Is Memory Hyperbolic? Maps, Yes. One Geometry, No.

Memory is hyperbolic.

It is almost too good a sentence. Three words turn memory from a filing cabinet into a navigable world. Facts become places, abstraction becomes distance, and a tree of concepts finds room in curved space. For an AI engineer, it sounds like an architecture diagram waiting to be drawn.

It also hides the decisive question: what, exactly, is curved?

A room has a shape. A single neuron has a patch of the world it responds to. A whole population of neurons, firing together, traces out a surface of possible activity states. Items that merely feel similar can be ranked, and — if you convert that ranking carefully — turned into distances. Memories can link into a graph. Five different objects, five different things that could be "curved." They are not interchangeable.

Once they are separated, the evidence becomes narrower—and more useful. The hippocampal–entorhinal system supports map-like organization in spatial and task-relevant representations. One rat CA1 analysis found that a sampled three-dimensional hyperbolic model matched its neuron-correlation topology better than the Euclidean models tested. Olfactory studies add mostly indirect evidence. None establishes a universal geometry of memory.

The durable conclusion is simpler: neuroscience motivates the map; it does not yet determine the curvature.

TL;DR

• Hippocampal and entorhinal research supports navigable, relational representations. It does not select a single metric.
• A 2023 CA1 study favored a three-dimensional hyperbolic model for Betti curves derived from rat neuron correlations. Its points were neurons, not memories or places.
• Odor evidence moves from natural-mixture statistics to peripheral sensing and preliminary central alignment; it does not establish a hyperbolic odor code.
• For AI memory, negative curvature is a testable candidate for branching hierarchy, not a biological mandate.

Before asking whether memory is hyperbolic, name the object

Representational geometry is the structure of distances, neighborhoods, or transformations assigned to a specified set of represented objects. Before accepting a geometry claim, run a four-part object check:

1. Name the points. Are they locations, stimuli, neurons, population states, concepts, events, or stored records?
2. Define the distance. Is it supplied by the task, measured behaviorally, or derived from similarity? Turning similarity into distance is a modelling choice, not a measurement: it counts as evidence about a geometry only if the conversion behaves like a real distance — never negative, symmetric, and no shortcut through a third point.
3. Identify the task and domain. What did the subject do, and is the claim about an environment, receptive fields, a population-state manifold, or relations among represented items?
4. State the inferred property. Did the analysis infer topology, metric structure, or both? Dimensionality and curvature answer different questions again.

This checklist prevents evidence from changing objects mid-sentence. Zhang and colleagues treated CA1 neurons as points. Gardner and colleagues studied joint activity states within a grid module. Zhou and colleagues analysed volatile compounds and human odor descriptors. None directly measured a geometry whose points were memories.

Hyperbolic geometry is curved space that gets roomier the further out you go — which is exactly what a branching tree needs. That fit makes it attractive for hierarchical knowledge. It says nothing about which geometry biological memory actually uses.

Cognitive maps establish the map, not the curvature

A cognitive map is an internal organization of relations that supports flexible navigation rather than a fixed response chain. Tolman motivated the construct through latent learning and flexible navigation; he did not specify a neural mechanism or metric (Tolman, 1948).

Neural evidence made the map less metaphorical. Place-selective hippocampal neurons tied activity to location (O'Keefe and Dostrovsky, 1971), while hippocampal lesions impaired place navigation (Morris et al., 1982). Entorhinal grid cells showed triangular or hexagonal periodic firing (Hafting et al., 2005). These findings establish spatial selectivity and functional relevance. A hexagonal lattice remains compatible with a Euclidean plane.

The mapping principle also travels beyond literal position. Hippocampal and entorhinal circuits formed fields along a learned sound-frequency axis (Aronov, Nevers and Tank, 2017). Human hippocampal–entorhinal activity reflected abstract graph relations (Garvert et al., 2017). CA1 activity occupied a low-dimensional task manifold carrying position and accumulated evidence (Nieh et al., 2021). The system can organize task-relevant relations in map-like forms without using one universal map or metric.

Hippocampal indexing belongs beside this evidence, not inside the geometric inference. The framework says that the hippocampus binds an index of distributed cortical activity and that partial cues can reactivate the wider pattern (Teyler and DiScenna, 1986; Teyler and Rudy, 2007). Indexing is binding/reactivation, not geometric evidence. Calling an index an “address” does not supply an address geometry.

