Source
Zhang, Rich, Lee & Sharpee (2022 online; Nature Neuroscience 26, 2023), LIT-876, Figs. 2–3 and Extended Data Figs. 1–10, as read in NOTE-668.
What was actually shown
Each neuron’s distance to another is its negated spike correlation over ±1 s. Clique topology turns the matrix into Betti curves, the counts of 1-, 2- and 3-cycles as the edge threshold sweeps from sparse to complete; the curves depend only on the rank order of the correlations. For each candidate geometry, as many points as recorded cells are sampled uniformly, their distances (with 5% noise) give model Betti curves, and 300 such samples give a distribution.
In three rats on a novel 48-m track, two 2.5-m track sessions and seven 1.8-m box sessions (34–56 active cells in the sessions shown), the data fell inside the 3D hyperbolic distribution and outside the Euclidean cube distributions on integrated Betti values and L1 distances; shuffled spike trains fell outside the hyperbolic one. The test could reject: it rejected the Euclidean cubes and the shuffled data, and it rejects hyperbolic balls of the wrong radius or dimension. 3D fitted better than the other hyperbolic dimensions compared.
The radius grew with log exploration time across days (box, r = 0.85), and within seconds on the long track when estimated from the exponential rate of place-field sizes (r = 0.50). Behavioural covariates did not account for it, and four hours away from the track did not advance it.
What this does not say
- Not that the metric is hyperbolic as against flat. The nulls are uniform cubes. Uniform sampling in a hyperbolic ball and an exponential distribution of a scale coordinate are close to the same thing, and the paper’s own construction (field centre in the plane, field size as height) is the half-space picture of it. A flat model with exponentially distributed field sizes was not tested.
- Not that the code is a tree. No branching structure is inferred. “Hierarchical” means fields of many sizes, with smaller ones more numerous, and larger ones containing smaller ones.
- Not that the radius is optimal. The match between the measured radius and the Fisher-optimal one for CA1’s cell count is an extrapolation of a simulated curve under independent Poisson neurons.
- Not general. Dorsal CA1 of rats, spatial tasks, a few environments; the title names the sessions. Whether the match replicates, and whether it survives a flat multiscale null, is QUESTION-153. Nothing here says that non-spatial or cortical codes are hyperbolic, or that semantic or categorical hierarchies are represented this way.