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Three-dimensional head-direction coding in the bat brain



Navigation requires a sense of direction (‘compass’), which in mammals is thought to be provided by head-direction cells, neurons that discharge when the animal’s head points to a specific azimuth. However, it remains unclear whether a three-dimensional (3D) compass exists in the brain. Here we conducted neural recordings in bats, mammals well-adapted to 3D spatial behaviours, and found head-direction cells tuned to azimuth, pitch or roll, or to conjunctive combinations of 3D angles, in both crawling and flying bats. Head-direction cells were organized along a functional–anatomical gradient in the presubiculum, transitioning from 2D to 3D representations. In inverted bats, the azimuth-tuning of neurons shifted by 180°, suggesting that 3D head direction is represented in azimuth × pitch toroidal coordinates. Consistent with our toroidal model, pitch-cell tuning was unimodal, circular, and continuous within the available 360° of pitch. Taken together, these results demonstrate a 3D head-direction mechanism in mammals, which could support navigation in 3D space.

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Figure 1: Representation of head direction in the bat presubiculum: one-dimensional tuning to azimuth, pitch, and roll.
Figure 2: Single cells encode one, two, or three Euler angles of head direction, and are organized according to a functional–anatomical gradient.
Figure 3: Azimuth tuning in inverted bats is consistent with a toroidal model of head-direction coding.
Figure 4: Pitch cells are tuned unimodally and continuously to a cyclical range of 360°, as predicted by the toroidal model.
Figure 5: Head-direction cells in freely flying bats.


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We thank A. Treves, J.-M. Fellous, M. Okun, A. Wallach, M. Geva-Sagiv and M. M. Yartsev for comments on the manuscript; B. Pasmantirer and G. Ankaoua for mechanical designs; S. Kaufman for bat training, assistance in neural recordings, and illustrations; T. Eliav and T. Tamir for help in experiments; A. Tuval and M. Weinberg for veterinary oversight; G. Brodsky for graphics; R. Eilam and C. Ra’anan for histology; and M. P. Witter for advice on reconstruction of tetrode-track locations and anatomical delineations. This study was supported by research grants to N.U. from the European Research Council (ERC–NEUROBAT), the Human Frontiers Science Program (HFSP RGP0062/2009-C), the Israel Science Foundation (ISF 1017/08 and ISF 1319/13), and the Minerva Foundation; by a Clore predoctoral excellence fellowship to A.F.; and by an MIT-Israel (MISTI) student exchange internship to J.N.F. D.D. is the incumbent of the David and Inez Myers Career Advancement Chair in Life Sciences.

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Authors and Affiliations



A.F., D.D. and N.U. designed the experiments and the analyses. A.F. performed the experiments, with contributions by D.D. and L.L. to some of the surgeries and tetrode recordings. A.F. and A.R. developed the toroidal model. A.F. and J.N.F. developed algorithms. A.F. analysed the data, and discussed with D.D, A.R., L.L. and N.U. the results and interpretations. A.F. and N.U. wrote the manuscript with input from D.D., A.R., J.N.F. and L.L.

Corresponding author

Correspondence to Nachum Ulanovsky.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Experimental methods for behavioural setup number 1, recording locations, and example of spike-sorting.

