Combined recordings of cerebellar activity and head movements were obtained in 16 freely moving rats (Figure 1A). Spontaneous exploratory behavior produced a wide variety of head positions and movements. Our inertial device captured the same parameters as vestibular organs: rotations and accelerations in a head-bound reference frame. In average, head rotations occurred more frequently and swiftly along the pitch and yaw axes than along the roll axis (Figure 1—figure supplement 1A,B), with typical angular speed in the 18–287 °/s range (average 2.5–97.5% percentiles calculated for velocities >15 °/s, n = 16 rats). Angular velocity (Ω) signals displayed multi-peaked power spectra spanning frequencies up to 20 Hz (Figure 1—figure supplement 1C) and showed strong temporal autocorrelation over a short timescale (<0.2 s, Figure 1—figure supplement 2A). The acceleration signal (A) was composed of a gravitational (A^G) and a non-gravitational (A^nG) component. The gravitational component could be described as a vector a^G with a constant norm (1 g) and a fixed orientation in the earth reference frame, but whose coordinates varied in the sensor (head-bound) reference frame during changes of head orientation relative to gravity (head tilt). The direction of a^G in the sensor frame thus reflected head tilt (see Video 1). We separated the two components of acceleration using an orientation filter algorithm (Figure 1—figure supplement 1D; see Appendix and Madgwick et al., 2011). A^G accounted for almost all (99%) of the power of the acceleration below 2 Hz, and for only 9% of it in the 2–20 Hz range (n = 16 rats, Figure 1—figure supplement 1E), indicating that the low-frequency component of acceleration (<2 Hz) mostly contained head tilt information. Consistent with this, A^G displayed temporal autocorrelation over long timescales (<5 s, Figure 1—figure supplement 2B). A^nG varied at the same timescale as Ω and exhibited the same order of magnitude, and temporal correlation pattern with Ω , as linear tangential acceleration predicted from head rotations (see Figure 1—figure supplement 2C–E and Appendix). This suggests that, in our conditions, A^nG signals arose primarily from head rotations.

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Video 1

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A total of 86 units were recorded (Figure 1A,B and Figure 1—figure supplement 3A–C) and classified into putative Purkinje cells (90%), Golgi cells (5%) and mossy fibers (5%) using established criteria (Van Dijck et al., 2013, see Figure 1—figure supplement 3E and Appendix). Putative Purkinje cells exhibited irregular inter-spike intervals (ISI) at rest (average CV: 0.95 ± 0.58, calculated for periods of immobility isolated from 76 cells), resulting in sharp fluctuations of the instantaneous firing rate even during immobility (Figure 1—figure supplement 3D).

We first examined to which extent these firing rate fluctuations could be explained by head movements. Simple linear models may be inadequate for describing nonlinear regimes of responses of vestibular afferents observed during naturalistic head movements (Schneider et al., 2015) or for capturing complex receptive fields reflecting a nonlinear remapping of head kinematics into a gravitationally polarized reference frame (Green and Angelaki, 2007). We therefore designed a model-free approach based on a jackknife resampling technique (see Appendix), which makes no assumption on the nature of the link between inertial parameters and firing rate, except that similar inertial configurations yield similar firing rate. At each time point of a recording, this method identifies the corresponding values of the inertial parameters (e.g. Ω alone, or Ω + A, etc.), and computes the average of instantaneous firing rates observed at other time points with similar values of inertial signals (with an adjustable time delay between the firing rate and inertial parameters, see Appendix and Figure 1C). The resulting time series represents an estimate of firing rate modulations that can be attributed to the influence of this set of parameters and is equivalent to the global output expected from a population of cells that would share the corresponding sensitivity profile. The square of the Pearson correlation coefficient (R²) between estimated and observed firing rates was taken as a measure of the fraction of instantaneous firing rate fluctuations that could be explained by a particular set of inertial parameters (firing rate predictability, Figure 1D).

Overall, we found that firing rate predictability was greater when considering Ω combined with either A or A^G (mean R² = 0.17 ± 0.13 in both cases, n = 86 units; Figure 1E). Considered alone, all inertial parameters (Ω, A, A^G or A^nG) explained significantly smaller fractions of firing rate fluctuations than combinations of Ω and A or A^G (p<0.005, n = 86 units). R² values were always greater for A^G-based than for A^nG-based estimates (p<8e-9 for both A^G vs. A^nG and Ω + A^G vs. Ω + A^nG, n = 86 units), showing that gravitational information dominated the effect of acceleration on firing rate. R² values for firing rate estimates obtained with Ω or Ω + A^nG were similar (p=0.20, n = 86 units), consistent with a redundancy of these parameters due to their coupling (Figure 1—figure supplement 2E). This analysis suggests that cerebellar units preferentially exhibited a mixed sensitivity to head rotations and head tilt.

To assess the robustness of our method, we examined the correlation between independent firing rate estimates computed using non-overlapping (alternating) portions of the same recordings (see Appendix). The Pearson correlation coefficients (R) between independent estimates were mostly above 0.5 (mean R = 0.65 ± 0.23, n = 86, top panel of Figure 1F) while the null distribution computed using shuffled spike trains exhibited significantly smaller R values centered near zero (mean R = 0.14 ± 0.10, n = 86 units; p=2.2 × 10⁻¹⁶, bottom panel of Figure 1F) showing the ability of our method to consistently capture the link between head movements and firing rate modulations.

