The visual representation of 3D object orientation in parietal cortex

Abstract

An accurate representation of three-dimensional (3D) object orientation is essential for interacting with the environment. Where and how the brain visually encodes 3D object orientation remains unknown, but prior studies suggest the caudal intraparietal area (CIP) may be involved. Here, we develop rigorous analytical methods for quantifying 3D orientation tuning curves, and use these tools to the study the neural coding of surface orientation. Specifically, we show that single neurons in area CIP of the rhesus macaque jointly encode the slant and tilt of a planar surface, and that across the population, the distribution of preferred slant-tilts is not statistically different from uniform. This suggests that all slant-tilt combinations are equally represented in area CIP. Furthermore, some CIP neurons are found to also represent the third rotational degree of freedom that determines the orientation of the image pattern on the planar surface. Together, the present results suggest that CIP is a critical neural locus for the encoding of all three rotational degrees of freedom specifying an object's 3D spatial orientation.

Figure 1.

Anatomical localization of recording sites. A, Inflated cortical surface of monkey X illustrating the location of area CIP and neighboring regions. The vertical white line shows the approximate anterior–posterior location of the coronal section shown in B. B, C, Coronal MRI sections showing approximate recording locations in the left hemisphere of monkey X and the right hemisphere of monkey U. Recording locations were distributed over 2.4 mm (monkey X) or 3.2 mm (monkey U) anterior–posterior and projected onto a single section. White symbols indicate positions of unresponsive cells. Magenta symbols with blue borders indicate positions of surface orientation selective neurons (monkey X, N = 22; monkey U, N = 36).

Figure 2.

Slant–tilt representation of planar surface orientation. A, The slant and tilt of a planar surface describes the direction of its surface normal, n̂. i, A frontoparallel plane, whose normal is aligned with the Z axis, serves as a reference for describing spatial orientation. ii, Slant (s) describes the first rotation, which occurs about the Y axis and ranges between 0° and 90°. A slant of 22° is illustrated. iii, Tilt (t) describes the second rotation, which occurs about the Z axis and ranges between 0° and 360°. A tilt of 90° is illustrated. Note that tilt is undefined when s = 0° since rotations about the Z axis do not affect the normal. B, For the purpose of visualization, planar surface orientation can be organized on a disc where slant is the radial variable and tilt is the angular variable. C, Tilt-tuning curves measured at different slants for a single CIP neuron are plotted on the left. The response to the frontoparallel plane is plotted as a horizontal line with the error bar offset slightly to the right. Responses are baseline subtracted and error bars are SEM. On the right, the same data are organized as in B to construct a joint slant-tilt tuning curve with firing rate color coded. Dotted grid lines show the sampling of slant and tilt, with intersection points corresponding to presented 3D surface orientations.

Figure 3.

Geometric description of the Bingham function. A, The configuration space of planar slant-tilt is the surface of a unit sphere with antipodal symmetry. Slant varies along the elevation and tilt varies along the azimuth. The Bingham function is formed by taking the intersection of an ellipsoid with a unit sphere. Because an ellipsoid is axially symmetric, the Bingham function is antipodally symmetric. The orientation of the ellipsoid's major axis (blue line), set by s* and t*, determines the preferred slant-tilt. The length of the ellipsoid along the major axis, set by G, determines the response amplitude. The parameter φ (yellow arrow) rotates the intermediate and minor axes (red and green lines) about the major axis (i.e., the preferred slant-tilt). The parameter λ₁ determines the aspect ratio of the ellipsoid across the intermediate and minor axes, allowing the function to be broader along one of these axes than the other (i.e., more elliptical). The parameter λ₂ sets how broad the ellipsoid is across the intermediate and minor axes, determining the tuning bandwidth. All seven of the Bingham function parameters are summarized in the table beneath the illustration (the DC offset was not illustrated). B, Three examples showing the correspondence between ellipsoids (first column), the Bingham function on the sphere (second column), and the Bingham function projected onto the slant-tilt disc as in Figure 2B, C (third column) for visualization.

Figure 4.

Slant–tilt tuning curves of four CIP neurons. A–D, Tilt tuning curves measured at different slants are plotted along with the joint slant-tilt tuning curve (as in Fig. 2C) and Bingham function fit (as in the right column of Fig. 3B). Responses are baseline subtracted, error bars on the tilt tuning curves are SEM, and firing rate is color coded in the joint tuning curves. Parameter fits are provided for comparison with the population histograms (Figs. 6, 7). A, Cell preferring small slants (near frontoparallel planes). Fit parameters: s = 2°, t = 221°, λ₁ = −0.21, λ₂ = 1.30, and φ = 103°. B, Cell preferring planar surfaces with the lower left side closest to the monkey. Fit parameters: s = 39°, t = 231°, λ₁ = −1.34, λ₂ = 1.37, and φ = 63°. C, Cell preferring large slants with narrow tilt tuning. Fit parameters: s = 87°, t = 19°, λ₁ = −0.34, λ₂ = 1.21, and φ = 102°. The appearance of two peaks reflects the cell's large slant preference, and that points on opposite sides of the disc correspond to similar orientations (see Materials and Methods). D, Cell preferring large slants with broad tilt tuning. Fit parameters: s = 86°, t = 261°, λ₁ = −1.35, λ₂ = 0.40, and φ = 100°.

