the patterns of physiological place-cells. Only few output units (19%) show
patterns of activity without clear structure (Fig. 1 B, unit 100). The size of
the resulting place fields is similar for most units and comparable to the size
of the smallest grid-cell fields, but it also depends on the number of grid cell
inputs: more inputs lead to more localized output fields, while too few inputs
can increase the number of fields per output unit.
There are different ways of achieving sparseness and localized place fields.
We have used ICA here and have obtained similar results by maximimizing peak
activity; Competetive learning schemes would presumably also work. There are
other linear transformations, however, that do not lead to localized place fields.
As controls we have applied random mixtures, principal component analysis
(PCA), and slow feature analysis (SFA, Wiskott and Sejnowski (2002)) to the
grid cell input. The latter has been chosen because Wyss et al. (2006) have
presented a model based on the slowness principle that was able learn localized
place cells. As one would expect with random rotations of the input, the re-
sults retain some grid structure but are less regular than the input (Fig. 1 C),
but no unit has one single or two peaks of activity. With PCA the first units
(i. e. those with highest variance) are highly structured and have large am-
plitudes, much like the grid cells themselves, while the later low-variance units
have low amplitudes and are noise-like (three representative examples are shown
in Fig. 1 D). None of these units had a single or two peaks of activity. From the
temporal slowness objective we’d expect patterns with low spatial frequencies
first, and high-frequency поп-localized patterns later, when outputs are sorted
by slowness (Fig. 1 E). Only 2% of these outputs have one or two peaks of
activity. None of these three alternative linear transforms (Fig. 1 C-E) lead to
localized place fields, which in case of SFA is somewhat inconsistent with the
results from Wyss et al. (2006).