The cerebellar cortex is divided into narrow functional zones, the microzones (Apps and Hawkes, 2009; Dean et al., 2010), to which neighboring units (recorded simultaneously by a tetrode) likely belong. The instantaneous firing rate of neighboring units (Figure 2A) indeed displayed positive correlations (R = 0.14 ± 0.26, n = 35 pairs, p=0.0014, one-sampled Wilcoxon test) that were lost if the spike train of one unit was time-reversed (R = 0.00 ± 0.03, p=0.49, one-sampled Wilcoxon test, Figure 2C). To test whether these correlations were due to a similar dependency on inertial parameters (versus a shared, movement-independent entrainment of neighboring units), we isolated the movement-dependent part of the firing rate with our resampling method (Figure 2B) and examined their correlations. Correlations were higher (p=0.0074, paired Wilcoxon test) when comparing firing rate estimates (R = 0.31 ± 0.47, p=0.0013, one-sampled Wilcoxon test, Figure 2C) than when comparing instantaneous firing rates (Figure 2D), showing that neighboring unit tended to share similar sensitivities to head movements.

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We then examined whether changing experimental conditions affected the units’ sensitivity by comparing independent firing rate estimates obtained by resampling from the same or from different conditions (see Appendix); if changing the condition has little effect on the firing, these estimates should exhibit similar degrees of correlation. Overall, the presence or absence of light did not change the correlation between independent estimates (p=0.87, paired Wilcoxon test, n = 23, Figure 2E,F), showing a limited influence of visual cues on the units’ sensitivity to head movements. We also tested whether passively-applied whole-body movements (which neither recruit proprioceptive inputs nor produce a motor efferent copy) drove the units similarly compared to the active situation. These passive movements explored a subset of the inertial configurations observed in the active condition (Figure 2—figure supplement 1); we could therefore use our resampling method to estimate the firing rate in the passive session using observations from the active session. The resulting estimates significantly differed from estimates obtained using observations from the passive session (p=0.0021, paired Wilcoxon test, n = 17, Figure 2G,H), suggesting that in many cases the units’ coding schemes differed when movements were self-generated or passively experienced.

According to our model-free approach, most units exhibited a mixed gravitational and rotational sensitivity (Figure 1E), but a fraction of them appeared to be mostly tuned to Ω or A^G. We then first examined the nature of the link between the firing rate and these inertial parameters in these cells. We isolated 6 Ω-selective and 12 A^G-selective units by picking cases for which the firing rate predictability was at least eightfold greater for one parameter than the other (Figure 3A).

Figure 3

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The sensitivity of Ω-units was examined by computing ‘inertio-temporal receptive fields’ (average instantaneous firing rate timecourse around specific angular velocity values at lag 0; see Appendix). These plots showed a bidirectional modulation of the firing rate by specific combinations of rotations, since the firing rate was increased or decreased at a lag close to 0, that is, close to the occurrence of specific rotation values (Figure 3B). 3D plots representing the firing rate as a function of the three components of Ω (calculated for the lag showing the strongest modulation of firing rate) revealed clear firing rate gradients along specific directions of rotation, showing that these units were tuned to specific 3D rotations (Figure 3C and Video 2).

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To quantify the sensitivity of Ω-units to angular velocity, we used a simple regression model in which the instantaneous firing rate is described as a combination of the roll, pitch and yaw velocities. The coefficients of the fit define the coordinates of a rotation sensitivity vector, whose norm, in Hz/(°/s) or °⁻¹, represents the gain of the unit’s response, and whose direction represents the unit’s preferred rotation (Figure 3D). For each unit, the linear fit was calculated with a variable time delay (lag) between the firing rate and angular velocity; the presence of a peak in the sensitivity vs. lag curve (see Figure 3E for average gain vs. lag curves) allowed us to identify an optimal lag at which the unit’s gain was maximal. The sensitivity vector calculated at the optimal lag was defined as the unit’s optimal rotational sensitivity vector (ω_opt). The gain calculated at the optimal lag was greater for Ω-units (0.28 ± 0.15 °⁻¹, n = 6) than for A^G-units and other units with mixed sensitivity (0.03 ± 0.02 °⁻¹, n = 7, p=0.0012 and 0.07 ± 0.05 °⁻¹, n = 53, p=0.00022, respectively; calculated only for units with a significant ω_opt vector; see Appendix). The optimal lag of Ω-units was centered around zero (0.5 ± 21.8 ms, n = 6) while the one of mixed units was greater (29.8 ± 142.4 ms, n = 53), although not significantly (p=0.072). Ω-units corresponded to putative mossy fibers (n = 2) and putative Purkinje cells (n = 4), and exhibited preferred rotation axes clustered around the excitatory direction of semi-circular canals (Figure 3F). Units with mixed sensitivity corresponded in majority (96%) to putative Purkinje cells.

The tuning of A^G-units was examined by computing their average firing rate as a function of head tilt; head tilt was defined as the orientation of the gravity vector a^G in head coordinates (Video 1), which could be mapped on a sphere (Figure 3G) and then in two dimensions using an equal-area projection (Figure 3H). The resulting plots confirmed that A^G-units were strongly modulated by head tilt with simple receptive fields (region of increased firing rate; Figure 3H), contrarily to Ω-units which were not modulated by head tilt (the CV of firing rate across head tilts was higher for A^G-units than for Ω-units: 2.28 vs. 0.29, p=0.0047, Figure 3I). A^G-units were classified as putative mossy fibers (n = 1), Golgi cells (n = 3) and Purkinje cells (n = 8). Overall, these data show that a fraction (20%) of caudal cerebellar units displayed selective tuning to either head angular velocity or head tilt; most of our putative granular layer units (6/8) belonged to these categories.

