What are the axes of this olfactory space? Or, in other words, do odors associated with certain parts of this space have different perceptual or physicochemical properties? Previous studies found axes that correlated with perceptual odor pleasantnes and physicochemical properties such as molecular boiling point and acidity . We checked for associations with all of these properties, and supporting previous findings found that points corresponding to pleasant and unpleasant odors occupied different parts of the space. We note that this analysis of pleasantness rankings was based only on odor components from tomato and strawberry data sets, for which these rankings were available. Thus, the pleasant-unpleasant odor axis can be identified even solely using fruit odor components. The direction most associated with a change in pleasantness value is marked by the red line in Fig. 4 . For odor mixtures produced by individual fruit samples, we use the “overall liking” rating assigned by humans to fruit samples as a measure of pleasantness . To test how well the identified pleasantness axes can predict measured pleasantness rankings for novel samples,procona valencia buckets we regenerated this axis using only strawberry samples and use it to predict pleasantness ranking for tomato samples. The correlation was significant, with correlation coefficient R = 0.34 and P = 0.01 .
The pleasant- ness values could also be assigned to individual odor components based on the correlation between the odor concentration in a mixture and mixture pleasantness computed over samples . This measure of pleasantness produced an even stronger correlation between pleasantness and odor coordinates within the space,R = 0.66 and P = 3×10−7 . We also computed these correlation values using different odor components from those used to generate the pleasant-unpleasant axis for odor components in Fig. 4C. Specifically, we used two-thirds of randomly selected odor components to generate this axis; we computed the correlation value using the remaining components. The pleasantness axis for odor components had a similar orientation to the one for mixtures . In addition to the pleasantness axis, we could also find axes that were strongly associated with two other properties: molecular boiling point—which is probably a reflection of volatility—and acidity, both of which showed significant correlations . We assigned acidity for individual odors as the correlation coefficient be- tween its concentration and fruit sample acidity measurement. Be- cause the space is essentially 2D, the three axes of odor pleasantness, acidity, and molecular boiling point are not independent. In other words, knowing the coordinates along the molecular boiling point and acidity axes, one can predict the position along the pleasantness axis. That is, the identified mapping to a sphere in a hyperbolic space makes it possible to predict, with correlation R = 0.34 for natural mixtures and with R = 0.66 for monomolecular odors, how perceptually pleasant these odors are based on their projections on the acidity and volatility axes.
The observation that the odor mixtures can be mapped onto a continuous metric space is consistent with a previous vector-based model of human olfactory perception . This model posits that the perception of odor mixtures is based on a combination of the mixture components or, in other words, that there is an underlying set of coordinates that can represent olfactory mixtures. Previous analysis of the Dravnieks database containing human perceptual descriptions of > 120 monomolecular odors showed that the perceptual space is likely to be curved. Qualitatively, the points were found to form a “potato-chip” surface . This can be a signature of the hyperbolic space; potato-chip or saddle-like surface have a negative curvature and serve as an everyday example of hyperboloid surfaces. To quantitatively test whether the perceptual space is described by hyperbolic geometry, we applied the Betti curve method to the Dravnieks database. First, we found that Euclidean spaces were not consistent with measured Betti curves . The first Betti curve could not be matched to the data in terms of its area for any dimensionality of the Euclidean space . Second, we found that the full hyperbolic space of varying dimensions could match the area of the first Betti curve. However, only hyperbolic spaces with small number of dimensions could also simultaneously match the area of the second Betti curve . The 3D hyperbolic space produced the best fit, with larger dimensions yielding increasing deviations. Hyperbolic spaces with dimensions nine and above could be excluded with P < 0.034. The third Betti curve was essentially zero and is not shown here.
One may notice that the first and second Betti curves were not as regular as in the case of odorants and contained multiple peaks. It turns out that the biphasic nature of the Betti curve could be explained by the nonuniform distribution of points across the two angles . Unlike in the case of olfactory stimulus spaces that are sampled approximately uniformly, here, the distribution of points obtained using MDS is not uniform and clusters in one-half of the space. Sampling points from this embedding yields biphasic Betti curves that match those derived from perceptual data . Specifically, P values for L1 differences between Betti curves derived from data and MDS fits were P = 0.32 , P = 0.20 , P = 0 , and P = 0.06 . The MDS distances also better correlated with perceptual distances when we carried out MDS in the hyperbolic space compared to Euclidean space . Our results highlight the importance of hyperbolic curved geometry for understanding how natural odors are represented in the nervous system. Overall, we find that both the statistics of natural odor mixtures and human odor perception can be mapped onto hyperbolic spaces. In the natural environment, hierarchical biochemical networks produce odor components. Hierarchical networks can often be approximated by trees and, therefore, by hyperbolic spaces . We find that most natural odor components fall near the boundary of the observed hyperbolic space, corresponding to leaves of the trees . At the perceptual level, we also found hyperbolic organization. However, in this case, the odors selected for the Dravnieks database did not sample the human perceptual space uniformly . Hyperbolic perceptual organization is likely to be general across different sensory modalities. There are two reasons for this. First, neural networks that give rise to perception are hierarchically organized, and as we have seen in Fig. 1, this can lead to hyperbolic geometry. Second, individual neurons have limited response ranges. Because of response saturation, small changes in neural responses near their limit correspond to exponentially large changes in the input values. This compressive mapping is similar to the Poincare disk representation of the hyperbolic space. There is evidence that visual, haptic,procona buckets and auditory perceptual spaces are all hyperbolic . Adding olfactory perception to this list could help explain why humans can map odors to auditory pitch and to colors. Noteworthy is the low dimensionality of both the physical odor space and perceptual odor spaces. In both cases, the curved space contains approximately three dimensionsdespite the fact that the data vary in > 50 dimensions associated with different samples of natural odor mixtures and according > 100 perceptual descriptors. The low dimensionality of the environmental odor space could be a general property of natural odors because it occurred for odors as diverse as fruit and mammalian urine odors. Note that all four natural odor data sets were described by the same 3D hyperbolic space with exactly the same radius . This property could make it easier to represent data from different data sets within the same space. For example, odors from strawberry and tomato could be represented jointly within a single 3D space .
