Geometric Phase Transition Enables Extreme Hippocampal Memory Capacity

Prashant C. Raju

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Geometric Phase Transition Enables Extreme Hippocampal Memory Capacity

Prashant C. Raju

arXiv:2605.17199v1May 16, 2026q-bio.NCcond-mat.dis-nnphysics.bio-ph

TL;DR

The paper asks how biological memory can scale despite similar neural hardware and tests whether population geometry, rather than neuron allocation, explains caching birds’ superior spatial memory. Using comparative recordings, circuit analyses, and computational modeling, it finds that rigid crystalline geometry supports distributed, high-capacity coding but requires substantial redundancy and remains subject to modeling and sampling limits.

Problem
Biological memory systems can store different amounts of information with apparently similar hardware, but the network principles supporting high-capacity storage without catastrophic interference remain poorly understood.
Method
The study compares caching and non-caching birds, analyzes excitatory-inhibitory population geometry and allocation metrics, and evaluates topological capacity across computational configurations.
Results
Combined excitatory-inhibitory coding expands intrinsic dimensionality from D=7.5 to D=9.8, while distributed crystalline geometry supports more than 100-fold greater reliable capacity than unstructured coding under identical hardware constraints.
Takeaways & Limitations
The findings shift the focus of biological memory capacity from proliferating neurons or assigning dedicated ensembles toward engineering the geometry of a distributed neural code.
Takeaways & Limitations
The E-I analysis uses 13 sessions, and the computational model is restricted to a one-dimensional circular track because direct two-dimensional extension was intractable.

Abstract

from arXiv · show

Memory systems can store vastly different amounts of information despite similar hardware constraints. Here, we show that superior spatial memory emerges from a discrete stiffening of hippocampal population geometry-a transition from disorganized to crystalline collective coding. Comparing food-caching chickadees to non-caching zebra finches, we found that the caching hippocampus maintains a topologically rigid, "crystalline" geometry with significantly higher geometric stability (Shesha 0.245 v 0.166) and nearly two-fold greater temporal coherence (Shesha 0.393 v 0.209), while the non-caching hippocampus resembles a disorganized "mist." This stability is actively constructed by synergistic circuit dynamics: excitatory neurons form the spatial scaffold while inhibitory populations contribute orthogonal decorrelation, a circuit motif in which excitatory and inhibitory populations occupy largely non-overlapping representational subspaces. A double dissociation with Valiant's Stable Memory Allocator, a model predicting that dedicated neuron ensembles underlie each memory, confirms this advantage reflects continuous topological organization rather than discrete neuron allocation: caching networks exhibit near-zero split-half allocation reliability despite their geometric superiority. Computational modeling across 10k configurations reveals topological rigidity as the mathematical prerequisite for scale: crystalline codes sustain high-fidelity readout beyond M=1k locations while mist codes fail below M=10, a >100-fold capacity advantage. This capacity requires a 169fold representational redundancy: a "geometric tax" stabilizing the manifold against biological noise. These results establish geometric stability as a candidate organizing principle of biological memory: evolution achieves high-capacity memory not by proliferating neurons, but by engineering the geometry of the neural code itself.

Results

Caching chickadees exhibit a more geometrically organized and stable hippocampal code than non-caching finches, with excitatory-inhibitory structure supporting distributed spatial representations. Modeling shows that topological organization improves capacity across loads, though it carries substantial redundancy and depends on estimator-sensitive measurements.

Population geometry: Chickadee population activity preserves physical-spatial structure, whereas finch activity is comparatively featureless under identical spatial sorting.The chickadee RDM shows nearby-location similarity and distant-location dissimilarity; the finch RDM lacks this organized pattern.
Controls: Circular shifting reduced stability from 0.241 to 0.154 in chickadees and from 0.171 to 0.072 in finches, supporting dependence on authentic population continuity.The control preserves individual-neuron statistics while disrupting the population code.
Population geometry: Split-half population-vector correlation showed no species difference, while geometry-sensitive measures captured greater chickadee rigidity.PV correlation was 0.058 in chickadees versus 0.167 in finches (p=0.894); independent CCA stability was 0.554 versus 0.481.
Excitatory-Inhibitory dissociation: Excitatory cells carried nearly 20-fold more spatial information per spike than inhibitory cells, while inhibitory stability remained high and negatively coupled to excitatory stability.Excitatory and inhibitory codes therefore do not simply duplicate the same spatial signal.
Capacity scaling: At 100 stored memories, crystal error was 0.285 versus 0.529 for mist and 0.996 for noise, while crystal codes remained below threshold through 1,000 memories.Mist and noise exceeded the 0.640 threshold at the smallest tested load of 10 memories.
Capacity scaling: Across 10,000 configurations, topology advantage was governed primarily by sparsity, rising to a median ΔError of approximately 0.183 for ρ≥0.11.Chickadee sparsity, ρ≈0.15, fell within this saturated high-advantage regime.
Geometric tax: Caching networks showed higher redundancy than finches, but the filtered comparison was directional and sensitive to the limited filtered finch sample.Filtered means were 19.3 for chickadees and 5.7 for finches, with p=0.057; the unfiltered median comparison was 14.5 versus 2.2, p=0.041.