Figure 5.

Tuning curves of cells not selective for planar slant-tilt. Six tuning curves illustrating the range of observed responses that were not tuned for a unique slant-tilt. Responses are baseline subtracted and firing rate is color coded. A, Tuning curves of three neurons for which sensitivity to the orientation of the frontoparallel checkerboard image was not tested. Like the slant-tilt tuning curves plotted in Figure 4, the response at the origin was to a frontoparallel plane with a checkerboard image with θ = 0° (Fig. 9A). B, Tuning curves of three neurons for which sensitivity to the orientation of the frontoparallel checkerboard image was tested. The average of these responses is plotted at the origin. Tilt tuning curves measured at different slants are shown for each of these three neurons (as in Fig. 2C). Error bars are SEM.

Figure 6.

Area CIP encodes a uniform distribution of slant-tilts. A, Slant–tilt is a spherical coordinate system describing the direction of a planar surface's unit normal vector. For a frontoparallel plane, the normal vector (red arrow), n̂, aligns with the Z axis of the reference frame (compare Fig. 2Ai). The pair of antipodal yellow points connected by a yellow line segment represents a unique 3D surface orientation. The red curve traces the path that the plane's normal would travel from a frontoparallel orientation to the orientation specified by the pair of yellow points assuming motion along the first coordinate (slant) followed by motion along the second (tilt). The projection cylinder preserving area in the joint distribution (a Lambert-like projection) is also shown, with blue arrows illustrating the transformation. B, The joint distribution of slant-tilts following this transformation. C, Marginal distribution of slant tuning preferences. D, Marginal distribution of tilt tuning preferences. In B–D, tilt is plotted directly in the angular variable and slant is plotted in the cosine of the angle.

Figure 7.

Population histograms of additional Bingham function tuning parameters. A, Three examples illustrating how the φ, λ₁, and λ₂ parameters affect the shape of the Bingham function (plotted on the slant-tilt disc as in the right column of Fig. 3B). B, Population histogram for the φ parameter that rotates the tuning curve about the preferred slant-tilt. The peak at 90° indicates that the intermediate and minor axes of most tuning curves were aligned with the azimuth (tilt axis) and elevation (slant axis) of the sphere, respectively (see Fig. 3). Variation in tuning bandwidth thus tended to occur across the tilt and slant (as opposed to oblique) axes. C, Population histogram of the λ₁ parameter that sets the aspect ratio of the tuning curve, determining its “ellipticalness.” This parameter allows tuning to be broader along one axis (e.g., tilt) than the other (e.g., slant). The greater the magnitude, the more elongated the tuning curve. D, Population histogram of the λ₂ parameter, which determines the tuning bandwidth. Smaller values result in broader tuning.

Figure 8.

CIP slant-tilt tuning curves are often anisotropic. A, Slant–tilt tuning curves and Bingham function fits for two CIP neurons plotted on the slant-tilt disc. The tuning curves (left column) were fit with two versions of the Bingham function: (1) a simplified five parameter Bingham function (B5) constrained to have equal slant and tilt tuning bandwidths (middle column) and (2) the unconstrained seven parameter Bingham function (B7) as in Figure 4 (right column). Pearson correlations between the data and models are shown. Note that in both cases, the tuning curves were elongated along either the slant (top row) or tilt (bottom row) axis, and that these anisotropies cannot be captured by the five parameter Bingham function. B, Scatter plot of correlation coefficients between the data and two models. Akaike's Information Criterion corrected for finite sample size was used to perform model selection. For 88% of CIP slant-tilt selective neurons, the seven parameter model outperformed the five parameter model (B7>B5; blue points) in describing the data. For 12% of the slant-tilt selective neurons, the simpler five parameter model outperformed the seven parameter model (B5>B7; red points).

Figure 9.

CIP neurons can also encode the third rotational degree of freedom. A, A frontoparallel plane with a checkerboard image on its surface was presented at three different orientations: θ = 0°, θ = 30°, and θ = 60°. Although image orientation on the face of a planar surface is generally 360° periodic, the cross-hatching of a checkerboard pattern has a periodicity of 90°. B, Tuning curves for the orientation of the checkerboard image plotted for significantly tuned responses only (ANOVA, p < 0.05; N = 18). Fourteen of these cells were selective for a unique slant-tilt (cyan; Fig. 4) and four were not (gray; Fig. 5). Responses are baseline subtracted. Consistent with a population that represents all spatial orientations, the peak response varied across cells. C, Cumulative distribution of the percentage of slant-tilt selective cells showing significant tuning for frontoparallel image orientation as a function of preferred slant (red). The derivative of the cumulative curve (blue) shows that the greatest increases in the percentage of tuned cells occurred at intermediate slant preferences.