Optimal Population Codes for Space: Grid Cells Outperform Place Cells

Abstract

Rodents use two distinct neuronal coordinate systems to estimate their position: place fields in the hippocampus and grid fields in the entorhinal cortex. Whereas place cells spike at only one particular spatial location, grid cells fire at multiple sites that correspond to the points of an imaginary hexagonal lattice. We study how to best construct place and grid codes, taking the probabilistic nature of neural spiking into account. Which spatial encoding properties of individual neurons confer the highest resolution when decoding the animal's position from the neuronal population response? A priori, estimating a spatial position from a grid code could be ambiguous, as regular periodic lattices possess translational symmetry. The solution to this problem requires lattices for grid cells with different spacings; the spatial resolution crucially depends on choosing the right ratios of these spacings across the population. We compute the expected error in estimating the position in both the asymptotic limit, using Fisher information, and for low spike counts, using maximum likelihood estimation. Achieving high spatial resolution and covering a large range of space in a grid code leads to a trade-off: the best grid code for spatial resolution is built of nested modules with different spatial periods, one inside the other, whereas maximizing the spatial range requires distinct spatial periods that are pairwisely incommensurate. Optimizing the spatial resolution predicts two grid cell properties that have been experimentally observed. First, short lattice spacings should outnumber long lattice spacings. Second, the grid code should be self-similar across different lattice spacings, so that the grid field always covers a fixed fraction of the lattice period. If these conditions are satisfied and the spatial "tuning curves" for each neuron span the same range of firing rates, then the resolution of the grid code easily exceeds that of the best possible place code with the same number of neurons.

Spatial representation in the hippocampal formation: a history

“…7c). The scale change across successive grid modules could be described as a geometric progression with a constant scale factor 154 , confirming the prior predictions 91,172 , as well as theoretical analyses pointing to nested and modular organizations as the most efficient code for representing space at the highestpossible resolution with the lowest-possible cell number 173,174 .…”

“…Read-out Position can be decoded from grid cells and place cells, with greater accuracy in grid cells than place cells if the population is multimodular and scaled in particular ways 159,173,174,276 . Whether neural circuits decode information in the same way remains to be determined, however.…”

Heterogeneity and Efficiency in the Brain

“…The constraint on resolution then gives m ¼ log r R, so that we seek to minimize NðrÞ ¼ dr log r R 1 with respect to r: the solution is r ¼ e. This gives a second prediction: the ratio of adjacent grid periods should be close to r ¼ e. Repeating this analysis in 2-D (where resolution will be set by a ratio of areas 2 1 =l 2 m ), predicts a constant period ratio of ffiffi e p between adjacent modules, each arranged in a triangular lattice, for a grid system that minimizes the number of neurons required to achieve a given resolution [52]. Similar conclusions concerning optimality have emerged from [58], [59]. The analysis above made various simplifying assumptions, but the result is robust to relaxing these conditions [52].…”

Grid cells and cortical representation

“…At the same time, the number of cells per module decreases. Theoretical analyses suggest that such an organization may be optimal for obtaining maximal spatial resolution from a minimal number of grid cells 49,50 . The emergence of an architecture that maximizes information from a limited pool of neurons is reminiscent of the balance between the number of ON and OFF cells in the retina, which has been shown to match the statistical structure of common visual scenes 51 .…”