingly, surgical procedures and details of task design have been described in detail in the prior publication.

Reaching task

Briefly, after headplating, mice recovered for 2 days, then were food restricted to 80–90% of their original weight for reach training. Mice were trained to sit in a headfixed setup, and were presented with food pellets (20 mg, BioServ #F0163). The position of each pellet was moved further from the face until the mice reached with their forelimb to retrieve the pellet. To encourage the animal to reach with their right hand, the pellet was positioned slightly to the right of center at a consistent position for each mouse ( ~ 1.2–2.5 cm from starting position). Sessions lasted for 20 successful reaches or 30 minutes. The training process lasted 15 days and mice were considered proficient once they achieved a 50% success rate for 3 consecutive days.

Kinematic tracking

The position of the hand was tracked in real time using an infrared-based machine-vision motion-capture system (6 Optitrack Slim3U Cameras mounted with LED ring arrays, Motive software) at 120 frames-per-second as previously described^13,20. The cameras were positioned in front of and to the right of the mouse and focused on the space that covered the right forelimb reach trajectory. For camera calibration and kinematic tracking, retroreflective markers (1 mm) were affixed to the animal’s hand. To set the position and orientation of the cameras, a custom-built calibration wand and ground plane were used, in Optitrack Motive software. Motion tracking was recalibrated periodically to correct for drift. The spatial origin was defined as the center of the bar the mice held during rest. Spatial blocking and camera detection threshold were adjusted to prevent erroneous tracking of other minimally infrared-reflective objects.

High density electrophysiological recordings

Craniotomies were made over the cerebellum ipsilateral to the reaching arm of the mice and a custom-made recording chamber was implanted over the craniotomy targeting Lobule 4/5 and medial Simplex. The brain was covered with triple-antibiotic ointment (GlobePharma, Inc), and the recording chamber was sealed with Qwik-sil silicone (World Precision Instruments) to facilitate multiple recording sessions. During recordings, a Neuropixels 1.0 probe was lowered into the brain using a motorized micromanipulator (Sensapex uMp micromanipulator). Once the electrode shank spanned the PC layer, we waited 15 min for tissue to stabilize before recording. Electrophysiology data were acquired using an OpenEphys system (https://open-ephys.org). Data were sorted offline in Kilosort2⁸⁰ and manually curated in Phy (https://github.com/cortex-lab/phy).

Data analysis

Purkinje cell identification and inclusion criteria

Units were classified as PCs or putative PCs (collectively termed PCs) as in Calame et al.¹³, after spike sorting. Briefly, one of two criteria were met. In the first, we identified complex spikes (CS), by cross correlating units with low firing rate with high-firing units and looking for the presence of CS-related simple spike pause, as well as characteristic simple spikes and CS waveforms. With our recording method, the CS was not always present in the recording. In these cases, PCs were identified based on recording location and electrophysiological criteria using the firing rate, CV2 and median absolute difference from the median interspike interval (MAD) as described in Hensbroek et al.⁸¹. Specifically, cortical units with firing rate >40 sp/s, CV2 > 0.2 and MAD < 0.008 were labeled as PCs (see Extended Data Fig. 2 in Calame et al. 2023). This method was found to correctly identify 95% of CS-confirmed PCs¹³. To mitigate concerns that high density probes and associated sorting algorithms may inflate synchrony measurements by sampling and sorting the same unit twice, we only analyzed units that did not share contacts on the recording probe.

Time-resolved PC firing during task

To examine how the PC firing rate changes in time around reach endpoint, we computed peri-event time histograms (PETH). For each trial, we aligned spikes to the time of maximal outward position (endpoint), spanning −1 s before to 750 ms after endpoint. Data were binned at 1 ms (1 if it fired; 0 if not). We computed the average probability of firing in 1-ms bins across trials and multiplied by 1000 to achieve an estimate of rate (spike/second (sp/s))⁸². We smoothed the resultant function with a 51-point Gaussian kernel (matlab’s conv2.m), normalized by its sum such that it did not change the overall firing rate of the PC.

Hierarchical clustering of PCs

We classified PC activity using hierarchical clustering^83,84. First, we computed pairwise correlation coefficients across all (n) PC PETHs using matlab’s corrcoeff.m function, yielding an n x n matrix containing the correlation value for each pair of units. We then applied matlab’s linkage.m function to this correlation matrix to generate hierarchical linkage values ($Z\in {\mathbb{R}}^{nx3}$). Finally, we used matlab’s cluster.m function to partition the linkage tree into clusters (Fig. 1b).

