Probable nature of higher-dimensional symmetries underlying mammalian grid-cell activity patterns

  1. Alexander Mathis  Is a corresponding author
  2. Martin B Stemmler
  3. Andreas V M Herz
  1. Harvard University, United States
  2. Ludwig-Maximilians-Universität München, Germany


Lattices 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 a 3D 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.

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Author details

  1. Alexander Mathis

    Department of Molecular and Cellular Biology, Harvard University, Cambridge, United States
    For correspondence
    Competing interests
    The authors declare that no competing interests exist.
  2. Martin B Stemmler

    Bernstein Center for Computational Neuroscience, Ludwig-Maximilians-Universität München, München, Germany
    Competing interests
    The authors declare that no competing interests exist.
  3. Andreas V M Herz

    Bernstein Center for Computational Neuroscience Munich, Ludwig-Maximilians-Universität München, München, Germany
    Competing interests
    The authors declare that no competing interests exist.

Reviewing Editor

  1. Mark S. Goldman, University of California at Davis, United States

Publication history

  1. Received: December 9, 2014
  2. Accepted: April 23, 2015
  3. Accepted Manuscript published: April 24, 2015 (version 1)
  4. Version of Record published: June 4, 2015 (version 2)


© 2015, Mathis et al.

This article is distributed under the terms of the Creative Commons Attribution License permitting unrestricted use and redistribution provided that the original author and source are credited.


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  1. Alexander Mathis
  2. Martin B Stemmler
  3. Andreas V M Herz
Probable nature of higher-dimensional symmetries underlying mammalian grid-cell activity patterns
eLife 4:e05979.

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