  Encoding information about continuous variables using noisy computational
units is a challenge; nonetheless, asymptotic theory shows that combining
multiple periodic scales for coding can be highly precise despite the
corrupting influence of noise (Mathis et al., Phys. Rev. Lett. 2012). Indeed,
cortex seems to use such stochastic multi-scale periodic `grid codes' to
represent position accurately. We show here how these codes can be read out
without taking the asymptotic limit; even on short time scales, the precision
of neuronal grid codes scales exponentially in the number N of neurons. Does
this finding also hold for neurons that are not statistically independent? To
assess the extent to which biological grid codes are subject to statistical
dependencies, we analyze the noise correlations between pairs of grid code
neurons in behaving rodents. We find that if the grids of the two neurons align
and have the same length scale, the noise correlations between the neurons can
reach 0.8. For increasing mismatches between the grids of the two neurons, the
noise correlations fall rapidly. Incorporating such correlations into a
population coding model reveals that the correlations lessen the resolution,
but the exponential scaling of resolution with N is unaffected.