As shown above, most units displayed a mixed sensitivity to rotational and gravitational information and were not classified as Ω-units or A^G-units. A direct examination of inertio-temporal receptive fields for different head orientations showed that the rotational sensitivity of these units was indeed often highly tilt-dependent. Figure 4A–F shows two example units for which the apparent sensitivity to yaw angular velocity (ω_z) increased (Figure 4A–C) or even reversed (Figure 4D–F) for nose down vs. nose up head situations (i.e. positive vs. negative values of a_x^G). Inertio-temporal receptive fields (Figure 4A,D) clearly showed changes in the modulation of the cells by rotation as a function of lag between the nose up and nose down situations; this is clearly seen as a change of the sensitivity of the cells to ω_z (i.e. slope of the firing rate vs. ω_z linear regression) for different time lags (Figure 4B,E). At the optimal lag (that maximizes the sensitivity for each cell), the modulation of firing rate by angular velocity progressively changed as a function of the head elevation angle (Figure 4C,F). The occurrence of such head tilt-dependent modulation of rotational sensitivity was quantified by a stability index (σ) quantifying the effect of a_x^G (σ_x) or a_y^G (σ_y) on the direction of rotational sensitivity vectors (a maximal value of 1 indicating identical directions for opposite head orientations; see Appendix). This index was lower for non-Ω-units (σ_x = 0.36 ± 0.32 and σ_y = 0.41 ± 0.29, n = 60) than for Ω-units (σ_x = 0.73 ± 0.13 and σ_y = 0.88 ± 0.06, n = 6; p=0.0027 and p=0.00011, respectively, Figure 4G,H), confirming that the rotational tuning of the largest fraction of our cell population was head tilt-dependent.

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The above data suggest that some units employed a rotational coding scheme that takes into account the direction of gravity, raising the possibility that some of them might encode head rotations in a reference frame aligned with the earth-vertical direction. To explore this, we computed rotational sensitivity vectors (ω_opt) for all head tilt angles explored by the animal, using angular velocity signals expressed either in an internal (head-bound), or in an external (earth-bound), reference frame (Figure 5B, see Appendix). We reasoned that ω_opt vectors of units encoding rotations in one reference frame (internal of external) should remain aligned (tilt-independent) only when calculated in this particular frame. As a means to visualize the effect of head tilt on these vectors, we plotted them on a sphere at the coordinates corresponding to the head tilt (i.e. the coordinates of gravity in the head reference frame, see Figure 3G and Video 1) for which they were calculated (Figure 5A). Figure 5C,D shows two example units with different tuning properties. In the left unit (Figure 5C), ω_opt vectors appeared more consistently aligned when calculated in external (vs. internal) coordinates, while the opposite was observed for the right unit (Figure 5D, see Video 3 for a 3D version of these plots). This was confirmed by examining the collinearity of ω_opt vectors as a function of the angular distance between their localization on the sphere: collinearity decreased with angular distance for internal but not external ω_opt vectors in the left unit (Figure 5E), while it decreased for external but not internal ω_opt vectors in the right unit (Figure 5F). Other units exhibited no clear preferred orientation for internal vs. external ω_opt vectors, and collinearity curves that decayed similarly for the two reference frames (Figure 5—figure supplement 1A,B). When plotted for all units, the difference between collinearity curves of external and internal ω_opt vectors revealed a continuum of properties (Figure 5G). Units exhibiting a stronger tuning toward an external or internal coding scheme were isolated by setting a cutoff (difference between external and internal collinearity > 0.5 or < −0.5 for angular distances ranging 80–100 °). All these units corresponded to putative Purkinje cells. The preferred rotation axes of external-coding cells did not appear to cluster around specific directions (Figure 5H).

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Video 3

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As shown above, the mixed sensitivity of most units to rotational and gravitational information, captured by our model-free analysis, was confirmed by an empirical approach consisting in describing the units’ rotational sensitivity as a linear tuning to a preferred rotation axis which depends on head tilt. This empirical approach showed that the units’ sensitivity to angular velocity is highly dependent on head tilt, allowing in some cases the encoding of head rotations in a reference frame aligned with gravity.

We then compared the performance of different approaches to account for the observed firing rate modulations: ‘local’ (head-tilt dependent) linear models, and ‘global’ linear models (which simply assume a linear dependency of the firing rate on particular combinations of inertial parameters: rotational velocity, noted $\mathrm{\Omega }$, angular acceleration, noted ${\displaystyle \dot{\mathrm{\Omega }}}$, gravitational acceleration, noted ${\text{A}}^{\text{G}}$, and jerk of the gravitational acceleration, noted ${\displaystyle {\dot{\mathrm{A}}}^{\text{G}}}$). These approaches were compared by computing their correlation with the observed instantaneous firing rate. As shown in Figure 6A (for the linear model based on $\mathrm{\Omega }$, ${\displaystyle \dot{\mathrm{\Omega }}}$ and ${\text{A}}^{\text{G}}$) and in Figure 6—figure supplement 1 (for all other models), ‘global’ linear models always produced poorer predictions than our model-free approach. In contrast, the firing rate predictability calculated with our model-free approach was not significantly different from the coefficient of determination calculated from ‘local’ linear models used to generate tilt-dependent rotational sensitivity maps (p=0.24, paired Wilcoxon test, n = 86 units, Figure 6B). This highlights the presence of a nonlinearity in the way that caudal cerebellar units are tuned to head inertial parameters. This nonlinearity appears to be mainly due to a dependency to head tilt of the (linear) rotational sensitivity, since a series of ‘local’ linear fits of the unit's rotational sensitivity (head-tilt dependent ω_opt) collectively provide a description of the firing rate as performant as the model-free approach (Figure 6B).

