Cerebellar reencoding of selfgenerated head movements
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Abstract
Head movements are primarily sensed in a reference frame tied to the head, yet they are used to calculate selforientation relative to the world. This requires to reencode head kinematic signals into a reference frame anchored to earthcentered landmarks such as gravity, through computations whose neuronal substrate remains to be determined. Here, we studied the encoding of selfgenerated head movements in the rat caudal cerebellar vermis, an area essential for graviceptive functions. We found that, contrarily to peripheral vestibular inputs, most Purkinje cells exhibited a mixed sensitivity to head rotational and gravitational information and were differentially modulated by active and passive movements. In a subpopulation of cells, this mixed sensitivity underlay a tuning to rotations about an axis defined relative to gravity. Therefore, we show that the caudal vermis hosts a reencoded, gravitationally polarized representation of selfgenerated head kinematics in freely moving rats.
https://doi.org/10.7554/eLife.26179.001Introduction
Selforientation is largely dependent on modalities that document head movements, including neck proprioception, optic flow and, most notably, vestibular inputs (Angelaki and Cullen, 2008). Vestibular signals are essential for stabilizing gaze (du Lac et al., 1995) and for computing head direction, spatial maps and navigation trajectories (Stackman et al., 2005; Yoder and Taube, 2014; Wallace et al., 2002; Klier and Angelaki, 2008; Rochefort et al., 2011; Rochefort et al., 2013; RondiReig et al., 2014). These signals originate from two categories of skullanchored inertial sensors: gyroscopelike structures (semicircular canals), which transduce head angular velocity, and accelerometerlike structures (otolith organs), which are activated indifferently by accelerated linear motion and by gravity. Gravity provides an absolute directional cue on the external world and is effectively derived from vestibular inputs in the vestibular system (Merfeld et al., 1999; Angelaki et al., 1999), allowing the brain to align the axes of eye rotations with the direction of gravity (Hess and Angelaki, 1997; Hess and Angelaki, 2003). Head direction cells are also anchored to a reference frame aligned with gravity rather than to the animal’s locomotor plane (Taube et al., 2013; Finkelstein et al., 2016; Wilson et al., 2016; Olson et al., 2017); their activity, which relies on the temporal integration of head angular velocity signals (Song and Wang, 2005), thus also requires information on head orientation relative to gravity (head tilt) (Yoder and Taube, 2009). Indeed, identical activations of semicircular canals may affect the azimuth and elevation of the head in very different ways depending on head tilt: for example, a rotation about the interaural axis will lead opposite changes of azimuth if the head is tilted with the left or right ear down. Therefore, understanding how the brain computes head direction requires to identify the neuronal substrate of the operations transforming skullbound angular velocity into changes of azimuth and elevation.
Lesion data suggest that the caudal cerebellar vermis, a brain region receiving multimodal sensory cues related to head kinematics and orientation (Quy et al., 2011; Yakusheva et al., 2013), plays a pivotal role in the discrimination of gravity (e.g. Kim et al., 2015; Tarnutzer et al., 2015; Lee et al., 2017). Moreover, a distinct population of caudal cerebellar Purkinje cells in monkeys dynamically reports head tilt during passive wholebody movements (Yakusheva et al., 2007; Laurens et al., 2013a; Laurens et al., 2013b). We therefore hypothesized that this structure might also host a representation of head rotations anchored to the direction of gravity.
A considerable literature has described the responses of caudal vermis Purkinje cells to passively experienced head movements (reviewed in Barmack and Yakhnitsa, 2011). However, these movements only covered the lower range of frequencies and amplitudes observed during active selfmotion (Carriot et al., 2014; Carriot et al., 2015; Pasquet et al., 2016; Carriot et al., 2017). Moreover, despite the remarkable linearity of early vestibular information processing (Bagnall et al., 2008; but see Massot et al., 2011 and Sadeghi et al., 2007), the high amplitude of active movements might recruit vestibular afferents in a nonlinear way (Hullar et al., 2005; Schneider et al., 2015). In addition, studies in mice and monkeys have revealed that active and passive head movements are processed in fundamentally different ways within the vestibular nuclei (McCrea et al., 1999; Cullen and Roy, 2004; Roy and Cullen, 2004), which are highly interconnected with the caudal cerebellar vermis. Thus, the principles of vestibular coding in passive conditions might not apply to the active condition. We therefore decided to study the encoding of head movements in the caudal cerebellar vermis in freely moving rats, while monitoring the movements of their head using a miniature inertial sensor.
Results
Kinematics of selfgenerated head movements
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 headbound 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 multipeaked 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 nongravitational (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 (headbound) 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 lowfrequency 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.
Caudal cerebellar units exhibit a mixed sensitivity to head angular velocity and gravitational acceleration
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 interspike 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 modelfree 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 (R^{2}) 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 R^{2} = 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). R^{2} values were always greater for A^{G}based than for A^{nG}based estimates (p<8e9 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. R^{2} 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 nonoverlapping (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 influence of head movements on firing rate is shared by neighboring units and is independent of visual cues, but varies when movements are selfgenerated or passively experienced
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, onesampled Wilcoxon test) that were lost if the spike train of one unit was timereversed (R = 0.00 ± 0.03, p=0.49, onesampled Wilcoxon test, Figure 2C). To test whether these correlations were due to a similar dependency on inertial parameters (versus a shared, movementindependent entrainment of neighboring units), we isolated the movementdependent 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, onesampled Wilcoxon test, Figure 2C) than when comparing instantaneous firing rates (Figure 2D), showing that neighboring unit tended to share similar sensitivities to head movements.
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 passivelyapplied wholebody 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 selfgenerated or passively experienced.
Subsets of caudal cerebellar units are specifically tuned to either rotational or gravitational information
According to our modelfree 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 Ωselective 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).
The sensitivity of Ωunits was examined by computing ‘inertiotemporal 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).
To quantify the sensitivity of Ωunits 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 Ωunits (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 Ωunits 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). Ωunits corresponded to putative mossy fibers (n = 2) and putative Purkinje cells (n = 4), and exhibited preferred rotation axes clustered around the excitatory direction of semicircular 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 equalarea 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 Ωunits which were not modulated by head tilt (the CV of firing rate across head tilts was higher for A^{G}units than for Ωunits: 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.
The rotational sensitivity of most caudal cerebellar units is tiltdependent
As shown above, most units displayed a mixed sensitivity to rotational and gravitational information and were not classified as Ωunits or A^{G}units. A direct examination of inertiotemporal receptive fields for different head orientations showed that the rotational sensitivity of these units was indeed often highly tiltdependent. 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}). Inertiotemporal 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 tiltdependent 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 nonΩunits (σ_{x} = 0.36 ± 0.32 and σ_{y} = 0.41 ± 0.29, n = 60) than for Ωunits (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 tiltdependent.
A fraction of caudal cerebellar units encodes head rotations in a gravitycentered reference frame
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 earthvertical 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 (headbound), or in an external (earthbound), 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 (tiltindependent) 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 externalcoding cells did not appear to cluster around specific directions (Figure 5H).
The sensitivity of caudal cerebellar units can be described as a head tiltdependent tuning of linear responses to head angular velocity
As shown above, the mixed sensitivity of most units to rotational and gravitational information, captured by our modelfree 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’ (headtilt 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 $\dot{\mathrm{\Omega}}$, gravitational acceleration, noted ${\text{A}}^{\text{G}}$, and jerk of the gravitational acceleration, noted $\dot{\mathrm{A}}}^{\text{G}$). 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 $\mathrm{\Omega}$, $\dot{\mathrm{\Omega}}$ 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 modelfree approach. In contrast, the firing rate predictability calculated with our modelfree approach was not significantly different from the coefficient of determination calculated from ‘local’ linear models used to generate tiltdependent 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 (headtilt dependent ω_{opt}) collectively provide a description of the firing rate as performant as the modelfree approach (Figure 6B).
Discussion
Our findings reveal that the caudal cerebellar vermis hosts gravitationally polarized representations of head rotations in freely moving rats. We found that most putative Purkinje cell in the posterior cerebellum indeed exhibit receptive fields encoding head rotations about 3D axes anchored to the direction of gravity. This type of rotational sensitivity requires a complex reencoding of headcentered sensory cues and might subserve downstream computations such as the encoding of head direction in the earthhorizontal plane.
