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Encoding information about continuous variables using noisy computationalunits is a challenge; nonetheless, asymptotic theory shows that combiningmultiple periodic scales for coding can be highly precise despite thecorrupting influence of noise (Mathis et al., Phys. Rev. Lett. 2012). Indeed,cortex seems to use such stochastic multi-scale periodic `grid codes' torepresent position accurately. We show here how these codes can be read outwithout taking the asymptotic limit; even on short time scales, the precisionof neuronal grid codes scales exponentially in the number N of neurons. Doesthis finding also hold for neurons that are not statistically independent? Toassess the extent to which biological grid codes are subject to statisticaldependencies, we analyze the noise correlations between pairs of grid codeneurons in behaving rodents. We find that if the grids of the two neurons alignand have the same length scale, the noise correlations between the neurons canreach 0.8. For increasing mismatches between the grids of the two neurons, thenoise correlations fall rapidly. Incorporating such correlations into apopulation coding model reveals that the correlations lessen the resolution,but the exponential scaling of resolution with N is unaffected.

Multi-Scale Codes in the Nervous System: The Problem of Noise Correlations and the Ambiguity of Periodic Scales