What the hyperbolic CA1 study actually found

Zhang, Rich, Lee and Sharpee retrospectively analysed dorsal CA1 spike recordings from rats on linear tracks and in square arenas. The key 48-metre track set retained 113 putative active pyramidal cells from three rats; two other animals were excluded for having fewer than 30 active cells. Other data brought the analysed total to 12 sessions and 464 active cells. A session is not an independent animal (Zhang et al., 2023).

The route from spikes to geometry matters:

1. Compute pairwise spike-train correlations among neurons.
2. Use the rank order of correlations as an inverse-distance ordering.
3. Threshold that ordering at increasing edge densities and construct clique, or order, complexes.
4. Compute Betti curves as those complexes grow.
5. Compare the observed curves against curves produced by points scattered evenly inside Euclidean cubes and hyperbolic balls, across a range of dimensions and radii. (The model distances — not the recorded ones — were given a fixed amount of random noise, so the comparison is not unfairly clean.)

A three-dimensional hyperbolic model gave the best match. The tested Euclidean models up to ten dimensions matched worse. The fitted radius increased approximately with the logarithm of exploration time, and the authors connected the organization to place-field scales and efficient positional coding (Zhang et al., 2023).

That empirical result deserves neither dismissal nor inflation. The defensible conclusion is that the Betti-curve summary induced by ranked correlations among the analysed CA1 neurons was consistent with a three-dimensional hyperbolic model within the tested comparison. The physical environments were not hyperbolic. The points were neurons, not positions, events, concepts, or memories. Subjective distances were not measured, curvature was not manipulated, and no behavior specific to negative curvature was tested. Human, episodic, semantic, working, and procedural memory were outside the study's scope.

Why a best fit is not a geometry verdict

Model comparison answers a bounded question: which tested candidate best reproduces the selected statistic? It does not identify the generator uniquely.

The comparison left whole families of candidates untested: Euclidean spaces with uneven density, plain trees and graphs, systems built from a few overlapping modules, spaces that mix curvatures, and patchworks of local maps stitched together. Betti signatures themselves vary with sample size, dimension, metric, and hyperbolic radius. In simulations, low-rank modular systems can produce Euclidean-like, hyperbolic-like, spherical-like, and random regimes as module count changes (Caputi, Pidnebesna and Hlinka, 2025). A hyperbolic-like signature is evidence within a comparison, not a unique measurement of intrinsic curvature.

A thresholded random Gaussian process can also capture place-field arrangement, shape, and topology across rodents and bats in one-, two-, and three-dimensional environments (Mainali, Azeredo da Silveira and Burak, 2025). This does not refute Zhang. It shows that relevant place-field heterogeneity is compatible with random or highly unstructured input.

Human evidence is mixed too. Waraich and Victor compared constant-curvature models across five perceptual and semantic domains. Modest spherical curvature fit best, while the more semantic domains became more tree-like without becoming negatively curved: hierarchy did not imply hyperbolicity (Waraich and Victor, 2024). In the other direction, a 2024 preprint built a hyperbolic space for 1,854 object concepts and found that its distances predicted memorability and prototypicality better than Euclidean distances. Its points were concepts constructed from 49 feature dimensions—not individual memories or neural states—and its ALBATROSS method was described as in preparation (Lee et al., 2024).

As of August 2026 we found no independent replication: no second team, new animals, rival geometries named in advance. Not finding one is not proof that none exists. A 2026 preprint supplies a conditional mathematical and computational account — simulations, not new biological recordings (Wu et al., 2026 preprint).

The logarithm needs the same discipline. The fitted CA1 radius grew roughly with the logarithm of exploration time — a striking pattern, and a tempting one, because hyperbolic space also grows exponentially outward. But that growth is a single number changing over time, and any smooth one-to-one relabelling of one quantity can be undone by relabelling it back — so on its own it carries no geometric information at all. Growth that looks exponential is not the same as a space that is curved. The logarithm alone does not establish a shared geometric substrate.

The grid-cell torus is not a contradiction

Gardner and colleagues analysed 7,671 medial entorhinal and parasubicular units in four recordings from three rats and identified six grid modules. Persistent cohomology showed that joint activity within a module occupied a two-dimensional torus. The structure persisted across environments and was detected in most modules during sleep (Gardner et al., 2022).

