Figure 2. The grids-to-places transformation is generated by a large class of networks.

When synaptic weights are assigned to periodic basis function (N= 100) with random spacings and spatial offsets according to Eq. 15 (A), place-specific activation emerges in both 1-d (B) and 2-d (C). Place fields can be identified by thresholding the activation of the hippocampal neuron shown here (see Methods). Place-specific activation also appear, if the relationship between weights and normalized offsets are a step function (D–F) or linear functions with added Gaussian noise (G–I). The derived solution is robust in that large deviations in the weights do not disrupt the transformation. J,K, Summary of robustness to Gaussian noise in the weights across 1000 simulations. The noise is quantified by the explained variance (R²). Shown is the fraction of simulations, in which a place field emerged in 1-d (J) and 2-d (K) for different network sizes N=10 (magenta), N=20 (cyan), N=50 (red), N=100 (green), and N=200 (blue). The dashed lines in K show results from simulations in which all grids shared the same orientation.