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A key choice in our study was to approach the neuronal activity in the caudal cerebellum in freely moving animals, which express a large repertoire of spontaneous motor activity. The vestibular system has been mostly approached so far in head-fixed animals submitted to passive, externally applied, movements; however, studies in the vestibular and cerebellar nuclei during self-generated movements have provided strong indications that passive vs. active movements elicit different firing modulations in the same neurons (Cullen et al., 2011; Medrea and Cullen, 2013; Cullen and Brooks, 2015; Brooks et al., 2015). The use of spontaneously-generated movements yields specific constraints for the analysis: the movements have a high dimensionality (rotations and non-gravitational acceleration along 3 axes, head tilt, etc.), identical movements and head postures are not repeated over and over (as in passive conditions), and the repertoire of movements may not explore all the possible combinations: for example, we found that the data contained little purely translational acceleration (i.e. non gravitational acceleration which was not coupled to head rotations), which precluded the analysis of the influence of translations on the cells’ firing. To deal with the high-dimensionality of the data, we used a two-step analysis: the first step involved a model-free approach, designed to identify (1) the fraction of instantaneous firing rate variability which could be explained by inertial parameters (head movement/tilt) and (2) which parameters best explained the observed firing rate fluctuations. Since the best prediction of firing rate for most units, particularly putative Purkinje cells, required to combine the knowledge of both head tilt (i.e. gravitational acceleration in the head reference frame) and angular velocity, we used a model where the firing rate is linearly tuned to a preferred rotation axis which depends on head tilt. We found that this model was as successful as the model-free approach to capture the units’ sensitivity to inertial parameters.

We also identified a small set of units tuned to either head tilt or head angular velocity. Most of our granular layer units belonged to these categories, thus potentially reflecting the activity of otolithic and semi-circular mossy fibers or Golgi cells directly driven by these fibers. A fraction of head tilt-selective units (n = 8) was identified as Purkinje cells, and might correspond to previously identified static roll-tilt Purkinje cells (Marini et al., 1976; Yakhnitsa and Barmack, 2006), or to tilt-selective Purkinje cells dynamically extracting head tilt information through multisensory integration (Laurens et al., 2013a; Laurens et al., 2013b).

Our analysis is based on the units’ instantaneous firing rate (rather than on spike times), as used previously in the cerebellum (e.g. Ohtsuka and Noda, 1995; Pasalar et al., 2006; Medina and Lisberger, 2007; Yakusheva et al., 2007). The rationale for this is that each units recorded may be viewed as a sample of a population with similar receptive fields, which shall converge on the same postsynaptic target; therefore the average behavior of a unit is similar to the instantaneous behavior of the population. The action of a single Purkinje cell on its postsynaptic target is likely to be small (Bengtsson et al., 2011) so that the intracellular potential of the target neuron is conditioned by the average activity of tens to a hundred Purkinje cells (Person and Raman, 2011). In the cerebellum, Purkinje cells are thought to be organized in microzones (for review, see Apps and Hawkes, 2009; Dean et al., 2010) which exhibit a narrow medio-lateral and large antero-posterior extension, which converge onto their downstream targets (Sugihara et al., 2009) and which share a similar climbing fiber teaching signal (as climbing fibers are thought to shape receptive fields). Indeed in the caudal vermis, the collaterals of climbing fibers follow the geometry of microzones (Ruigrok, 2003). We found that the instantaneous firing rate of many units exhibited strong fluctuations even during complete immobility. As a result, the instantaneous firing rate was only loosely connected to the rapid component of head inertial signals, consistent with a coding of information at the level of Purkinje cell populations (Herzfeld et al., 2015). In support of this interpretation, we found that neighboring putative Purkinje cells exhibited positive correlations of their firing rate and of the component of firing rate explained by inertial parameters (isolated with our model-free approach), indicating a similarity of the inertial receptive fields in cells likely belonging to the same microzone.

The majority of units in our sample exhibited tilt-dependent rotational tuning. Such units may intervene in the transformation of head-bound angular velocity signals (as sensed by the semi-circular canals) into externally referenced angular velocity signals, as observed in the forebrain (Laurens et al., 2016), or in azimuthal information within the head-direction system (Taube et al., 2013; Finkelstein et al., 2016; Wilson et al., 2016). Indeed, head tilt affects the correspondence between head-bound and earth-bound rotations (see Video 4). Therefore, cells tuned to rotations about an earth-bound axis are expected to exhibit a tilt-dependent sensitivity to head-bound rotations. How the activity of putative Purkinje cells is decoded in vestibular nuclear neurons shall largely depend on the (unknown) other inputs to these cells; however, as noted previously by Yakusheva et al. (2008), sets of Purkinje cells (which are inhibitory neurons) encoding rotations relative to gravity (such as displayed in Figure 5C) could subtract the component of rotations that changes head orientation relative to gravity from semi-circular canal inputs in vestibular nuclear neurons; this operation would isolate an earth-horizontal component of head angular velocity suitable for the computation of azimuthal heading in head direction cells (Finkelstein et al., 2016).

