(2015) Encoding of Action by the Purkinje Cells of the Cerebellum

LETTER doi:10.1038/nature15693

Encoding of action by the Purkinje cells of the cerebellum

David J. Herzfeld1, Yoshiko Kojima2, Robijanto Soetedjo2 & Reza Shadmehr1

Execution of accurate eye movements depends critically on the Purkinje cells project to the caudal fastigial nucleus (cFN), where cerebellum1–3, suggesting that the major output neurons of the cere- about 50 Purkinje cells converge onto a cFN neuron17. For each bellum, Purkinje cells, may predict motion of the eye. However, this Purkinje cell we computed the probability of a simple spike in 1-ms encoding of action for rapid eye movements (saccades) has remained time bins during saccades of a given peak speed, averaged across all unclear: Purkinje cells show little consistent modulation with respect to saccade amplitude4,5 or direction4, and critically, their discharge lasts longer than the duration of a saccade6,7.Herewe a 400° per s saccade 650° per s saccade 400° per s saccade 650° per s saccade analysed Purkinje-cell discharge in the oculomotor vermis of behav- 500° per s 8,9 200 ing rhesus monkeys (Macaca mulatta) and found neurons that N12 increased or decreased their activity during saccades. We estimated the combined effect of these two populations via their projections to 100 the caudal fastigial nucleus, and uncovered a simple-spike population response that precisely predicted the real-time motion of 0 the eye. When we organized the Purkinje cells according to each –1000 100 0 100 0 100 0 100 cell’s complex-spike directional tuning, the simple-spike population b 500° per s response predicted both the real-time speed and direction of saccade multiplicatively via a gain field. This suggests that the cerebellum N23 predicts the real-time motion of the eye during saccades via the 100 combined inputs of Purkinje cells onto individual nucleus neurons. 50 A gain-field encoding of simple spikes emerges if the Purkinje cells 0 Firing rate (Hz) Firing rate (Hz) that project onto a nucleus neuron are not selected at random but –1000 100 0 100 0 100 0 100 share a common complex-spike property. Time relative to saccade onset (ms)

Previous studies have focused on bursting activity of Purkinje cells ce400° per s saccade 650° per s saccade 6,10,11 Burst during saccadess and found no consistent modulation with sac- 80 60 4,5 5–7 4 12 End cade amplitude , speed or direction . A recent simulation sug- 40 Start 60 Burst gested that Purkinje cells that pause during saccades may be important 20 in understanding the responses observed in the deep cerebellar nucleus Pause 0 neurons. The main question that we wished to address was how the 40 –20 Mean firing rate (Hz) Change in firing rate

Purkinje cells encode the real-time motion of the eye. of individual cells (Hz) Pause 20 –40 We analysed simple-spike activity of 72 Purkinje cells in the oculo- 10 15 –100 0 100 –50 0 100 motor vermis (OMV, cerebellar lobules VI and VII) of five monkeys Saccade magnitude (deg) Time relative to peak speed (ms) during saccades. The population included cells that exhibited increased d f activity (bursting; n 5 39, Fig. 1a) or decreased activity (pausing; n 5 33, 800 Speed (deg per s) 5,6,10 Burst Fig. 1b). Consistent with previous reports , most neurons were poorly Simple 600 80 4,000 spikes Speed modulated by saccade amplitude (Fig. 1c and Extended Data Fig. 1); 400 however, the mean firing rate of burst cells (but not pause cells) increased 60 210 200 significantly with saccade peak speed (Fig. 1d, P , 10 ). Previous work 3,000 40 Pause had demonstrated that the population response encoded additional sac- 0 cade-related information that was not reliably present in the responses of Mean firing rate (Hz) 20 2,000 6,13,14 400 650 Rate of spikes generated by a –100 0 100 –50 0 100 individual neurons . To examine the population response, we mea- population of 50 cells (spikes per s) sured change in firing rates (from baseline) for the bursting and pausing Saccade peak speed (deg per s) Time relative to peak speed (ms) cells during slow (400u s21) and fast (650u s21) saccades (Fig. 1e), pooled Figure 1 | Population of burst and pause Purkinje cells together predict eye across all directions. The onset of change in firing rates in both popula- speed in real time. a, b, Perisaccade histograms for a bursting (a) and tions generally led saccade onset bymorethan50ms.Theterminationof pausing (b) Purkinje cell during saccades of various speeds and directions (red activity was also significantly arrow). The trace on the top row is saccade speed. The grey arrow indicates 21 c d later than the saccade: a 650u s saccade was 38 6 1.2 ms in duration saccade end. , , Mean firing rates over the duration of saccade computed across all directions. Changes in speed produced an increase in the firing rate of 6 (mean s.e.m.), whereas activity of burst and pause cells persisted for the burst cells but not the pause cells. e, Change in firing rates (with respect more than 100 ms. Given that the cerebellum is thought to have a critical to baseline) of the bursting and pausing Purkinje cells for two saccade 15,16 role in termination of ipsiversive saccades , it is unlikely that separate speeds. Grey bars are onset and termination of the saccade (width is s.e.m.). populations of burst or pause Purkinje cells control the motion of the eye, f, The total rate of simple spikes produced by a random selection of 50 since their activity persists for much longer than the saccade. Purkinje cells.