To determine the appropriate number of clusters, we performed principal component analysis on the PETHs. Using the first 3 principle components (accounting for 83% of variance) we calculated the ratio of the inter- versus intra-cluster Euclidian distances (Suppl. Fig. 1a) for each PC, comparing the centroid of the corresponding cluster (intra) or centroids of other clusters (inter) as a clustering metric. Values above 1 suggest good clustering (Suppl. Fig. 1). As the number of clusters increased from 1-15, the clustering metric plateaued which informed the appropriate number of clusters. The asymptote of the curve in Suppl. Fig. 1b was found by fitting an exponential function $f\left(x\right)=A{e}^{-bx}+C$ on the curve with C corresponding to the asymptote value (=1.4), and interpolation along the cluster axis to determine the corresponding number of clusters ( = 4.19 clusters using interp1.m in matlab). Thus, we limited the number of clusters to 4 (Supplementary Fig. 1b).

Joint peri-event time histogram

We quantified event-related changes in joint firing between pairs of PCs using jPETHs as described previously^1,26. Briefly, we first computed PETHs for each neuron aligned to reach endpoint (or maximal velocity where indicated). For each trial, we then computed a matrix in which the (i,j)^th bin represented the number of coincident events where one unit fired in time bin i and the second unit fired in time bin j. Single trial matrices were summed across trials to obtain the raw jPETH (jPETH_raw). To estimate the expected coincident firing due to rate modulation alone, we computed a shift predictor based on the reach-aligned PETHs of each neuron. The predicted matrix (jPETH_predict) was defined as the outer product of the single-neuron PETHs, such that the (i,j)th bin was equal to PETH1(i)*PETH2(j), where PETH1 and PETH2 were the reach-aligned responses of units 1 and 2, respectively. We then obtained a corrected jPETH by subtracting the predicted from raw matrix (jPETH_raw − jPETH_predict). The jPETH matrices were computed using a bin size of 1 ms and then smoothed with a two-dimensional Gaussian kernel (σ = 3 ms).

Cross-correlograms

Cross-correlograms (CCGs; Fig. 2) were calculated as in Herzfeld et al.⁵⁰. Briefly, spikes from one PC (PC1) were aligned to the spike times of a second PC (PC2) to construct the raw CCG. To correct for apparent co-firing arising from local rate fluctuations, we estimated the coincident firing expected by chance over a three-spike interval. Specifically, we analyzed observed spike triplets (T1, T2, T3) from PC1 and assumed that the middle spike (T2) could have occurred at any point within the window from halfway between T1 and T2 (½*(T2-T1) to halfway between T2 and T3 (T2 to ½*(T3-T2). This resulting estimate of the conditional firing probability was formalized as:

$${\widehat{\mathrm{PC}}}_1\left(t\right)=\frac{2\mathrm{\Delta }t}{T_{i+1}-T_{i-1}}$$ 1

where$\mathrm{\Delta }t$ represents the bin width.

The rate-corrected CCG was then obtained by subtracting the null CCG from the raw CCG and normalized by the bin width $\left(\mathrm{\Delta }t=1ms\right)$.

$$\mathrm{CCG}\left(\mathrm{t}\right)=\frac{\mathrm{Pr}\left({\mathrm{PC}}_1\left(\mathrm{t}\right)\mid {\mathrm{PC}}_2\left(0\right)=1\right)-\mathrm{Pr}\left({\widehat{\mathrm{PC}}}_1\left(\mathrm{t}\right)\mid {\mathrm{PC}}_2\left(0\right)=1\right)}{\mathrm{\Delta }\mathrm{t}}$$ 2

This procedure isolates millisecond-timescale synchrony while accounting for slower covariation in firing rate.

Synchrony-behavioral correlation

To quantify the relationship between neuronal co-firing and behavior, we correlated the diagonal of the jPETH matrix with kinematic variables stratified by quartile (visually described in Supplementary Fig. 8). For analyses of peak velocity and deceleration, jPETHs were aligned to peak velocity for both peak velocity. For, endpoint position and deceleration error, jPETHs were aligned to reach endpoint. We then computed the Pearson correlation coefficient (matlab corr.m) between the 1-ms-resolved jPETH with kinematic measurement across quartiles across different time bins.

Shuffling methods

To control for chance synchrony between PCs, we employed a variety of shuffling methods85. The shuffling process for each pair was repeated >100 times, unless stated otherwise.

