The name is absent

To show that place fields can be derived from grid-cells by Sparsification we
simulated a linear two-layer network. The input units were IOO simulated grid-
cells of a virtual rat with activity patterns synthesized by Gaussians arranged
on a hexagonal grid (Fig. 1 A). Some positional 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 function 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(~(t')'). To achieve sparseness we applied
independent component analysis (ICA) Hyvarinen (1999b) on a set of 200.000
time points from this input 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 (Hyvarinen, 1999a) or sim-
ply 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 Cj
were added to ensure nonnegative values. This yielded an affine transformation
with matrix T producing 100 output signals y_j∙(t) := P_i Tj_ix_i(t) + c_j∙ that are
maximally independent and significantly sparser than the input signals (kurtosis
increased on average from 2.9 for the input units to 28.6 for the output units).
The output-unit activities as a function of location are pj(r) := P_i Tj_ig_i(r) + Cj
and show localized place fields (Fig. 1, B). 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 unit’s maximum activity. A large proportion of
output units (76%) show a single spot of activity (Fig. 1 B, units 1, 25, 50, 75),
some units (5%) show few spots (Fig. 1 B, unit 87), both being consistent with

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