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The rotational sensitivity of caudal cerebellar units was maximal for positive lags relative to angular velocity, suggesting that it was driven by sensory cues rather than by motor commands. By comparing the firing rate predicted from sessions recorded in different conditions, we found no evidence for a crucial role of visual inputs, but instead found differences in the sensitivity to active vs. passive movements. This result is reminiscent of the lower sensitivity of certain vestibular nuclear neurons to active vs. passive movements (Cullen and Roy, 2004), and might reflect a similar mechanism of attenuation of self-generated inputs. Alternatively, the passive movements used in our study may have imperfectly sampled the inertial configurations encountered during active sessions, thereby complicating the comparison between the two conditions; indeed, self-generated movements often occurred at higher frequencies than our passive movements and may have induced smaller Purkinje cell modulations (Yakusheva et al., 2008). Further studies are required to identify the cause of differences of neuronal sensitivity to active vs. passive movements.

Deciphering how Purkinje cell receptive fields may be tuned to rotations about fixed directions relative to gravity is a complex topic. One challenge is to explain how rotational sensitivity is gated by gravity. Such an operation might be performed in the granular layer, by first computing an estimate of gravitational acceleration and then combining it with rotational (vestibular, visual or proprioceptive) informations. We found that the low-frequency (<2 Hz) component of acceleration during free movements mainly contains gravitational information. The granular layer of the caudal vermis contains a high amount of unipolar brush cells (excitatory interneurons intercalated between mossy fibers and granule cells; Mugnaini et al., 2011) which may smooth otolithic signals over hundreds of milliseconds (van Dorp and De Zeeuw, 2014; Borges-Merjane and Trussell, 2015; Zampini et al., 2016) and thus provide a proxy of gravitational signals to granule cells. Granule cells receiving convergent inputs (Huang et al., 2013) carrying gravitational and rotational information could then operate as coincidence detectors (Chadderton et al., 2004; Chabrol et al., 2015) and signal the occurrence of specific combinations of rotation and head tilt to Purkinje cells.

Purkinje cell receptive fields are shaped by the activity of climbing fibers which determine the sensitivity to subsets of granule cells (parallel fiber) afferents (e.g. Dean et al., 2010). Vestibular climbing fibers emanate from inferior olive neurons of the β-nucleus, which are controlled by dorsal Y-group and parasolitary nucleus afferents. These nuclei carry rotational and low-pass filtered otolithic signals but are also under the influence of vestibulo-cerebellar Purkinje cells (Barmack and Yakhnitsa, 2000; Barmack, 2003; Wylie et al., 1994). Therefore, the teaching signal sent to Purkinje cells by way of olivary neurons shall result from a complex interplay between external afferents and the action of Purkinje cells themselves, leading to the observed tilt-dependent rotational sensitivity.

In conclusion, our work reveals that the dominant coding scheme of natural head kinematics in the caudal cerebellum is a tilt-dependent representation of head rotations. In a subset of cells, the mixed sensitivity to gravitational and rotational information was tuned in a way that allowed the encoding of head rotations in a gravity-centered reference frame. As emphasized previously (Green and Angelaki, 2007), the transformation of head-bound peripheral vestibular signals into an earth-bound representation of head kinematics involves complex nonlinear computations. In particular, the detailed mechanisms underlying the computation of gravitationally polarized rotational receptive fields remain to be elucidated. The presence of such receptive fields in the caudal cerebellar cortex, as shown here, as well as the key anatomical position of this structure immediately downstream of the vestibular organs and upstream of the vestibular nuclei, warrant a detailed examination of how the cerebellar cortical microcircuit might re-express head movements relative to gravity.

Sixteen adult male Long-Evans rats (aged 3–4 months, 250–300 g at surgery, RRID:RGD_60991) were used in this study. Animals were housed individually in standard homecages maintained in standard laboratory conditions (12 h day/night cycle, $\simeq$21°C with free access to food and water). Experimental procedures were conducted in conformity with the institutional guidelines and in compliance with French national and European laws and policies. All procedures were approved by the ‘Charles Darwin’ Ethics Committee (project number 1334).

Our inertial measurement unit (IMU) hosts a 9-axis digital inertial sensor containing a 3-axis accelerometer, 3-axis gyroscope and 3-axis magnetometer (MPU-9150, Invensense) soldered onto a custom printed circuit board (PCB, 9 × 7 mm) designed using EAGLE (CadSoft Computer GmbH). The sensor and additional passive components were soldered in-house using a reflow-soldering oven (FT-2000, CIF). Powering and communication with the IMU were performed using four wires that were soldered directly into wirepads present on the PCB. The IMU was connected to an I²C interface (USB-8451, National Instruments) through a motorized commutator used for electrophysiological recordings (Tucker-Davis Technologies). Data were acquired at 250 Hz using a custom LabVIEW program. During experiments, the IMU was secured onto the rat’s head using double row socket connectors (Mill-Max) as shown in Figure 1A.

Our microdrive was designed based on the architecture proposed by Anikeeva et al. (2011). The drive consisted of an M2.5 × 0.2 adjustment screw (F2D5ES15, Thorlabs) that was drilled along its axis (hole diameter: 1 mm) and placed inside an M2.5 × 0.2 adjustment knob (F2D5ESK1, Thorlabs). The knob and screw were inserted inside a custom aluminum housing machined in-house. The knob was machined such that it could be maintained in place while rotating using two interference pins. The screw was machined such that the housing prevented its rotation (thus rotation of the knob resulted in a translation of the screw, as in Anikeeva et al., 2011). The tetrode was threaded into a series of stainless steel tubings with increasing diameter and glued inside the screw such that at least 5 mm of tetrode protruded from the base of the microdrive with the screw in its upper position. The microdrive was designed to provide at least 3.5 mm of travel distance and weighted less than 3 g.

