[Paper Review] Actionable Neural Representations: Grid Cells from Minimal Constraints

William Dorrell, Peter E. Latham+2 more|arXiv (Cornell University)|Sep 30, 2022

Neural dynamics and brain functionNeuroscience17 citations

TL;DR

This paper proposes that grid cells in the brain form 'actionable neural representations'—internal models that encode not just spatial position but also the predictable outcomes of actions, such as movement in 2D space. Using group and representation theory, it shows that under minimal biological and functional constraints (non-negative, bounded, precise firing), multiple hexagonal grid modules emerge as the optimal solution, explaining key grid cell phenomena and predicting novel features like grid alignment to environment geometry and optimal lattice spacing.

ABSTRACT

To afford flexible behaviour, the brain must build internal representations that mirror the structure of variables in the external world. For example, 2D space obeys rules: the same set of actions combine in the same way everywhere (step north, then south, and you won't have moved, wherever you start). We suggest the brain must represent this consistent meaning of actions across space, as it allows you to find new short-cuts and navigate in unfamiliar settings. We term this representation an `actionable representation'. We formulate actionable representations using group and representation theory, and show that, when combined with biological and functional constraints - non-negative firing, bounded neural activity, and precise coding - multiple modules of hexagonal grid cells are the optimal representation of 2D space. We support this claim with intuition, analytic justification, and simulations. Our analytic results normatively explain a set of surprising grid cell phenomena, and make testable predictions for future experiments. Lastly, we highlight the generality of our approach beyond just understanding 2D space. Our work characterises a new principle for understanding and designing flexible internal representations: they should be actionable, allowing animals and machines to predict the consequences of their actions, rather than just encode.

Motivation & Objective

• To identify the representational principles that enable flexible, zero-shot inference in biological and artificial systems.
• To formalize 'actionability' as a normative principle for internal representations, where actions transform representations via linear matrices.
• To derive optimal neural representations by combining actionability with biological constraints (non-negative, bounded, precise firing) and functional efficiency (maximal discriminability).
• To explain why grid cells in the medial entorhinal cortex exhibit hexagonal firing patterns, multiple modules, and scale/orientation diversity.
• To extend the framework beyond 2D space to angles, spheres, and 3D space, showing its generality.

Proposed method

• Formalize actionability using group and representation theory: each action (e.g., step north) corresponds to a matrix that linearly updates the neural representation.
• Define a constrained optimization problem: minimize representation error under non-negative firing, bounded activity, and precise coding, while maximizing discriminability between spatial points.
• Use Bessel functions and Fourier analysis to derive the optimal base frequencies for grid modules, showing that zeros of J₁ determine optimal lattice wavelengths.
• Simulate neural representations under these constraints to verify that hexagonal, multi-module grids emerge as optimal.
• Apply the framework to 1D, 2D, 3D, and spherical spaces, demonstrating that place cells or grid-like patterns arise naturally under low-frequency bias.
• Perform ablation studies to show that removing the actionability constraint leads to unstructured, multimodal tuning, as seen in 3D entorhinal recordings.

Experimental results

Research questions

• RQ1 Why do grid cells in the medial entorhinal cortex fire in hexagonal lattices across multiple modules with varying scales and orientations?
• RQ2 How can neural representations support zero-shot inference, such as predicting the outcome of an untraveled path?
• RQ3 What constraints—biological, functional, or structural—lead to the emergence of grid-like representations in 2D space?
• RQ4 Why do grid modules align with environmental boundaries, such as room geometry?
• RQ5 Can the same framework explain representations in higher-dimensional or non-Euclidean spaces, such as 3D space or spheres?

Key findings

• The optimal representation under actionability, non-negative firing, bounded activity, and maximal discriminability is a multi-module grid system with hexagonal tuning, matching biological observations.
• Grid cells align with environmental boundaries due to the actionability constraint, which enforces consistent transformation rules across space.
• The optimal wavelengths of grid modules are determined by the zeros of the Bessel function J₁, predicting specific multiples of the circle diameter: 0.82, 0.45, 0.31, etc.
• Larger grid modules are predicted to follow the Bessel-based wavelength rule more closely, while smaller modules focus on non-harmonic lattice parameters.
• In 3D space, the framework predicts dense, gridded representations that are absent when actionability is removed—matching the multimodal, unstructured cells seen in bat and mouse recordings.
• The framework generalizes beyond 2D: it predicts head-direction cells on rings, place cells on spheres, and 3D grid-like patterns, with non-actionable representations failing to produce structured tuning.