Optimal dynamic coding by mixed-dimensionality neurons in the head-direction system of bats

Abstract

Ethologically relevant stimuli are often multidimensional. In many brain systems, neurons with "pure" tuning to one stimulus dimension are found along with "conjunctive" neurons that encode several dimensions, forming an apparently redundant representation. Here we show using theoretical analysis that a mixed-dimensionality code can efficiently represent a stimulus in different behavioral regimes: encoding by conjunctive cells is more robust when the stimulus changes quickly, whereas on long timescales pure cells represent the stimulus more efficiently with fewer neurons. We tested our predictions experimentally in the bat head-direction system and found that many head-direction cells switched their tuning dynamically from pure to conjunctive representation as a function of angular velocity-confirming our theoretical prediction. More broadly, our results suggest that optimal dimensionality depends on population size and on the time available for decoding-which might explain why mixed-dimensionality representations are common in sensory, motor, and higher cognitive systems across species.

Fig. 1

Head-direction coding by mixed-dimensionality neurons in the bat brain. a Schematic illustrating that a multidimensional stimulus (e.g., a 2D stimulus), can be represented with sub-populations of pure cells that are tuned to only one dimension of the stimulus, or by a population of conjunctive cells that encode the different dimensions of the stimulus jointly. Because pure cells have larger receptive fields they can tile the stimulus space more densely, compared to a population comprising the same number of conjunctive cells. Therefore, when naively considering a two-dimensional variable such as a position of a rook on a chessboard, one would expect to need only 2 × N pure cells (N cells encoding the X dimension and N cells encoding the Y dimension) in order to reach the same representational accuracy as N × N conjunctive cells (with X × Y tuning). However, conjunctive cells provide information about both dimensions of the stimulus at the same time, whereas decoding the activity of pure cells requires co-firing of both pure X and pure Y cells, and thus can be compromised at short decoding times. b, c Examples of 1D tuning curves of head-direction cells that we recorded in the bat dorsal presubiculum: a pure azimuth cell (b) and a pure pitch cell (c), overlaid with von-Mises fits (black). Top insets in b and c illustrates schematically the directional tuning of these pure cells in the 2D space of solid angles (360° azimuth × 360° pitch). d An illustration of a conjunctive cell with 2D tuning to a specific combination of azimuth × pitch angles. The existence of both pure and conjunctive neurons in the same brain region suggests a mixed-dimensionality coding

Fig. 2

Decoding accuracy of a 2D stimulus for finite N and T: fast decoding from many neurons versus slow decoding from fewer neurons. a, b Decoding error for a 2D stimulus (head-direction in azimuth and pitch) estimated using a maximum likelihood (ML) decoder, as a function of decoding time (T) and the number of neurons that participate in the task (N). a ML decoder applied to the responses of a population of pure cells; b ML decoder applied to a population of conjunctive cells. Here the tuning curves are normalized such that the total number of spikes emitted by each population is the same (on average). The error (in degrees) is color-coded, and we also plot several contour lines (solid lines) and lines for which the total average number of spikes is fixed (dashed green lines; note the log-log scale). Deviations of the solid lines from the dashed lines represent regions in phase-space where the magnitude of the error is determined by the number of spikes as a function of N and T independently (and not only as a function of N × T). These deviations indicate that although the overall number of spikes is the same, fast decoding from many neurons differs from slow decoding from fewer neurons—and that pure and conjunctive cells perform differently under these conditions (compare a versus b)

Fig. 3

Relative coding accuracy of pure and conjunctive cells representing a multidimensional variable. a Relative decoding performance of pure versus conjunctive cells representing a 2D variable. We identify three regimes: regime #1, where ${ϵ}_{\mathrm{pure}}\mathrm{∕}{ϵ}_{\mathrm{conj}}=\sqrt{2}$, as predicted by the Fisher information (FI) calculation (white area); regime #2, where the relative performance of the pure cell population improves as compared to regime #1; and regime #3 (blue), where the conjunctive cells outperform the pure cells even beyond the FI calculation $\left({ϵ}_{\mathrm{pure}}\mathrm{∕}{ϵ}_{\mathrm{conj}}>\sqrt{2}\right)$. When the number of neurons is moderate (regime #2) we find a specific subregion below a critical value of N (dashed line), for which the pure cells start to outperform the conjunctive cells also in absolute terms $\left({ϵ}_{\mathrm{pure}}\mathrm{∕}{ϵ}_{\mathrm{conj}}<1\right)$. b Same as (a) computed for neurons with a wider tuning curve (90° width at half-height). The color bar (bottom) indicates the error ratio between pure and conjunctive cells for a 2D stimulus. c The critical value of N, denoted N_cr, is defined to be the population size for which the pure and conjunctive populations have the same average errors at a long decoding time, T = 10s (i.e., for N < N_cr. pure cells outperform the conjunctive cells in absolute value, corresponding to the dashed lines in a, b). Shown is N_cr (y-axis) for different stimulus dimensions D, as function of the tuning width. N_cr becomes larger for narrower tuning and for higher dimensionality of the stimulus. Together, this supports the notion that in regime #2 the performance of the conjunctive population is degraded due to loss of coverage of the stimulus space by the available N neurons—which can happen either due to narrower tuning or due to higher dimensionality of the stimulus space. Green square and circle corresponds to the tuning widths (at 2D) for which we plot the error ratio in N − T space (a, b). Brown symbols correspond to the plots in 5D (Supplementary Fig. 2)

