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Lattices abound in nature - from the crystal structure of minerals to the honey-comb organization of ommatidia in the compound eye of insects. Such regular arrangements provide solutions for optimally dense packings, efficient resource distribution and cryptographic schemes, highlighting the importance of lattice theory in mathematics and physics, biology and economics, and computer science and coding theory. Do lattices also play a role in how the brain represents information? To answer this question, we focus on higher-dimensional stimulus domains, with particular emphasis on neural representations of the physical space explored by an animal. Using information theory, we ask how to optimize the spatial resolution of neuronal lattice codes. We show that the hexagonal activity patterns of grid cells found in the hippocampal formation of mammals navigating on a flat surface lead to the highest spatial resolution in a two-dimensional world. For species that move freely in a three-dimensional environment, the firing fields should be arranged along a face-centered cubic (FCC) lattice or a equally dense non-lattice variant thereof known as a hexagonal close packing (HCP). This quantitative prediction could be tested experimentally in flying bats, arboreal monkeys, or cetaceans. More generally, our theoretical results suggest that the brain encodes higher-dimensional sensory or cognitive variables with populations of grid-cell-like neurons whose activity patterns exhibit lattice structures at multiple, nested scales., Comment: 12 pages, 6 figures

Encoding information about continuous variables using noisy computational units is a challenge; nonetheless, asymptotic theory shows that combining multiple periodic scales for coding can be highly precise despite the corrupting influence of noise (Mathis et al., Phys. Rev. Lett. 2012). Indeed, cortex seems to use such stochastic multi-scale periodic `grid codes' to represent position accurately. We show here how these codes can be read out without taking the asymptotic limit; even on short time scales, the precision of neuronal grid codes scales exponentially in the number N of neurons. Does this finding also hold for neurons that are not statistically independent? To assess the extent to which biological grid codes are subject to statistical dependencies, we analyze the noise correlations between pairs of grid code neurons in behaving rodents. We find that if the grids of the two neurons align and have the same length scale, the noise correlations between the neurons can reach 0.8. For increasing mismatches between the grids of the two neurons, the noise correlations fall rapidly. Incorporating such correlations into a population coding model reveals that the correlations lessen the resolution, but the exponential scaling of resolution with N is unaffected., Comment: 11 pages, 9 figures