12. The Space of Olfaction is δ-Hyperbolic

(Epistemic status: barely even half-baked - but unique, intriguingly plausible, and anyway no one has any better ideas.)

Vision, hearing, the numerous aspects of touch, taste, and smell: of these, smell - or olfaction - is by far the worst-understood, even if we try to tease out the role that olfaction plays in flavor, separating it from the gustation and chemoception that strict-sense taste encompasses. As Convergent Research puts it, “We can’t yet replicate animal olfaction synthetically as a sensing and classification modality. We currently lack a comprehensive model explaining how biological systems decode and classify chemical signals through olfaction. Understanding this process is critical for applications ranging from flavor science to disease diagnostics to understanding and harnessing animal communication.” This past weekend, I briefly attended a “gap mapping” research hackathon organized by YJK; my thanks both to him and to DK who invited me.

While I couldn’t hope to build a full olfaction decoding model, nor fully map odorant-receptor binding, nor even give a robust and comprehensive working theory of how to replicate olfaction in the few hours I had, I thought it prudent to at least clean up my existing thoughts on the subject, given how they’re informed heavily by both my experience in geometric group theory - far removed from the life sciences - as well as my experience as a skilled home chef, sometime perfume blender, and possessor of a keen sense of smell as linked to a keener phenomenology. With any luck, the added insight from the model I sketch out of how olfaction might work will prove a useful map for others more skilled in more central approaches to the question of olfaction; I believe the model to be a plausible one, given a few established facts about both the biochemical basis and subjective experience of olfaction.

Let me start by defining some terms carefully and laying out premises in the language that those terms scaffold. I’ll use “smell” to describe a direct olfactory percept, like the experience of exposure to (+)-limonene, or to ammonia with minor adjuncts, or to a blend of citronellol, geraniol, rose oxide, and beta-damascenone. I’ll use “scent” to mean the olfactory experience a person might have on being exposed to that smell in some concentration or set of concentrations; respectively: orange, stale cat urine, and rose.

For some established facts, we first note that olfactory receptors come in many different varieties, each highly selective to a single small molecule, or to a small set of chemically similar small molecules. Additionally, every such receptor has a band of sensitivity in terms of (say) parts per billion, below which the smell is imperceptible and above which the receptor either tops out or else no longer fires at all (consider the infamous case of hydrogen sulfide); we can rescale that range to the open interval (0, 1) as a fraction of maximal perception strength. As a minor fact, chiral molecules generally smell very different from each other, and don’t cancel each other out: the scent of (+)-limonene closely corresponds to the smell of oranges, but (-)-limonene’s scent better approximates pine; (+)-carvone smells like caraway or dill, while (-)-carvone’s scent is much more like the smell of spearmint. Meanwhile, we may make two mysterious observations: that given one scent, the addition of any amount of any other scent will be smothered by it, blend with or mutate it into a different scent, or stand out against it altogether; and that when moving through a room with a single (complex) smell source present, the resulting scent perceived can nonetheless change with factors including position with respect to the source, air currents, and even different individuals’ olfactory keenness or disabilities.

My major premise is this: arbitrary combinations of smells can be observed, but any two scents built up from smells - even the same list of smells, in some cases - differ greatly from each other, and this suggests that olfaction is best understood as having a treelike or hyperbolic structure to it. (This is notably unlike audition, which may be modeled as involving something like a Fourier decomposition with some added spatial information from timing differences, and unlike vision, which may be loosely modeled as having some pixel-like structure with three-dimensional Euclidean (color) coordinates for each pixel.) In addition, just as the color gamut is limited to only a part of Euclidean space, so too is the scent gamut limited to a tiny sliver of the possible (high-dimensional) hyperbolic space, given the nonexistence of anti-scents - though the nature of hyperbolic space is such that unlike with color, we barely notice the lack.

To understand why this is so plausible, it will be necessary to explain the concept of a δ-hyperbolic space. A δ-hyperbolic space is a metric space in which for any triangle ABC that we might draw, every point on the side AC is distance at most δ from some point on one of the sides AB, BC, and likewise for the other two sides Put another way, the entirety of each of the sides is relatively nearby to the other two sides of the triangle. (This is the picture at the head of the page.) The ordinary hyperbolic plane can be calculated to be ~0.88-hyperbolic, and at δ = 0, we find trees - note that for any three points in a tree, if they’re not part of a single path, then there exists a unique vertex which all three sides of the triangle contain.

The δ-thin hyperbolic model of olfaction then goes like this:

• Each olfactory receptor has some band it receives best in; we can rescale this to an open interval like (0, 1).

• Smells are best transcribed as a list of olfactory receptors, ordered from strongest to weakest (rescaled) response; something like (ABCDEF…)

• A pair of scents is similar in quality exactly when the two scents agree for a large number of initial receptors, that is, we have something like a natural word metric on the space.

• Each such receptor defines a vector in the hyperbolic space, with most pairs of scents orthogonal to each other and a rare few pairs closer-linked. We can consistently define something like cosine similarity between subjective responses to pure stimulation of individual olfactory receptors, but this would need to be measured empirically.

• Obviously, chiral and algebraic inverse are not the same; spearmint and caraway do not cancel out, even if their rescaled percept strengths are similar enough for the receptors to be adjacent in the scent-list.

• Slight differences in scent perception can occur if the strengths of the components of the smell vary slightly (perhaps due to air/fluid circulation), especially if some pair of smell components provokes similarly strong rescaled responses. This might lead to bistable smells.

• The set of all perceptible smells, having this treelike structure, is δ-hyperbolic, and we could (empirically) measure this δ; scent perception is based on dividing up this hyperbolic space into contiguous directions, possibly Voronoi cells on the surface of some segment of a hyperbolic hypersphere, surely of varying size.

• The experience of more than one different scent occurs when the components of a smell can be most naturally grouped into two or more recognizable clusters with small intra-cluster and large inter-cluster metric distances.

This model suggests a few ex ante predictions/explanations and proposes associated measurements and tests.. 

• There should be a bistable scent that can be generated by the right mix of smells - probably three of them, where going from (ABC) to (ACB) produces notably different scents. That scent (or pair of scents?) should give us good bounds on the just-noticeable-difference level for relative concentrations from swapping the concentrations of the subdominant smell components.

• There should be some overlapping chain of smells which makes two otherwise unrelated-smelling scents blend. Similarly, there should be a scent (probably a mix of at least four or five smells) from which the removal of some “central” scent breaks the scent apart into two unrelated-smelling scents.

• There should be a “base perfume” blend that one could turn into numerous importantly different scents through the addition of a small amount of chosen extra odorants - specifically not just “neutral-ish base enriching chosen scents”.

• The phenomenon of synthetic scents feeling “flat” but only rarely wrong likely comes from a natural scent having a form like (ABCDEF…) and the synthetic form having a form like (ABCD) - truncated, but still nearby.

• The (now sadly-well known) phenomenon where some sufferers of COVID-19 smell nothing but garbage is likely partially explained by total loss or at least of some of the olfactory receptors, collapsing most of the space of scents down to their garbage-like components by dropping the other coordinates. It’s also possible that these receptors have instead been scrambled such that their signals cannot be adequately received or interpreted, much as some sufferers of nerve damage have reported with various forms of touch.

• Actually do the empirical measurements for subjective cosine similarities between olfactory receptor activations - something like the process that brought us word2vec.

• Likewise, actually measure the subjective metric distances between nearby scents, and probe the borders of each projective scent subspace.

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