Fig. 4

Conjunctive coding is more robust at short decoding times. a Error ratio in decoding 1D versus 2D stimulus for pure and conjunctive cells. At short decoding times T, the accuracy of pure cells in decoding of a 2D stimulus drops as compared to a 1D stimulus (red line: 2D/1D error ratio increases at short T)—whereas for conjunctive cells this ratio is independent of T (blue line: relatively flat). b Distribution of the decoding error in azimuth and pitch for pure cells (top) and conjunctive cells (bottom), for different decoding times T (columns). At short T (left column), pure cells have larger spread in the 1D error magnitude for azimuth or pitch, which may compromise the estimation of the combined 2D stimulus. As T increases, the error variance of pure cells becomes symmetric (circular) in both dimensions, resulting in a more accurate combined estimate. For the conjunctive cells, the variance is symmetric in both dimensions for all decoding times (compare top row to bottom row). c Ratio of the decoding error of a 2D stimulus (azimuth × pitch) by pure versus conjunctive cells, as a function of T. The ratio is plotted for three different population sizes (N). For short T, pure cells fail to integrate the two dimensions of the stimulus (failure of coincidence-detection)—and therefore the relative decoding performance of conjunctive cells improves as T gets shorter. At longer T, the relative decoding performance by pure and conjunctive cells converges to a fixed ratio, and as N increases this ratio asymptotically approaches the ratio of $\sqrt{2}$ predicted by the Cramér–Rao bound (dashed black line). Inset: the ratio of the pure and conjunctive decoding errors ${ϵ}_{\mathrm{pure}}\mathrm{∕}{ϵ}_{\mathrm{conj}}$ can be estimated from the theory by dividing the error for a given value of the FI by the error for twice that value of the FI (see Supplementary Note 1). The predicted ratio exceeds $\sqrt{2}$ for small T, similar to the simulation results, but fails to capture the differences in the maximum value of the ratio for different values of N (N = 1000 and N = 2000)

Fig. 5

Bat orientation behavior consists of two distinct modes—navigation and maneuvering—that are characterized by different angular speeds. a Left, a typical night-flight of an Egyptian fruit bat from its roosting cave (green ellipse) to distal foraging sites; scale bar, 2 km. Right, a zoom-in on the natural orientation behavior, showing epochs of navigation (commuting)—characterized by relatively straight flights during which there was little modulation of heading-direction, interspersed by periods of intensive maneuvering around fruit-trees (indicated by white arrows)—characterized by rapid turns (imagery produced using desktop version of Google Earth Pro). b Distribution of the combined (azimuth × pitch) angular velocity versus horizontal displacement of the bat. Maneuvering mode and navigation mode were classified according to the threshold (vertical dashed black line) at the minimum of the marginal distribution of the horizontal displacement. The marginal distribution of combined angular velocity is shown for all data (grey), and separately for navigation (red) and maneuvering modes (blue). c Angular-velocity distribution computed separately for pitch versus azimuth during navigation (left) and maneuvering (right). Inset (middle) shows the angular velocity distribution for the entire session. Note that during maneuvering (right), angular velocities in azimuth and pitch were correlated, suggesting that bats’ maneuvers were composed of rapid rotations in both azimuth and pitch. The data in b and c were pooled over all bats and flights that we recorded (45 bats with one nightly flight for each bat)

Fig. 6

Dynamic shifts from pure to conjunctive tuning in head-direction cells as a function of angular velocity. a Top view of the trajectory of a bat crawling on the floor of a horizontal arena, color-coded according to combined (azimuth × pitch) angular velocity (AV). Left, trajectory from an entire session; scale bar, 10 cm. Right, 60 s trajectory from the same session. During crawling, there were frequent transitions between epochs of low and high angular velocity. b Distribution of the combined (azimuth × pitch) angular velocity in crawling bats. A cutoff of 10 degrees per second (dashed line) was used to separate between low and high angular velocity. c Peak-firing rates of untuned, pure, and conjunctive cells during low or high angular velocity. There was no significant difference in the peak-firing rate computed at different angular velocities. For both low and high angular velocity, conjunctive cells had higher peak-firing rate than pure cells. Error bars, mean ± s.e.m.; ^*P < 0.05, ^**P < 0.01, using Student’s t-test. d Fractions of untuned cells and cells with pure or conjunctive tuning to azimuth and pitch, plotted separately for low (red) versus high angular velocity (blue). e Ratio between the percentages of cells in high/low angular velocity, plotted separately for each of the 3 cell classes (3 bars). The percentage of conjunctive cells increased fourfold in high versus low angular velocities. f Examples of 3 neurons recorded in the dorsal presubiculum of crawling bats with different tuning properties. Shown are 2D rate-maps as a function of azimuth and pitch, computed separately for low (left) and high angular velocity (right). The significant dimensions to which each cell was tuned, under low or high angular velocities, is indicated above the map. Color scale: zero (blue) to maximal firing rate (red), values in Hz are indicated. g Proportions of tuning-type transitions between low and high angular velocities, for cells with pure (left) or conjunctive tuning (right). For example, the right pie chart shows—for cells with conjunctive tuning at high angular velocity—what percentage of these cells had pure, conjunctive, or no tuning at low angular velocity