a, Schematic illustration of the behavioural arena and camera position used in the crawling experiments (setup number 1). b, Illustrations of the device used for computing the head-direction Euler angles, which was based on a 3D non-coplanar arrangement of four LEDs. This four-LED device was mounted on the recording headstage, and allowed measuring the Euler angles using one overhead camera (see Methods and Extended Data Figs 11 and 12 for details of the algorithm). c, Schematic camera views of the four-LED headstage at different pitch angles. d, Nissl-stained sagittal brain section through the hippocampal formation of an Egyptian fruit bat, including the dorsal presubiculum (same brain section as in Fig. 1c). Electrolytic lesions were done at the end of the experiment. Red lines denote the borders of the presubiculum. Scale bar, 1 mm. eh, Sagittal Nissl-stained brain sections showing representative recording sites in the presubiculum of 4 bats. Tetrode tracks are marked by red arrowheads (the arrowheads point to lesion sites in bats number 0016 and 6847; while in bats number 7812 and 1594, the arrowheads point to the end and to the middle of the tetrode-track, respectively). Sections in d and e are from the same animal (bat 0016); the section in e shows the track of a different tetrode than the one seen in d. Scale bar in eh, 0.5 mm. i, Waveforms of four different neurons (different colours, rows) recorded simultaneously on the 4 channels of a single tetrode (columns). Scale bar, 100 µV; waveform duration, 1 ms. j, Energy displays (cluster plots) for the data in i, showing the energy of spikes (dots) on two of the tetrode’s four channels (left and middle panels), or on three of the tetrode’s four channels (right panel). Colours match the waveforms in i; grey dots, small spikes or noise that crossed the voltage threshold but were not classified as single units.

Extended Data Figure 2 Head-direction cells maintain their preferred azimuthal direction when the bat is being moved passively in the upright position.

a, Azimuth tuning of four example cells in three upright sessions: active session number 1 (top row), passive session (middle), and active session number 2 (bottom). Note that the preferred direction in the passive session stayed similar to the preferred direction of the active sessions, despite the reduction in cells’ firing rate during the passive session. b, Population average tuning curve for active (blue) versus passive (black) sessions, for all azimuth-significant cells (n = 63 cells); shading, mean ± s.e.m. Prior to averaging, the tuning-curve of each neuron was centred around its peak, and the firing rate was normalized to the mean firing rate of the neuron across all sessions (active and passive pooled together). Note that, on average, presubiculum neurons exhibited a lower firing rate during the passive session as compared to the active sessions. c, Peak firing rate of azimuth-tuned neurons (n = 63 cells, that is, 126 sessions) in the active versus passive sessions, showing a significantly lower firing rate during the passive condition (sign test, P < 0.001; number of sessions above and below the diagonal are indicated on the graph). The peak firing rates of these neurons ranged up to 12 Hz, and were generally lower than peak firing rates in rats, similar to what we found previously for hippocampal place-cells and entorhinal grid-cells in crawling bats9,16,54, and consistent with the slow crawling velocity of bats, which reduces the firing rate36,40. Importantly, even cells with low firing rates exhibited stable directional tuning across sessions (see panel a). df, Changes in tuning properties during inversion are not caused by the passivity of the movement in the inverted session. d, Distribution of angular differences in the preferred azimuthal head direction between the two active behavioural sessions (S1 and S2). The histogram was plotted for cells with significant tuning in the upright active sessions that also had significant tuning in the inverted session. Peak around 0° indicates stability of the azimuthal tuning across the two active behavioural sessions. e, Angular difference between the upright sessions and the inverted session. Peaks at around ± 180° indicate that inversion of the bat upside-down resulted in a 180° shift in the preferred azimuthal direction for the azimuth cells. f, Angular difference between the active upright sessions and the passive (upright) session. Peak around 0° indicates that passive movement in itself does not induce a change in the preferred direction of the neurons (see also examples in a).

Extended Data Figure 3 Bat presubiculum neurons are modulated by angular velocity but exhibit weak spatial tuning.

