If Grid Cells are the Answer, What is the Question? A Review of Normative Grid Cell Theory

TL;DR

This review synthesizes mechanistic, perturbation, and normative literatures to argue that grid cells implement a high-fidelity, path-integrating code for space under biological constraints. It contends that grid cells arise from combining nonlinear encoding with path-integration demands, yielding multiple axis-aligned modules, rather than from pure efficient coding alone. While task-optimised networks can reproduce many grid-cell phenomena, the precise velocity-update mechanism and conjunctive coding remain areas of active debate. The work highlights the broader utility of normative modelling for neural computation and the careful integration of theory with experiment to address complex brain-function questions.

Abstract

For 20 years the beautiful structure in the grid cell code has presented an attractive puzzle: what computation do these representations subserve, and why does it manifest so curiously in neurons. The first question quickly attracted an answer: grid cells subserve path-integration, the ability to keep track of one's position as you move about the world. Subsequent work has only solidified this link: bottom-up mechanistic models that perform path-integration match the measured neural responses, while experimental perturbations that selectively disrupt grid cell activity impair performance on path-integration dependent tasks. A more controversial area of work has been top-down normative modelling: why has the brain chosen to compute like this? Floods of ink have been spilt attempting to build a precise link between the population's objective and the measured implementation. The holy grail is a normative link with broad predictive power which generalises to other neural systems. We review this literature and argue that, despite some controversies, the literature largely agrees that grid cells can be explained as a (1) biologically plausible (2) high fidelity, non-linearly decodable code for position that (3) subserves path-integration. As a rare area of neuroscience with mature theoretical and experimental work, this story holds lessons for normative theories of neural computations, and on the risks and rewards of integrating task-optimised neural networks into such theorising.

Figures (6)

• Figure 1: Structure of the grid cell code. A: Neurons are tuned to a hexagonal lattice of positions in 2D space. B: They are grouped into modules: neurons in the same module have translated (but not rotated) receptive fields, and across a module they uniformly sample the phases (translations). C: There are only a handful of modules in one animal, each with its own lattice, and $\sim$ 1000s neurons covering the possible phases. D: For each grid module there is a population of grid cells that are conjunctively tuned to both the underlying grid of the module, and a particular heading direction. E: These conjunctive neurons can implement path-integration by pushing the bump of neural activity around the module burak2009accurate, like the ring attractor in the fly central complex hulse2020mechanisms, using a shifted connectivity pattern: pure spatial neurons project to conjunctive neurons with the same spatial tuning profile (red connections), which project back to the spatial neurons shifted by their velocity tuning (blue connections). When the rightward neurons are more active than the leftward, this will cause the activity bump to move rightwards on the ring, implementing path-integration.
• Figure 2: Path-integration with different codesA: Path-integrating with a place cell code is easy, current cell plus step uniquely determines next cell, but it is limited by the number of cells. B: Multifield cells improve the coding capacity but make path-integration more challenging, instead resources must be devoted to learning a mapping between unique combinations of cells. C: Within a grid moudle, current cell plus movement again uniquely determines the next cell: no matter which firing field of a grid cell you are in, thanks to the translational symmetry, you always know which cell to activate after a step. As such, grid cells elegantly combine the easy path-integration of place cells, with the higher capacity coding of multifield cells, and the path-integration mechanism generalises across space.
• Figure 3: Grid Cells via Bandpass Filtering. A: A Gaussian place cell code has a covariance whose frequency content is a smoothly-decaying Gaussian, left, but a difference-of-Gaussian code has covariance whose frequency content peaks at a non-zero frequency, figure from sorscher2019unified. B: The grid cells that result from nonnegative PCA on difference-of-Gaussian place cells are not translationally symmetric, each population contains grid cells whose axes are rotated relative to one another (for example, the left and rightmost grid cells from dordek have lattices rotated 30$^\circ$ relative to one another), figures from dordek2016extractingsorscher2019unified. C: We create a representation, ${\bm{g}}({\bm{x}})$, that contains a single frequency, and plot the conformal loss, \ref{['eq:pettersen']}, as a function of this single frequency for a few $\sigma$ values. This loss is minimised (dark blue) at an intermediate value of frequency: a bandpass filtering effect. D: Metric encoding also produces a population of grid cells that are rotated relative to one another, figure from pettersen2024self.
• Figure 4: Successor representation eigenvectors are poor models of grid cells, figure from stachenfeld2017hippocampus.
• Figure 5: A Space of Optimal Codes We optimise a nonnegative, unit-norm representation of position to minimise a similarity matching objective either linear, \ref{['eq:sim_match']}, or nonlinear, \ref{['eq:non_linear_similarity']}, with or without a path-integrating constraint, \ref{['eq:actionability']}. With more neurons than positions all choices lead to place cells (not shown). With few neurons and no path-integration (left column) we get place cells with a linear objective, and random multifields with a nonlinear objective (see also fig 15C, dorrell2023actionable). Adding a path-integration constraint leads to either one grid module for the linear similarity loss, or multiple under the nonlinear loss (for more discussion, see dorrell2023actionable).