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Modeling Grid Cells with a Recursive Growing Neural Gas

Jochen Kerdels, Gabriele Peters

Publications:

  • Jochen Kerdels and Gabriele Peters,
    A Survey of Entorhinal Grid Cell Properties,
    submitted to: Journal of Cognitive Neuroscience, 2017.
  • Jochen Kerdels and Gabriele Peters,
    Modelling the Grid-like Encoding of Visual Space in Primates,
    8th International Conference on Neural Computation Theory and Applications (NCTA 2016), Porto, Portugal, November 9-11, 2016.
    best paper award
  • Jochen Kerdels and Gabriele Peters,
    Noise Resilience of an RGNG-based Grid Cell Model,
    8th International Conference on Neural Computation Theory and Applications (NCTA 2016), Porto, Portugal, November 9-11, 2016.
    nominated for best paper award
  • Jochen Kerdels,
    A Computational Model of Grid Cells based on a Recursive Growing Neural Gas, Dissertation at the Faculty of Mathematics and Computer Science, University of Hagen, 2016.
  • Jochen Kerdels and Gabriele Peters,
    A New View on Grid Cells Beyond the Cognitive Map Hypothesis, 8th Conference on Artificial General Intelligence (AGI 2015), LNCS 9205, Springer, pp. 283-292, Berlin, Germany, July 22-25, 2015.
  • Jochen Kerdels and Gabriele Peters,
    A Computational Model of Grid Cells based on Dendritic Self-Organized Learning, International Conference on Neural Computation Theory and Applications (NCTA 2013), SciTePress, pp. 420-429, Vilamoura, Portugal, September 20-22, 2013.
    nominated for best paper award

All publications are available from Publications.

See also coverage of this project in Public Media and FernUni Media.


Illustration
Real and Simulated Grid Cell's Firing Fields. Circular-Shaped Images: Data and results from experiments with rats done by 2014 Nobel Prize laureates Edvard and May-Britt Moser. Square-Shaped Images: Data for and results from our simulations. Black Lines: Trajectories of a rat in a circular, respectively squared, environment with superimposed red dots, indicating a real, respectively simulated, grid cells' firing locations. Firing rate maps of a grid cell (real as well as simulated) range from dark blue (no activity) to red (maximum activity). - Circular images taken from Moser and Moser, A metric for space, Hippocampus, 2008. Trajectory data shown in squared image provided by Sargolini et al., described in Conjunctive representation of position, direction, and velocity in entorhinal cortex, Science, 2006.

Summary

The entorhinal cortex of rats contains neurons, called grid cells, that exhibit a very peculiar behavior. Discovered about a decade ago, the activity of these cells was found to correlate with the allocentric position of the animal by forming a regular, hexagonal lattice of firing fields across the entire environment. Due to this unusual behavior and the proximity of the entorhinal cortex to other brain regions that also contain cells with spatially correlated activity grid cells are commonly recognized as an important element of a neuronal system for navigation. Existing computational models of grid cells share this view and typically describe the behavior of grid cells as a path integration component of such a system.
Complementary to this view, we presented a new computational model of grid cells that does not assume that grid cells are a specialized component of a navigational system. Instead, it assumes that the activity of grid cells reflects a general principle by which neurons in higher order parts of the cortex process information. The proposed model extends the growing neural gas (GNG) approach by Bernd Fritzke into a recursive algorithm (RGNG) that describes the joint behavior of grid cells in a group as well as the processes within each individual cell. We demonstrated that this approach is able to model the characteristic behavior of grid cells. In addition, it is also able to model the behavior of other cells, which also exhibit grid cell-like firing patterns but whose activity does not correlate with the animal's location in the environment.


Recursive Growing Neural Gas (RGNG)

Our model describes a putative general principle by which neurons in higher-order parts of the cortex process information of arbitrary input spaces. We argue, that each neuron tries to represent its entire input space as well as possible while being in competition with its peers. To model this behavior we employed a novel, two-layer recursive growing neural gas to describe a group or module of grid cells. The top layer represents the interaction between individual grid cells, while the bottom layer models the processes within each cell.

Illustration
Two-Layer RGNG Model with 3 Units in the Top Layer (Representing 3 Grid Cells) and 3 Associated Sets of Connected Nodes in the Bottom Layer (illustrated by corresponding colors).

Simulation Results

1. Results from Baseline Experiments
We evaluated the RGNG in a series of baseline experiments for various configurations. The obtained results show, that the RGNG is able to model the core properties of observed grid cell phenomena quantitatively as well as qualitatively covering phenomena like the distributions of gridness scores, grid spacings, or grid orientations within and across grid cell modules.

Illustration
Comparison of Our Simulations with Natural Firing Rates Measured in Rat Brains. Results are compared across different grid spacings and number of firing fields. Upper Rows: Our results. Lower Rows: Natural rate maps provided by Krupic et al. as supplementary material to Neural representations of location composed of spatially periodic bands. Science, 2012.

2. Results from Simulations on Real Movement Data of Rats
In addition to the baseline experiments we tested our model with realistic, publicly available movement data of rats provided by Sargolini et al., described in Conjunctive representation of position, direction, and velocity in entorhinal cortex, Science, 2006. We could show that our model is able to describe the activity of grid cells in this case as well. Furthermore, we demonstrated that the model can emulate two prominent grid cell phenomena: grid rescaling and grid rotation.

Illustration
Grid Rescaling
Grid Rescaling
Right: Artificial rate maps of a top layer unit of an RGNG within a single simulation run demonstrating the phenomenon of grid rescaling. The rate map on the right was sampled during a time period where the hypothetical original environment shown on the left was shrunk by 50% in one direction while simultaneously the assumed speed signal in that direction doubled, which is expected if the speed signal is derived from, e.g., optical flow.


Illustration
Grid Rotation
Grid Rotation
Right: Artificial rate maps of a top layer unit of an RGNG and corresponding grid orientation distributions within a single simulation run demonstrating the phenomenon of grid rotation. The rate map and orientation distribution on the right were sampled during a time period where the input space coordinate system was rotated by 25 degrees from the 0 degree starting point shown on the left.

General Information Processing Scheme - Other Cells Than Grid Cells

Since the proposed model implements a general information processing scheme we are also able to model the behavior of non-grid cells that nevertheless exhibit hexagonal firing patterns. The activity of such cells does not correlate with the animal's allocentric position but, e.g., with their gaze position (Killian et al., A map of visual space in the primate entorhinal cortex, Nature, 2012). As far as we know our model is currently the only one that can describe the activity of such non-grid cell phenomena.

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