1Department of Biomedical Engineering, Laboratory for Computational Motor Control, Johns Hopkins University School of Medicine, Baltimore, Maryland 21205, USA. 2Department of Physiology and Biophysics, Washington National Primate Center, University of Washington, Seattle, Washington 98195, USA.

directions. We then chose 50 Purkinje cells at random and computed ab the total number of simple spikes generated by the population at each 90° millisecond, resulting in an estimate of the rate of presynaptic spikes 0.6 converging onto a cFN cell. The results (Fig. 1f) revealed a real-time Error 0.4 encoding of the speed of the eye: the peak of the activity preceded peak 0.2 speed, increased in magnitude when speed increased, and returned to 0o baseline just before saccade termination (R2 at the optimal delay, 21 2 222 21 2 243 400u s : R 5 0.52, P , 10 ; 650u s : R 5 0.62, P , 10 ). It N1 CS-on appeared that the simple spikes of the pause and burst cells combined Primary target Pr(CS) together to predict motion of the eye. Back-step target Let us hypothesize that the Purkinje cells that project to a nucleus 0 100 200 0 100 200 neuron are not selected randomly, but are organized by their inputs from the inferior olive18. That is, suppose that the olive projections 0 100 200 divide the Purkinje cells into clusters where each cluster of Purkinje cells Time relative to primary saccade termination (ms) projects onto a single nucleus neuron. The input from the olive produces complex spikes in the Purkinje cells. We found that if we organized the Figure 2 | Determination of complex-spike properties of Purkinje cells. simple spikes of the Purkinje cells based on each cell’s complex-spike a, Response of a Purkinje cell during the 250-ms period after completion of a properties, additional features of the population activity were unmasked. saccade (simple spikes are grey; complex spikes are red). CS-on was We measured complex-spike properties of each Purkinje cell by determined via a back-step paradigm in which the target was jumped (unfilled target to filled target) during saccade execution. Black arrow inducing a post-saccadic error through displacement of the target indicates saccade vector, red arrow indicates error vector. We computed the during the saccade, and then measured the probability of complex probability of complex spikes in the 50–200-ms period after saccade spikes as a function of the direction of this error (Fig. 2 and termination. b, The probability of a complex spike (Pr(CS)) as a function of Supplementary Information section 2). For each Purkinje cell, the the direction of the error vector. For this neuron, the highest probability (CS-on) direction of error that produced the largest probability of complex occurred when the error vector was in direction 245u. The direction of spikes during the 50–200-ms post-saccade period was labelled as CS-off for this cell was 135u. CS-on, and the opposite direction was labelled as CS-off9 (Extended Data Fig. 2). We then made the assumption that the Purkinje cells that repeated measures analysis of variance (ANOVA) with main effects projected onto a nucleus neuron all had the same CS-on direction 215 27 of peak speed, P , 10 ; CS-direction P , 10 ; and a speed by (Fig. 3a). Under this assumption, we computed the rate of presynaptic 215 CS-direction interaction, P , 10 ). simple spikes that a nucleus neuron would receive from the cluster of To examine the effects of saccade direction more closely, we plotted Purkinje cells (Supplementary Information section 3). We did this by the population response across saccade directions with respect to convolving each Purkinje cell simple-spike train with a 2.5 ms stand- CS-on (Fig. 4c). We found that the population response was highest ard-deviation normalized Gaussian, approximating the temporal for saccades made in the CS-off direction, with an encoding of direction characteristics of the inhibition produced in the nucleus neuron due that resembled a cosine function (Fig. 4d). Therefore, the combined to a simple spike in the Purkinje cell17,19. activity of burst and pause cells, but not the activity of either population Figure 3b shows the change in population response from the base- individually (Extended Data Fig. 4), aligned to CS-off, produced a line level when a saccade was made in the same direction as CS-off. The population response that exhibited gain-field encoding: the magnitude response rose above baseline before saccade onset, peaked before peak of the population response increased linearly with speed, and was speed, and then returned to near baseline. The peak response scaled cosine-tuned in direction, with a multiplicative interaction between robustly with saccade amplitude (Fig. 3c, R2 5 0.93, P , 1025). We speed and direction. The rate of simple spikes converging onto cFN, observed a strong correspondence between the real-time population represented by s(t), predicted in real-time motion of the eye response and the real-time speed (Fig. 3d, lower plot, and Extended (Supplementary Information section 4 and Extended Data Fig. 5) is: Data Fig. 3). The population response preceded eye speed by an average of 21.2 6 0.4 ms (correlation analysis in the CS-off direction, stðÞ~jjx_ ðÞtzD gðÞh, hCS zc mean 6 s.e.m.). Peak population response precisely predicted peak ð1Þ 2 27 gðÞh, h ~a cos (h{h )zb speed (Fig. 3e, R 5 0.98, P , 10 ). CS CS We took advantage of natural variability in saccades to test further In equation (1), jjx_ ðÞtzD represents the magnitude of the eye velocity the relationship between the population response and speed. We vector (the time derivative of eye position, x) at time t 1 D (where sorted all 10u saccades (direction CS-off) according to peak speed D 5 19 ms), b and c are baseline offsets, a is a scaling factor, h is saccade (Fig. 3f) and found that despite the constant amplitude, the population direction, and hCS is direction of CS-off for that cluster of Purkinje cells. response precisely predicts the actual peak speed of the eye (Fig. 3g, The resulting gain-field encoding of eye motion is depicted in Fig. 4e. R2 5 0.96, P , 1027). Therefore, when the simple spikes were orga- We next addressed the question of how the activity of individual nized according to each cell’s complex-spike directional preference, cells produced this directional encoding in the population response. the population response for saccades of constant direction predicted The main contributors were the pause cells, which started their pause nearly all of the variability in saccade peak speed. approximately 10 ms earlier when the saccade was in the CS-on dir- No previous work, to our knowledge, had revealed how direction of ection (Fig. 4f), a change that was independent of saccade speed a saccade is encoded in the activity of Purkinje cells. For the burst (Extended Data Fig. 6). This subtle shift in the timing of spikes pro- and pause cells, the mean and peak firing rates during the saccade duced an increase of the population response when saccade direction did not vary as a function of direction (Extended Data Fig. 4). changed from CS-on to CS-off (Fig. 4a). However, organizing the population response according to each We found that the anatomical distribution of Purkinje cells, as Purkinje cell’s complex-spike tuning preference revealed a clear labelled by their CS-off direction, was not random, but lateralized9 encoding of direction: the peak response was greater if the saccades (Extended Data Fig. 7), confirming previous anatomical studies were in the same direction as CS-off as compared to CS-on (Fig. 4a, suggesting that olivary projections are contralateral20,21. Purkinje cells t-test, P , 10216). Indeed, the peak population response rose linearly with rightward CS-off were more likely to be on the right side of the as a function of peak speed in both directions, but with a larger gain cerebellum (t-test, P , 1024). This indicates that saccades made in when the saccade direction was congruent with CS-off (Fig. 4b, the same direction as CS-off were typically ipsiversive, whereas saccades

