nucleation

THEORY-618: Neural circuits that must represent hierarchically organised states in few dimensions use a hyperbolic geometry, three-dimensional where it has been measured, because hyperbolic space holds exponentially many well-separated states within a small radius; the argument from Zipf's law to this geometry does not hold, and the evidence is two Betti-curve studies tested only against uniform Euclidean nulls

3 min read

Properties4
StatusProposed
Tagsneuroscience, complex-systems, mathematics
SourceLIT-875 — An argument for hyperbolic geometry in neural circuits, LIT-872 — Hyperbolic geometry of the olfactory space, LIT-876 — Hippocampal spatial representations exhibit a hyperbolic geometry that expands with experience
ExtendsTHEORY-192 — In twelve sessions of rat dorsal CA1 from three data sets, pairwise correlations among place cells have the Betti-curve signature of points sampled uniformly from a three-dimensional hyperbolic ball, not of a low-dimensional Euclidean cube, and the fitted radius rose with the logarithm of exploration time, THEORY-191 — In four natural odour sources (strawberry, tomato, blueberry and mouse-urine volatiles), the rank order of correlations between compound concentrations has the Betti-curve signature of points near the boundary of a three-dimensional hyperbolic ball, not of any uniform Euclidean cube tried, and human odour-descriptor profiles for 127 odorants have that of a full three-dimensional hyperbolic ball

Graph View

Source

Tatyana O. Sharpee (2019), “An argument for hyperbolic geometry in neural circuits”, Current Opinion in Neurobiology 58:101–104, LIT-875, read in NOTE-670. NOTE-670 filed no THEORY: the paper’s claims are conjectures and its one finding belongs to its ref. 4. The evidence this account argues from is that finding and the later test in neural activity:

What was actually shown

The argument (LIT-875). A hierarchy has exponentially many states at each further level. A low-dimensional hyperbolic space holds exponentially many well-separated points within a radius. So a system that must represent a hierarchy in few dimensions would be expected to use a hyperbolic metric. In hyperbolic random graphs, density growing exponentially with radius and degree falling exponentially with it give a power-law degree distribution (Krioukov et al., Fig. 1), the paper’s one derived link between the geometry and a power law.

Stimuli (LIT-872). In four natural odour sources the rank order of compound-concentration correlations gives Betti curves that match points near the surface of a 3D hyperbolic ball and reject uniform Euclidean cubes of any dimension tried; human descriptor profiles for 127 odorants match a full 3D hyperbolic ball.

Neurons (LIT-876). Pairwise correlations of a few dozen CA1 cells per session match points sampled uniformly from a 3D hyperbolic ball and fail to match low-dimensional uniform Euclidean cubes or shuffled spikes; the fitted radius grows with the logarithm of exploration time.

What could have come out otherwise. In both tests the Betti curves could have matched a Euclidean cube. They did not.

What this does not say

  • That Zipf’s law signals hyperbolic geometry. Eq. 2 rewrites P(s) ∝ 1/r(s) as a count of states growing as e^E. Any rank power law gives an exponential count, and so does any system with extensive entropy; the unit slope that makes Zipf special plays no part in the step to geometry, and no metric on a circuit’s states is proposed (NOTE-670).
  • That three dimensions are privileged. Mostow rigidity concerns the hyperbolic structure of finite-volume manifolds in every dimension from 3 up, and says nothing about the stability of a point embedding under noise. Three is what the two tests found.
  • That a tree was found. Neither test recovers the hierarchy: odour compounds do not cluster by chemical class, and the CA1 “hierarchy” is one of nested place-field sizes.
  • That hyperbolic beats every alternative. The only null geometry in either test is uniform Euclidean sampling. A Euclidean space with an exponential spread of scales is the confound THEORY-192 and NOTE-670 name.
  • That hyperbolic networks are “maximally responsive to perturbations”. The abstract says so; the body only sets two cited literatures side by side.

Connections

  • THEORY-191 (odour statistics) and THEORY-192 (CA1 correlations) are the two instances, filed as findings with their own limits. This account widens them to neural circuits in general, so it extends both, and falls with them.
  • Hyperbolic random graphs (THEORY-188, from LIT-865) derive the power law from two exponentials in the radius; here the same shape is asserted for state probabilities without a model.
  • Zipf without meaning (LIT-270, NOTE-243): Zipf-like curves arise in text with no semantic structure, so a Zipf curve alone does not identify a mechanism.
  • QUESTION-025 asks whether hierarchical attributes keep linear directions in an embedding. This account bears on it only through the general point that a hierarchy may live in a radial coordinate; NOTE-670 finds nothing toward either answer, so no answers is declared.

Backlinks