Inertial receptive fields in the caudal cerebellar vermis of freely moving animals
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 headfixed animals submitted to passive, externally applied, movements; however, studies in the vestibular and cerebellar nuclei during selfgenerated 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 spontaneouslygenerated movements yields specific constraints for the analysis: the movements have a high dimensionality (rotations and nongravitational 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 highdimensionality of the data, we used a twostep analysis: the first step involved a modelfree 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 modelfree 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 semicircular mossy fibers or Golgi cells directly driven by these fibers. A fraction of head tiltselective units (n = 8) was identified as Purkinje cells, and might correspond to previously identified static rolltilt Purkinje cells (Marini et al., 1976; Yakhnitsa and Barmack, 2006), or to tiltselective Purkinje cells dynamically extracting head tilt information through multisensory integration (Laurens et al., 2013a; Laurens et al., 2013b).
Cerebellar processing of inertial information
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 mediolateral and large anteroposterior 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 modelfree 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 tiltdependent rotational tuning. Such units may intervene in the transformation of headbound angular velocity signals (as sensed by the semicircular canals) into externally referenced angular velocity signals, as observed in the forebrain (Laurens et al., 2016), or in azimuthal information within the headdirection system (Taube et al., 2013; Finkelstein et al., 2016; Wilson et al., 2016). Indeed, head tilt affects the correspondence between headbound and earthbound rotations (see Video 4). Therefore, cells tuned to rotations about an earthbound axis are expected to exhibit a tiltdependent sensitivity to headbound 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 semicircular canal inputs in vestibular nuclear neurons; this operation would isolate an earthhorizontal component of head angular velocity suitable for the computation of azimuthal heading in head direction cells (Finkelstein et al., 2016).
Origin of the receptive fields in the caudal cerebellar vermis
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 selfgenerated 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, selfgenerated 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 lowfrequency (<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; BorgesMerjane 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 βnucleus, which are controlled by dorsal Ygroup and parasolitary nucleus afferents. These nuclei carry rotational and lowpass filtered otolithic signals but are also under the influence of vestibulocerebellar 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 tiltdependent rotational sensitivity.
Conclusion
In conclusion, our work reveals that the dominant coding scheme of natural head kinematics in the caudal cerebellum is a tiltdependent 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 gravitycentered reference frame. As emphasized previously (Green and Angelaki, 2007), the transformation of headbound peripheral vestibular signals into an earthbound 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 reexpress head movements relative to gravity.
Materials and methods
Animals
Sixteen adult male LongEvans 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).
Inertial measurement unit
Our inertial measurement unit (IMU) hosts a 9axis digital inertial sensor containing a 3axis accelerometer, 3axis gyroscope and 3axis magnetometer (MPU9150, 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 inhouse using a reflowsoldering oven (FT2000, 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 I^{2}C interface (USB8451, National Instruments) through a motorized commutator used for electrophysiological recordings (TuckerDavis 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 (MillMax) as shown in Figure 1A.
Tetrode drive
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 inhouse. 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.
Implantation surgery
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 povidoneiodine 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 selfcuring dental adhesive (SuperBond 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 anteroposterior coordinate ranging −3.4 to −5.0 mm, that is, over the cerebellar lobule VI). After removing the dura, a single quartzinsulated 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 lowmelting 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 (PiKuPlast HP 36, Bredent). A lightweight 3Dprinted headstage hosting the IMU pinconnectors and a 16channel 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 3Dprinted 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
Electrophysiological recordings were performed using a previously established protocol (Gao et al., 2012). Quartz insulated tetrodes (Thomas Recording) were goldplated 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 (EIB16, Neuralynx). Signals were referenced against a tungsten electrode positioned in the cerebellum 4 mm below the surface. The EIB was connected to a custommade differential amplifier through a commercial headstage (Tucker Davis) and a motorized 32channel commutator (Tucker Davis). Amplified signals were digitized at 30 kHz by a multifunction acquisition board (NI PCIe6353, 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).
Recording protocol
After a 1week postoperative 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.
Data analysis
Full details of the analysis procedures and statistical tests are provided in the Appendix. Unless mentionned otherwise, pvalues were obtained with an unpaired Wilcoxon test. All mean values are given with the standard deviation.
Appendix 1
Spike sorting and data postprocessing
To isolate spikes, continuous wideband extracellular recordings were first filtered offline with a Butterworth 1 kHz highpass 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 handclustered by polygoncutting in twodimensional projections of the parameter space using Xclust (Matt Wilson, MIT). The quality of clustering was evaluated by inspecting the units’ autocorrelograms, 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 steplike 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.
Classification of recorded units
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 (cutoff 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 cutoff 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 $p({\mathrm{I}\mathrm{S}\mathrm{I}}_{\mathrm{i}})$ 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 onesixth of the histogram’s standard deviation. The entropy was then defined as ($n$ being the number of bins):
Analysis of acceleration signals
Rationale of the analysis
The inertial parameters measured by the sensor and the animal’s vestibular organs can be decomposed in:
An angular velocity signal (Ω, sensed by semicircular canals and by the sensor’s gyroscopes).
An acceleration signal (A, sensed by otolith organs and the sensor’s accelerometers), which can be decomposed in gravitational acceleration (A^{G}), calculated using an orientation filter (as explained below), and nongravitational acceleration (A^{nG}). A^{nG} contains purely translational acceleration (A^{T}), as well as linear tangential acceleration (A^{LT}) and centripetal acceleration (A^{C}), both resulting from rotations.
Hence:
Ω, A^{G} and A^{T} are experienced identically at any point on a solid. Because the sensor was fixed to the skull via a connector embedded in dental cement and because vestibular organs are skullanchored structures, Ω, A^{G} and A^{T} were experienced identically by the sensor and the animal’s vestibular organs.
This section details the calculations for estimating A^{G} and for assessing the contribution of head rotations to the production of nongravitational acceleration signals. Because the quantification of A^{LT} and A^{C} requires to know the distance between the sensor and the axes about which rotations are executed, and because this distance can not be precisely determined, the calculations detailed below are only meant to provide an order of magnitude for these rotationdependent components of acceleration, and to assess their global contribution to A^{nG}. In other words, the goal of these calculations was not to quantify A^{LT} and A^{C} precisely, but to assess whether A^{nG} was likely to be dominated by pure translational acceleration signals or by acceleration signals produced by head rotations.
Estimation of gravitational acceleration
An estimate of gravitational acceleration was calculated using Sebastian Madgwick’s IMU sensor fusion algorithm (also known as ‘orientation filter’, Madgwick et al., 2011). For every time point, the algorithm computes a quaternion representation of a 3D rotation in the Earth frame that brings the sensor from an initial orientation to its current orientation (Figure 1—figure supplement 1D1). The converse rotation can be used to keep track of the orientation of the gravity vector in the sensor frame (Figure 1—figure supplement 1D2), and thus to calculate the gravitational components of acceleration along the 3 axes of the sensor (Figure 1B). Because gyroscopes provide measurements of angular velocity with a certain amount of noise, the temporal integration of gyroscope signals leads to accumulating error in the estimation of orientation. The purpose of the algorithm is to use acceleration signals (which, on long timescales, reflect mostly the effect of gravity) to compute the direction of this error at every time step, and to compensate for it by subtracting the magnitude of the noise of gyroscope measurement ($\beta $) along that direction (in our conditions, $\beta $ was in the order of 0.1 °/s, as per component specifications and according to recordings performed from an immobile IMU). This procedure ensures that the estimated orientation remains properly aligned with gravity, thereby providing accurate head tilt (attitude) information, but does not guarantee accurate heading information. In other words, the algorithm does not provide accurate information on the absolute orientation of the sensor in the Earth frame, but only on its orientation relative to gravity.
Linear tangential and centripetal acceleration resulting from head rotations
Linear tangential acceleration (A^{LT}) is produced in the tangential direction of movement, perpendicular to the axis of rotation, during a change of angular velocity (angular acceleration). Centripetal acceleration (A^{C}) is exerted in the radial direction toward the center of rotation. The magnitude of these accelerations depends on the distance d between the sensor and the axes about which head rotations are executed. Because d can not be precisely determined, the following calculations are only meant to provide an order of magnitude for A^{LT} and A^{C}. For these calculations, we made the following assumptions:
That most head rotations are pitch (up/down) and yaw (left/right) rotations (as shown in Figure 1—figure supplement 1B). Because roll rotations are executed about and axis that passes near the sensor, the contribution of this type of rotation to A^{LT} and A^{C} can be considered as minimal.
That pitch/yaw head rotations are executed about axes of rotation that pass through the animal’s neck, and that the distance d between the sensor and these rotations axes is in the order of several cm (as defined during the implantation surgery).