a, Distribution of spatial information (top) and sparsity (bottom) for all the 122 presubiculum neurons recorded in setup number 1. Note that, with a few exceptions, neurons in the presubiculum conveyed little information about the location of the bat in the environment and displayed weak spatial selectivity, as indicated by their low spatial-information index and high sparsity index. be, Examples of four neurons, depicting the head-direction fields (top) and place-fields (bottom); spatial information (SI) and sparsity for each neuron are indicated next to the positional plot; the cells were ordered according to spatial information values. Most neurons (bd) carried little spatial information and showed irregular spatial firing patterns, despite having significant directional responses. Only 3% of the neurons had spatial information > 0.4 bits per spike (for example, the cell in e), in addition to their directional tuning. This small minority of cells could possibly be border/boundary cells18,19,20, or place-by-direction cells37,58, or perhaps conjunctive grid × head-direction cells15,24 (we could not detect grid structure because of the small size of the arena, 50 × 50 cm). fn, Head-direction cell activity is modulated by angular velocity. Shown is the tuning to head direction (left) and angular velocity (right) for 9 examples from 8 neurons (panels f and i are from the same neuron). Top row (fh): azimuth and azimuth-velocity; middle row (ik): pitch and pitch-velocity; bottom row (ln): roll and roll-velocity. f, A neuron tuned to azimuth (left panels) that increases its firing rate in response to faster turning in the clockwise direction (right panels). g, A neuron that did not exhibit any tuning to head direction in azimuth (left panels), but nevertheless increased its firing rate with increasing head velocity in the clockwise direction (right panels); this neuron might be classified as a pure ‘angular velocity neuron’. h, A neuron tuned to azimuth that exhibited almost no modulation by azimuth-velocity; this neuron might be classified as a pure ‘head-direction neuron’. i, A neuron tuned to pitch, which increased its firing rate in response to fast decrease in pitch. Note that this is in fact a conjunctive azimuth × pitch neuron (same neuron as in f). jk, Pitch-tuned neurons that also fired preferentially to slow angular head-velocity in pitch. ln, Roll-neurons that were also modulated by anti-clockwise angular-velocity in roll (l), clockwise angular-velocity in roll (m), or slow-roll velocity (n). These diverse types of tuning to angular-velocity resemble the functional diversity of azimuth–velocity tuning reported for presubiculum neurons in the rat4,59, and also suggest that bat presubiculum neurons can integrate head angular-movements in rotation planes other than Earth’s horizontal plane.

Extended Data Figure 4 Functional–anatomical gradient of head-direction cells along the transverse axis of dorsal presubiculum.

a, Sagittal section through the presubiculum; red lines, presubiculum borders. Yellow line shows the axis used to determine the anterior–posterior distance of the tetrode-track (arrowhead) from the subiculum border. Scale bar, 0.5 mm. be, Percentage of head-direction cells in each tetrode-track, plotted against the anterior-posterior (‘A–P’) distance of that tetrode-track from the subiculum. Each dot represents one tetrode-track; shown are only tetrode-tracks with > 5 significant head-direction cells recorded per track. Percentages plotted separately for neurons with pure tuning to azimuth (b), pitch (c), or roll (d), and for neurons with conjunctive tuning to any angular combination (e); regression lines are also indicated. f, 2D unfolded map of dorsal presubiculum, showing the reconstructed recording location for all the 266 neurons recorded from the 4 bats in setup number 1. Colours indicate pure tuning to azimuth (blue), pitch (red), or roll (green), or neurons with conjunctive tuning to any angular combination (yellow); cells that did not pass the criterion for stability or directionality are also shown (empty circles). Note that the antero-posterior (A–P) distance is measured along the curved structure of the presubiculum (as shown by the yellow line in Fig. 2h), while the medio–lateral (M–L) distance is measured straight from midline. The multilinear regression line (black) was computed from the 2D maps in g, h. Dots were slightly jittered, for display purposes only, to prevent overlap. g, h, Two-dimensional unfolded maps of dorsal presubiculum from the 4 bats tested in setup number 1, showing the percentage of pure azimuth (g) and conjunctive cells (h), based on the recording locations of individual cells (M–L, medio–lateral distance from midline; Methods). These maps reveal a functional gradient along a diagonal axis, with pure azimuth cells dominating the anterolateral part and conjunctive cells the posteromedial part of dorsal presubiculum. From these maps, we calculated the multilinear regression line (black line in f), based on A–P and M–L positions as predictors. This line represents the axis of the maximal diagonal gradient of head-direction cells, and is computed as the slope of the multilinear regression coefficients, averaged for pure azimuth and for conjunctive cells, which together represent the majority of the head-direction cells. i, Percentages of all 4 neuronal types from be, binned here along the diagonal axis (regression line) of the functional gradient, which corresponds to the transverse axis of the presubiculum (Methods; graph based on all the significant head-direction cells from the 4 bats in setup number 1: n = 78 neurons). jo, Same as in gi, computed now for two individual bats in which at least 3 tetrodes were identified within the dorsal presubiculum, which is the minimum number of tetrode-tracks that allowed us to create a functional–anatomical map for an individual animal (jl, bat number 0016, n = 36 significant head-direction cells; mo, bat number 6847, n = 21 significant head-direction cells).