These were:

1. Trial shuffle: This method disrupts temporal correlations between units while preserving their relationship to behavior (Supplementary Fig. 4a–d). The activity of a given unit was cross correlated with the activity of a second unit from a different trial.

2. Spike resampling: This method preserves overall firing rate modulation while disrupting precise spike timing across a 75 ms time window (Supplementary Fig. 4e–h). Each trial was segmented into 75 ms windows. Spike times were then shuffled within each window.

3. Unit shuffle: To determine the relationship between firing rate and cofiring (as in Supplementary Fig. 5), we calculated the jPETHs between randomly chosen pairs of PC within a specific cluster. The shuffling procedure was run 1000 times.

4. Latency covariation control: To disambiguate the contribution of latency covariation on the jPETH^28,29, we implemented two complementary control procedures, roughly following Brody, 1999. In both approaches, spike trains from each PC pair were temporally shifted between −200 and 200 ms in 10 ms increments. For each shift, we computed the covariance between the units within a window spanning −0.15 to 0 s relative to reach endpoint. The latency producing the minimal covariance values was identified. Covariance was defined such that the first term captured the trial-by-trial probability of coincident spiking, and the second term represented the average firing probability across trials. After finding the relevant latency per trial, we applied two correction methods:

   1. Latency correction (Supplementary Fig. 6a–c): For each trial, the spike trains of both PCs were shifted by the latency value that minimized covariance (Brody, 1999).
   2. Latency-based trial shuffle (Supplementary Fig. 6d–f): For one unit in each pair, each trial was replaced by another trial with a latency that most closely matched that trial’s identified latency offset. We interpreted the preservation of jPETH structure following this method as evidence that latency covariation did not account for apparent cofiring.

Model of PC-nuclear integration

To model the consequences of PC cofluctuations across timescales we created PC pseudopopulations comprised of 20 pairs of simultaneously recorded Cluster 1-2 PCs. Trials were selected by having cofiring in the range of either high- or low-deceleration error (high: 150 ms pre-endpoint cofiring, between 93 and 153 sp²/s²; low: cofiring between 28.6 and 92.8 sp²/s². These ranges are within 3 standard deviations of the mean cofiring for each condition (Fig. 3o). For each pair a random 15 trials were chosen to construct a paired firing rate of the two units, and these were combined across pairs to construct PC population spike trains to test in the model.

We then used these trains to examine integration of IPSCs in a model of the cerebellar nuclei. Nuclear cells were modeled using a reduced model of neuronal spiking dynamics³⁰ which combines the simplicity of an integrate-and-fire model for spiking without explicit modeling of Hodgkin-Huxley type conductances. The spiking behavior of the model is set by a variety of parameters within a term U that determines the burstiness of the modeled neuron. Each PC spike within the population drove an IPSC with 2.5 ms decay time and 5 nS unitary conductance.

We then calculated the voltage of the nuclear target cell using the following parameters:

Table 1

Parameter	variable	Value
time-scale of the recovery parameter (U)	a	0.03 ms
sensitivity of U to subthreshold fluctuations of the voltage	b	−2
after-spike reset value of v	c	−50 mV
after-spike reset of U	d	2
IPSC decay time constant	tau	2.5 ms
Maximal inhibitory conductance	Gmax	5 nS
spike threshold	Vthresh	−40 mV
GABAA receptor reversal potential	Vreverse	−75 mV
membrane starting potential	Vm(0)	−60 mV
Membrane capacitance	Cm	89 pF
DC excitatory offset		700 nA

These parameters resulted in model neuron firing around 90 sp/s without inhibition, within the range of values seen in vitro^17,25,31 and in vivo³². Note that excitatory input was constant.

For each PC, the inhibitory current was calculated and summed, as follows:

$$I\left(t\right)=I\left(t\right)+G_i\left(t\right)*\left(V_{reverse}-V_m\left(t\right)\right)$$ 3

where i is the PC index (1…40), and V_m(t) is the membrane potential at time t.

The voltage was then modeled using:

$$C_m\frac{\mathrm{\Delta }{V}_m}{\mathrm{\Delta }T}=k*\left(V_m\left(t\right)-V_{reverse}\right)*\left(V_m\left(t\right)-V_{thresh}\right)-U\left(t\right)+I\left(t\right)$$ 4

where k is gain of the voltage nonlinearities30. In our model, we used k = 5/89 = 0.0562 pA/mV². U is the “membrane recovery parameter” in pA which captures the effect of the voltage gated K+ currents and Na+ current inactivation, and provides negative feedback to membrane voltage. It is calculated as:

$$\stackrel{°}{U}={\tau }_{decay}*a*\left(b*\left(V_m\left(t\right)-V_{reverse}\right)-U\left(t\right)\right)$$ 5

where a is the time scale of the recovery variable U; b is the sensitivity of the U to subthreshold membrane potential fluctuations.