For the calculation, we thus discarded ω_{x} values measured by the sensor, which concentrate most of the roll information, and assumed that pitch/yaw rotations move the head around a resting posture (pitched 35° nose down, as shown in Blanks and Torigoe, 1989). In this default orientation, the sensor is tilted as shown in Figure 1—figure supplement 2C1 and thus captures the yaw component of rotations (here defined as leftright rotations) along its x and z axes. The yaw component of angular velocity produces a linear tangential acceleration along the y axis of the sensor (Figure 1—figure supplement 2C1) that can be calculated as follows:
where $\theta $ is the tilt angle of the head relative to an upright orientation (Figure 1—figure supplement 2C1), d is the distance between the sensor and the center of rotation and a_{y}^{lt} is the linear tangential acceleration along y in m·s^{−2}. Pitch rotations produce linear tangential accelerations along the x and z axes of the sensor (Figure 1—figure supplement 2C1) which can be calculated as follows:
The norm of the linear tangential acceleration vector a^{lt} produced by pitch/yaw rotations (Figure 1—figure supplement 2D) was then calculated as:
The norm of the centripetal acceleration vector a^{c} (pointing toward the center of rotation) resulting from pitch/yaw rotations (Figure 1—figure supplement 2D) was calculated as:
Estimation of firing rate modulations that are due to head movements
Principle of the analysis
The goal was to implement a modelfree, datadriven method to test the intensity and robustness of the link between a set of external variables (here the inertial parameters of head movements) and the fluctuations of the instantaneous firing rate of recorded units. The core computation (detailed below) employs a jackknife resampling technique that calculates firing rate estimates based on the repeated occurrence of similar combinations of values in the external variables considered.
The square of the Pearson correlation coefficient between the estimated and observed firing rates is taken as a measure of the influence of these external variables on a unit’s activity. It accounts for the amount of firing rate variance that can be explained by these variables, and thus can be considered as a measure of how well these variables can predict firing modulations (firing predictability).
In practice, other factors may contribute to firing rate fluctuations (such as a high discharge irregularity at rest or other types of external variables) and may reduce the observed correlation. To test the consistency of our estimation method (i.e. the robustness of the link between the external variables and firing rate fluctuations), we used a crossvalidation approach consisting in calculating the Pearson correlation coefficient between independent firing rate estimates obtained using nonoverlapping portions of the same recording.
Steps of the analysis
For the sake of clarity, the different steps of the analysis are listed below. These different steps, which are described in detail in the next paragraphs, are performed independently for each cell and provide a metric of how well inertial parameters can explain fluctuations of the instantaneous firing rate (firing rate predictability), and of how reliable firing rate estimation can be with these parameters (robustness of the prediction).
Calculation of the instantaneous firing rate (FR, see ‘Spike sorting and data postprocessing’). FR and raw inertial signals are transformed into time series with the same sampling interval.
Estimation of the gravitational component of acceleration (A^{G}) using an orientation filter algorithm (see ‘Estimation of gravitational acceleration’).
Each category of signal (Ω, A, A^{G}, A^{nG}) is normalized (see ‘Firing rate predictability’).
Optimal time delays (lags) between FR and each category of signal are estimated by crosscorrelation (see ‘Firing rate predictability’).
For each point in the recording, a resampling procedure is used to calculate the expected firing rate at that point given firing rate values observed for points with similar combinations of inertial parameters (see ‘Firing rate predictability’). The operation is repeated for each time point to obtain an estimated firing rate time series. For this step, combinations of inertial parameters can be chosen arbitrarily.
The square of the Pearson correlation coefficient between the estimated firing rate time series and FR is taken as a measure of firing rate predictability using a specific combination of inertial parameters (see ‘Firing rate predictability’).
The robustness of the prediction is assessed by computing independent estimated firing rate time series using nonoverlapping portions of the recording (see ‘Robustness of the prediction’). The Pearson correlation coefficient between these independent estimates is taken as a measure of the robustness of the estimation.
Firing rate predictability
A set of $n$ timedependent external variables (${v}_{1},{v}_{2},\mathrm{\dots},{v}_{n}$), here representing the inertial parameters of head movements, and recorded within a time interval $[0,T]$, can be described as a timedependent vector $\mathit{}(t)=({v}_{1}(t),{v}_{2}(t),\mathrm{\dots},{v}_{n}(t))$, with $t\in [0,T]$. At each time ${t}_{0}$ in $[0,T]$, an estimate of the instantaneous firing rate can be computed as the average of firing rate values observed at times during which $\mathit{V}$ is similar to $\mathit{V}({t}_{0})$. In practice, the predicted firing rate is calculated using only a set of points where the Euclidean distance between $\mathit{V}$ and $\mathit{V}({t}_{0})$ is the smallest, discarding the points immediately preceding or following ${t}_{0}$. The procedure is detailed below.
Because the variables in $\mathit{V}$ may span very different values (accelerations, measured in $g$, are typically in the $[2$,$2]$ range while angular velocity values, measured in °/s, are typically in the $[200,200]$ range), we first rescaled the variables (grouped by type, either acceleration or angular velocity) by dividing them by their group SD, yielding normalized versions of the variables ${\mathit{V}}^{\mathit{n}\mathit{o}\mathit{r}\mathit{m}}=({v}_{1}^{norm},{v}_{2}^{norm},...,{v}_{n}^{norm})$. For example, the normalization of angular velocity values was performed as follows:
Before grouping the variables into the vector $\mathit{V}}^{\mathit{n}\mathit{o}\mathit{r}\mathit{m}$, we had to take into account the fact that there was potentially a lag between these variables and their effect on the firing rate. For each category of variable, the optimal lag $\tau $ was taken as the temporal delay at which the highest crosscorrelation was observed between the instantaneous firing rate and variables of that category. For angular velocity:
where $\omega}_{x}\text{}\ast \text{}\text{FR$ is the correlation function of ${\omega}_{x}$ and FR. The normalization and timeshifting steps above yield the following vector of variables:
where (${\omega}_{x},{\omega}_{y},{\omega}_{z}$) and (${a}_{x},{a}_{y},{a}_{z}$) represent respectively angular velocity and net acceleration values along the sensor’s axes, $\tau}_{opt}^{\mathrm{\Omega}$ is the optimal lag for angular velocity and $\tau}_{opt}^{a$ the optimal lag for net acceleration. Note that $\mathit{V}}^{\mathit{n}\mathit{o}\mathit{r}\mathit{m}$ can incorporate other variables calculated offline, such as gravitational and nongravitational acceleration (see ‘Analysis of acceleration signals’).
For each ${t}_{0}$ in $[0,T]$, we then calculated the Euclidean distance between ${\mathit{V}}^{\mathit{n}\mathit{o}\mathit{r}\mathit{m}}({t}_{0})$ and all other values of $\mathit{V}}^{\mathit{n}\mathit{o}\mathit{r}\mathit{m}$ encountered in $[0,T]$:
We then selected the $N$ smallest values of $d\{d({t}_{0},{t}_{1}),...,d({t}_{0},{t}_{N})\}$ ($N=200$, $500$ or $1000$), excluding time points that were too close to ${t}_{0}$ (exclusion window = $\pm 0.5$ s in our calculations, which corresponds to 100 points before and after ${t}_{0}$, given our sampling frequency of 200 Hz). The firing rate estimate at ${t}_{0}$ was then calculated as the average of firing rate values observed at the time points corresponding to these $N$ values of $d$:
The process was then repeated for every time point in $[0,T]$ to obtain a complete firing rate estimate ${\rho}_{V}$ over the whole recording. This firing rate estimate was then compared to a version of the instantaneous firing rate computed using the local average ISI value (taking into account ISIs occurring before and after each ISI). This smoothed version of the instantaneous firing rate (${\text{FR}}_{\text{smooth}}$) was calculated as follows. For each time point ${t}_{0}$ falling inside the $j}^{th$ ISI, the value of ${\text{FR}}_{\text{smooth}}$ was given by:
where $n$ is the number of ISIs before and after the $j}^{th$ ISI (in our calculation, $n=2$). ${\text{FR}}_{\text{smooth}}$ was calculated at 1 kHz and then convolved with a 10 ms Gaussian kernel (as in ‘Spike sorting and data postprocessing’). The square of the Pearson correlation coefficient (${R}^{2}$) between ${\rho}_{V}$ and ${\text{FR}}_{\text{smooth}}$ was taken as a measure of the fraction of firing rate modulation that could be explained by $\mathit{V}$ (firing rate predictability). We found empirically that comparing ${\rho}_{V}$ with ${\text{FR}}_{\text{smooth}}$ and not the instantaneous firing rate calculated from the inverse of the ISI ($n=1$, as defined in Spike sorting and data postprocessing) yielded about 40% greater ${R}^{2}$ values. The interpretation of these higher values is that ${\text{FR}}_{\text{smooth}}$ softens the fluctuations of the ISI (Figure 1—figure supplement 3D), smoothing them down to a timescale closer to the one of movements (Figure 1—figure supplement 1C).