Extended Data Figure 5 Azimuth tuning and stability analysis for neurons in the inverted session.

a, Distribution of tuning significance of azimuth-encoding cells (n = 63), in the inverted session (setup number 1). Red line, 95th percentile value from shuffled data. b, c, Stability analysis of azimuth cells in the inverted session. b, Example of azimuth tuning for a single cell. Left, tuning curve computed for the entire inverted session. Middle column, comparing the tuning curve computed for the first half versus the second half of the inverted session. Right column, comparing the tuning curve computed for odd versus even minutes of the inverted session. c, Distribution of correlation coefficients for azimuth cells with significant tuning in the inverted session (n = 24), parsed based on two partitioning conditions. Left, first half versus second half of the session; right, odd versus even minutes. d, Left, distributions of differences in preferred direction between sessions, for cells that have not reached tuning-significance criterion (that is, neurons to the left of the red line in a), but were nevertheless somewhat tuned in azimuth in the inverted session (‘weakly tuned’, n = 20; see Methods). Distribution of angular differences between each of the upright sessions and the inverted session (S1–inverted, S2–inverted) of these ‘weakly tuned’ neurons shows peaks near ± 180° for the inverted session, indicating 180°-shift of the preferred direction, similar to that observed in azimuth cells that were significantly tuned under inversion (Fig. 3b, bottom). Right, azimuth tuning of an example cell that did not pass the shuffling criterion for tuning under inversion, but was close to significance (P = 0.06); this neuron was considered ‘weakly tuned’ by our criteria, but nevertheless this neuron showed a shift of 180° in the inverted session (middle plot, blue) relative to the upright sessions (top and bottom). Magenta curve in the middle row: the tuning-curve of the inverted session, shifted by 180°. e, Left, same as in d, for the remainder of the cells that have not reached significance criterion and were indeed untuned in the inverted session (that is, these cells were to the left of the red line in panel a, and were also very clearly not directionally tuned: ‘untuned cells’, n = 11; Methods). Right, an example cell with very little tuning under inversion (test for tuning significance for this neuron: P = 0.29).

Extended Data Figure 6 Azimuth cells exhibit a 180°-shift in their preferred direction when bats position themselves upside-down on their own volition.