Rats received an injection of the opioid analgesic buprenorphine (0.05 mg/kg s.c.) and were placed in a stereotaxic apparatus (model 942, David Kopf Instruments) under isoflurane anesthesia (at induction: 4%, 2 L/min; during surgery: 0.5–1.5%, 0.4–0.5 L/min). Body temperature was maintained between 37.4 and 37.6°C during the whole procedure using a regulated temperature controller coupled to a rectal probe (CMA 450, CMA). The scalp was shaved and wiped with povidone-iodine followed by 70% ethanol. Lidocaine (2%) was injected subcutaneously before incising the scalp. The skull was gently scraped with a scalpel blade and cleaned with a 3% hydrogen peroxide solution. Horizontal alignment of Bregma and Lambda was checked and burr holes were drilled (two over the frontal, four over the parietal and two over the interparietal bone plates) in order to insert skull screws (#0–80 × 3/32’’ stainless steel screws, Plastics One). One of the screws was connected to a tungsten wire and used as a ground signal. A layer of self-curing dental adhesive (Super-Bond C&B, Sun Medical) was deposited over the skull except at the midline of the posterior half of the interparietal plate, where a craniotomy was drilled (interaural antero-posterior coordinate ranging −3.4 to −5.0 mm, that is, over the cerebellar lobule VI). After removing the dura, a single quartz-insulated tetrode (Thomas Recording) housed inside our custom microdrive and coated with DiI (Sigma) was lowered 4 mm below the surface of the brain, together with a 70 tungsten wire used as a reference. A drop of warm low-melting point agarose (1.5% in saline) was deposited around the tetrode, between the brain and the base of the microdrive to ensure brain mechanical stability. The base of the microdrive was then secured to the skull with dental cement (Pi-Ku-Plast HP 36, Bredent). A lightweight 3D-printed headstage hosting the IMU pin-connectors and a 16-channel electrode interface board (EIB, Neuralynx) was then placed over the microdrive and secured with dental cement. The four tetrode channels and the reference and ground wires were then connected to the EIB. The 3D-printed headstage contained a thin (3 mm) aluminium bar positioned horizontally along the interaural axis of the animal’s head, and protruding 1.5 cm toward the right side. This bar was cemented in place with the rest of the implant and was used to immobilize the head during passive recording sessions (see ‘Recording protocol’) using a custom clamping system. The skin ridges were sutured in front and at the rear of the implant and covered with antiseptic powder (Battle Hayward & Bower). A prophylactic injection of antibiotics was performed (gentamicin, 3 mg/kg, i.m.) and warm sterile saline was injected subcutaneously (2% of b.w.) to prevent dehydration. Rats were then allowed to recover under monitoring in their home cage placed over a heating pad. Gentamicin is known to produce vestibulotoxic effects when injected intratympanically or systemically over several days. Given the observations by Oei et al. (2004), who only observed functional vestibular deficits after ten daily intramuscular injections of gentamicin at 50 mg/kg, it is highly unlikely that our single injection of gentamicin may have significantly affected the animal’s vestibular organs.

Electrophysiological recordings were performed using a previously established protocol (Gao et al., 2012). Quartz insulated tetrodes (Thomas Recording) were gold-plated to reach an impedance of 150–200 kΩ at 1 kHz, mounted inside a custom microdrive (see ‘Tetrode drive’) and connected to an interface board (EIB-16, Neuralynx). Signals were referenced against a tungsten electrode positioned in the cerebellum 4 mm below the surface. The EIB was connected to a custom-made differential amplifier through a commercial headstage (Tucker Davis) and a motorized 32-channel commutator (Tucker Davis). Amplified signals were digitized at 30 kHz by a multifunction acquisition board (NI PCIe-6353, National Instruments) and acquired using a custom LabVIEW program.

Recordings were obtained from a volume of tissue comprised within the following interaural (IA) coordinates: 3.4–5 mm posterior to the IA point, ±5 mm around the midline and 2.5–5.4 mm above the IA point. This volume represents roughly less than 25% of the most caudal part of the vermis (lobules IX and X) according to Paxinos and Watson, 2007. Electrode tracks were examined using post hoc histology in eight rats. All tracks traversed lobules IX and X (Figure 1—figure supplement 3F,G).

After a 1-week post-operative recovery period, daily recording sessions were conducted as follows. The IMU and headstage were connected to the rat’s head and the animal was placed inside a rectangular arena (120 × 60 cm). The animal was temporarily removed from the arena everytime the experimenter decided to lower the tetrode (by 1/8th of a turn, that is, 12.5 μm). The recordings were targeting zones of dense neuronal activity which typically correspond to Purkinje cell layers. Once units were obtained, the signal was controlled for stability and quality and electrophysiological and IMU recordings were started. A first block of 5 mm of free activity was obtained in the dark, followed by a second block of the same duration in the presence of light (active blocks). In some recordings, the head of the animal was immobilized for less than 10 s at the begining of the first block using a custom fixation system (see Implantation surgery), in order to provide enough signal to calculate gyroscope and accelerometer offsets.