Note that the resampling procedure can be performed using arbitrary sets of external variables, and thus the influence of different sets of variables on firing rate can be tested and compared (Figure 1E). We used the following notation to differentiate firing rate estimates calculated using differents combinations of inertial parameters: ${\rho}_{\mathrm{\Omega}}$ when considering angular velocity only, ${\rho}_{{A}^{G}}$ when considering gravitational acceleration only, ${\rho}_{\mathrm{\Omega},{A}^{G}}$ when considering both, etc.
Robustness of the prediction
To test the robustness of the link between a set of external variables $\mathit{V}$ and firing rate modulations, we computed independent firing rate estimates using nonoverlapping portions of the same recordings. In practice, we split the interval $[0,T]$ of each recording into two separate ensembles by selecting alternatively 10 secondlong time intervals:
We then calculated two firing rate estimates (${\rho}_{{V}_{1}}$ and ${\rho}_{{V}_{2}}$) using time points (and the corresponding values of $\mathit{V}$) from either one or the other dataset (i.e. ${\rho}_{{V}_{1}}$, resp. ${\rho}_{{V}_{2}}$, was calculated for all time points of the recording using only data from ${\text{set}}_{1}$, resp. ${\text{set}}_{2}$). The Pearson correlation coefficient ($R$) between ${\rho}_{{V}_{1}}$ and ${\rho}_{{V}_{2}}$ was then calculated and taken as a measure of the robustness of the firing rate estimation procedure. The distribution of $R$ values for all units is shown in Figure 1F. To calculate these values, the best combination of inertial parameters (i.e. providing the greatest firing predictability) was selected on a cellbycell basis. A null distribution of $R$ values was obtained by randomly shuffling the ISIs of the units (Figure 1F). Values in this distribution are the average $R$ calculated over 10 iterations (each with an independent ISI shuffling step).
Comparing the robustness of firing rate predictions for different recording conditions
A similar crossvalidation strategy was used to evaluate the consistency of the link between $\mathit{V}$ and firing rate modulations when the experimental conditions were changed (e.g. when recordings were performed in the light or in the dark, or when head movements were selfgenerated or passively experienced). The logic was the following: if the link between $\mathit{V}$ and firing rate modulations remains unchanged between two blocks recorded in different conditions (${\text{block}}_{1}$ and ${\text{block}}_{2}$), then the similarity (i.e. the Pearson correlation coefficient) between two independent firing rate estimates in ${\text{block}}_{1}$ (${\rho}_{{V}_{1}}^{{\text{block}}_{1}}$ and ${\rho}_{{V}_{2}}^{{\text{block}}_{1}}$) should be equivalent to the similarity between the firing rate estimate in ${\text{block}}_{1}$ (${\rho}_{V}^{{\text{block}}_{1}}$) and a firing rate estimate in ${\text{block}}_{1}$ computed with the dataset from ${\text{block}}_{2}$ (${\rho}_{V}^{{\text{block}}_{2\to 1}}$).
For the comparison of blocks recorded in the light or dark (Figure 2E,F), we first noticed that the combination of inertial parameters yielding the best firing rate estimate for each cell ($\mathbf{V}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}$) was always the same for the two conditions. We then compared the similarity between independent firing rate estimates for the light block (${\rho}_{{V}_{best,1}}^{\text{light}}$ and ${\rho}_{{V}_{best,2}}^{\text{light}}$) to the similarity between the firing rate estimate for the light block (${\rho}_{{V}_{best}}^{\text{light}}$) and a firing rate estimate for the light block computed using the dark block dataset (${\rho}_{{V}_{best}}^{\text{dark}\to \text{light}}$).
For the comparison of blocks recorded in the presence of active or passive movements (Figure 2G,H), the best predictive model for each cell ($\mathbf{V}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}$) was first identified, irrespective of the condition (active or passive). Using this model, the similarity between independent firing rate estimates for the passive block (${\rho}_{{V}_{best,1}}^{\text{passive}}$ and ${\rho}_{{V}_{best,2}}^{\text{passive}}$) was compared to the similarity between the firing rate estimate for the passive block (${\rho}_{{V}_{best}}^{\text{passive}}$) and a firing rate estimate for the passive block computed using the active block dataset (${\rho}_{{V}_{best}}^{\text{active}\to \text{passive}}$).
Receptive fields of rotationselective units
Inertiotemporal receptive fields
These plots (Figure 3B and Figure 4A,D) were obtained by calculating the average timecourse of the instantaneous firing rate around specific values of angular velocity (ω_{x}, ω_{y} or ω_{z}). The variable considered (e.g. ω_{x}) was first divided into equal intervals (binwidth = 5 °/s). Every point in the recording was assigned to its corresponding interval and the instantaneous firing rate trace within a given time window around that point (lag of ±0.5 s) was saved. To build inertiotemporal receptive fields, firing rate traces were averaged for each interval. The resulting firing rate values were then color coded and represented as a function of the variable (center value of each interval) and the lag.
3D plots of rotational sensitivity
These plots (see for example Figure 3C and Video 2) were obtained by calculating the average instantaneous firing rate for specific combinations of the three components of angular velocity (ω_{x}, ω_{y} and ω_{z}) at a given lag. These variables were first divided into equal intervals (binwidth = 25 °/s). Every point in the recording was assigned to a specific combination of ω_{x}, ω_{y} and ω_{z} intervals and the instantaneous firing rate at the optimal lag (see ‘Linear fits’) relative to that point was saved. Firing rate values belonging to the same combination of intervals were averaged, color coded and represented in 3D as a function of angular velocity values (center values of the intervals).
Linear fits
The instantaneous firing rate of rotationselective units at time $t+lag$ ($lag$ defined by the user) was described as a linear combination of angular velocity values at time $t$ (see Figure 3D):
The coefficients of the fit define a sensitivity vector ${\mathit{\omega}}_{\mathit{l}\mathit{a}\mathit{g}}=(\alpha ,\beta ,\gamma )$ whose direction represents the unit’s 3D preferred axis and direction of rotation, and whose norm ${\mathit{\omega}}_{\mathit{l}\mathit{a}\mathit{g}}=\sqrt{{\alpha}^{2}+{\beta}^{2}+{\gamma}^{2}}$ represents the unit’s response gain (i.e. sensitivity, in Hz/(°/s) or °^{−1}) at the lag considered. The above linear fit formula is equivalent to:
where ${\mathit{\omega}}_{\mathit{l}\mathit{a}\mathit{g}}\text{}\bullet \text{}\mathit{\omega}(t)$ is the dot product of $\mathit{\omega}}_{\mathit{l}\mathit{a}\mathit{g}$ and the instantaneous angular velocity vector $\mathit{\omega}(t)=({\omega}_{x}(t),{\omega}_{y}(t),{\omega}_{z}(t))$. The optimal sensitivity vector ($\mathit{\omega}}_{\mathit{o}\mathit{p}\mathit{t}$) is obtained at the lag yielding the largest sensitivity (optimal lag). The direction of $\mathit{\omega}}_{\mathit{o}\mathit{p}\mathit{t}$ indicates the unit’s preferred 3D axis and direction of rotation. Its norm ${\mathit{\omega}}_{\mathit{o}\mathit{p}\mathit{t}}$ represents the unit’s sensitivity along its preferred rotation axis. To assess the significativity of rotational sensitivity, an average ${\mathit{\omega}}_{\mathit{l}\mathit{a}\mathit{g}}$ vs. lag curve was generated for each unit using shuffled spike trains (constructed from the original ISI distribution; $n=100$ iterations; lag range = $[0.5,0.5\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}]$). The average norm of ${\mathit{\omega}}_{\mathit{l}\mathit{a}\mathit{g}}+3.5\times SD$ calculated from this curve was taken as the significativity threshold. The sensitivity of a unit was considered significant if the $[0.5,0.5\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}]$ lag range contained ${\mathit{\omega}}_{\mathit{o}\mathit{p}\mathit{t}}$ values greater than the significativity threshold.