a, b, Schematic illustration of the vertical-ring apparatus (setup number 2), designed to compare the tuning properties of neurons during upright crawling on the arena floor (pose 1) with the activity of the same neurons during epochs in which the bat had positioned itself upside-down on its own volition along the inner side of a vertical ring (pose 2). The analysis was restricted to azimuth cells whose preferred direction on the arena floor was aligned to the cardinal axis of the ring (west–east), so that movement of the inverted bat would be either in the upright-preferred-direction of the neuron (case a), or in the exact opposite (180°) direction (case b). ce, Examples of azimuth cells, showing reversed firing in animals that positioned themselves upside down on their own volition. Left, azimuth tuning curves of individuals neurons during upright locomotion on the arena floor (pose 1). Right, mean firing rates of the same neurons during active inverted motion (pose 2), shown separately for west and east directions. c, d, Two example cells with west direction tuning on the arena floor (left, blue curves), showing a reversal of their firing direction in the inverted position (right, increase in firing to the east). e, Example cell with east direction tuning on the arena floor, showing a reversal of its firing direction in the inverted position (increase in firing to the west). f, Example cell, whose preferred direction on the floor was orthogonal to the cardinal axis of the ring (preferred direction was to the north). Note that this neuron showed no preference to either west or east directions in the inverted position (right). g, Pairwise comparison of mean firing rates of azimuth cells in the inverted position, for epochs in which the inverted bat moved in the upright-preferred-direction of the neuron (case a, as shown in panel a), versus epochs in which it moved in the opposite (180°) direction (case b, as shown in panel b). Notably, 92% of the cells (n = 11/12) increased their firing rate when the head-azimuth in the inverted position was 180° opposite to the upright-preferred-direction of the cells, as predicted by the torus model. **P < 0.01. h, Comparison of mean firing rates in the upright position versus the inverted position for the azimuth cells analysed in g, showing no significant change in the mean firing rate under inversion. Error bars, mean ± s.e.m.

Extended Data Figure 7 Spherical versus toroidal coordinate systems.

