Using hyperbolic geometry to map the olfactory space

In the natural environment, the sense of smell, or olfaction, is used to identify contaminants and assess nutritional value by utilizing the connections formed between chemicals during biological processes. As a result, the synthesis of a specific toxin by a plant or bacteria will be accompanied by the emission of specific sets of volatile chemicals, an animal being able to detect the presence of toxins in food by smelling it.

Scientists from the Salk Institute and Arizona State University have developed a method for organizing odor molecules based on how frequently they appear together in nature, which is where our sense of smell arose. They were subsequently able to map this data in order to identify regions of odor combinations that individuals find most enjoyable.

“We can arrange sound by high frequency and low frequency; vision by a spectrum of wavelengths and colors,” says Tatiana Sharpee. “But when it comes to olfaction, it’s been an unsolved problem whether there is a way to organize odors.”

**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. 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.

They utilized statistical methods to map odorant molecules identified in distinct samples of strawberries, tomatoes, blueberries, and mouse urine based on how frequently they appeared together in the four sets of samples. Those molecules that appeared together more frequently were clustered together.

“It’s akin to me telling you that Chicago is x number of miles from New York, Los Angeles and Melbourne. And if you mapped the cities, you’d find out is that Earth’s surface is curved, not flat—otherwise the distance from, say, Melbourne to Los Angeles wouldn’t add up. We did the same thing for odor.”

**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.

Using this method, the researchers discovered that odor molecules could also be mapped onto a curved surface in three dimensions. However, instead of being a sphere-like Earth, it was discovered to be the shape of a Pringles potato chip—a shape known to mathematicians as a hyperboloid.

When the researchers examined how molecules gathered on this surface, they discovered pleasant and unpleasant directions, as well as directions that corresponded with acidity or how quickly smells evaporate off surfaces. These findings make it easier to create pleasant odor combinations for use in artificial settings, for example (such as a space station).

“By revealing more about how odorant molecules and the brain interact, this work may also have implications for understanding why people with some diseases—like Parkinson’s—lose their sense of smell.”
Brian Smith

Hyperbolic geometry of the olfactory space, Yuansheng Zhou, Brian H. Smith and Tatyana O. Sharpee

Published: August 2018, Science Advances Vol. 4, no. 8, eaaq1458
DOI: 10.1126/sciadv.aaq1458

Source: Hyperbolic geometry of the olfactory space