Title: Diagnosing Manifold Collapse: Intervention-Based Topology in Low-Rank RNNs
Primary area: learning on graphs and other geometries & topologies
Keywords: Low-rank recurrent neural networks, Neural manifolds, Persistent homology, Topology

Abstract: Understanding how neural circuits give rise to low-dimensional manifolds remains a central challenge in neuroscience and AI. While toroidal and ring-like topologies have been observed, the mechanisms linking connectivity to emergent geometry are not fully understood. We propose an intervention-based topological framework that integrates low-rank recurrent neural networks, persistent homology, and curvature-aware scoring to study the formation, degradation and recovery of neural manifolds. Our analysis compares perturbed and unperturbed trajectories under targeted lesions (random, axis-aligned, low-norm), introducing the Causal Topological Intervention Score (CTIS) to quantify Betti number shifts with curvature weighting. We also develop a Dynamic Betti Fingerprint for anomaly detection and evaluate resilience via the Area Under Recovery Curve (AURC). Using synthetic velocity-driven trajectories and spike-train recordings from the CRCNS pfc-7 dataset, we show that structured low-rank connectivity yields toroidal dynamics, and that CTIS captures non-linear collapse thresholds aligned with recovery trends. This framework offers a principled tool for linking interventions on connectivity to interpretable topological signatures of circuit fragility.