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Geometric Stability of Neural Population Codes: Regional Variation, Behavioral Relevance, and Circuit Dependence
Prashant C. Raju
TL;DR
Existing reliability measures track centroid preservation but not whether stimulus distance structure reproduces within sessions. The paper formalizes geometric stability and finds it dissociates from temporal drift while relating to neural-behavioral coupling and recurrent excitation.
Problem
Centroid preservation does not assess how reliably stimulus-to-stimulus distance structure reproduces across observations.
Method
The paper evaluates split-half representational consistency and simulates an incomplete olfactory bulb-to-cortical projection in an attractor network.
Results
Geometric stability dissociates from centroid drift, with striatum most stable despite drifting most and neural-behavioral coupling at ρ = 0.18, p = 0.005.
Takeaways & Limitations
Geometric stability is a functionally relevant, circuit-dependent property of neural population codes associated with recurrent excitation.
Takeaways & Limitations
Neither the OB-versus-PCx comparison nor the TeLC manipulation reached conventional statistical significance.
Abstract
from arXiv · showhide
Current models of representational reliability in neural populations focus on temporal stability: whether population centroids are preserved across sessions and days. This framing leaves a fundamental question unanswered: how reliably does the pairwise distance structure among stimuli reproduce across independent observations within a session? We argue that this property, geometric stability, constitutes an independent axis of representational analysis that existing frameworks do not capture. We formalize geometric stability as the Spearman rank correlation between split-half representational dissimilarity matrices (Shesha) and show that it is empirically dissociable from both temporal stability and decoding accuracy. Across 229 area-session observations spanning 68 brain regions in a visual discrimination task (Steinmetz et al. 2019), geometric stability predicts trial-by-trial neural-behavioral coupling ($ρ= 0.18$, $p = 0.005$) while centroid drift does not ($ρ= 0.002$, $p = 0.976$). The regional hierarchy, with striatum most stable ($\bar{S} = 0.44$) and hippocampus least ($\bar{S} = 0.19$), runs roughly opposite to the temporal stability hierarchy. Directionally consistent olfactory data (Bolding \& Franks 2018) motivate an attractor network model in which recurrent excitatory coupling amplifies split-half RDM consistency by completing stimulus patterns from sparse feedforward input ($ρ= +0.64$, $p = 0.010$), providing a circuit-level account of how geometric stability emerges. These results establish geometric stability as a functionally relevant, circuit-dependent property of neural population codes, orthogonal to temporal drift measures and complementary to recent accounts of how recurrent connectivity balances representational stability with sequential dynamics in hippocampal circuits.
1 Introduction
The paper introduces geometric stability as the reliability of pairwise stimulus geometry across independent within-session observations, distinct from centroid preservation and decoding accuracy. Using Shesha, it tests regional variation, behavioral relevance, and dependence on recurrent circuit architecture.
- Conceptual distinction: Geometric stability measures whether pairwise distances among conditions reproduce across independent trial subsets, addressing a limitation of centroid-based temporal stability.Centroid preservation indicates maintained average population states but does not establish reproducible relational structure among stimuli.
- Behavioral relevance: ρ = 0.09, p = 0.19, n = 228: geometric stability and decoding accuracy were empirically orthogonal.Geometric stability predicted neural-behavioral coupling (ρ = 0.18, p = 0.005), whereas decoding accuracy did not (ρ = 0.01, p = 0.88).
- Measurement: Shesha quantifies geometric stability as the Spearman rank correlation between condition-averaged split-half RDMs.The measure compares data to data and operates on full relational structure rather than a centroid or single summary statistic.
- Datasets: Shesha was applied to Neuropixels recordings spanning 26 sessions and 68 brain regions during a visual discrimination task.The paper examines whether within-session geometric reliability varies systematically across brain regions and predicts behavior.
- Circuit dependence: The complementary Bolding & Franks (2018) PCX-1 dataset tests whether olfactory geometric stability depends on recurrent circuitry.This design addresses whether geometric stability varies with circuit architecture in the way predicted by an attractor account.
2 Results
Across 229 area-session observations from 68 brain regions, geometric stability varied regionally and predicted trial-by-trial neural-behavioral coupling, unlike centroid drift or session-level accuracy. Olfactory recordings and a recurrent network model supported a circuit-level account in which recurrent pattern completion increases geometric stability.