Tiltdependent rate maps
Tiltdependent 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 ${\mathit{a}}^{\mathit{G}}=({a}_{x}^{G},{a}_{y}^{G},{a}_{z}^{G})$ in the sensor frame (see Video 1). During rotations causing a reorientation of the head relative to gravity, $\mathit{a}}^{\mathit{G}$ 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 headtilt, 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 $\mathit{a}}^{\mathit{G}$ 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 equalarea projection (the center of the map corresponding to an upright head orientation, that is, with the headvertical axis aligned with $\mathit{a}}^{\mathit{G}$). Finally the polygons were filled with a color indicating the corresponding normalized average firing rate value (Figure 3H). The nonparametric 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).
Stability index
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 (${\mathit{\omega}}_{\mathit{l}\mathit{a}\mathit{g}}=({\alpha}_{lag},{\beta}_{lag},{\gamma}_{lag})$) 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).
Calculation of angular velocity in a reference frame aligned with gravity
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 righthanded 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 nasooccipital 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 externallyreferenced optimal rotational sensitivity vectors using a linear model (as in ‘Linear fits’).
Tiltdependent rotational sensitivity maps
To compute head tiltdependent optimal rotational sensitivity vectors, we used a strategy similar to the one employed for tiltdependent 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 (R^{2}) obtained by this method was then compared to the R^{2} calculated using the modelfree approach (see ‘Estimation of firing rate modulations that are due to head movements’ and Figure 6B).
Collinearity between rotational sensitivity vectors
The collinearity of pairs of rotational sensitivity vectors $({\mathit{\omega}}_{\mathit{o}\mathit{p}\mathit{t}\mathbf{,}\mathit{i}},{\mathit{\omega}}_{\mathit{o}\mathit{p}\mathit{t}\mathbf{,}\mathit{j}})$, 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 $\mathit{a}}^{\mathit{G}$ 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 $\mathit{a}}^{\mathit{G}$ orientations for which sensitivity vectors were calculated (among the 500 points selected over the sphere, see ‘Tiltdependent rotational sensitivity maps’). These values were then averaged as a function of $\phi $ and plotted for bins of π/16 (Figure 5E,F).
Global linear models
To compare the performance of our modelfree 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 tiltdependent rotational sensitivity maps, described in section ‘Tiltdependent 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 $\mathit{\omega}(t)$, $\dot{\mathit{\omega}}(t)$, ${\mathit{a}}^{\mathit{G}}(t)$ and ${\dot{\mathit{a}}}^{\mathit{G}}(t)$ are 3D vectors representing the instantaneous angular velocity, angular acceleration, gravitational acceleration and jerk (time derivative) of the gravitational acceleration, respectively, and $\mathit{\lambda}$, $\mathit{\xi}$, $\mathit{\alpha}$ and $\mathit{\beta}$ 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 modelfree 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).
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Decision letter

Jennifer L RaymondReviewing Editor; Stanford School of Medicine, United States
In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.
Thank you for submitting your article "Cerebellar reencoding of selfgenerated head movements" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and a Senior Editor. The following individuals involved in review of your submission have agreed to reveal their identity: Sergio Carmona (Reviewer #3).
The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.
Summary:
This manuscript describes an elegant set of experiments examining the reference frames and neural computations used to encode selfmotion from vestibular cues. The responses of cerebellar neurons in the caudal cerebellar vermis to natural vestibular stimuli in freely moving animals, and the recording in freely moving animals is a great strength of this approach. The results indicate that a majority of cerebellar Purkinje cells encode head rotations in a head tiltdependent manner, and a subset of cells encode head rotations in a gravity/earthcentered reference frame, rather than the headcentered reference frame of the vestibular periphery. These findings about how natural head movements are represented in the cerebellum suggest a possible mechanism for computing head direction for spatial orientation, and thus should be of broad interest.
Essential revisions:
1) The data analysis is complex, and needs to be more clearly described in the main text and justified. In particular, the model free analysis should be more clearly described and justified.
The reviewers struggled to determine whether the parameters (weights and time delays) were determined from the average of instantaneous firing rate of each cell, average from all cells, or changed from case to case. More generally, there was concern about the unstated assumptions, and potential noise and accumulation of small errors associated with the multiple layers of analysis (data smoothing, effectively data binning, multidimensional fitting algorithms across both time and parameters whose amplitudes are normalized, calculating correlation coefficients and then taking correlations of correlation coefficients).
The description around Figure 1F, which assesses the robustness of the analysis, is confusing. For 80% of the units, the Pearson correlation coefficient (R) between independent estimates was greater than 0.6. How was this value of 0.6 selected? What would the Pearson coefficients look like for shuffled data, or some type of null distribution? What about the other 20% of the neurons? If this method is one to confirm the veracity of the analysis, doesn't this imply that for 20% of the neurons the method does not work or that the unit was not the same one throughout the recording? What influenced the decision to keep such units (particularly the 8 or so units with low or no correlation)?
Does the modelfree approach do significantly better than more traditional approaches? For smooth pursuit eye movements, the instantaneous firing rate of Purkinje cells does a great job of representing the kinematics as a linear sum of velocity, acceleration and the position of the eye. Does this more standard approach fail for the current data?
The value of looking at sensitivity vs. lag (Figure 3E) is not clear.
In Figure 4A and D, wouldn't the degree of nose up/down tilt effect the yaw sensitivity? The significance of Figure 4B and E should be clarified or those panels not included.
2) There is currently considerable debate as to whether one can use the firing of cerebellar neurons to identify them as a specific cell type. The authors should use additional criteria to confirm cell type identity, or back off on their claims about which cell types were recorded.
3) The "passive whole body movements condition" is very poorly described. How well matched are the vestibular stimuli that the animals experience in the passive and active condition? If they are not similar in the spatial or temporal dynamics, it seems this might explain the poorer correlations in Figure 2H. Could the smaller amount of data for the passive condition be contributing to the discrepancy?
4) Once subdivided, some of the groups have Ns of less than 5. Can the number of recordings increased to generate more confidence in these divisions and perhaps unravel additional ones which might have been missed?
5) The analysis appears to require knowledge of the distance d, between the IMU and the point around which yaw and pitch movements are made. It is not clear whether the absolute distance is important, and if so, how it is determined. If (as it appears) the nongravitational linear acceleration is not important for understanding the behavior of the neurons, this should be made clearer (subsection “Cerebellar units exhibit a mixed sensitivity to head angular velocity and gravitational acceleration”).
https://doi.org/10.7554/eLife.26179.018Author response
Essential revisions:
1) The data analysis is complex, and needs to be more clearly described in the main text and justified. In particular, the model free analysis should be more clearly described and justified.
We are grateful to the reviewers for pointing out the lack of clarity of the main text. We have modified our manuscript to provide a clearer description and justification of each step of the analysis.
The data analysis is mostly directed at assessing the link between the firing rate and the inertial parameters of head movements. In brief, we first look at how much of the instantaneous firing rate fluctuations can be explained by the inertial parameters and at how consistent this link is across conditions (these questions are addressed using a modelfree approach in Figure 1 and 2). We then explore this link using linear models (Figure 3 to 6) and show that, except for a small subset of units which show a rather simple tuning to angular velocity or head tilt (Figure 3), the sensitivity of most units can be modeled as a headtiltvarying linear dependency to head rotations (i.e. with a “preferred” rotation that depends on head orientation relative to gravity; Figure 4 and 5). We show that this model performs as well as the modelfree approach (but, as we now show in the revised version of the manuscript, gives better results than simple linear models of the instantaneous firing rate; see Figure 6 and Figure 6—figure supplement 1).
The modelfree approach was used to determine how much (and which set of) inertial parameters influence the instantaneous firing rate. The choice of a modelfree approach as an initial method for analyzing the data was motivated by the fact that nonlinearities in the responses of primary vestibular afferents have been reported during naturalistic movements (Schneider et al., 2015), and nonlinearities in the cells’ sensitivity to head inertial signal are expected if head movements are reencoded in an external reference frame (Green and Angelaki, 2007). This approach makes the assumption that similar values of inertial parameters yield (with an optional delay) similar instantaneous firing rate values; it does not hypothesize any a priori model. The approach yields an estimate of the average instantaneous firing rate driven by a given set of inertial parameters; this estimate can also be viewed as the instantaneous output of a population of neurons sharing similar sensitivity. An important caveat for the use of this approach is its sensitivity to the sampling of the parameter space by the behaving animal during the session, which determines whether the “neighbors” in the parameter space used to average the firing rate are close enough to build a consistent estimate. We found in general a good consistency (robustness) of the estimation as shown by the correlation coefficient between independent firing rate estimates (Figure 1F), indicating that in most cases, the sampling of the parameter space was sufficient. Finally, this approach was used to compare the units’ sensitivity across conditions (e.g. light and dark) without making any assumptions on this sensitivity: firing rate estimates for a given session may be generated using either the same or another session as a “training” session (i.e. neighbors in the inertial parameter space may be searched in the same or in another session), and compared (Figure 2). A noticeable point is the computational cost of this approach which typically require several hours of CPU per unit.