a Pause Figure 3 | A cluster of Purkinje cells, organized cell by their complex spikes, produced a popula- Purkinje cells with similar complex spike field n( = 50) tion response that predicted in real time the motion of the eye. a, Hypothesized organization of the oculomotor vermis. To compute a Population response (simple spikes) Inferior olive neurons with Burst cell similar error encoding population response, we measured the simple Midline Caudal fastigial nucleus (cFN) neuron spikes of each Purkinje cell as a function of saccade Change in population response (spikes per s) Change in population response (spikes per s) b d Change in population response (spikes per s) f direction with respect to the CS-on direction of All saccades (direction CS-off) 1,500 All saccades (direction CS-off) 10° saccades (direction CS-off) 2,250 that cell. For the Purkinje cells shown here, the CS- 1,500 650 650 on is an error vector to the left (arrow). b, Change

750 in population response (with respect to baseline) 1,000 as a function of saccade amplitude in 0.5u bins, for 13 14

500 saccades in the CS-off direction. Data in the 0 12 500 550 600 500 550 600 amplitude axis were smoothed by a first-order 6 –750

0 Savitzky–Golay filter with a width of three bins . 0 11 15 Peak speed (deg per s) Peak speed (deg per s) 450 Saccade magnitude (deg) Bottom plot shows the population response for Start End 10 400 450 400 three representative amplitudes. c, Peak population –100 –50 0 50 100 –100 –50 0 50 100 –100 –50 0 50 100 response increased linearly with saccade amplitude. P values indicate significant linear

600 650° per s 15° Speed (deg per s) Population Speed 1,500 correlation. d, Population response as a function 2,250

