topological-grid-cell-decoding-codes

name	topological-grid-cell-decoding-codes
description	Topological decoding of grid cell activity via path lifting to covering spaces. Uses TDA to extract toroidal coordinates from grid cell populations and reconstructs spatial trajectories from a single module without training data or external position info. Activation: grid cells, topological data analysis, toroidal manifold, path lifting, spatial navigation, neural manifolds, continuous attractor network, entorhinal cortex, path integration, covering space
metadata	{"arxiv_id":"2510.16216","published":"2025-10-17","updated":"2026-07-08","authors":"Yuxing Jared Yao, Iris H. R. Yoon","tags":["grid cells","topological data analysis","toroidal manifold","path lifting","spatial navigation","neural manifolds","entorhinal cortex","covering spaces"]}

Topological Decoding of Grid Cell Activity via Path Lifting to Covering Spaces

Core Concept

Grid cells in the medial entorhinal cortex encode spatial position on a toroidal manifold due to their periodic firing patterns. This paper introduces a purely topological framework to decode spatial trajectories from grid cell population activity without training data, external position labels, or supervised learning — using only the intrinsic topological structure of the neural manifold.

Key Innovations

1. Toroidal Coordinate Extraction via TDA

• Use topological data analysis (persistent homology) to identify the toroidal structure in high-dimensional grid cell population activity
• Extract toroidal coordinates (angular parameters on the torus) from the neural manifold
• Works on both simulated continuous attractor networks (CANs) and experimental recordings

2. Path Lifting to Covering Spaces

• The toroidal manifold is periodic — a single point on the torus corresponds to infinitely many physical positions
• Path lifting to the covering space (Euclidean plane ℝ²) resolves this ambiguity
• Reconstructed trajectories differ from ground truth only by an affine transformation (rotation, scaling, translation)
• Demonstrates that a single grid cell module contains sufficient information for path integration

3. Training-Free Decoding

• No neural network training, no calibration, no external position reference
• Entirely based on the topological structure of the population code
• Validated on both CAN simulations and real experimental grid cell recordings

Methodology

Step 1: Extract Neural Manifold

1. Collect grid cell population activity vectors (firing rates across cells at each time point)
2. Apply dimensionality reduction (e.g., PCA, UMAP) to identify low-dimensional embedding
3. Verify toroidal topology via persistent homology (two independent 1-cycles = torus)

Step 2: Extract Toroidal Coordinates

1. Map neural activity to angular coordinates (θ₁, θ₂) on the torus
2. Use circular statistics to track the position on each cycle of the torus
3. Handle wrapping/discontinuities at torus boundaries

Step 3: Path Lifting

1. Starting from an initial position, integrate angular changes over time
2. Lift the path from the torus T² to the covering space ℝ²
3. Track unwrapping events when the path crosses torus boundaries
4. Result: reconstructed trajectory in physical space (up to affine transform)

Step 4: Validation

• Compare reconstructed trajectory to ground truth via Procrustes analysis
• Measure correlation, angular error, and path similarity
• Validated on: (a) CAN simulations with known ground truth, (b) experimental recordings

Key Findings

• Single-module sufficiency: One grid cell module contains enough information for reliable path reconstruction
• Affine equivalence: Reconstructed paths match ground truth up to rotation + scaling + translation
• No training needed: Purely topological, no supervised learning or calibration
• Robust to noise: Works on experimental data with biological variability
• Path integration mechanism: Suggests how the brain performs path integration from grid cell activity alone

Applications

• Spatial navigation research: Decoding animal trajectories from neural recordings
• BCI for navigation: Brain-computer interfaces for spatial state estimation
• Neuroscience theory: Understanding how grid cell population codes represent space
• Robotics: Bio-inspired navigation systems using topological representations

Implementation Considerations

Data Requirements

• Grid cell population recordings (≥ ~50 cells recommended for reliable TDA)
• Sufficient spatial coverage (animal should explore environment thoroughly)
• Single module identification (cells with similar grid spacing/orientation)

TDA Parameters

• Persistence threshold: filter out topological noise
• Window size for sliding window analysis
• Choice of distance metric for point cloud construction

Pitfalls

• Multiple modules: If multiple grid modules are mixed, the manifold structure becomes more complex (higher-dimensional torus)
• Boundary effects: At environment boundaries, grid cell firing patterns may distort
• Temporal resolution: Path lifting requires sufficient temporal sampling to resolve wrapping events
• Ambiguity: Path lifting recovers trajectory up to affine transform — absolute position and orientation cannot be determined without additional information

Related Concepts

• Continuous attractor networks (CANs) for grid cells
• Persistent homology / topological data analysis
• Covering spaces and path lifting in algebraic topology
• Path integration in the entorhinal-hippocampal system
• Neural manifold analysis

References

• Paper: arXiv:2510.16216 (October 2025, updated July 2026)
• Authors: Yuxing Jared Yao, Iris H. R. Yoon
• Categories: q-bio.NC, math.AT