The alternative method used to analyze the data is composed by multiple “local” linear models, assuming a linear tuning to a preferred rotation which depends on the head orientation (Figure 5). This method seems as good as the methodfree approach to capture the fraction of firing rate fluctuation due to the inertial parameters considered. This is now more clearly stated in the manuscript and highlighted with Figure 6B (formerly Figure 5—figure supplement 1C).
We made substantial efforts to justify and describe more thoroughly the modelfree approach in the revised manuscript and in the revised Appendix section and to better justify the steps of analysis.
The reviewers struggled to determine whether the parameters (weights and time delays) were determined from the average of instantaneous firing rate of each cell, average from all cells, or changed from case to case. More generally, there was concern about the unstated assumptions, and potential noise and accumulation of small errors associated with the multiple layers of analysis (data smoothing, effectively data binning, multidimensional fitting algorithms across both time and parameters whose amplitudes are normalized, calculating correlation coefficients and then taking correlations of correlation coefficients).
We apologize for any lack of clarity of the description of the procedures. We added a summary of all the analysis steps for the modelfree approach in the revised Appendix section. We summarize here the key points.
There is no use of weights in the analysis; on the other hand, we use lags to take into account that fact that the firing rate may not be instantly linked to the inertial parameters, but that there may be a delay between the two. The lags are computed for each cell in each recording block. In the case of the modelfree approach, the lag is taken as the time delay that maximizes the crosscorrelation between the firing rate and each category of inertial parameter (for reason of computational cost, we did not attempt to optimize this choice by seeking the lags giving the largest correlation between the estimated and observed instantaneous firing rates). In the case of linear models, the lag is chosen to yield the largest coefficients (hence, the largest sensitivity) in a simple linear regression predicting the firing rate from angular velocity signals (although this simple linear model does not perform as well as the combination of linear models calculated for different head tilts, shown in Figure 5C and D, we felt it would not make much sense to use different lags for different head tilts). Average sensitivity vs. lag curves are shown in Figure 3E.
We summarize below the main steps of the analysis.
In the modelfree analysis, the calculations are:
1) Downsampling of inertial signal: some inertial data were acquired at 200 Hz and the other were thus downsampled from 250 Hz to 200 Hz; we felt this 200 Hz frequency acceptable in view that 1) the time constant of head movements is longer than 50 ms (see our response to point 1 in the minor revisions), 2) the transmission delay between the sensor and the acquisition system exhibited a small jitter of a 1–2 milliseconds (Pasquet et al., 2016), therefore limiting the temporal precision of the measures under 5 ms.
2) Computing an instantaneous firing probability. We calculated a time series representing an estimate of the instantaneous firing probability as a function of time. Each time point of the time series was defined as the inverse of the ISI during which this point fell.; the resulting steplike function was then convolved with a 10 ms Gaussian distribution to soften the transitions between successive ISIs (see minor point 1 below for more justification on this procedure).
3) Normalizing the different types of inertial parameters (angular velocity, acceleration). This is needed to allow the identification of neighboring points within the high dimensional space of inertial parameters. Indeed, the modelfree resampling procedure requires, for a given (“current”) time point, to find similar combinations of inertial parameters at other time points in the recording (taking into account time lags, as discussed above) and to average the corresponding firing rate values. Similar combination of inertial parameters are identified by the fact that their Euclidean distance to the “current” combination is minimal (see Appendix section). Given the fact that angular velocity and acceleration signals span ranges of values that differ by two orders of magnitude ( ± 200 °/s and ± 2 g, respectively), the calculation of this Euclidean distance only makes sense if both types of signals are normalized.
4) The optimal time delay (lag) used for the resampling procedure (described above) was calculated for each cell using a simple crosscorrelation analysis (see above). In other words, we calculated an optimal lag for each category of signal (angular velocity, total acceleration, gravitational acceleration, nongravitational acceleration) for each cell in each recording block.
5) We then run the resampling procedure. For a given time point in the recording, this step consists in identifying the nearest neighbors in the parameters space (time points with a combination of inertial parameters similar to the point considered) and calculating an estimated (predicted) firing rate value for this point by averaging the corresponding instantaneous firing rate values. This step is repeated for all points in the recording to obtain a time series representing the estimated firing rate.
6) We then compare this estimated firing rate with the instantaneous firing rate; we found that the correlation was low, largely because the instantaneous firing rate exhibits fast fluctuations due to irregularities in the units’ discharge (CV of ISI ~1); we therefore chose to compare the estimated firing rate with a smoothed version of the observed instantaneous firing rate, obtained by averaging a few ISIs around each time point (as explained Appendix section). This smoothed instantaneous firing rate provides a more consistent measure of the instantaneous firing probability (e.g. more stable instantaneous firing rate values at rest) at the cost of a slower time course (which is however adaptive: the instantaneous firing rate evolves faster when all ISIs are short than when all ISIs are long). Indeed, we found empirically that linear fitting and modelfree approaches were generally better at predicting an instantaneous firing rate computed by averaging a few ISIs (which is therefore smoother but is still modulated at the time scales of variations of the inertial parameters); the number of ISIs used (2 on each side, see Appendix section) was chosen empirically as the one giving the best match between the estimated (by the modelfree approach) and the smoothed instantaneous firing rate.
For the tiltdependent rotational sensitivity maps:
1) We compute an optimal lag from a global linear model (firing rate as a linear combination of the angular velocity components, see Appendix section)
2) The data are binned by headtilt.
3) Within each headtilt bin, we compute a “local” (i.e. headtilt dependent) linear model linking the firing rate to the angular velocity (using the optimal lag defined above), which provides a headtilt dependent preferred rotation (see Appendix section). No normalization is involved.
Other reviewers concerns:
1) Data binning was used to calculate inertiotemporal receptive fields (Figure 3B, Figure 4A and D), 3D plots of rotational sensitivity (Figure 3C), tiltdependent rate maps (Figure 3H) and tiltdependent rotational sensitivity maps (Figure 5C and D). No binning takes place in the modelfree approach. Inertiotemporal receptive fields and tiltdependent rate maps are used for data inspection only (and illustration purposes); the choice of bin is a trade off between resolution and noise. In the case of rotational sensitivity maps, we found that using less than 500 time points in the recording yielded poor estimates of the unit’s preferred rotation; the choice of the “bins” of head tilt (in which to compute the preferred rotation) was a trade off between this requirement and obtaining this preferred rotations for a large fraction of the different head tilts explored by the animal (i.e. for smaller bins, only a few head positions would be visited frequently enough to provide the 500 values needed to compute the preferred rotation).
2) “multidimensional fitting algorithms across both time and parameters whose amplitudes are normalized”: this may refer to the modelfree approach (which is not exactly a fitting procedure since it simply calculates the average instantaneous firing rate for comparable combinations of inertial signals, and then compares it to the original instantaneous firing rate). Justification for the normalization is provided above. The fitting procedures in the manuscript are otherwise linear regression but they do not operate over normalized variables.