The fixation system contained a small lightweight platform to which the body of the animal was strapped after immobilizing the head. To produce passive whole body movements, the platform was held by the experimenter and rotated about the roll, pitch and yaw axes, and about combinations of these axes. Two blocks were acquired in this condition (passive blocks), one in the dark and one in the light. A total of 17 units were recorded in both the active and passive condition (Figure 2H). Because several units could be isolated in the same site, these recordings corresponding to a total of 11 acquisition blocks in each condition. Because passive movements were produced by the experimenter, their kinematics differed from the rat’s natural head kinematics. Nevertheless, the range of angular velocities was relatively similar in the passive vs. active condition (Figure 2—figure supplement 1A). Passive movements contained proportionally more low frequencies than active movements (Figure 2—figure supplement 1B,C), and the range of head orientations relative to gravity in the passive condition was slightly smaller (Figure 2—figure supplement 1D). Overall, recording sessions (in the active and passive condition) never lasted more than 3 hr in the same day.

Full details of the analysis procedures and statistical tests are provided in the Appendix. Unless mentionned otherwise, p-values were obtained with an unpaired Wilcoxon test. All mean values are given with the standard deviation.

To isolate spikes, continuous wide-band extracellular recordings were first filtered offline with a Butterworth 1 kHz high-pass filter. Spikes were then extracted by thresholding the filtered trace and the main parameters of their waveform were extracted (width and amplitude on the four channels). The data were hand-clustered by polygon-cutting in two-dimensional projections of the parameter space using Xclust (Matt Wilson, MIT). The quality of clustering was evaluated by inspecting the units’ auto-correlograms, as in Gao et al. (2012).

A time series representing the instantaneous firing rate (FR) was obtained by taking the inverse of the local interspike interval (ISI) every millisecond. The resulting step-like function was convolved with a 10 ms Gaussian to soften the transitions between successive ISIs. This smoothing had no deleterious incidence on the correlations calculated between FR and inertial parameters given the fact that head movements occurred with significantly longer time scales (>50 ms, see Figure 1—figure supplement 1C). The instantaneous firing rate time series was then downsampled at 200 Hz to match the times of inertial measurements.

To classify the recorded units, we used criteria derived from Van Dijck et al. (2013) (Figure 1—figure supplement 3E). Most units exhibited a spiking entropy between 6 and 8, consistent with a Golgi or Purkinje cell profile. Among them, units with low firing rate (cut-off set at 12 Hz) were classified as putative Golgi cells (n = 4) and units with a higher firing rate were classified as putative Purkinje cells (n = 78), consistent with observations in the posterior vermis of monkeys. Because tetrodes pick many units at the same time, and because of the high level of activity in the Purkinje cell layer of awake rats, we could distinguish complex spikes from simple spikes only in a few recordings (Figure 1—figure supplement 3H). Putative mossy fibers (n = 4) were distinguished based on their characteristic low entropy (Van Dijck et al., 2013) and separated from other units using a cut-off of 6.3.

Spike entropy characterizes the regularity of firing (the higher the less regular). It was computed as previously described (Van Dijck et al., 2013) from the histogram ${\displaystyle p({\mathrm{I}\mathrm{S}\mathrm{I}}_{\mathrm{i}})}$ of the natural logarithm of interspike intervals (in milliseconds) taken with a binwidth of 0.02 and smoothed with a gaussian kernel with a standard deviation of one-sixth of the histogram’s standard deviation. The entropy was then defined as ($n$ being the number of bins):

Tilt-dependent rate maps indicate normalized average firing rate values calculated for different head orientations (Figure 3H). Head orientation relative to gravity (head tilt) was identified by the direction of the gravity vector ${\displaystyle {\mathit{a}}^{\mathit{G}}=(a_x^G,a_y^G,a_z^G)}$ in the sensor frame (see Video 1). During rotations causing a reorientation of the head relative to gravity, ${\displaystyle {\mathit{a}}^{\mathit{G}}}$ describes a trajectory contained over a sphere of radius 1 g in the sensor frame. To examine how the instantaneous firing rate was modulated by head-tilt, we selected 500 points evenly distributed over that sphere (Figure 3G). We then calculated the average instantaneous firing rate for all time points during which ${\displaystyle {\mathit{a}}^{\mathit{G}}}$ fell within 20° of each of these points (discarding points for which less than 200 time points were found in the recording). The resulting average firing rate values were normalized as follows:

The sphere was tiled with polygons centered around these points using a Delaunay triangulation and the resulting spherical map was projected in 2D using the Lambert azimutal equal-area projection (the center of the map corresponding to an upright head orientation, that is, with the head-vertical axis aligned with ${\displaystyle {\mathit{a}}^{\mathit{G}}}$). Finally the polygons were filled with a color indicating the corresponding normalized average firing rate value (Figure 3H). The non-parametric coefficient of variation of these values (Figure 3I) was given by:

where $Q(0.84)$ and $Q(0.16)$ are the quantiles of the normalized average firing rate distribution corresponding to the 0.84 and 0.16 probabilities (the probabilities corresponding to $\pm 1SD$ in a gaussian distribution).

To quantify the effect of head tilt on the direction of preferred rotational sensitivity vectors over a defined lag range, we first computed sensitivity vectors (${\displaystyle {\mathit{\omega }}_{\mathit{l}\mathit{a}\mathit{g}}=({\alpha }_{lag},{\beta }_{lag},{\gamma }_{lag})}$) for negative and positive values of $a_x^G$ or $a_y^G$ for lag values comprised between $-0.5$ and 0.5. Sensitivity vectors were normalized as follows:

The stability index $\sigma$ (Figure 4G,H) was calculated as follows:

The above normalization ensures that $\sigma$ is bounded between $-1$ and $1$ (1 indicating strictly identical directions and $-1$ indicating strictly opposite directions over the lag range).