12.5° 1,500

response 400 10° 525° per s of saccade peak speed. Bottom plot shows

200 representative responses with their corresponding

0 per s 500 speed traces. e, Peak population response increased 0 0 0 linearly with saccade peak speed. f, Population Change in population Change in population response (spikes per s) response (spikes per s) response Change in population response (spikes per s) response 10° saccades response for 10u saccades (61u), as a function of

–1,500 –100 –50 0 50 100 –100 –50 0 50 100 –100 –50 0 50 100 saccade peak speed. Bottom plot shows the Time relative to peak speed (ms) Time relative to peak speed (ms) Time relative to peak speed (ms) population response for slow, medium and fast ce g saccades of 10u amplitude. g, Peak population R2 P –5 R2 P –7 R2 P –7 response increased linearly as a function of 1,800

= 0.93 ( < 10 ) = 0.98 ( < 10 ) 2,700 = 0.96 < 10 )

1,750 peak speed even for a fixed magnitude saccade. Error bars are s.e.m. 2,100 1,500 1,500 1,500 1,200 1,200 response (spikes per s) response (spikes per s) response (spikes per s) response 900 Peak change in population Peak change in population Peak change in population All saccades (direction CS-off)900 All saccades (direction CS-off) 10° saccades (direction CS-off)

900 10 11 12 13 14 15 400 450 500 550 600 650 400 450 500 550 600 650 Saccade amplitude (deg) Peak saccade speed (deg per s) Peak saccade speed (deg per s) congruent with CS-on were contraversive. In contrast, pause and burst cells, all with a common complex-spike tuning preference (Fig. 3a). cells were uniformly distributed across the cerebellum (P . 0.4). If, contrary to our hypothesis, pause and burst cells organized into Our results rely critically on our hypothesis that Purkinje cells separate clusters, the population response would not predict the real- organize into clusters with roughly equal numbers of pause and burst time motion of the eye (Fig. 1e). Similarly, if each cluster was not

270

acb 900 2,000 1,200 response (spikes per s) End Change in population

Start CS-off CS-off 180 1,500

CS-off 450 600 90 1,000 300 900 CS-on 0 Change in population CS-on (spikes per s) response 0 response (spikes per s) response Saccade direction (deg) 0 Peak change in population

CS-on End 500 Start –90 –300 –100 –50 0 50 100 400 450 500 550 600 650 –100 –50 0 50 100 Time to peak speed (ms) Peak saccade speed (deg per s) Time to peak speed (ms)

deSimple Complex f

45 End Probability of complex spike Start spikes spikes Population response (spikes per s)

1,800 Burst 30 Burst (CS-on) 0.2 (CS-off)

1,200 15

0 Pause (CS-off) 0.1 CS-off 600 Pause –15 (CS-on) response (spikes per s) -On 0 Peak change in population 600 750 900 1,050 –500 0 500 –30 Contribution of single Purkinje CS-Off CS-On CS-Off cells to population response (Hz) 0 eye velocity (deg per s) Vertical –180 –900 90 180 –750 0 750 –100 –50 0 50 100 Angle relative to CS-on (deg) Horizontal eye velocity (deg per s) Time relative to peak speed (ms) Figure 4 | Population response of Purkinje cells predicted saccade speed and simple spikes) as a function of saccade direction with respect to CS-on. The direction in real time as a gain field. a, Population response for saccades in brown curve shows probability of observing a complex spike as a function of direction CS-on and CS-off. The population response is larger when the the angle relative to each neuron’s CS-on. The black curve indicates cosine saccade is in the CS-off direction. b, Peak population response grew linearly fit of the peak population response. e, Gain-field encoding by a cluster of with saccade speed, but had a higher gain for saccades in the CS-off direction. Purkinje cells whose CS-off direction is to the right (equation (1)). f, Contribu- c, Real-time population response as a function of saccade direction relative tion of single Purkinje cells to the population response. A change in direction to CS-on. Data smoothed as in Fig. 3b. d, Peak population response (labelled as coincides with a shift in timing of the pause cells. Error bars are s.e.m.