To determine whether covariance changes plausibly account for excess deceleration in high-gain reaches, we predicted the deceleration error from modeled nuclear firing rates. We used the relationship between IntA firing rates and reach outward deceleration established in Becker and Person, 2019 to estimate the firing rate change that would account for excess deceleration in high gain reaches. We then found difference in CbN firing rates generated by high and low covariance PC populations that had been recorded during high- and low-gain reaches20. Briefly, the deceleration values were found on a trial-by-trial basis, and the firing rates were estimated as:

$$\mathrm{FR}=\frac{\mathrm{\Delta }\mathrm{Vel}-0.24}{-0.012}$$ 6

with the following units: firing rate (FR), sp/s; ∆Vel (velocity), cm/s; 0.24 is the intercept, cm/s; -0.012 is the slope in units of (cm/s)/(sp/s).

Using Eq. 6, we estimated CbN firing based on the total deceleration per reach, within a 100 ms epoch. Because deceleration and deceleration error are not identical, we next computed the estimated firing rate associated with a given deceleration error per reach (Fig. 4j, dots), and computed the least-squares regression for each session (e.g., Fig. 4j black line). The average regression across sessions (Fig. 4j red line) was defined by Eq. 7.

$$\mathrm{decel\; error}=1.1\mathrm{xFR}-148.6$$ 7

with the following units: FR, sp/s; deceleration error cm/s²; −148.6 is the intercept in units of cm/s²; 1.1 is the slope in units of (cm/s²)/(sp/s).

To calculate the effect of covariance on nuclear firing and estimated deceleration error across timescales, we jittered PC spike times within increasing windows for PC pairs with original CCGs that corresponded to cofiring related to high or low deceleration error reaches. Specifically, the contribution (F) was calculated as:

$$\mathrm{F}=\frac{{\mathrm{FR}}_{\mathrm{high}}^{\mathrm{jit}}-{\mathrm{FR}}_{\mathrm{low}}^{\mathrm{jit}}}{{\mathrm{FR}}_{\mathrm{high}}^{\mathrm{est}}-{\mathrm{FR}}_{\mathrm{low}}^{\mathrm{est}}}$$ 8

where the numerator is the difference between the CbN firing changes after jittering between high and low deceleration error, and the denominator is the difference in the CbN firing before jittering.

To relate the in silico model to dynamic clamp experiments, we used a different method to generate PC population activity. We first generated a kernel of time-varying firing rates in a population of 40 Purkinje cells from a random draw of 40 of our recorded PCs in clusters 1 + 2. This time-varying average firing rate was then used as the rate parameter passed to a Poisson random number generator (poissrnd.m in matlab). This process resulted in a collection of 40 simulated PCs, which had time-varying firing rates matching the average population activity, but with spike times randomized for each cell.

To match the in vivo calculated CCGs with synchrony in the modeled population (Fig. 2), we organized 40 units into 20 randomly assigned pairs. For each pair, we drew a CCG target value from the distribution of CCGs, then adjusted spiketimes such that the spiketrain matched the CCG value. To impose synchronize, we chose a random non-synchronous spike then moved a spike from the other unit to synchronize with it. This was performed as many times as necessary to match the CCG. Note that this process also preserves the overall firing rate. To test the relationship of nuclear firing to PC CCG peak values, we used the same procedure but systematically varied CCG peak values in modeled populations.

Acute brain slice preparation

Tissue preparation

Postnatal day 32–47 C57BL/6 mice of either sex (n = 22 mice, 10 females 12 males) were deeply anesthetized with isoflurane before transcardial perfusion with warmed Tyrode’s solution containing (in mM) 123.75 NaCl, 26 NaHCO₃, 10 glucose, 3.5 KCl, 1.25 NaH₂PO₄, 1.5 CaCl₂, and 1 MgCl₂. Solutions were oxygenated with 95% O2, 5% CO₂. Following perfusion, tissue was harvested and 300 µm thick parasagittal slices were made using a Leica VT1000 S vibratome (Leica, Wetzlar, Germany). Slices were incubated at 34 °C for 1 h before being stored at room temperature until recording (up to approx. 5 h).