Discussion

The paper argues that high-capacity spatial memory depends on a rigid, distributed population geometry rather than simply more neurons. This organization stabilizes recall but incurs substantial representational redundancy and remains bounded by empirical sampling and model scope.

Theoretical interpretation: A topological phase transition, rather than greater neural allocation, is proposed to explain how caching birds bypass standard memory-capacity limits.The proposed transition separates a rigid crystalline regime from an unstructured mist-like regime.
Circuit mechanism: In the crystalline code, excitatory neurons provide the spatial scaffold while inhibitory populations enforce orthogonal decorrelation between representations.Combined excitatory-inhibitory coding increases intrinsic dimensionality from D=7.5 to D=9.8.
Distributed coding: Caching memories are distributed across continuous lattice geometry rather than isolated neurons, with near-zero split-half allocation reliability despite high geometric stability.This double dissociation distinguishes topological organization from Valiant’s discrete allocation model.
Capacity and variation: A greater than 100-fold capacity expansion is associated with crystalline geometry, which supports high-fidelity readout at loads that collapse unstructured networks.The rigid scaffold also leaves flexibility for episodic details without compromising the baseline spatial coordinate system.
Limitations: The empirical comparison uses one publicly available dataset, while the E-I analysis includes only N=13 sessions and the computational model uses a 1D circular track.Larger datasets with simultaneous E-I recordings and broader model geometries are identified as future directions.

Materials and Methods

The study analyzes previously published in vivo electrophysiological recordings from caching and non-caching birds, with custom analysis and simulation code available publicly.

Data source: The dataset contains hippocampal single-unit recordings from black-capped chickadees and zebra finches during open-field foraging.The methods summary identifies 39 sessions from 9 chickadees and 8 sessions from 9 zebra finches.
Reproducibility: All custom analysis code, including computational parameter sweeps, is available on GitHub.The repository is provided as the study’s code-availability resource.

A Methods

The study uses publicly available extracellular recordings from the avian hippocampal formation to compare food-caching chickadees with non-caching zebra finches.

Caching birds: 755 single units were recorded from black-capped chickadees across 39 sessions and 9 birds.The chickadee recordings represent the food-caching group.
Non-caching birds: 238 single units were recorded from zebra finches across 8 sessions and 9 birds.The zebra finch recordings represent the non-caching comparison group.

A.2 Cell-type classification

Cell types were classified from standardized waveform features using Ward clustering, followed by spatial-selectivity analysis based on shuffled null distributions and activity-quality filters.

Feature extraction: Two waveform features—spike width and peak-to-peak ratio—were z-scored before Ward hierarchical clustering.The clustering matrix used standardized unit-level waveform measurements.
Cluster assignment: The dendrogram was cut at k=2, assigning the lower-mean-spike-width cluster as putative inhibitory and the other cluster as putative excitatory.The classification achieved approximately 99% concordance with published labels in chickadees.
Spatial selectivity: Spatial selectivity was assessed using each unit’s spatial information against 200 bin-shuffled surrogate maps.This procedure compared observed information with a session-specific null distribution.
Spatial representation: For population-map analysis, arenas were discretized into a 40 × 40 grid, while unvisited bins remained NaN and pairwise distances used pairwise deletion.Each firing-rate map was represented as a vector over S=1,600 spatial bins.
Quality control: Zero-variance units were assigned unit standard deviation, and bins were retained only when enough neurons showed non-zero firing rates.Sessions with fewer than 30 active bins were excluded.

A.4 The Shesha metric: geometric stability of the neural manifold

Shesha measures geometric stability by comparing representational dissimilarity structures across neuron or spatial-bin splits, while masking invalid or unvisited data. The framework also tests whether neural geometry preserves physical spatial relationships.

