The Algebra of Spatial Navigation

Recurrent networks for path integration over finite groups

Overview • Install • Notebooks

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How can a recurrent neural circuit integrate a sequence of local, egocentric movements into a global, allocentric representation of position and orientation?

This repository studies path integration as sequential group composition. A recurrent network receives an allocentric population code together with egocentric transformations and must maintain the allocentric code of their cumulative product:

$$ \left(x_{\mathrm{allo}},; g_1 \cdot x_{\mathrm{ego}},\ldots,g_T \cdot x_{\mathrm{ego}}\right) \longmapsto (g_T\cdots g_1) \cdot x_{\mathrm{allo}}. $$

The group $G$ specifies the geometry of the navigated space. Circular groups model head direction, product groups model periodic translations, and semidirect products model coupled rotations and translations in two and three dimensions.

We then use finite-group Fourier analysis to derive QuadraticRNN weights that solve the task exactly when all irreducible representations are included.

Overview

Algebraic formulation

For a finite group $G$, an encoding $x\in\mathbb R^{|G|}$ is equivalently a scalar function $x:G\to\mathbb R$. Group elements act by permuting its coordinates through the regular action.

The recurrent model uses a squared-ReLU activation,

$$ \sigma(z)=\mathrm{ReLU}(z)^2, $$

and updates

$$ \begin{aligned} h_1 &= \sigma \left(W_{\mathrm{in}}x_{\mathrm{allo}} +W_{\mathrm{drive}}(g_1 \cdot x_{\mathrm{ego}})\right),\\ h_t &= \sigma \left(W_{\mathrm{mix}}h_{t-1} +W_{\mathrm{drive}}(g_t \cdot x_{\mathrm{ego}})\right),\\ y_t &= W_{\mathrm{out}}h_t. \end{aligned} $$

The closed-form construction decomposes the computation into modules indexed by irreducible representations of $G$. The same representation-theoretic quantities are used to analyze networks learned by gradient descent.

Navigation groups

Group	Interpretation
$C_n$	Circular variable or head direction
$C_n\times C_m$	Periodic planar translations
$\mathbb Z_n^2\rtimes C_m$	Discrete planar rigid motion (Discrete SE(2))
$\mathbb Z_n^3\rtimes O$	Discrete volumetric motion with 24 proper cubic rotations (Discrete SE(3))

Installation

Prerequisite

• Conda or Miniconda

Setup

git clone [anonymous/grids-and-groups]
cd grids-and-groups

conda env create -f conda.yaml
conda activate grids
poetry install

Notebooks

Notebooks give analytically constructed networks. See notebooks/README.md for a short navigation note.

Notebook	Purpose
rnn_constructed_cnxcn.ipynb	Exact and Fourier-truncated translation RNNs on $C_n\times C_n$
rnn_constructed_discrete_se2_c6.ipynb	C6 construction, regular actions, and naturalistic rollout
rnn_constructed_discrete_se2_c6_tuning.ipynb	Empirical and theoretical tuning comparisons
rnn_constructed_discrete_se2_c6_manifolds.ipynb	Fixed-point module manifolds and persistent homology
rnn_constructed_discrete_SE3.ipynb	Exact and cost-aware truncated QuadraticRNNs on $\mathbb Z_n^3\rtimes O$

License

This project is licensed under the MIT License. See LICENSE.