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Flexible mapping of abstract domains by grid cells via self-supervised extraction and projection of generalized velocity signals
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Proposes a self-supervised encoder-decoder network that extracts low-dimensional, state-independent velocity signals from abstract domains (stretchy blobs/birds, moving blobs, frequency modulation) using next-state prediction and a loop-closure constraint, so that grid-cell-like integrators can map these domains.
Problem
How the brain builds metric maps of abstract, non-spatial continuous domains with grid cells requires extracting a faithful, low-dimensional velocity signal from high-dimensional sensory inputs, without known coordinates or velocity dimensionality.
Approach
An MLP encoder-decoder is trained self-supervisedly on frame triplets with next-state prediction plus a loop-closure loss (velocities along closed loops sum to zero, performed by a neural integrator), plus shortcut and isotropy auxiliary losses, yielding state-independent low-dimensional velocity estimates that drive grid-cell-like hexagonal tuning.
Key results
- Our model achieves transformation errors of 0.02-0.07 across five domains, versus 0.21-0.63 for PCA and up to ~2 for MCNet on 2D tasks (Table 1).
- On the 1D Frequency Modulation task, PCA, Isomap and UMAP yield errors around 2.0 while the proposed model achieves 0.02.
- In the Moving Blobs task, about 24 PCA components are needed to capture 95% of variance, whereas the model's integrated representation lies in a 2D plane.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the self-supervised velocity extraction model instead of the strongest listed baseline, Isomap | abstract velocity estimation · Stretchy Blob (2D) | representation error: 0.42 -> 0.05 ± 0.01 | Helped |
| Use the self-supervised velocity extraction model instead of PCA | abstract velocity estimation · Stretchy Bird (2D) | representation error: 0.21 -> 0.07 ± 0.03 | Helped |
| Use the self-supervised velocity extraction model instead of PCA | abstract velocity estimation · Stretchy Bird (3D) | representation error: 0.31 -> 0.07 ± 0.02 | Helped |
| Use the self-supervised velocity extraction model instead of UMAP | abstract velocity estimation · Moving Blobs (2D) | representation error: 0.62 -> 0.02 ± 0.01 | Helped |
| Use the self-supervised velocity extraction model instead of PCA | abstract velocity estimation · Frequency Modulation (1D) | representation error: 1.97 -> 0.02 ± 0.02 | Helped |
| Add loop-closure loss to next-state prediction loss (ablation) | Our Model · Moving Blobs (2D) | representation error: 0.134 -> 0.044 | Helped |
| Add shortcut estimation and isotropy losses to next-state prediction and loop-closure losses (ablation) | Our Model · Moving Blobs (2D) | representation error: 0.044 -> 0.02 | Helped |
| Use next-state prediction only, removing loop closure and auxiliary losses (ablation) | Our Model · Moving Blobs (2D) | representation error: 0.02 -> 0.134 | Hurt |
| Train with 50% loop trajectories and 50% independent random walks rather than only loop trajectories (ablation) | Our Model · Stretchy Bird (2D) | transformation error: 0.035 | No clear effect |
| Vary velocity randomly at each time step and predict the next state from the current state (ablation) | Our Model · Moving Blobs (2D) | transformation error: 0.02 | No clear effect |
Surprising findings
- Training on trajectories where 50% are not closed loops still achieves comparable error (0.035), so loops are not strictly required.
Limitations
- Auxiliary shortcut and isotropy losses are less biologically supported
- Assumes velocities commute, so non-Euclidean spaces are not directly handled
- Evaluated only on synthetic procedurally generated domains
Experimental setup
Datasets: procedurally generated Stretchy Blob Stretchy Bird (2D) Stretchy Bird (3D) Moving Blobs (2D) Frequency Modulation (1D)
Benchmarks: Stretchy Blob (2D) Stretchy Bird (2D) Stretchy Bird (3D) Moving Blobs (2D) Frequency Modulation (1D)
Metrics: normalized mean-squared error after best-fit linear transfor fraction of variance in PCA
Baselines: PCA Isomap UMAP Autoencoder MCNet
Builds on: Tolman-Eichenbaum Machine continuous attractor grid cell model
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