Mathis, Alexander; Herz, Andreas V. M.; Stemmler, Martin B.
Optimal Population Codes for Space:
Grid Cells Outperform Place Cells.
In: Neural Computation, Vol. 24, No. 9: pp. 2280-2317
Rodents use two distinct neuronal coordinate systems to estimate their
position: place fields in the hippocampus and grid fields in the entorhinal cortex. Whereas place cells spike at only one particular spatial lo-
cation, grid cells fire at multiple sites that correspond to the points of
an imaginary hexagonal lattice. We study how to best construct place
and grid codes, taking the probabilistic nature of neural spiking into account. Which spatial encoding properties of individual neurons confer the highest resolution when decoding the animal’s position from the neuronal population response? A priori, estimating a spatial position from a
grid code could be ambiguous, as regular periodic lattices possess translational symmetry. The solution to this problem requires lattices for grid
cells with different spacings; the spatial resolution crucially depends on
choosing the right ratios of these spacings across the population. We compute the expected error in estimating the position in both the asymptotic
limit, using Fisher information, and for low spike counts, using maximum
likelihood estimation. Achieving high spatial resolution and covering a
large range of space in a grid code leads to a trade-off: the best grid code
for spatial resolution is built of nested modules with different spatial periods, one inside the other, whereas maximizing the spatial range requires
distinct spatial periods that are pairwisely incommensurate. Optimizing
the spatial resolution predicts two grid cell properties that have been
experimentally observed. First, short lattice spacings should outnumber
long lattice spacings. Second, the grid code should be self-similar across
different lattice spacings, so that the grid field always covers a fixed fraction of the lattice period. If these conditions are satisfied and the spatial
“tuning curves” for each neuron span the same range of firing rates, then
the resolution of the grid code easily exceeds that of the best possible
place code with the same number of neurons.