a, In a spherical coordinate system, which describes the direction of a vector in 3D space, 3D head direction is defined by its azimuth and pitch angles. Each line of longitude on a sphere corresponds to a specific azimuth. The position of the head along this line of longitude is given by its pitch angle (ranging from –90° to +90°). For example, the east longitude (red-coloured half-ring) corresponds to different pitch angles along the 0° azimuth. The west longitude (purple-coloured half-ring) corresponds to different pitch angles along the 180° azimuth. b, The azimuth and pitch angles of the head in spherical coordinates define its absolute direction in 3D, regardless of whether the bat is upright or inverted. For example, the spherical coordinates of the bat furthest on the right in this plot (0° azimuth, 0° pitch), correspond to an upright bat with its head parallel to the horizontal plane facing the east direction but also to an inverted bat facing east (also parallel to the ground). Thus, the spherical representation is ambiguous with respect to the upright versus the inverted state. c, In the toroidal coordinate system, the two angles that determine the direction of the animal’s head in 3D (azimuth and pitch) represent two independent cyclic degrees of freedom, both having a range of 360° (blue circle, azimuth; red and purple circles, pitch). Importantly, the toroidal azimuth does not represent the direction of the animal’s nose (rostro–caudal axis) relative to a distal point in space, but rather the direction of the inter-aural axis. The toroidal pitch, in contrast, is anchored to the direction of the rostro–caudal axis. In this representation, any rotation in pitch does not change the azimuth, as defined in toroidal coordinates (Methods). Specifically, if the bat pitches its head to angles greater than +90° pitch (resulting in flipping from upright to inverted position), the toroidal azimuth still remains the same. Importantly, the upright and inverted positions are represented continuously, but can be distinguished according to the pitch angle: The outer surface of the torus (white) corresponds to all upright positions (–90° < pitch < +90°), whereas the inner part of the torus (grey) corresponds to all inverted positions (+90° < pitch < +180° or −180° < pitch < −90°). d, Detailed depiction of two different azimuthal directions on the torus (shown as red and purple rings in c). Right panel (0° azimuth): for an upright bat facing east (0° azimuth, 0° pitch), any change of the pitch angle (red ring) will not change the toroidal azimuth, which will remain 0°. Left panel (180° azimuth): analogously, for an upright bat facing west (180° azimuth, 0° pitch), any change of the pitch angle (purple ring) will not change the toroidal azimuth, which will stay 180°. Note, that this set of positions, corresponding to 180° azimuth (purple ring), is mapped onto the opposite side of the torus, relative to the set of positions corresponding to 0° azimuth (red ring). Therefore, unlike the ambiguous representation of head direction in spherical coordinates (a, b), there is no ambiguity in the toroidal representation: each point on the torus describes a unique orientation of the bat, and defines not only its head direction in 3D, but also whether the bat is in the upright or in the inverted position. ej, Example of construction of the toroidal representation and angular transformations of the 2D rate-maps for the inverted session, for one pure azimuth neuron. e, f, 2D directional rate maps for the upright session (e) and inverted session (f) computed initially in spherical coordinates. gi, Angular transformations of the 2D rate map of the same inverted session (shown in f), done in order to represent it in toroidal coordinates: g, 180° shift in azimuth; h, flipping the pitch axis (to represent it continuously and allocentrically); i, both transformations together (that is, 180° shift in azimuth and flipping the pitch axis). For each of the maps in fi, the value of the 2D Pearson correlation coefficient, r, of this map with the upright map in e, is also indicated. j, The toroidal representation is constructed by concatenating the 2D directional map of the upright session (shown in e) with the map of the inverted session after both transformations (shown in i). k, Difference in 2D correlations (Δ corr) of the firing maps in the upright and inverted sessions, before (f) versus after the various angular transformations (examples shown in gi). Bars, various angular transformations (see panels gi), which included shifting the azimuth of the inverted session by 180°, or flipping the pitch in order to represent it allocentrically, or both. Error bars, mean ± s.e.m.; *P < 0.05. Comparisons are shown for pure azimuth cells (k, upper panel, n = 84 (42 cells × 2 sessions)), pure pitch cells (middle, n = 14), and conjunctive azimuth × pitch cells (lower panel, n = 28). Consistent with the toroidal model, a 180° azimuth-shift of the inverted map increased substantially the correlation with the upright-map for pure azimuth cells (k, upper panel-left bar; t-test, P < 0.05), but not for pure pitch cells (middle panel left bar: t-test, NS). Conversely, flipping the pitch increased substantially the correlation for pure pitch cells (middle panel middle bar; t-test, P < 0.05), but not for pure azimuth cells (upper panel middle bar: t-test, NS). For the azimuth × pitch conjunctive cells, both shifting the azimuth by 180° and flipping the pitch resulted in significant increase in correlation values (lower panel; t-test: P < 0.05, for all bars), as expected from cells encoding both azimuth and pitch. l,Similar analysis for azimuth × roll 2D maps did not show a significant effect of roll on the correlations between the upright and the inverted sessions, suggesting that the roll dimension is not crucial for 3D head-direction representation in bats. m, Population averages of 2D correlations of the firing maps in the upright versus inverted session, for all the head-direction cells, including azimuth, pitch and roll cells (n = 156 (78 cells × 2 sessions)), when represented in either spherical coordinates (left), or azimuth × pitch toroidal coordinates (middle), or azimuth × roll toroidal coordinates (right). These results suggest that an azimuth × pitch torus, but not the alternative models such as the sphere or an azimuth × roll torus, captures well the activity of head-direction cells in the bat presubiculum. Error bars, mean ± s.e.m.; **P < 0.01; ***P < 0.001.

Extended Data Figure 8 Toroidal representation of head-direction cells.

a, 2D directional rate-maps of the upright (top) and inverted session (bottom), for a pure azimuth cell (left), pure pitch cell (middle), and a conjunctive azimuth × pitch cell (right) ; same neurons as in Fig. 3f of the main text. b, Same three cells as in a, shown in the toroidal representation from different viewing angles. Each cell is shown from two horizontal viewing angles (rotated 90° horizontally (azimuthally) with respect to each other) and from three different pitch viewing angles. The tori were constructed by plotting on the outside half of the torus the 2D directional rate map for the upright session, and on the inside half of the torus plotting the rate map of the inverted session in toroidal coordinates; see Extended Data Fig. 7e–i and Methods for the details of the angular transformations.