3) “calculating correlation coefficients and then taking correlations of correlation coefficients”. Correlations coefficients are used to evaluate the performance of the modelfree approach. The correlation between the estimated and the observed firing rates is a measure of how well a set of inertial parameters can explain the observed firing rate fluctuations (they provide a measure of firing rate predictability, Figure 1E). We did not specifically calculate correlations of correlation coefficients, but the reviewers’ comment may refer to correlation between independent firing rate estimates (these estimates are rather averages than correlations); these computations are used to measure of how robust the estimation is (Figure 1F). The robustness of firing rate estimates were compared (using a Wilcoxon test) when an experimental condition was changed, e.g. when recordings were performed in the presence or absence of light (Figure 2F) or when movements were actively or passively generated (Figure 2H). The rationale for this analysis is the following: the modelfree approach provides an estimate of the fraction of firing rate variance determined by the inertial parameters; for a given set of inertial signals we can use different recording sessions to perform the prediction of the firing rate. We can then compare the two firing rates predicted from the two sessions by computing their correlation coefficients; if this coefficient is 0, it means that the link between the firing rate and movements in the two sessions (at least as assessed by our method) is entirely uncorrelated. However, two predictions obtained from the same condition might also differ to a certain degree if the overall contribution of inertial parameters to firing rate fluctuations is small (e.g. in the presence of other factors influencing the firing rate or because of differences in sampling of the parameter space); therefore, to assess whether a low correlation coefficient between two firing rates predicted from two different conditions is due to a change of sensitivity between the two conditions, we compared this correlation coefficient to the correlation between two firing rates estimates predicted from the same condition: if both are low, this indicates that there is globally a poor predictability of the cell, while a lower correlation across conditions versus within condition will indicate a change of sensitivity (which appears to be the case in the active vs. passive condition; see our response to point 3 of the essential revisions).
We hope that our efforts for providing a clearer explanation and justification of our analysis in the revised manuscript and Appendix section methods will dispel the reviewers’ concerns about the analysis framework.
The description around Figure 1F, which assesses the robustness of the analysis, is confusing. For 80% of the units, the Pearson correlation coefficient (R) between independent estimates was greater than 0.6. How was this value of 0.6 selected? What would the Pearson coefficients look like for shuffled data, or some type of null distribution? What about the other 20% of the neurons? If this method is one to confirm the veracity of the analysis, doesn't this imply that for 20% of the neurons the method does not work or that the unit was not the same one throughout the recording? What influenced the decision to keep such units (particularly the 8 or so units with low or no correlation)?
We thank the reviewers for pointing out this confusing presentation. Figure 1F shows the distribution of correlation coefficients between independent firing rate estimates (the measure of the robustness of the estimation, as explained above in our response to point 1.1) calculated for each cell using the modelfree approach. The value of 0.6 indicated in the text was given for descriptive purpose, as the point was to simply illustrate the fact that most units had a large correlation coefficient. This has been removed.
To clarify the significance of the values of this correlation coefficient (noted R), and as suggested by the reviewer, we calculated the distribution of R values obtained from shuffled spike trains (“null distribution”): for each cell, an average value of R was calculated over 10 iterations, each performed using an independent spike shuffling step. The resulting null distribution is shown in the revised Figure 1F in red. It is significantly different from the original distribution (p = 2.2 x 10^{16}, Wilcoxon test), with a mean value of 0.14 ± 0.10 (vs. 0.65 ± 0.23 for the original distribution). [NB: the original distribution (i.e. in gray) is slightly different in this revised panel as we had failed in the previous version to properly select R values corresponding to the optimal combination of inertial parameters (i.e. the one that gives the greatest predictability). This was corrected in the current version of the plot.]
As noted by the reviewers, a few neurons indeed display R values that lie within the limits of the null distribution. This is unlikely to result from unstable recording or from successive appearance of two units in the same recording: the calculation of R is in principle insensitive to these situations since R is computed from firing rate estimates computed using short nonoverlapping portions of the recording that are interleaved throughout the whole recording block (see Appendix section). Hence these two estimates should be equally sensitive to changes in the recording across time, and their correlation should not be affected by these changes (we verified explicitly that cells with small R values did not show large Gini index on the ISIs, which is a measure of the stability of the units’ firing rate along time). On the other hand, small R values are found for cells with little of no tuning to head movements or with limited sampling (animal less active). We kept these units for the subsequent analyses, which largely involved linear models, as the linear fit may be less sensitive to reduced sampling of head movements than the modelfree approach (i.e. fewer points may suffice to extract a linear trend), and may still be able to pick up correlations between the firing rate and head movements. The significativity of sensitivity of the cells provided by the linear models was independently assessed (see the bootstrap analysis for the significativity of linear models in the Appendix section); a few of the cells with low R value indeed exhibited a low, but significant sensitivity to head rotations.
The part of the main text describing this part of the results was rewritten to take into account the above comments, and the calculation of the null distribution is detailed in the revised version of the Appendix section).
Does the modelfree approach do significantly better than more traditional approaches? For smooth pursuit eye movements, the instantaneous firing rate of Purkinje cells does a great job of representing the kinematics as a linear sum of velocity, acceleration and the position of the eye. Does this more standard approach fail for the current data?
The use of a modelfree approach needed to be better justified. We thank the reviewer for point this out.
As explained in the response to point 1 of the essential revisions, the original motivation for using the modelfree approach is that it may pick up nonlinearities that could be missed by linear models. These nonlinearities could be for example a nonlinear transfer function of primary vestibular afferents in the amplitude range covered by naturalistic movements (as shown by Schneider et al., 2015), or nonlinearities associated to a change of reference frame as observed for acceleration in primate cerebellum (i.e. coordinate transformations, as explained by Green and Angelaki, 2007).
As suggested by the reviewers, we now provide a comparison of the performance of the modelfree approach with the one of “global” linear models. The terms of the models were different combinations of angular velocity, angular acceleration, gravitational acceleration (head tilt) and the jerk (time derivative) of gravitational acceleration. When comparing the coefficient of determination of these linear regressions with the R2 of the modelfree approach computed with angular velocity and gravitational acceleration, we found that the modelfree approach was always better at explaining firing rate fluctuations than these models. These results are detailed in a new paragraph in the main text of the revised manuscript (last section of the results) and shown in a new Figure 6 and a new Figure 6—figure supplement 1. The calculation of global linear models is detailed in the Appendix section.
Strikingly, the series of “local” linear models of rotational sensitivity (used to estimate the preferred rotation as a function of head tilt in “tiltdependent rotational sensitivity maps”) performed as well as the modelfree approach (Figure 6B, formerly in Figure 5—figure supplement 1C). This suggests that, for each head tilt, the cells are linearly tuned to the head’s angular velocity, but this linear dependency depends on head tilt in a nonlinear way. The similarity of the performance of the modelfree approach and of this model (that assumes a linear dependency to head rotation changing with headtilt) suggests that this later model is close to the best prediction possible in most cells in our conditions.
The value of looking at sensitivity vs. lag (Figure 3E) is not clear.
This plot is related to the exploration of the cells’ sensitivity to angular velocity using a linear model (as shown in Figure 3D and explained in the Appendix section). The coefficients of the linear fit provide a measure of a cell’s sensitivity (in Hz/(deg/s) for a given lag chosen by the user. By running the linear model for each cell using different lags, we identified “optimal lags” (i.e. that maximize the calculated sensitivity) on a cellbycell basis. By showing how the calculated sensitivity of units is strongly modulated by the choice of the lag used, Figure 3E illustrates the importance of identifying optimal lags and shows differences in the average sensitivity and optimal lag value for the 3 categories of units distinguished using the modelfree approach (gravitational, rotational and mixed, identified in Figure 3A). [NB: we show the average tuning curve in Figure 3E, but the optimal lag was computed for each cell].
To better explain the importance of this graph, we modified the corresponding text in the revised manuscript.
In Figure 4A and D, wouldn't the degree of nose up/down tilt effect the yaw sensitivity? The significance of Figure 4B and E should be clarified or those panels not included.
The overall aim of Figure 4 is to illustrate how the sensitivity of the firing rate to head angular velocity is affected by head tilt, taking into account the temporal dimension. We were therefore facing the challenge of representing firing rate as a (nonlinear) function of multidimensional data: angular velocity along the three axes, gravitational acceleration along the three axes and temporal lag. We focused on characteristic examples, in which the sensitivity of a cell to the yaw rotation axis is modulated by pitch tilt, reducing the analysis to 4 dimensions: yaw velocity, pitch tilt, firing rate and temporal lag. Despite this reduction in the dimensions of the analysis, the information can only be displayed using multiple graphs.
Figure 4A and D are inertiotemporal receptive fields (as in Figure 3B) that show how the firing rate varies in time around specific yaw velocity values, and that were calculated for positive and negative values of the gravitational acceleration along x (reflecting the pitchtilt angle); we therefore plotted the firing rate (color value) as a function of the time delay (abscissa) and yaw velocity at zero lag (ordinate) by grouping respectively all noseup (right graph in panels A and D) and nosedown (left graph in panels A and D) orientations. This plot shows qualitatively different inertiotemporal receptive fields for the headup and headdown cases (which we feel are useful to show), but indeed the pooling of all noseup and nosedown orientations prevents a proper assessment of the effect of different degrees of pitch tilt on the sensitivity of these example units. To examine how the sensitivity to yaw velocity changes with the degree of pitch tilt, we plotted the firing rate (color value) as a function of pitch tilt (abscissa) and yaw velocity (ordinate) at the optimal lag in panels C and F; these plots thus do not show the time dependency of panels A and D. Panel C indeed shows that the firing rate of this example cell is much increased (respectively decreased) by leftward (respectively rightward) rotations when the nose is up while much fewer modulation takes place when the nose is down. Panel F instead shows that the firing rate of another cell is increased by leftward rotations when the nose is down while it is decreased by similar rotations when the nose is up; the progressive change of sensitivity (i.e. firing rate modulation along the ordinate axis) as a function of headtilt (abscissae axis) is indeed visible on these panels.