15 OCTOBER 2015 | VOL 526 | NATURE | 441 G2015 Macmillan Publishers Limited. All rights reserved RESEARCH LETTER composed of roughly equal numbers of pause and burst cells, the 6. Thier, P., Dicke, P. W., Haas, R. & Barash, S. Encoding of movement time by populations of cerebellar Purkinje cells. Nature 405, 72–76 (2000). population response could not predict the real-time speed of the 7. Kase, M.,Miller, D. C. & Noda, H. Discharges of Purkinje cells and mossy fibres in the eye (Extended Data Fig. 8 and Supplementary Information section 5). cerebellar vermis of the monkey during saccadic eye movements and fixation. The fact that burst and pause cells were distributed uniformly across the J. Physiol. (Lond.) 300, 539–555 (1980). recording locations, and not lateralized as we found with the complex- 8. Kojima, Y., Soetedjo, R. & Fuchs, A. F. 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Received 24 February; accepted 8 September 2015. Supplementary Information is available in the online version of the paper. Acknowledgements These data were collected in the laboratory of A. Fuchs. The 1. Xu-Wilson, M., Chen-Harris, H., Zee, D. S. & Shadmehr, R. Cerebellar contributions authors are grateful to his generosity. The authors would like to thank S. du Lac for to adaptive control of saccades in humans. J. Neurosci. 29, 12930–12939 (2009). comments. The work was supported by NIH grants R01NS078311, R01EY019258, 2. Barash, S. et al. Saccadic dysmetria and adaptation after lesions of the cerebellar R01EY023277 and F31NS090860. cortex. J. Neurosci. 19, 10931–10939 (1999). 3. Kojima, Y., Soetedjo, R. & Fuchs, A. F. Effects of GABA agonist and antagonist Author Contributions Y.K. and R.So. conceived, designed and performed all injections into the oculomotor vermis on horizontal saccades. Brain Res. 1366, experiments. D.J.H. and R.Sh. formed the conceptual model. D.J.H. analysed the data 93–100 (2010). and made all figures. R.Sh. and D.J.H. wrote the paper. 4. Ohtsuka, K. & Noda, H. Discharge properties of Purkinje cells in the oculomotor vermis during visually guided saccades in the macaque monkey. J. Neurophysiol. Author Information Reprints and permissions information is available at 74, 1828–1840 (1995). www.nature.com/reprints. The authors declare no competing financial interests. 5. Helmchen, C. & Bu¨ttner, U. Saccade-related Purkinje cell activity in the oculomotor Readers are welcome to comment on the online version of the paper. Correspondence vermis during spontaneous eye movements in light and darkness. Exp. Brain Res. and requests for materials should be addressed to D.J.H. ([email protected]) or R.S. 103, 198–208 (1995). ([email protected]).

METHODS saccade by taking the average of the firing rate from the start to the end of the No statistical methods were used to predetermine sample size. saccade. We determined the peak response of a Purkinje cell as the maximum We analysed extracellular recordings from Purkinje cells of the oculomotor firing during the saccade period. 8 vermis in five rhesus monkeys (B, F, W, K, KO) while they made saccades to visual The response of an individual Purkinje cell is quite variable , with some neurons targets8,9 (Supplementary Information section 1). Each cell was well isolated for an showing a combination of bursting and pausing activity in the time period near average of 3,000 saccades. Briefly, a scleral search coil was surgically attached to the saccade onset. Therefore, to categorize neurons exclusively as pausing or bursting, eye of each monkey, allowing measurement of eye position26 while the animal’s we compared the mean firing rate of each cell from the period 200–100 ms before head was restrained. After surgery, the monkeys were trained to make saccades to saccade onset of all recorded trials to a 150-ms period centred on saccade peak targets of varying amplitudes and directions. We identified Purkinje cell activity in speed. Neurons that reduced their rate during this extended period were classified OMV by their saccade-related change in the simple-spike response, as well as the as ‘pausing’, whereas neurons that significantly exceeded this rate were classified as presence of complex spikes. Neurophysiological data were sampled at 50 kHz. All ‘bursting’. We tested this categorization statistically via a paired t-test with a cutoff experiments were performed in accordance with the Guide for the Care and Use of of P 5 0.05. Only two neurons (both bursting) did not pass this statistical test. We Laboratory Animals (1997) and exceeded the minimal requirements recom- included these two neurons in the bursting population. mended by the Institute of Laboratory Animal Resources and the Association To establish confidence intervals on the population responses, we performed a for Assessment and Accreditation of Laboratory Animal Care International. All bootstrap analysis in which we randomly sampled 50 neurons from the available animal procedures were approved by the local committee at the University of pool of 72 (with replacement), which simulated the approximate number of Washington. Purkinje cell inputs that project onto a nucleus neuron17. Error bars show mean The CS-on direction for each cell was determined using the standard intra- 6 s.e.m. of 50 bootstrapped Purkinje cell populations. In cases where we show the saccadic step paradigm27, in which the target was displaced during the initial responses of the bursting/pausing populations separately, we report the mean 6 saccade (Fig. 2). This error resulted in complex spikes during the period following s.e.m. across neurons in the respective population without bootstrapping. the saccade, when the monkey observed the error. For every cell, we determined To compute the population response of a cluster of 50 Purkinje cells, we first the CS-on direction as the error direction which elicited the largest probability of convolved the simple spikes of each Purkinje cell with a kernel (normalized complex spikes during the 50–200 ms following the primary saccade. For n 5 39 Gaussian of 2.5 ms s.d.), approximately representing the time course of post- cells we were able to maintain excellent isolation of the Purkinje cell throughout synaptic inhibition induced by the simple-spike train17. We then computed the the experiment, allowing us to perform automated identification of complex spikes change from baseline for each cell, and finally the sum of changes across the on every trial. This allowed us to compute the probability of complex spikes as a population, resulting in a population response that had units of spikes s21, com- function of error direction. For the remaining n 5 33 cells, the CS-on direction was puted at each millisecond of time. determined via analysis of the initial 50 trials for each direction of error8. CS-off was defined as CS-on 1180u. 26. Fuchs, A. F. & Robinson, D. A. A method for measuring horizontal and vertical eye We computed firing rates by determining the inverse of the time between two movement chronically in the monkey. J. Appl. Physiol. 21, 1068–1070 (1966). 28 27. McLaughlin, S. C. Parametric adjustment in saccadic eye movements. Percept. consecutive simple spikes and then convolved the resulting time series with a Psychophys. 2, 359–362 (1967). normalized Gaussian kernel with a standard deviation of 2.5 ms, which is signifi- 28. Lisberger, S. G. & Pavelko, T. A. Vestibular signals carried by pathways subserving cantly shorter than conventional kernels6. This guarded against overestimating the plasticity of the vestibulo-ocular reflex in monkeys. J. Neurosci. 6, 346–354 duration of a population response. We calculated the mean firing rate during the (1986).