Torus versus hyperbolic space is a false contest. Gardner studied topology in grid-module population-state space. Zhang studied similarity ordering among CA1 neurons and compared a topological summary with sampled metric models. Different cells, points, and properties can yield different geometries. Both findings can be true, which argues for plural description rather than one geometry at every level.

Does the brain represent odors in hyperbolic space?

Odor shows how easily a claim grows as its object changes.

Evidence level	What was measured	What it supports	What remains missing
Environment and perception	Natural volatile co-occurrence and human verbal descriptors	Tested 3D hyperbolic models fit better than tested Euclidean alternatives	No neural activity was recorded
Peripheral sensing	Plant mixtures and summed bee-antenna responses	Hyperbolic coordinates predicted some variables better than PCA comparisons	Peripheral activity is not a central representation
Preliminary central alignment	Fly neural distances versus an external source embedding	Mushroom-body distances aligned modestly with the environmental model	Intrinsic neural curvature was not inferred
Odor navigation	Human fMRI in a constructed 6×6 odor-intensity grid	Navigation-like sixfold modulation can apply to odor variables	The task space was an ordinary 2D grid

Zhou, Smith and Sharpee supplied the first row: natural-mixture and human descriptor data were consistent with sampled three-dimensional hyperbolic models, while tested Euclidean alternatives fit worse (Zhou et al., 2018). Human descriptors may include linguistic and category structure. “Model not rejected” is not “model proved.”

Bee electroantennography then brought the analysis to summed peripheral activity (Ghaninia et al., 2022). A fly preprint moved centrally: mushroom-body distances showed a modest association, approximately Spearman rho 0.23, with an external hyperbolic embedding of natural-source co-occurrence (Yang et al., 2023). That is preliminary central alignment, not evidence that the neural manifold has intrinsic negative curvature.

The navigation result also stops short of curvature. Twenty-five people imagined moving along two axes: how banana-like a smell was, and how pine-like. The study reported sixfold effects across entorhinal, ventromedial prefrontal, and anterior piriform analyses; the entorhinal result used multivoxel patterns, while the other two were univariate (Bao et al., 2019). The experiment imposed an ordinary two-dimensional grid. Recent virtual-navigation work in mice found that lateral entorhinal cortex preferentially mapped an olfactory gradient, medial entorhinal cortex a visual space, and CA1 the behaviorally relevant sensory or first-target space. This supports plural sensory maps, not negative curvature (Davoudi et al., 2026).

World representation may need a geometric atlas

What follows is an engineering hypothesis, not a neuroscience finding. Read that way, the evidence is at least compatible with a plural working architecture. Hyperbolic structure may serve hierarchy and containment; Euclidean or spherical coordinates may serve physical domains; ordered time and toroidal phase may retain their own structure. Product spaces or an atlas of local charts can preserve those differences instead of forcing every relation through one curvature.

Its merit is testability: hierarchy, cycles, local metrics, and boundary conditions can be measured separately.

The rule that follows is the same for anyone: a geometry must earn its place on the engineering evidence, not on the biology. A hyperbolic backend should not win because CA1 once produced hyperbolic-like Betti curves. It should win when actual memory graphs show the relevant structure and parameter- and compute-matched tests improve retrieval, navigation, or update behavior. The distinction between a hypergraph and a hyperbolic graph is a useful warning too: adjacent names do not make structures interchangeable.

We hold one such bet ourselves — tensor-hyperbolic graphs, research-stage — and it is bound by the same rule. Nothing in this article tests it, and nothing here should be read as support for it.

How to falsify hyperbolic memory claims

A useful hypothesis needs a way to lose. The program has two tracks.

Neuroscience:

1. Preregister the rivals. Include uniform and nonuniform Euclidean, spherical, hyperbolic, toroidal, tree or graph, low-rank modular, product or mixed-curvature, and local-atlas models.
2. Score held-out prediction. Fit training animals or sessions, then compare likelihood, distance prediction, decoding, and behavioral prediction—not only in-sample Betti summaries.
3. Collect independent data. Use new multi-lab CA1 recordings and test spatial, non-spatial, episodic, and semantic tasks without presuming a shared geometry.
4. Demand a distinctive consequence. Negative curvature must predict a neural or behavioral observation its alternatives do not, with causal perturbation or environment manipulation where possible.

