Research ArticleNEUROSCIENCE

Hyperbolic geometry of the olfactory space

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Science Advances  29 Aug 2018:
Vol. 4, no. 8, eaaq1458
DOI: 10.1126/sciadv.aaq1458
  • Fig. 1 Hyperbolic spaces approximate hierarchical networks.

    (A) Example hierarchical description of data and (B) its equivalent representation using Venn diagrams. (C) Venn diagrams can be mapped onto points in a 3D space, forming approximately a tree. The metric in the resulting 3D space is hyperbolic (7). The hyperbolic aspects of the metric are illustrated by the fact that the shortest path between nodes in the tree goes upward and then descends back to the target node. (D) Discrete approximation to a half-space model of the hyperbolic space in 2D. Red solid and dashed lines show the discrete and continuous shortest paths between point a and b within the half-space model of the hyperbolic space (8).

  • Fig. 2 Topological organization of the natural odor space.

    (A to C) Illustration of the topological algorithm for identifying spaces consistent with correlation statistics. (A) Example correlation matrix for five odors in strawberry data set. (B) Correlation matrix after applying a threshold of 0.25. (C) A nonzero value represents an edge connecting the two elements. The resulting complex has one 1D cycle and edge density of 0.5. (D and E) Betti curves with the number of cycles in one (yellow), two (red), and three (blue) dimensions plotted as a function of edge densities. Data from Betti curves (dashed) are compared with predictions using model geometry (solid lines) of 3D hyperbolic space in (D) or Euclidean space (E). Insets show comparisons between integrated Betti values from data (black triangles) compared with models. The error bars show 95% confidence intervals (from 2.5% to 97.5%) from 300 models with the same number of odors as the data, and the colored squares show the medium values of the models.

  • Fig. 3 Visualization of the natural olfactory space using nonmetric MDS.

    Because the variation in radius is small, data points are shown on the surface of a sphere with circles/rectangles for points falling on the near/far side of the sphere. The RGB color scales were proportional to the XYZ coordinates of points.

  • Fig. 4 Axes associated with pleasantness and odor physiochemical properties.

    (A) We represented 108 fruit samples (54 tomatoes and 54 strawberries) using normalized linear combinations of odor coordinates in the space, with the weights proportional to odor concentrations in the samples. The color indicates human rating for the overall liking; circles/squares represent points from the front/back sides of the sphere. The red line shows the direction most associated with the pleasantness ratings. (B) The correlation is significant, with correlation value R = 0.34 and P = 0.01. (C) Visualization of 144 individual monomolecular odors. The red, green, and blue lines showed the directions of pleasantness, boiling point, and acidity, respectively. Two-thirds of monomolecular odors were used to determine the directions associated with perceptual or physicochemical properties of odors, and the rest one-third were projected onto the directions as validation sets to evaluate the correlations. In the 144 monomolecular odors, only 62 of them have available boiling points and were used to find the boiling points direction. (D) Correlation between odor pleasantness with projection onto the pleasantness axes. (E) Correlation between molecular boiling point, a measure of odor volatility, with projection on the associated axes. (F) Correlation between acidity value and the associated axes.

  • Fig. 5 Hyperbolic organization of the human olfactory perception.

    The (A) first and (B) second Betti curves of the perceptual data set (dotted line) (20) compared to Betti curves of the 3D hyperbolic space (solid line) and 3D Euclidean space (dashed line). Euclidean and hyperbolic spaces of other dimensions provided a worse fit. Insets compare integrated Betti values from data (horizontal lines) and 300 repeated models in different dimensions with Euclidean or hyperbolic metrics. The error bars show 95% confidence intervals; the number of repeated computations of model curves was 300. (C) Visualization of odors in human olfactory perception space using nonmetric MDS in a 3D hyperbolic space. The sizes of points are proportional to their radii. The radius distribution is shown in bottom right inset. (D) The multimodal aspects of Betti curves derived from data (dotted lines) can be accounted for by the nonuniform distribution of points within the 3D hyperbolic space. Sampling points from (C) produces multimodal first (yellow) and second (red) Betti curves (solid lines). Inset shows comparison of L1 distances between Betti curves derived from data and those derived from 100 different MDS fits. Black open triangles represent the distance between data and model mean, and colored bar plots show the range of values, where data curves are substituted by different MDS fits.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/8/eaaq1458/DC1

    Fig. S1. No indications of hyperbolic geometry in shuffled odor data sets.

    Fig. S2. Alternative ways of evaluating differences between Betti curves also support hyperbolic geometry of natural odor spaces.

    Fig. S3. Error bar plots of Betti curves statistics for the hyperbolic model of different dimensions.

    Fig. S4. Test of the nonmetric multidimensional scaling algorithm in the hyperbolic space on synthetic data.

    Fig. S5. Odors within the identified space do not cluster by functional group.

    Fig. S6. Comparison between embedded geometric distances and reported perceptual distances.

    Fig. S7. Analysis of sensitivity of integrated Betti value to noise in the input distances.

    Table S1. Statistical tests (P values) for consistency with hyperbolic models based on integrated Betti values.

    Table S2. Statistical tests (P values) for consistency with hyperbolic models based on L1 distances between Betti curves.

    Table S3. Statistical tests (P values) for evaluating consistency of experimental Betti curves with respect to 3D hyperbolic model or optimal optimal Euclidean model.

    Table S4. Statistical tests (P values) for evaluating consistency of Betti curves computed based on logarithm of odor concentrations with respect to hyperbolic model.

    Table S5. P values of hyperbolic and Euclidean model using integrated Betti values for perceptual data set.

  • Supplementary Materials

    This PDF file includes:

    • Fig. S1. No indications of hyperbolic geometry in shuffled odor data sets.
    • Fig. S2. Alternative ways of evaluating differences between Betti curves also support hyperbolic geometry of natural odor spaces.
    • Fig. S3. Error bar plots of Betti curves statistics for the hyperbolic model of different dimensions.
    • Fig. S4. Test of the nonmetric multidimensional scaling algorithm in the hyperbolic space on synthetic data.
    • Fig. S5. Odors within the identified space do not cluster by functional group.
    • Fig. S6. Comparison between embedded geometric distances and reported perceptual distances.
    • Fig. S7. Analysis of sensitivity of integrated Betti value to noise in the input distances.
    • Table S1. Statistical tests (P values) for consistency with hyperbolic models based on integrated Betti values.
    • Table S2. Statistical tests (P values) for consistency with hyperbolic models based on L1 distances between Betti curves.
    • Table S3. Statistical tests (P values) for evaluating consistency of experimental Betti curves with respect to 3D hyperbolic model or optimal optimal Euclidean model.
    • Table S4. Statistical tests (P values) for evaluating consistency of Betti curves computed based on logarithm of odor concentrations with respect to hyperbolic model.
    • Table S5. P values of hyperbolic and Euclidean model using integrated Betti values for perceptual data set.

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