Extracting grid cell characteristics from place cell inputs using non-negative principal component analysis

Abstract

Many recent models study the downstream projection from grid cells to place cells, while recent data have pointed out the importance of the feedback projection. We thus asked how grid cells are affected by the nature of the input from the place cells. We propose a single-layer neural network with feedforward weights connecting place-like input cells to grid cell outputs. Place-to-grid weights are learned via a generalized Hebbian rule. The architecture of this network highly resembles neural networks used to perform Principal Component Analysis (PCA). Both numerical results and analytic considerations indicate that if the components of the feedforward neural network are non-negative, the output converges to a hexagonal lattice. Without the non-negativity constraint, the output converges to a square lattice. Consistent with experiments, grid spacing ratio between the first two consecutive modules is −1.4. Our results express a possible linkage between place cell to grid cell interactions and PCA.DOI: http://dx.doi.org/10.7554/eLife.10094.001

A single-cell spiking model for the origin of grid-cell patterns

“…Also note that the smallest grid scales in our model are obtained with negative-mean temporal filters (Fig 4). Yet our results agree with the ones of Dordek et al [65] in that the non-linearity introduced by imposing non-negative synaptic weights is sufficient for a triangular symmetry to emerge.…”

“…The authors discuss that such a zero-mean constraint could originate either from lateral inhibition or from a zero-mean temporal filter controlling the output activity of the neuron. In the latter case, the model by Dordek et al [65] is analogous to the present one. We note, however, that effectively zero-mean inputs are neither necessary nor sufficient for grid patterns to emerge.…”

“…In a different model, Dordek et al [65] obtain Mexican-hat correlations by constraining the input activity to be effectively zero-mean. The authors discuss that such a zero-mean constraint could originate either from lateral inhibition or from a zero-mean temporal filter controlling the output activity of the neuron.…”

Spatial Navigation and the Central Complex: Sensory Acquisition, Orientation, and Motor Control

“…The exact source of positional information and thus the relationship between place cells and grid cells is still unknown, however there is physiological evidence supporting interactions between the two populations of spatial cells (Witter and Amaral, 2004; Langston et al, 2010; Wills et al, 2010; Bonnevie et al, 2013). How grid cells might rely on place cells and vice versa , is still not clear (Bush et al, 2014; Dordek et al, 2016). …”

Grid coding, spatial representation, and navigation: Should we assume an isomorphism?

“…2 | GRID CODING AS AN UNDERLYING REPRESENTATIONAL METRIC OF SPACE: THE NAVIGATIONAL AND EXTENDED PERSPECTIVES As a rat navigates an environment, often by being offered rewards for exploring the environment as comprehensively as possible, grid cells fire in a relatively evenly spaced manner that covers much of the environment much like a piece of graph paper (Hafting et al, 2005;Sargolini et al, 2006). One of the key features of grid cells is that the grid often shows sixfold symmetry; in other words, the lattice surrounding each "node" in the grid forms a hexagon (Doeller, Barry, & Burgess, 2010;Dordek, Soudry, Meir, & Derdikman, 2016). Some studies have also found evidence that grid coding extends to other scales of neural recordings, such as functional magnetic resonance imaging, intracranial EEG, and MEG recordings (Doeller et al, 2010;Julian, Keinath, Frazzetta, & Epstein, 2018;Kunz et al, 2015;Maidenbaum, Miller, Stein, & Jacobs, 2018;Stangl et al, 2018;Staudigl et al, 2018).…”