The orientation filter (see ‘Estimation of gravitational acceleration’) allowed us to compute the direction of the gravity vector a^G in the sensor frame. An external right-handed reference frame was defined using the following constraints: a z axis aligned with a^G, and an x axis within the plane defined by a^G and the head’s naso-occipital axis and pointing toward the animal’s nose. The components of the instantaneous angular velocity vector, originally expressed in the sensor frame, were calculated in this external reference frame and used to compute externally-referenced optimal rotational sensitivity vectors using a linear model (as in ‘Linear fits’).

To compute head tilt-dependent optimal rotational sensitivity vectors, we used a strategy similar to the one employed for tilt-dependent rate maps. We first selected 500 points evenly distributed over a sphere and calculated ω_opt vectors (see ‘Linear fits’) using only observations during which directions of a^G fell within 20° of each of these points (discarding points for which less than 500 observations in the recording were found). These vectors were computed using the internal or external (see ‘Calculation of angular velocity in a reference frame aligned with gravity’) components of angular velocity signals and represented on a sphere at the coordinates corresponding to the direction of a^G for which they were calculated (Figure 5C,D and Video 3).

To quantify the performance of this approach, we identified for each time point the corresponding head tilt and thus the corresponding ω_opt vector. This allowed us to generate a firing rate prediction for each time point, and to correlate this prediction with the observed firing rate. The square of the Pearson correlation coefficient (R²) obtained by this method was then compared to the R² calculated using the model-free approach (see ‘Estimation of firing rate modulations that are due to head movements’ and Figure 6B).

The collinearity of pairs of rotational sensitivity vectors ${\displaystyle ({\mathit{\omega }}_{\mathit{o}\mathit{p}\mathit{t}\mathbf{,}\mathit{i}},{\mathit{\omega }}_{\mathit{o}\mathit{p}\mathit{t}\mathbf{,}\mathit{j}})}$, calculated for two different orientations of the gravity vector $({\mathit{}}_{\mathit{}}^{\mathit{}}$, ${\mathit{}}_{\mathit{}}^{\mathit{}})$ in the sensor frame, was assessed by calculating the dot product:

To compute collinearity profiles (average collinearity vs. angular distance $\phi$ between ${\displaystyle {\mathit{a}}^{\mathit{G}}}$ orientations), $S_{i,j}$ was calculated for all pairs $(i,j)$ with $i\&j\in \{1,\mathrm{\dots },N\}$ and $i\ne j$, where $N$ is the number of different ${\displaystyle {\mathit{a}}^{\mathit{G}}}$ orientations for which sensitivity vectors were calculated (among the 500 points selected over the sphere, see ‘Tilt-dependent rotational sensitivity maps’). These values were then averaged as a function of $\phi$ and plotted for bins of π/16 (Figure 5E,F).

To compare the performance of our model-free approach (see ‘Estimation of firing rate modulations that are due to head movements’) with simple linear regressions, we designed ‘global’ linear models. These models are called ‘global’ because they are calculated using the whole recording, and not a subset defined for example by specific head orientations relative to gravity (like in tilt-dependent rotational sensitivity maps, described in section ‘Tilt-dependent rotational sensitivity maps’). The linear model described in section ‘Linear fits’ is a particular case of such model. In these models, the instantaneous firing rate (FR) is described as a linear combination of different categories of inertial parameters, with a lag defined by the user. Optimal lags were calculated by testing lag values comprised between –0.5 and 0.5 s. For each model, the optimal lag was defined as the one that maximized the model’s coefficient of determination. The following models were used:

where ${\displaystyle \mathit{\omega }(t)}$, ${\displaystyle \dot{\mathit{\omega }}(t)}$, ${\displaystyle {\mathit{a}}^{\mathit{G}}(t)}$ and ${\displaystyle {\dot{\mathit{a}}}^{\mathit{G}}(t)}$ are 3D vectors representing the instantaneous angular velocity, angular acceleration, gravitational acceleration and jerk (time derivative) of the gravitational acceleration, respectively, and ${\displaystyle \mathit{\lambda }}$, ${\displaystyle \mathit{\xi }}$, ${\displaystyle \mathit{\alpha }}$ and ${\displaystyle \mathit{\beta }}$ are 3D vectors representing the corresponding coefficients of the fit. The coefficients of determination of these linear models (at their optimal lag, defined for each unit) were compared to the firing rate predictability calculated by our model-free approach (see ‘Estimation of firing rate modulations that are due to head movements’) using angular velocity and gravitational information (see Figure 6A and Figure 6—figure supplement 1).

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank J Laurens, D Bennequin, A Afgoustidis and D Popa for helpful discussions on the project, and L Rondi-Reig and C Rochefort for initial discussions. We thank M Pasquet, B Barbour, R Proville, Y Cabirou and G Paresys for technical assistance. This work was supported by grants from France’s Agence Nationale de la Recherche (ANR) to CL (ANR-12-BSV4-0027 and ANR-15-CE37-0001) and to BG (ANR-15-CE37-0007), and has received support under the program ‘Investissements d’Avenir’ launched by the French Government and implemented by the ANR, with the references ANR-10-LA-54 MEMO LIFE and ANR-11-IDEX-0001–02 PSL${}^{\star }$ Research University.

Animal experimentation: Experimental procedures were conducted in strict conformity with the institutional guidelines and in compliance with French national and European laws and policies. All procedures were approved by the "Charles Darwin" Ethics Committee (project number 1334).

© 2017, Dugué et al.

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