ACTIONABLE NEURAL REPRESENTATIONS: GRID CELLS FROM MINIMAL CONSTRAINTS

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.

1. INTRODUCTION

Animals should build representations that afford flexible behaviours. However, different representation make some tasks easy and others hard; representing red versus white is good for understanding wines but less good for opening screw-top versus corked bottles. A central mystery in neuroscience is the relationship between tasks and their optimal representations. Resolving this requires understanding the representational principles that permit flexible behaviours such as zero-shot inference. Here, we introduce actionable representations, a representation that permits flexible behaviours. Being actionable means encoding not only variables of interest, but also how the variable transforms. Actions cause many variables to transform in predictable ways. For example, actions in 2D space obey rules; north, east, south, and west, have a universal meaning, and combine in the same way everywhere. Embedding these rules into a representation of self-position permits deep inferences: having stepped north, then east, then south, an agent can infer that stepping west will lead home, having never taken that path -a zero-shot inference (Figure 1A ). 



Figure 1: A 2D space is defined by rules, e.g. at all positions north = -south. B Left: Entorhinal grid cells are hexagonally tuned cells (orange). Different cells within a module are translated copies (orange/blue/green). Right: Different modules have different lattice scale (pink/blue/green).