We could not combine data from other data sets because, for example, there were no overlapping components between fruit odors and mouse urine data sets. It is possible that representing all possible natural odors will increase the dimensionality of the overall space. Another possibility is that introducing odors from different sources will “fill in” the inner part of the hyperbolic space. The natural odors considered here mapped onto a surface of a hyperbolic space. Odors produced by biochemical pathways of different complexity are likely to map to surfaces with a different radius, filling in the space. This possibility is especially interesting because it would provide a link to the filled 3D hyperbolic space that we find for perceptual data, which was obtained using diverse classes of odors. At the same time, the perceptual odor mapping reveals that odors tested so far concentrate on one side of the space , whereas natural odor components cover their respective space rather uniformly . These analyses thus suggest perceptual coordinates that are yet to be explored. The match in dimensionality between the environmental and perceptual spaces would not have been expected a priori. The matching dimensionality between the input and perceptual spaces can help avoid nonlinear distortions that would necessarily arise when mapping two nonlinear spaces of different dimensionality. These distortions are known to exist in vision where we perceive distances in a compressed way: The moon appears disproportionately closer to us than would be based on the actual Euclidean distance . We also plot equidistant and parallel lines differently, which is one of the key signatures of the hyperbolic space. Similar distortions arise in the haptic space . The matching geometry between the input and perceptual spaces in olfaction may therefore serve to minimize these distortions in odor perception. Overall, the ability of the perceptual system to resolve points in the low-dimensional odor space would depend on the number and tuning properties of sensory receptors . We followed procedures from to generate Betti curves for samples taken from spaces with different geometries. This renders the algorithm’s results invariant under monotonic transformation of values, for example, due to nonlinearities introduced at the measurement stage. However, this property can also be used to assign a distance between points based on the correlation in the activity of two units in a network or, as in our case, between two odors across different samples. All monotonic functions will yield the same result. We chose Di j = −|Ci j|, where Di j is the assigned distance between odors, and |Ci j| is the absolute value of the correlation coefficient of odor concentrations among a set of points. This definition ensures that stronger correlations corresponded to tighter connections and smaller geometric distances, as in. The first three Betti curves turn out to be quite sensitive measures of the distance matrices and can be used to find underlying geometries consistent with the data. In addition to random spaces, we screened two kinds of geometric structures: Euclidean spaces of different dimensionality and hyperbolic space [we used the hyperbolic ball model with curvature z = 1] with different parameters. In each space, we uniformly sampled points based on the metric of the space. In a d-dimensional Euclidean space, the points were uniformly distributed in a d-cube with Euclidean distance. For a d-dimensional hyperbolic ball model, we used partial space by setting the minimal radius Rmin and maximal radius Rmax for the ball. This choice of the model was motivated by the fact that hyperbolic space approximates hierarchical tree-like networks, with odors reflecting leaves—the neighborhood of the surface. We use the differences between descriptions across odors, because in this case the absolute value of the descriptor matter, unlike in the case of odors where correlations were a more appropriate measure. When fitting the data using geometric models, no noise was added to distances in models. We also tested the sensitivity of the Betti curves to noise in pairwise perceptual distances between odors. This was done by computing perceptual distances based on randomly selected subset of 120 out of the total 146 descriptors. The variability in the resultant distance values was proportional to the mean distance . Importantly, the relative error in the integrated Betti values across these samples was the same as the relative error of the distances themselves . Thus, although the Betti curve construction evaluates data structure globally, it is not driven by variability in larger distances. In the case of the perceptual dataset, we found that the full hyperbolic space better described the data rather than a shell, and therefore the minimal radius was set to zero. We optimized maximal radius of the hyperbolic model, which is a measure of its curvature, to fit the integrated Betti value of the first Betti curve.