1. Computational and Systems Biology
  2. Neuroscience

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
https://doi.org/10.7554/eLife.05979
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  1. Alexander Mathis
  2. Martin B Stemmler
  3. Andreas V M Herz
(2015)
Probable nature of higher-dimensional symmetries underlying mammalian grid-cell activity patterns
eLife 4:e05979.
https://doi.org/10.7554/eLife.05979
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Abstract

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
    amathis@fas.harvard.edu
    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.

Copyright

© 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
(2015)
Probable nature of higher-dimensional symmetries underlying mammalian grid-cell activity patterns
eLife 4:e05979.
https://doi.org/10.7554/eLife.05979

Share this article

https://doi.org/10.7554/eLife.05979

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Further reading

    1. Neuroscience

    The self-organization of grid cells in 3D

    Federico Stella, Alessandro Treves
    Research Article Updated

    Do we expect periodic grid cells to emerge in bats, or perhaps dolphins, exploring a three-dimensional environment? How long will it take? Our self-organizing model, based on ring-rate adaptation, points at a complex answer. The mathematical analysis leads to asymptotic states resembling face centered cubic (FCC) and hexagonal close packed (HCP) crystal structures, which are calculated to be very close to each other in terms of cost function. The simulation of the full model, however, shows that the approach to such asymptotic states involves several sub-processes over distinct time scales. The smoothing of the initially irregular multiple fields of individual units and their arrangement into hexagonal grids over certain best planes are observed to occur relatively quickly, even in large 3D volumes. The correct mutual orientation of the planes, though, and the coordinated arrangement of different units, take a longer time, with the network showing no sign of convergence towards either a pure FCC or HCP ordering.