Extended Data Figure 9 Torus topology predicts that tuning to pitch is allocentric and distinct between upright and inverted positions.

a, According to the toroidal representation, pitch is computed in a world reference frame (allocentric) and not in body reference frame (egocentric). In the upright position, the two reference frames are indistinguishable. For example, when a bat pitches its head towards the moon (positive allocentric pitch) it also raises its head away from its chest (positive egocentric pitch). However, in the inverted position, allocentric head pitch is flipped with respect to the egocentric one. When the bat is upside-down and looks towards the moon (positive allocentric pitch), it now brings the head towards the chest (negative egocentric pitch). To test which of these reference frames is most consistent with our neural data, we computed the correlation between the pitch 1D tuning curves of the upright sessions versus the inverted session, in the two reference frames (for the experiments in setup number 1). Correlation of pitch tuning-curves between the upright and inverted positions was higher when the inverted session was plotted in allocentric coordinates (n = 42 (21 cells × 2) upright sessions; we included in the analysis only cells that were significantly tuned to pitch). *P < 0.05. b, A toroidal representation implies that pitch has a continuous representation, where every pitch angle corresponds to a unique orientation along a 360° ring of possible pitch angles (see Fig. 3d, red ring). This implies that if a neuron is active mostly at extreme pitch angles during the upright session (‘extreme-pitch’ neuron), it is likely to be active also at the contiguous pitch in the inverted session. Shown here are examples of two pitch cells with tuning to non-zero pitch angles in the upright session (cell 1, positive pitch; cell 2, negative pitch). 1D tuning to pitch is plotted for the average neuronal activity of the cell during the two upright sessions (‘upright’, left), and for the inverted session (‘inverted’, right). Note that the two cells exhibit contiguous firing in the inverted and upright sessions. ce, The toroidal model generates a prediction, that such a continuity between the upright and the inverted session (as shown in b), should occur for cells tuned to ‘extreme pitch’ (see example in d), but not for cells tuned to horizontal pitch (example in c). More specifically, in the toroidal model, neurons with preferred pitch at around 0°, an angle at which the head of an upright bat is parallel to the ground (‘horizontal pitch’ cells), are not expected to fire when the bat is inverted with its head being parallel to the ground, because these two situations are topologically distinct in the toroidal but not in the spherical representation (Fig. 3d vs 3c and Extended Data Fig. 7d vs 7b). In contrast, neurons tuned to an extreme pitch angle in the upright position, are likely to fire to some extent also in the contiguous part of the inverted session, as the ‘patch of activity’ on the ‘external side’ of the torus (which corresponds to upright position) is likely to extend also onto the ‘inner side’ of the torus (corresponding to inverted position). Therefore, according to the toroidal (but not the spherical) model, the correlations between the upright and inverted sessions for cells tuned to ‘extreme pitch’ are expected to be higher than for cells tuned to ‘horizontal pitch’. This prediction was tested here, and was indeed confirmed (see below). c, Upper panel, schematic representation of an azimuth × pitch cell, exhibiting pitch tuning to 0° (a ‘horizontal pitch’ neuron). Lower panel, example of an actual neuron exhibiting pitch tuning to 0°, similar to the schematic. Note that in both the schematic and in the real neuron, no directional field is present in the inverted session (that is, no firing on the inner (grey) part of the toroidal manifold), as predicted above. d, Upper panel, schematic representation of an azimuth × pitch cell, tuned to positive pitch (an extreme pitch neuron). Lower panel, example of an actual neuron exhibiting tuning to positive pitch, similar to the schematic. In this case, the activity of the neuron in the upright session is in fact correlated with its activity in the inverted session, as predicted above. e, Differences in 2D correlations between the upright and inverted session, for all the pitch-tuned neurons recorded in setup number 1 (both pure and conjunctive), computed similarly to the correlation analysis in Extended Data Fig. 7k (see Methods). This correlation was significantly larger for pitch cells that were tuned to extreme pitch (pitch ≤ –35° or pitch ≥ + 35°; ‘extreme pitch’, n = 22 cells × sessions), compared to pitch cells tuned approximately to zero pitch (between −35° and +35°; ‘horizontal pitch’, n = 20 cells × sessions). Error bars, mean ± s.e.m.; ***P < 0.001.