a 320 600

300 550

280 500

260 450 Mean speed (deg/s) Peak speed (deg/s) 240 400 10 15 10 15 Saccade amplitude (deg)

Modulation of P-cell discharge Modulation of P-cell discharge b Amplitude modulation c Speed modulation Mean response Peak response Mean response Peak response 1

1 0.1

0.1 0.01

0.01 10-3 p p 10-4 10-3

10-5 10-4 -6

10 Significance Significance Burst Burst Pause Pause 10-5 -0.2 0.2 0.6 1 -10 0 10 -20 0 20 -0.20 0.2 0.4 0.6 Slope (Hz/deg) Slope (Hz/(deg/s)) Extended Data Figure 1 | Firing rates of individual Purkinje cells as a their responses (negative slope). However, mean and peak firing rates of a function of saccade amplitude and peak speed. a, Increase in saccade majority of neurons (47 of 72) were not significantly modulated with saccade amplitude produced robust increases in mean and peak saccade speed (mean: amplitude. As a result, activity of neither the burst nor the pause cells showed a 2 24 29 R 5 0.86, P , 10 ; peak: R2 5 0.99, P , 10 ). Error bars indicate s.e.m. significant modulation with saccade amplitude (Fig. 1c, main manuscript). b, For each neuron, we correlated the average firing rate and the peak firing rate c, Most neurons (45 of 72) had a significant linear relationship between (computed over the saccade duration and averaged over all directions) with firing rates and peak saccade speed. In particular, mean and peak response of saccade amplitude. Some neurons increased their firing rates with burst cells showed a significant increase with peak speed (Fig. 1d). increasing saccade amplitude (positive slope) and some neurons decreased

a Error dir. CS-on CS-on CS-off CS-off Saccade dir. error+0o error+180o error+180o error+0o

Saccade Error

0 100 200 0 100 200 0 100 200 0 100 200 Time relative to primary saccade termination (ms)

b 0.25

0.2

0.15 n=39

0.1

0.05 Probability of complex-spike 0 Error dir. on on off off o o o o Saccade dir. error+0 error+0 error+180 error+180 Extended Data Figure 2 | Complex spikes encode direction of error, not statistics from n 5 39 Purkinje cells in which the probability of complex spikes direction of saccade that preceded that error. a, The response of the same cell was quantified as a function of direction of the error vector and direction of shown in Fig. 2 as a function of direction of saccade and direction of error. the saccade that preceded that error. Probability of complex spikes depended on The probability of complex spikes is high when the direction of error is at 2 45u, direction of error, not direction of saccade. despite the fact that saccade direction may be at 2 45u or 1 135u. b, Population

Speed (deg/s) 500 600 400 200 0 450 All saccades (dir CS-off) 100 Peak saccade speed (deg/s) 400

0.6 0.5 0.7

) Speed (R speed saccade and response

). Error bars in all panels indicate s.e.m. across bootstrapped populations.