- Regional variation: 229 area-session observations across 68 brain regions showed striatum had the highest geometric stability (S̄ = 0.44), whereas hippocampus had the lowest (0.19).Motor cortex and visual cortex were intermediate, with 0.38 and 0.36, respectively.
- Temporal comparison: 0.95, [0.92, 0.97]: thalamus showed the least centroid drift, while striatum showed the most (0.83, [0.76, 0.89]), opposite the geometric-stability hierarchy.Observed centroid similarity was 0.924, [0.915, 0.934], below the shuffled expectation of 0.995, [0.995, 0.996].
- Behavioral relevance: ρ = 0.18, p = 0.005: Shesha predicted trial-by-trial neural-behavioral coupling across area-session observations, whereas centroid drift did not (ρ = 0.002, p = 0.976).The coupling score correlated neural state magnitude with trial outcome.
- Behavioral relevance: ρ = 0.087, p = 0.191: Shesha did not predict mean task accuracy at the session level, indicating behavioral relevance was trial-to-trial rather than session-to-session.Accuracy change over the session was also unrelated to Shesha (ρ = −0.079, p = 0.701).
3 Materials and Methods
The study combines Neuropixels recordings from a visual discrimination task with olfactory PCx recordings and a recurrent rate-network model. Geometric stability is estimated from split-half RDM consistency, alongside centroid-drift and neural-behavioral coupling measures.
- Geometric stability: Shesha was computed as the Spearman correlation between upper-triangular split-half RDM vectors, using cosine distances between condition centroids.Trials were divided into odd and even series, and population vectors were L2-normalized before RDM computation.
- Comparison measures: Centroid drift was measured as early–late cosine similarity after splitting each session at the median trial, while neural-behavioral coupling used population-vector L2 norm versus trial outcome.These per-area-session values were correlated with each stability metric.
- Recurrent network model: The model used N = 200 rate units with 20% sparse random recurrent connectivity and 70% independent input-channel dropout across nine stimulus conditions.Recurrent excitation was swept through J while inhibition was held fixed, and the network operated in a stable fixed-point regime across the tested range.
4 Discussion
The discussion establishes geometric stability as distinct from centroid drift and decoding accuracy, with regional rankings roughly inverted relative to centroid preservation and behavioral coupling linked to geometry. Olfactory results motivate recurrent pattern completion as a mechanism, while whole-brain and cross-dataset limitations qualify the generality of this account.
- Dissociation from temporal stability: Striatum combines the most centroid drift with the most stable geometry, whereas hippocampus shows the opposite pattern, demonstrating that the measures are nearly orthogonal.Striatum has centroid similarity 0.83 and Shesha 0.44; hippocampus has centroid similarity 0.94 and Shesha 0.19.
- Dissociation from decoding: ρ = 0.09, p = 0.19, n = 228: mean decoding accuracy is uncorrelated with Shesha, separating information content from representational reliability.Decoding concerns linear separability, whereas Shesha concerns reproducibility of pairwise distances across independent trial subsets.
- Limitations: The olfactory comparisons use separate experiments, Shesha is global rather than stimulus- or subspace-specific, and the visual and olfactory datasets differ across species, tasks, recording technologies, and conditions.A larger reversible manipulation is needed to confirm the TeLC effect, and the shared recurrent-stabilization principle remains a cross-dataset connection rather than a direct test.
- Circuit dependence and limits: ρ = −0.30, p = 0.27, n = 15 non-hippocampal areas: the whole-brain recurrence gradient does not predict Shesha, implicating sensory-drive strength and reliability in regional variation.The authors therefore treat recurrent stabilization as a circuit mechanism rather than a sufficient explanation of the Steinmetz hierarchy.
Code and Data Availability
The study’s analysis and computational-model code are publicly available, including the geometric stability metric as a PyPI package. Both datasets are publicly available, with repository support for downloading them.
- All code needed to reproduce the analyses and computational models is publicly available.
- The geometric stability metric is implemented in the shesha-geometry Python package on PyPI.Installation: pip install shesha-geometry; Raju (2026b).
- Both datasets used in the study, Steinmetz et al. (2019) and Bolding & Franks (2018), are publicly available.
- The GitHub repository provides code for automatically downloading the Steinmetz data and instructions for downloading the Bolding and Franks data.