AI engineering:

5. Measure AI memory first. Estimate delta-hyperbolicity, hierarchy, and cycle structure in actual memory graphs before selecting a manifold.
6. Run matched ablations. Compare the candidate manifold against Euclidean, graph-only, product-manifold, mixed-curvature, and atlas baselines under matched parameter and compute budgets.

The rejection rules are explicit. Reject a general neural-curvature claim if a preregistered alternative matches or exceeds its held-out predictions, or if the curvature-specific prediction fails. Reject a system-wide curvature claim — ours included — if a simpler baseline performs as well, or if the advantage disappears under matched resources. If hyperbolic structure helps only within branching subgraphs, use it there. A local win does not establish a universal geometry of memory.

“Memory is hyperbolic” is beautiful because it collapses a world of relations into one image. Science begins by unfolding it again. Maps are real. Negative curvature may organize some of them. The question worth keeping is not whether memory is hyperbolic, but where a hyperbolic model predicts something its alternatives cannot.

Common questions

Is human memory hyperbolic?

That has not been established. A 2024 preprint linked a constructed hyperbolic space of object concepts to memorability, while the main neural result here is a model fit to rat CA1 correlations. Neither establishes the geometry of human memory as a whole.

What did the hyperbolic hippocampus study actually measure?

Zhang and colleagues ranked correlations between rat CA1 spike trains, built clique complexes across thresholds, and compared their Betti curves with sampled geometric models. The inferred points were neurons, not memories or places.

Do grid cells imply hyperbolic geometry?

No. Hexagonal grid firing is compatible with a Euclidean plane. A grid module can also have toroidal population topology while a separate CA1 neuron-similarity analysis favors a hyperbolic model within the comparison it tested; the results concern different objects.

Does the brain represent smells in hyperbolic space?

That has not been established. Natural odor mixtures and human descriptors fit a tested hyperbolic model better than the tested Euclidean alternatives, while peripheral bee data and preliminary fly alignment narrow the gap without showing intrinsic negative curvature in a central odor representation.

What does this mean for AI memory systems?

Neuroscience supports organizing relations as navigable maps, but it does not select one metric for AI memory. Any curved-space design — including our own research-stage tensor-hyperbolic graphs — has to beat geometric and graph alternatives on matched benchmarks, not on a brain analogy.

What experiment would test whether memory is hyperbolic?

Preregister competing geometries, fit them on training animals or sessions, and compare held-out neural and behavioral predictions on new data. Negative curvature must make a distinctive prediction that survives the alternatives.

Sources

• Tolman, “Cognitive Maps in Rats and Men” (1948)
• O'Keefe and Dostrovsky, place-selective hippocampal neurons (1971)
• Morris et al., hippocampal lesions and place navigation (1982)
• Hafting et al., grid cells in entorhinal cortex (2005)
• Teyler and DiScenna, hippocampal indexing theory (1986)
• Teyler and Rudy, hippocampal indexing theory update (2007)
• Aronov, Nevers and Tank, a non-spatial hippocampal map (2017)
• Garvert et al., abstract relational knowledge (2017)
• Nieh et al., abstract learned knowledge in CA1 (2021)
• Zhang et al., hyperbolic geometry in CA1 neuron correlations (2023)
• Caputi, Pidnebesna and Hlinka, modular structure can mimic curvature signatures (2025)
• Mainali, Azeredo da Silveira and Burak, place-field generator (2025)
• Waraich and Victor, perceptual and semantic curvature (2024)
• Lee et al., hyperbolic object-concept memorability (2024 preprint)
• Wu et al., conditional computational account (2026 preprint)
• Gardner et al., toroidal grid-cell population activity (2022)
• Zhou, Smith and Sharpee, natural and perceptual odor geometry (2018)
• Ghaninia et al., odor mixtures and bee antenna responses (2022)
• Yang et al., fly olfactory alignment (2023 preprint)
• Bao et al., grid-like coding in a constructed odor space (2019)
• Davoudi et al., olfactory and visual maps (2026)

Related

• Tensor-Hyperbolic Graphs — our own research-stage bet, for readers who want to see one such hypothesis stated in full.
• Hypergraph vs Hyperbolic Graph — two structures with different jobs.
• Knowledge Graph Memory for AI Agents — relational memory without one assumed geometry.
• Agent Memory Consolidation — how stored experience becomes usable memory.
• Episodic vs Semantic Memory — distinctions a universal claim must survive.

Edward Izgorodin — last reviewed August 2026.

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