Panels B and E are quantifications of panels A and D. They represent the slope of the firing rate vs. yaw velocity relation for discrete lag values (i.e. taken along vertical lines of the 2D histograms in panels A and D). They allow to visually confirm the change of sensitivity to yaw velocity for positive vs. negative pitch tilt angles. We edited panels B and E to include a measure a statistical significance using a bootstrap method (the gray areas represent the limits of the mean sensitivity slope ± 2SD calculated using shuffled spike trains).
We hope that these explanations will convince the reviewer that these plots are complementary and help provide the reader with a view of the multiple dependencies of the firing rate.
2) There is currently considerable debate as to whether one can use the firing of cerebellar neurons to identify them as a specific cell type. The authors should use additional criteria to confirm cell type identity, or back off on their claims about which cell types were recorded.
We thank the reviewer for raising this point. The activity was recorded presumably in the Purkinje cell layer which is noticeable for its dense activity. Indeed, the firing features of most units recorded fit with the classicallyobserved characteristics of Purkinje cells (Figure 1—figure supplement 3E; van Dijck et al., 2013). However, despite our best efforts, complex spikes proved very hard to distinguish from simple spikes except for a few exceptions (Figure 1—figure supplement 3H). Therefore, the compelling argument to prove that most of our units are Purkinje cells is lacking. The text has been amended accordingly by reminding the reader that units’ identification is tentative (the identity of the units is always preceded by the word “putative”). We still single out the few units which exhibit firing characteristics more consistent with those of Golgi and mossy fibers, since these units tend to show simpler receptive field (A^{G}selective and Ωselective) as displayed in Figure 3, and thus contrast with the main population of putative Purkinje cells recorded in our study.
3) The "passive whole body movements condition" is very poorly described. How well matched are the vestibular stimuli that the animals experience in the passive and active condition? If they are not similar in the spatial or temporal dynamics, it seems this might explain the poorer correlations in Figure 2H. Could the smaller amount of data for the passive condition be contributing to the discrepancy?
We provide a more detailed explanation of how passive wholebody movements were produced in the revised manuscript (paragraph “Recording protocol” of the Material and methods). Because these movements were produced by the experimenter, who held the platform to which animals were fixed, their detailed statistics necessarily (despite his best efforts) differed from the movements produced spontaneously by the animals in the freely moving condition.
To quantify these differences, we calculated the statistics of active vs. passive movements in recording blocks that corresponded to the 17 units that were recorded in the two conditions (Figure 2H). The results are shown in a new supplementary figure (Figure 2—figure supplement 1). Although passive movements were produced by the experimenter, the ranges of angular velocities were in fact quite similar (the only significant difference was found for the range of roll velocity values as shown in Figure 2—figure supplement 1A). Passive movements contained proportionally more low frequencies than active movements (Figure 2—figure supplement 1B), and the range of head orientations relative to gravity (head tilts) was slightly smaller in the passive condition (Figure 2—figure supplement 1C). Therefore, movements in the passive vs. active conditions were not perfectly matched, but did not differ drastically (perfectly matching passive movements with active movements would require a robotic device able for example to replay the range of active movements; this technically challenging approach was not possible in the present study). Overall, the passive conditions appeared to explore a subset of the inertial configurations explored in the active condition; this is the reason why we only attempted to predict (and compare) the firing rate of the passive condition using the passive and active sessions (Figure 2H): there would be too many inertial configurations in the active condition without a correspondence in the passive session to successfully predict the active firing rate by sampling the passive session. However, we can still not exclude that the difference between active and passive sessions revealed by our modelfree approach results from differences between the movements; this is now more clearly stated in the text.
4) Once subdivided, some of the groups have Ns of less than 5. Can the number of recordings increased to generate more confidence in these divisions and perhaps unravel additional ones which might have been missed?
The main result of this manuscript is the demonstration of the existence of a large group of cells showing a headtiltvarying sensitivity to head rotation. Few cells show discharge properties which are atypical for Purkinje cells and have been classified as putative Golgi cells and putative mossy fibers (see point 4). These cells could indeed be removed from the presentation, but we feel their inclusion indeed enriches the (small) set of cells showing simple sensitivities (to rotation or to gravity) shown in the Figure 3; moreover, it allows us to introduce several analysis procedures on simple cases (inertiotemporal receptive fields, preferred rotational sensitivity vector, spherical presentation of head tilt) which are then reused in cells exhibiting more complex receptive fields (Figure 4 and 5). Overall several relatively small groups, “A^{G}–selective” units, “Ω–selective” units, “internal” units, “external” units size (n = 12, 6, 5, 7) appear as extreme cases of somewhat continuous properties.
We apologize for not being able to add new data, but we have used in this manuscript all the recordings available and generating new data is a task which would require much more time than allotted for this revision.
5) The analysis appears to require knowledge of the distance d, between the IMU and the point around which yaw and pitch movements are made. It is not clear whether the absolute distance is important, and if so, how it is determined. If (as it appears) the nongravitational linear acceleration is not important for understanding the behavior of the neurons, this should be made clearer (subsection “Cerebellar units exhibit a mixed sensitivity to head angular velocity and gravitational acceleration”).
Because the sensor is attached on the animal’s skull, the angular velocity and the gravitational component of acceleration are experienced identically at the level of the sensor and of the animal’s vestibular organs and do not require to know the distance d between the sensor and the rotation axes. As shown by our modelfree approach (Figure 1E) and confirmed by our tiltdependent rotational sensitivity maps (see Figure 6 and our response to point 1.3 of the essential revisions), these are the two key parameters that modulate the units’ responses; the non gravitational acceleration does not seem to play an important role (Figure 1E); therefore changing the value of d would not change our understanding of the units’ behavior.
In Figure 1—figure supplement 2, we calculated an estimation of the magnitude of two types of nongravitational acceleration signals that are coupled to rotations (linear tangential and centripetal acceleration) and thus depend on the value of d. The purpose was only to estimate their order of magnitude and their contribution to nongravitational acceleration signals. Because d cannot be precisely determined, we assumed that its value is in the cm range, which is a realistic assumption given the placement of the sensor on the animal’s head. Our calculation shows that a large fraction of nongravitational acceleration most probably arises from linear tangential acceleration rather than centripetal or purely translational acceleration (Figure 1—figure supplement 2D). Therefore, the repertoire of behavior spontaneously expressed by the rats in our condition may not be most adequate for studying the dependency of firing rate on translational acceleration.
We edited the text in the revised Appendix section to provide clearer explanations.
https://doi.org/10.7554/eLife.26179.019Article and author information
Author details
Funding
Agence Nationale de la Recherche (ANR15 CE370007)
 Boris Gourévitch
Agence Nationale de la Recherche (ANR12BSV40027)
 Clément Léna
Agence Nationale de la Recherche (ANR10LABX54 MEMO LIFE)
 Clément Léna
Agence Nationale de la Recherche (ANR11IDEX000102 PSL Research University)
 Clément Léna
Agence Nationale de la Recherche (ANR15CE370001)
 Clément Léna
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We thank J Laurens, D Bennequin, A Afgoustidis and D Popa for helpful discussions on the project, and L RondiReig 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 (ANR12BSV40027 and ANR15CE370001) and to BG (ANR15CE370007), and has received support under the program ‘Investissements d’Avenir’ launched by the French Government and implemented by the ANR, with the references ANR10LA54 MEMO LIFE and ANR11IDEX0001–02 PSL${}^{\star}$ Research University.
Ethics
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).
Reviewing Editor
 Jennifer L Raymond, Stanford School of Medicine, United States
Publication history
 Received: February 21, 2017
 Accepted: June 9, 2017
 Accepted Manuscript published: June 13, 2017 (version 1)
 Version of Record published: June 28, 2017 (version 2)
Copyright
© 2017, Dugué et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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