J Comput Neurosci (2007) 22:297–299 DOI 10.1007/s10827-006-0013-7 BRIEF COMMUNICATION From grids to places M. Franzius · R. Vollgraf · L. Wiskott Received: 28 August 2006 / Revised: 3 November 2006 / Accepted: 13 November 2006 / Published online: 29 December 2006 C  Springer Science + Business Media, LLC 2007 Abstract Hafting et al. (2005) described grid cells in the dorsocaudal region of the medial entorhinal cortex (dMEC). These cells show a strikingly regular grid-like firing-pattern as a function of the position of a rat in an enclosure. Since the dMEC projects to the hippocampal areas containing the well-known place cells, the question arises whether and how the localized responses of the latter can emerge based on the output of grid cells. Here, we show that, starting with sim- ulated grid-cells, a simple linear transformation maximizing sparseness leads to a localized representation similar to place fields. Keywords Place cell . Grid cell . Hippocampus . Entorhinal cortex As reported by Hafting et al. (2005) grid cells in the dMEC show spatial firing patterns in the form of hexagonal grids with frequencies within one octave (39 to 73 cm mean dis- tance), random phase shifts, and random orientations. The firing patterns of place cells in the hippocampus, on the other hand, are localized spots (Muller, 1996). Our hypothesis is that the latter can be generated from the former simply by sparsification, which is consistent with evidence that firing Action Editor: Alessandro Treves M. Franzius () · L. Wiskott Institute for Theoretical Biology, Humboldt-University, Berlin, Germany e-mail: m.franzius@biologie.hu-berlin.de L. Wiskott e-mail: l.wiskott@biologie.hu-berlin.de R. Vollgraf Department of Computer Science, Technical University of Berlin, Germany e-mail: vro@cs.tuberlin. de patterns in hippocampal regions CA1 and CA3 are sparser than in entorhinal cortex (O’Reilly and McClelland, 1994). To show that place fields can be derived from grid-cells by sparsification we simulated a fully connected linear two- layer network. The input units were 100 simulated grid-cells of a virtual rat with activity patterns synthesized by Gaus- sians arranged on a hexagonal grid (Fig. 1(A)). Some posi- tional jitter, random anisotropy, and amplitude variation of the Gaussians was introduced and white noise was added to qualitatively match the slightly irregular experimental data. Let g i ( r ) denote the activity of grid-cell g i as a func- tion of location  r . Given a virtual path  r (t ) of a rat within the enclosure, the input into the hippocampus coming from the grid-cells is x i (t ):= g i ( r (t )). To achieve sparseness we applied independent component analysis (ICA) (Hyv¨ arinen, 1999b) on a set of 200.000 time points on the full set of 100 inputs by subtracting the mean and using the CuBICA algorithm, which attempts to diagonalize the tensors of third and fourth order cumulants (Blaschke and Wiskott, 2004), but we have obtained similar results with other sparsification algorithms, such as FastICA (Hyv¨ arinen, 1999a) or simply maximizing peak activity under a unit variance, zero mean, and decorrelation constraint. The sign of each output unit, which is arbitrary for ICA, was chosen such that the value with the largest magnitude is positive, and then constants c j were added to ensure nonnegative values. This yielded an affine transformation with matrix T producing 100 out- put signals y j (t ):= ∑ i T ji x i (t ) + c j that are maximally in- dependent and significantly sparser than the input signals (kurtosis increased on average from 2.8 for the input units to 27.3 for the output units). The output-unit activities as a function of location are p j ( r ):= ∑ i T ji g i ( r ) + c j and show localized place fields (Fig. 1(G)). We measured the number of peaks in a unit’s output by counting the number of distinct contiguous areas containing pixels with at least 50% of the Springer