2 650°/s 525°/s 400°/s Correlation between population population between Correlation R population response with respect to saccadetemporal speed shift as that computed maximized the by cross-correlation. finding the population response and the temporally shiftedas eye speed trace (measured 0 650 -50 All saccades (dir CS-off) All saccades 600 Population response

Time relative to peak speed (ms) Time relative to peak speed

2000 1000 0 -100 550

Macmillan Publishers Limited. All rights reserved Change in population response (spk/s) response population in Change 2015 500 , Temporal lead of the G a b 450 All saccades (dir CS-off) Peak saccade speed (deg/s) , Population response for saccades in 400

24 22 20 18 16

respect to motion of the eye (ms) eye the of motion to respect Population respose lead with lead respose Population The simple-spike population response of Purkinje | bc LETTER RESEARCH direction CS-off for three different peak speeds. cells, organized by their complex-spike propertieswith (Fig. motion 3a), of correlated the eye in real time. a Extended Data Figure 3 LETTER RESEARCH

a b All saccades 200 100 Burst n.s. Peak 80 150

Burst 60 Pause 100 n.s. Mean 40 Firing rate (Hz) 50 Mean Mean firing rate (Hz) 20

Min Pause 0 0 -180 -90 0 90 180 CS-on off on off Saccade direction relative to CS-on (deg) Saccade direction

cdAll saccades 120 200 CS-on CS-on 100 Burst (max) 150 CS-off 80 CS-off 100 60 Firing rate (Hz)

Mean firing rate (Hz) 50 40 Pause (min)

Pause 20 0 400 450 500 550 600 650 400 450 500 550 600 650 Saccade peak speed (deg/s) Saccade peak speed (deg/s) Extended Data Figure 4 | Mean and peak/trough firing rate of the burst and pause cells as a function of saccade speed, for saccades in the CS-off and CS-on pause cells were poorly modulated by saccade direction. a, Maximum, directions. We asked whether the maximum response of the burst cells or minimum and mean firing rates averaged across burst or pause cells with the minimum response of the pause cells was significantly modulated by respect to saccade direction, relative to CS-on direction of each cell. b, Mean direction. Separate repeated measures ANOVAs showed that for the burst cells, firing rates of the burst and pause cells, as measured across all saccades, were peak activity increased as a function of saccade peak speed (P , 0.001), but this not significantly different for saccades in the CS-on versus CS-off direction relationship was unaffected by saccade direction (P . 0.4). For the pause (burst P . 0.10, pause P . 0.05). c, Mean firing rates of the burst and pause cells, the response was not affected by saccade speed (P . 0.6), and this cells as a function of saccade speed, for saccades in the CS-on versus CS-off relationship was not modulated by saccade direction (P . 0.4). We found that direction. Saccade speed modulated the mean firing rates of the burst cells, saccade direction did not significantly alter the encoding of peak speed in but there were no significant interaction between saccade direction and speed either the mean or minimum/maximum activity of Purkinje cells. Error bars in (P . 0.6), nor a significant effect of saccade direction (P . 0.7). d, Peak all panels represent s.e.m. across neurons. (maximum) firing rates of the burst cells and the minimum firing rate of the

a 600

500

400

300 predicted actual 200

100 Population time lead = +19ms Speed of the eye (°/s) 0 MSE = 4070 deg2/sec2

-100 -1000 100-100 0 100-100 0 100 b

600

500

400

300 actual

200 Individual cells Speed of the eye (°/s) predicted 100 time lead = 0±0.002ms MSE = 9977±428 deg2/sec2 0 -1000 100-100 0 100-100 0 100 c 600

500

400 actual

300 predicted 200 Population (burst cells only)

Speed of the eye (°/s) 100 time lead = +11ms MSE = 5941 deg2/sec2 0

-1000 100-100 0 100-100 0 100 d 600

500

400 actual 300

200 predicted Population (pause cells only) 100 time lead = -9ms

Speed of the eye (°/s) 2 2 0 MSE = 6593 deg /sec

-100 -1000 100-100 0 100-100 0 100

Time to peak speed (ms)

Extended Data Figure 5 | A population of Purkinje cells, organized by their predicted and actual eye trajectory at the optimal value of D. b, The result of complex spike properties, predicted the real-time speed of the eye better fitting equation (S2) (see Supplementary Information) to the response of than activity of individual cells. a, We used equation (S2) (see Supplementary individual neurons. c, The result of fitting equation (S2) (Supplementary Information for details) and used the measured population response s(t)of Information) to the discharge of a population composed exclusively of burst

Purkinje cells to predict the real-time speed of the eye x^_ ðÞtzD . The plot cells. d, The result of fitting equation (S2) (Supplementary Information) to the shows the predicted speed for saccades of 400, 525 and 650u s21. The predicted discharge of a population composed exclusively of pause cells. speed led the actual speed by 19 ms. MSE is the mean squared error between the

20 -4

-6 15 CS−off -8 10 -10 5 -12

0 -14

CS−on onset (CS−on − CS−off, ms)

