The Sign Rule and Beyond: Boundary Effects, Flexibility, and Noise Correlations in Neural Population Codes
In our larger neural population, the sign rule governs optimal noise correlations only when these correlations are forced to be very small in magnitude; for stronger correlations, optimized noise correlations have a diverse structure.
Here we investigate the structure of the optimized noise correlations obtained in Fig. 4; we do this for three examples with increasing correlation strength, indicated by the labels in that figure. First (ABC) show scatter plots of the noise correlations of the neural pairs, as a function of their signal correlations (defined in Methods Section “Defining the information quantities, signal and noise correlations”). For each example, we also show (DEF) a version of the scatter plot where the signal correlations have been rescaled in a manner discussed in Section “Parameters for Fig. 1, Fig. 2 and Fig. 3”, that highlights the linear relationship (wherever it exists) between signal and noise correlations. In both sets of panels, we see the same key effect: the sign rule is violated as the (Euclidean) strength of noise correlations increases. In (ABC), this is seen by noting the quadrants where the dots are located: the sign rule predicts they should only be in the second and fourth quadrants. In (DEF), we quantify agreement with the sign rule by the statistic. Finally, (GHI) display histograms of the noise correlations; these are concentrated around 0, with low average values in each case.