Neuropixels Population Geometry Bracket Norm Ollivier-Ricci Curvature
Systems neuroscience has a measurement problem. Brain-wide recordings produce high-dimensional population activity, and standard similarity metrics give contradictory answers about which regions matter. We apply geometric causal discovery to cut through the contradiction.
Bracket Norm
Bracket Norm Identifies Causally Important Brain Regions From Population Geometry
Existing metrics for identifying causally important brain regions — CKA, Procrustes distance, decoding accuracy, explained variance — all collapse when you control for the number of recorded neurons. A region with more electrodes looks more important, regardless of its actual causal role. All 19 geometric metrics from the neural population literature fail this test.
Bracket norm is a geometric invariant that does not have this problem. BN/sqrt(n) is stable across a 25x range of population sizes (CV = 2.0%). Top-3 and bottom-3 regions ranked by bracket norm match optogenetic silencing results (p = 0.0006). Directly perturbing a region (photoinhibition) increases BN/sqrt(n) while degrading behavior, suggesting the metric tracks per-neuron geometric expressiveness rather than computational quality. Validated on human ECoG data during speech production with zero electrode-count confound.
Neural Population Geometry
Neural Geometry Is Not Metric-Neutral: Dimensionality, Dissociation, and Causal Subspaces in Brain-Wide Neuropixels Recordings
CKA and Procrustes distance give opposite answers about which brain regions are similar — and both are widely used. The resolution: dimensionality mediates the dissociation. High-dimensional regions look similar under CKA but dissimilar under Procrustes, and vice versa. The anti-correlation is strong (rho = -0.90; IBL replication: -0.94).
Structured VAE finds causal subspaces 3.4x stronger than LDA, and LDA is anti-correlated with optogenetic importance (rho = -0.73) — discriminative methods find the wrong subspaces in neural data.
Geometric Bottlenecks in Brain Subspace Graphs
Ollivier-Ricci Curvature Reveals Geometric Bottlenecks in Brain-Wide Causal Subspace Graphs
A weighted graph over brain regions uses Grassmannian geodesic distances between causal subspaces as edge weights, then computes Ollivier-Ricci curvature on this graph. Negatively curved regions correspond to geometric “bottlenecks” where linear dimensionality reduction methods (LDA) are most unreliable. Comparing LDA-weighted and VAE-weighted curvature maps reveals sign flips at thalamo-cortical and prefrontal-hippocampal interfaces — precisely the regions where the main paper’s subspace bias is most severe. Curvature on the VAE graph correlates with optogenetic silencing effects, while LDA curvature does not, providing an independent geometric signature of causal relevance.