Mathis, AlexanderStemmler, Martin BHerz, Andreas VM2015-07-132015Mathis, Alexander, Martin B Stemmler, and Andreas VM Herz. 2015. “Probable nature of higher-dimensional symmetries underlying mammalian grid-cell activity patterns.” eLife 4 (1): e05979. doi:10.7554/eLife.05979. http://dx.doi.org/10.7554/eLife.05979.2050-084Xhttp://nrs.harvard.edu/urn-3:HUL.InstRepos:17295604Lattices abound in nature—from the crystal structure of minerals to the honey-comb organization of ommatidia in the compound eye of insects. These arrangements provide solutions for optimal packings, efficient resource distribution, and cryptographic protocols. Do lattices also play a role in how the brain represents information? We focus on higher-dimensional stimulus domains, with particular emphasis on neural representations of physical space, and derive which neuronal lattice codes maximize spatial resolution. For mammals navigating on a surface, we show that the hexagonal activity patterns of grid cells are optimal. For species that move freely in three dimensions, a face-centered cubic lattice is best. This prediction could be tested experimentally in flying bats, arboreal monkeys, or marine mammals. More generally, our theory suggests 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. DOI: http://dx.doi.org/10.7554/eLife.05979.001en-USbatspatial representationgrid cellhippocampusface-centered cubic latticenested grid codehumanmouseratotherJournal Article2015-07-1310.7554/eLife.05979