Shesha metric: Shesha correlates split-half representational dissimilarity matrices to quantify stability of the hippocampal spatial code.Feature splits partition neurons, whereas sample splits assess robustness across spatial subsamples.
Distance computation: Masked cosine distances compare spatial-bin activity patterns using only neurons valid at both bins.Pairs with fewer than two mutually valid neurons are excluded.
Missing-data handling: Unvisited bins remain NaN for distance calculations, preventing shared zeros from artificially inflating similarity.NaN replacement is used only for spatially continuous place-field detection before smoothing.
Score aggregation: The final Shesha score is the mean Spearman correlation across 100 random neuron partitions with all active bins included.The documented defaults use cosine distance and random seed 320.
Spatial correspondence: A Mantel test compares neural distances with Euclidean distances between arena-grid coordinates to assess preservation of physical layout.Significance uses 1,000 permutations of spatial-bin labels.

A.6 Valiant’s Stable Memory Allocator theory and empirical operationalization

Valiant’s Stable Memory Allocator theory predicts controlled, continuous, and sufficiently distinct neural allocations for stable memory. The paper operationalizes these predictions with place-field and split-half allocation metrics.

Theory: The Stable Memory Allocator proposes that hippocampus identifies cortical neuron sets for new memory chunks while keeping allocation sizes within a narrow range.Uncontrolled allocations could either vanish or fill the cortex, destabilizing hierarchical memory allocation.
Theory: SMA stability requires output activity density to remain near a target across a broad range of input densities.The formal definition constrains output density within p ± ε over the input range [q,s].
Theory: Continuity preserves similarity for similar inputs, whereas orthogonality keeps substantially different inputs sufficiently distinct.Together these properties provide noise tolerance and reduce catastrophic interference.
Biological correspondence: The theory maps onto hippocampal circuitry through entorhinal compression and dentate gyrus–CA3–CA1 intrinsic processing.The proposed construction uses effectively random connectivity, consistent with decorrelated neighboring place fields.
Operationalization: Empirical tests quantify field-size consistency, population-overlap consistency, and split-half allocation reliability.Low variability and high split-half correlation support stable dedicated allocation under the SMA framework.
Operationalization: Place fields are smoothed, thresholded at 30% of peak rate, and filtered to remove fragments smaller than four bins.Neurons below 0.5 Hz peak firing are classified as non-spatial.

A.7 Alternative geometric stability benchmarks

The study benchmarks Shesha against population-vector, canonical-correlation, and Procrustes-based stability measures. These alternatives are used to test directional concordance and characterize temporal or anatomical variation.

Concordance: Two additional stability metrics were implemented to test whether Shesha’s species pattern agrees directionally with established alternatives.The primary concordance set comprises Shesha, population-vector correlation, and CCA stability.
Population-vector benchmark: Population-vector stability correlates aggregate spatial preference profiles across random neuron halves without geometric assumptions.The score averages 100 split correlations over bins with sufficient activity.
CCA benchmark: CCA stability averages the top adaptive number of canonical correlations between activity matrices from neuron halves.The number of components is capped at three and reduced for small halves to avoid unusable splits.
Procrustes benchmark: Procrustes stability aligns split-half pairwise distance structures and reports one minus the mean normalized residual.This benchmark directly evaluates full geometric reproducibility without dimensionality reduction or embedding.
Statistical comparisons: Species differences for the three primary metrics are tested with one-sided Mann–Whitney tests using the chickadee-greater-than-finch direction.Temporal stability of individual rate maps is separately assessed with first-half versus second-half cross-correlation.
Anatomical analysis: Within chickadees, Shesha is also compared across histologically confirmed hippocampal subdivisions along the anterior–posterior axis.This tests whether geometric stability varies by recording-site subdivision.

A.10 Neuron-count-matched downsampling control

The neuron-count-matched analysis tests whether chickadee–finch differences in Shesha could be explained by unequal recorded population sizes. Negative controls separately disrupt inter-neuron relationships or map continuity.

Matched downsampling: Chickadee populations are downsampled to the zebra-finch median session size before recomputing Shesha.Only chickadee sessions with at least the target neuron count are included.
Matched downsampling: The matched analysis directly evaluates whether species differences in Shesha persist after equalizing recorded neuron counts.The target count is the median number of neurons per zebra-finch session.
Negative controls: Circularly shifting each neuron’s rate map preserves marginal firing statistics and smoothness while destroying inter-neuron spatial relationships.Twenty shifted populations are generated per session for the control score.
Negative controls: Independent bin shuffling removes spatial continuity within each neuron’s map and disrupts manifold smoothness at single-neuron and population levels.Twenty shuffled populations are generated per session.
Interpretation: Original Shesha scores should exceed both controls if the species effect reflects coordinated population geometry rather than single-neuron statistics.Collapse under either control would implicate single-neuron properties instead.

