Research map / Probabilistic and scientific modeling / Dynamical systems
Binding in hippocampal-entorhinal circuits enables compositionality in cognitive maps
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Proposes a normative model of hippocampal-entorhinal spatial coding that encodes position as complex phasor vectors in a residue number system, bound by component-wise multiplication and factorized by a modular resonator attractor network.
Problem
How the hippocampal formation can represent space compositionally with high coding range, noise robustness, and carry-free path integration, and how grid modules and hippocampus could implement this.
Approach
Combines a residue number system with complex-valued phasor vector binding (Hadamard product) and a resonator network whose modules correspond to grid cell modules, yielding superlinear scaling of coding range with dimension, hexagonal carry-free coding via triangular frames, and path integration by binding velocity vectors.
Key results
- Number of stored component patterns scales superlinearly with dimension D, with empirical exponents about 2.05, 1.45 and 1.23 for 2, 3 and 4 moduli.
- Triangular frame encoding yields 3m^2-3m+1 states versus m^2 for Cartesian frames, with hexagonal grid fields.
- Path integration with intrinsic noise (kappa=2) strongly limits drift relative to no MEC correction; sensory cues correct extrinsic-noise drift.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the RNS modular attractor network with 2, 3, or 4 moduli | RNS modular attractor network · Coding range versus pattern dimension | coding-range scaling exponent: 2.05; 1.45; 1.23 | Helped |
| Use triangular rather than Cartesian frames | 2D spatial coding · 2D spatial coding | number of distinct states: m^2 -> 3m^2 - 3m + 1 | Helped |
| Decode sub-integer positions with κ=4 | RNS modular attractor network · Sub-integer decoding | spatial precision: Δx = 0.07 | Helped |
| Increase noise in the input, updates, or codebook | RNS modular attractor network · Position recovery under noise | maximum coding range decoded with high accuracy | Hurt |
| Use codebook noise rather than input noise | RNS modular attractor network · Position recovery under noise | high-accuracy coding range | Hurt |
| Add correction by MEC modules during path integration | RNS modular attractor network · Path integration with intrinsic noise | trajectory integration accuracy | Helped |
| Tether path integration to external sensory cues | RNS modular attractor network · Path integration with extrinsic noise | position accuracy | Helped |
| Increase the number of superposed sensory patterns | Heteroassociative sensory denoising · Superposition of patterns | denoising accuracy | Hurt |
| Denoise FashionMNIST patterns rather than random binary patterns | Heteroassociative sensory denoising · FashionMNIST | denoising accuracy | Hurt |
| Add extrinsic noise to velocity inputs during sequence retrieval | RNS modular attractor network · Sequence retrieval | pattern retrieval accuracy | Hurt |
Surprising findings
- Coding range scaling exponent alpha_K decreases with the number of moduli K, so maximal superlinearity occurs at K=2.
Limitations
- High-level abstraction; mapping of phasor states to spiking neural population activity is not straightforward.
- Only heteroassociative projection to LEC is learned; most HF plasticity not captured.
- Sensory input assumptions are simplified and a proof of concept.
Experimental setup
Datasets: FashionMNIST MNIST
Metrics: coding range factorization accuracy decoded information in bits path integration error denoising accuracy scaling exponent alpha
Baselines: oblique lattice grid models circularly symmetric receptive field models
Builds on: Rademacher-Hopfield network resonator network random Fourier features vector symbolic architectures
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