-5 Within neuron difference in pause -16

onset (saccade - pause onset, ms) 400 450 500 550 600 650 400 450 500 550 600 650 Timing of pause onset relative to saccade Saccade peak speed (deg/s) Saccade peak speed (deg/s) Extended Data Figure 6 | Change in saccade direction was associated with a saccade onset. b, Within-neuron measure of pause onset for saccade in change in the timing of the reduction of discharge in the pause cells (that is, direction CS-on, minus onset from saccades in direction CS-off. pause onset) (see Fig. 4f). a, Timing of pause onset with respect to saccade Negative numbers indicate that the pause onset occurred earlier for onset for saccades of various speeds and directions. We computed the pause saccades in the CS-on direction. Error bars in all panels indicate s.e.m. across onset as the time when the neuron’s response reached 20% of its minimum neurons. response. Positive numbers indicate that the pause onset occurred before

a Left Right

CS-off CS-off pause burst pause burst P-cell Midline cFN cell Mossy NRTP fibers

b CS-off direction

c 0.6 left right 0.4 up/down 0.2

0 -3-2-1 0 1 2 3 d 0.6 Pause

0.4 Burst

0.2 Probability Probability 0 -3-2-1 0 1 2 3 left Recording location right (chamber coordinates, mm) Extended Data Figure 7 | Complex-spike-dependent organization of the Purkinje cells. a, Hypothesized anatomical organization of the oculomotor vermis (OMV). Bursting and pausing Purkinje cells are organized into clusters, with the cells in each cluster sharing a common complex-spike direction. Neurons on the right side of the OMV project to right cFN neurons and have CS-off directions to the right. b, Distribution of the CS-off directions from recorded neurons in chamber coordinates. Vertical dotted line shows the line that best separates rightwards CS-off direction neurons (blue) from leftwards CS-off direction neurons (red). c, Probability of having a rightwards (blue), up/ down (green), or leftwards CS-off direction as a function of chamber coordinates. Purkinje cells with CS-off to the left were more probable on the left side of the cerebellum. Purkinje cells with CS-off to the right were more probable on the right side of the cerebellum. d, Pause (red) and burst (blue) Purkinje cells were equally likely at all recorded locations.

percent burst cells

All saccades

2250 100% 90%

1500 70% Measured

750 (54%) 0 50% 30%

-750 10% 0% -1500 Change in population response (spk/s) start end

-100 -50 0 50 100 Time relative to peak speed (ms) Extended Data Figure 8 | The population response was sensitive to the fraction of pause and burst cells that composed a cluster of Purkinje cells. In our data set, 54% of the population was composed of burst cells. We computed the population response under the assumption that the membership of a cluster was 54% burst cells. Here, we tested how sensitive the population response was to this membership ratio. The vertical lines indicate saccade onset and offset for all saccades pooled across direction and speed. As the percentage of burst cells in the cluster becomes larger than 70%, or smaller than 50%, the population response no longer returns to baseline at saccade offset.

a pause cell

Purkinje cells with similar simple spike field

burst cell Population response (spikes/s) caudal fastigial nucleus neuron

b All saccades (dir max. response) All saccades (dir max. response) 1500 1500 525 °/s

1000 400 °/s 1400 650 °/s 1300 500 1200 0 1100

−500 1000 R2 = 0.01 (p > 0.3) −1000 900 Peak change in pop. response (spk/s)

Change in population response (spk/sec) −100 −50 0 50 100 400 450 500 550 600 650 Time to peak speed (ms) Peak saccade speed (deg/s) c All saccades (dir max. modulation) All saccades (dir max. modulation) 1500 1400

1000 525 °/s 1200 400 °/s 500 650 °/s 1000

0 800

−500 600 R2 = 0.001 (p > 0.7) −1000 400 Peak change in pop. response (spk/s)

Change in population response (spk/sec) −100 −50 0 50 100 400 450 500 550 600 650 Time to peak speed (ms) Peak saccade speed (deg/s) Extended Data Figure 9 | Gain-field encoding of saccade kinematics in the for various saccade peak speeds. The peak population response was not population response of the Purkinje cells disappeared if the Purkinje modulated with saccade speed. Error bars are boot-strap-estimated s.e.m. cells were organized by their simple-spike activity. a, In this analysis we c, The population response for saccades made in the direction of maximal assumed that a collection of 50 Purkinje cells projected onto a single cFN modulation. For burst cells, this was the direction for which the Purkinje cell neuron, with the property that all the Purkinje cells shared a similar simple- showed the largest mean firing rate, whereas for pause cells, this was the spike preferred direction. Therefore, the cluster was organized based on the direction associated with the minimum activity (largest pause). The peak simple-spike properties of the Purkinje cells, not their complex-spike population response was not modulated with saccade speed when clusters were properties. b, The population response for saccades made in the direction for organized based on the direction of maximal simple-spike modulation. which each Purkinje cell showed the largest mean firing rate (simple spikes),