Data acquisition

Slices were transferred to a recording chamber and were continuously perfused with oxygenated Tyrode’s solution at 32 °C - 34 °C which included synaptic blockers SR95531 (gabazine, 10 µM), 6,7-dinitroquinoxaline-2,3-dione (DNQX, 10 µM), and (R)-CPP (10 µM). All drugs were purchased from Tocris (Bristol, United Kingdom). Glass patch pipettes (Sutter Instruments BF150-86-7.5) were pulled using a Sutter P97 (Sutter Instrument, Novato, CA, USA) to produce tip resistances of 2 – 4 MΩ and were filled with a K-gluconate intracellular solution (290 mOsm, pH 7.37) consisting of 132 K-gluconate, 4.4 Na-gluconate, 4.4 NaCl, 2.2 MgCl2, 1.1 EGTA, 11 HEPES, 22 sucrose, 14 TRIS creatine phosphate, 4 Mg-ATP, 0.3 and TRIS-GTP. Stable whole-cell recordings were acquired at 100 kHz in pClamp 10.3 and pClamp 10.7 (Molecular Devices, San Jose, CA, USA) via an Axon 700B multiclamp amplifier and Digidata 1440 digitizer. Cells were visually identified and included for recording based on an access resistance below 20MΩ, the demonstration of a robust fast inward sodium current in voltage clamp, and overshooting action potentials in current clamp.

Dynamic Clamp

Dynamic clamp experiments were conducted based on hardware and software designs described previously⁸⁵. IPSGs were simulated with characteristics described previously17, specifically, 5 nS peak conductance with a 100 µsec rise time and 2.5 msec decay time. IPSC commands were conveyed to the dynamic clamp from the digitizer via TTL pulses with a 10-microsecond pulse width. The 100 kHz sampling rate allowed us to simulate synaptic integration at instantaneous rates of up to 50 kHz and thus allowed us to model a pool of 40 convergent Purkinje cells with 1 millisecond synchrony. Purkinje cell trains were constructed based on previously published PC population firing rates during reaching movements¹³. Briefly, interspike intervals of a PC “seed” trace were established to reproduce a PETH from a given reach velocity range (0−20%; 60–80% and 80–100% peak speed). Spike times for 39 additional “PCs” within that “pool” were then generated from the seed trace, where new spike times were calculated by jittering each spike time as a random draw from a normal distribution with varying standard deviations, varied between 9 and 2 ms. This method allowed us to modulate population synchrony dynamically throughout the reach epoch by modulating the standard deviation of these jitter Gaussians, maintaining the net time-varying population activity observed in Calame et al.¹³. Finally, for experiments modulating PC rate and synchrony, PC pools with mean rates of 40–100 Hz in 10 Hz intervals were generated. Within these pools, mean covariance was swept from 0 to 2 spks/s.

Statistical analysis

Data presented in the manuscript treat each neuron as an independent sample. All data were tested for normality using Kolmogrov–Smirnov test to choose the proper statistical test. All t-tests or Wilcoxon’s signed rank are paired, unless stated otherwise.

No statistical tests were used to predetermine required sample size. Randomizations and controls are described in the main text and methods. The experimenters were not blind to conditions during experiments and outcome assessments. Nonparametric tests were used when the dataset violated normality assumptions. Corrections for multiple comparisons were used to adjust significance thresholds where noted.

All data analyses were made in Matlab (Mathworks, 2020). Raw data and waveform analysis were made in Python and Visual Studio.

Reporting summary

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

Author contributions

A.N. and A.L.P. conceptualized the study. D.J.C. acquired the neural data. M.S.S. performed the in vitro experiments and analyzed the in vitro data. A.N. analyzed the in vivo data and constructed models. A.N. and A.L.P. wrote the paper.

Peer review

Peer review information

Nature Communications thanks Igor Delvendahl, Eun Jung Hwang and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

This study was supported by the Life Science Research Foundation (and Amgen) and the Edmond and Lily Safra Campus Post-doctoral fellowship to A.N.; F31-NS113395 to D.J.C.; and NS114430 and NS131839 to A.L.P.

Data availability

Source data are provided with this paper. Processed data required to replicate figures in this study are available through GitHub (https://github.com/AbedNashef/Synchrony_paper). Raw data are available upon reasonable request to the corresponding author. Source data are provided with this paper.

Code availability

All code required to replicate figures in this study are available through GitHub (https://github.com/AbedNashef/Synchrony_paper).

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-74576-x.