The self-organization of grid cells in 3D

  1. Federico Stella  Is a corresponding author
  2. Alessandro Treves
  1. Scuola Internazionale Superiore di Studi Avanzati, Italy


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 FCC and 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 fast, 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.

Article and author information

Author details

  1. Federico Stella

    Cognitive Neuroscience, Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy
    For correspondence
    Competing interests
    The authors declare that no competing interests exist.
  2. Alessandro Treves

    Cognitive Neuroscience, Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy
    Competing interests
    The authors declare that no competing interests exist.

Reviewing Editor

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

Version history

  1. Received: December 5, 2014
  2. Accepted: March 25, 2015
  3. Accepted Manuscript published: March 30, 2015 (version 1)
  4. Version of Record published: June 4, 2015 (version 2)


© 2015, Stella & Treves

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. Federico Stella
  2. Alessandro Treves
The self-organization of grid cells in 3D
eLife 4:e05913.

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