A.12 Excitatory-inhibitory circuit analysis

The analysis separates excitatory and inhibitory contributions to hippocampal geometry, testing whether their representations overlap and whether inhibition stabilizes the combined manifold.

Population decomposition: Shesha was computed separately for excitatory, inhibitory, and combined populations to compare their geometric stability.The analysis used session-level population matrices for each subpopulation and the full population.
Complementary analyses: Residual, coordination, spectral, dimensionality, and temporal analyses isolate unique, shared, and complementary E/I contributions to the population code.These analyses include subtracting opposite-type maps, correlating session-level Shesha, comparing power spectra, estimating PCA dimensionality, and computing sample-split stability.
Subspace organization: Principal angles between excitatory and inhibitory subspaces quantify whether their spatial representations are shared or orthogonal.Angles approaching 90° indicate non-overlapping subspaces, whereas small angles indicate shared structure.
E/I synergy: The paired full-population versus E-only test evaluates whether inhibitory neurons increase geometric stability beyond the excitatory population alone.The comparison uses within-session Shesha differences and a one-sided Wilcoxon signed-rank test.
Controls: Anti-correlated, cross-session-randomized, and noise-injected inhibitory populations test whether observed E/I effects depend on coordinated within-session dynamics.The controls respectively impose maximal conflict, disrupt pairing while preserving marginal statistics, and grade the degradation caused by added noise.

A.13 Computational model of topological capacity

The computational model generates spatial population codes with tunable topology, then measures noisy location readout and capacity across memory loads and control regimes.

Code construction: Synthetic populations contain N=500 neurons with Gaussian tuning curves on a circular track, while topology strength controls how many neurons retain spatial organization.The standard configuration uses sparsity ρ=0.15 and adds Gaussian functional noise before L2 normalization.
Model regimes: Crystal, mist, and noise regimes correspond to topology strengths τ=1.0, τ=0.5, and τ=0.0, respectively.These regimes model the chickadee, zebra finch, and null controls.
Readout: Four decoders—noisy nearest-neighbor, Ridge, SVR, and ideal Bayesian—estimate location from noisy population vectors.Nearest-neighbor performance is measured by misclassification rate for finite stored-location sets.
Capacity measure: Critical capacity is the memory load at which nearest-neighbor misclassification exceeds a threshold calibrated between crystal and noise performance at M=100.Loads span M∈{10,20,30,…,1000}, with interpolation refining the crossing point.
Controls and robustness: Topology is tested against matched-random, anti-topological, and dimension-matched controls, alongside additive, multiplicative, and correlated noise.The controls separate topology from overall distance structure, inverted spatial similarity, and effective dimensionality.

A.14 10,000-configuration parameter sweep

A 10,000-configuration sweep tests whether topology benefits depend on population size, trial count, sparsity, or a particular stability metric.

Outcome measure: The topology advantage is defined as Error_random − Error_crystal, with larger values indicating greater benefit from organization.The reported sweep uses a fixed memory load and compares topological and unstructured codes.
Robustness: The crystal < mist < noise ordering remains robust across all 10,000 configurations and is supported by a pre-specified concordance set of stability metrics.Shesha, PV correlation, and CCA stability were tested independently for directional agreement.
Statistical framework: Species comparisons use directional or two-sided Mann–Whitney U tests, while within-chickadee E/I analyses use bootstrap procedures and 95% confidence intervals.Fixed random seeds support exact reproducibility, and figures show individual data points alongside summaries.
Controls: Supplementary controls include neuron-count matching, spatial permutations, alternative stability benchmarks, and E/I knockout analyses.These analyses test whether geometric differences reflect population structure, authentic spatial continuity, metric choice, or intact inhibitory dynamics.

A. Chickadee Place Cells (Crystalline Map)

Representative firing maps contrast heterogeneous, multi-field chickadee tuning with diffuse or untuned zebra finch activity.

Chickadee place cells: Chickadee excitatory neurons show heterogeneous, multi-field spatial tuning characteristic of the crystalline code.Peak firing rate and spatial information are shown for eight highly informative cells.
Zebra finch comparison: Zebra finch neurons show diffuse, single-field, or untuned firing that lacks the sharp boundaries associated with a rigid high-capacity manifold.This contrast holds even among the finch population’s most spatially selective cells.
Single-unit correlates: Single-unit signatures in chickadees are consistent with the population-level crystalline geometry identified by Shesha analysis.The figure compares 755 chickadee units with 238 zebra finch units.