Extended Data Figure 10 Pitch cells are narrowly tuned and span a range of 360°, similar to azimuth cells.

a, b, Example azimuth cells recorded on the arena floor in setup number 1 (panel a) and pitch cells recorded on the vertical-ring in setup number 2 (panel b), showing that preferred directions of both neuronal types span the entire range of 360° (from 0° to 360° of azimuth for azimuth cells, and from –180° to +180° of pitch angles for pitch cells). Cells were sorted according to their preferred azimuth (top) or preferred pitch (bottom), highlighting the similarity of the tuning properties of azimuth cells and pitch cells. c, Pitch cells that were recorded on the vertical ring in two separate sessions exhibited a stable unimodal tuning. d, Tuning width to azimuth and to pitch. Left, population average tuning width. Right, average tuning widths for each individual animal (one bar per animal: azimuth, first 4 bars, coloured blue; pitch, last 2 bars, coloured red). The first 4 bars represent the tuning widths of azimuth cells recorded from the 4 bats in the upright-crawling experiment (setup number 1), and the last 2 bars represent the tuning widths of pitch cells recorded from the 2 bats in the vertical-ring experiment (setup number 2). Error bars, mean ± s.e.m.

Extended Data Figure 11 Device used for measuring the 3 Euler angles of the bat’s head, based on a 3D non-coplanar arrangement of four LEDs.

a, b, Schematic illustrations of the device used for computing the head-direction angles using the top-view camera. This device was mounted on the recording headstage, and allowed measuring the Euler angles using one overhead camera (Methods). Shown are several camera-views of a schematic illustration of the four-LED headstage, rotated in azimuth (a) or rotated at different combinations of pitch and roll (b). Central panels in both a and b: zero pitch and no roll (‘flat head’ position). The azimuth of the device was defined as the absolute direction of the red LED along the green–red direction. Pitch and roll were kept constant in all the plots in a; azimuth was kept constant in all the plots in b.

Extended Data Figure 12 Definitions of intermediate angles used during the computation of the final Euler angles of the head.

In order to compute the final Euler angles (in arena coordinates), we first computed intermediate Euler angles with respect to the plane of the camera, and then transformed them into arena coordinates based on the xy position of the animal inside the arena (see Methods for the detailed computation and full definitions of these angles). a, b, Illustration of the 3D tetrahedral arrangement of 4 LEDs, including the relevant geometry and angles. c, Illustration of measurement as seen by the camera. The distances relevant for the measurement of angles have been labelled a1, a2, b1, b2. d, Coordinate frames of the arena (‘A’) and camera-view (‘C’). Shown are the arena frame in blue and an example of the camera frame in red, for a particular location of the bat. Note that the alignment of the camera frame with respect to the arena frame changes as a function of the position of the LED device (position of the bat’s head) within the arena; the algorithm described in the Methods section disambiguates these changes.

Supplementary information

Freely-flying bats maneuver strongly in azimuth and pitch, but almost never roll

This video shows the natural flight maneuvers of 4 bats, captured using two high-speed cameras; slowed down 10×, for clarity. As seen in the video, the bats maneuvered extensively in azimuth and pitch during takeoff and landing, as well as during intermediate flight epochs – including 360° maneuvers in azimuth and 360° full circle in pitch (cumulatively over landing and takeoff). In contrast, the roll angle of the head hardly changed: Even during sharp maneuvers in azimuth and in pitch, the roll remained unaltered, close to 0°. (MP4 13895 kb)

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Finkelstein, A., Derdikman, D., Rubin, A. et al. Three-dimensional head-direction coding in the bat brain. Nature 517, 159–164 (2015).

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