Robust information propagation through noisy neural circuits
Fig 4
Family of optimal covariance matrices.
For all panels, green arrows indicate the signal direction, f′(s). Magenta ellipses indicate the noise in the first layer (with corresponding covariance matrix Σξ), and grey ellipses indicate the effective noise in the second layer (with corresponding covariance matrix Σy). (A) The covariance ellipse in the first layer has its long axis aligned with the signal direction; this configuration (which corresponds to differential correlations) optimizes information robustness for any distribution of second layer noise. (B) The covariance ellipse in the first layer does not have its long axis aligned with the signal direction. However, the covariance ellipse of the effective noise in the second layer, Σy, has the same shape as the covariance ellipse in the first. In this case, the blue “good” projection—which is aligned both with a low-variance direction of the first-layer distribution (magenta), and with the signal curve (green), and thus is relatively informative about the stimulus (see text)—is corrupted by relatively little noise at the second layer. This “matched” noise configuration is among those that optimize robustness to noise. The optimal family of covariance matrices interpolates between the configurations shown in panels A and B. (C) Again the covariance ellipse in the first layer does not have its long axis aligned with the signal direction. But now the “good” projection is heavily corrupted by noise at the second layer. In this configuration, all projections are substantially corrupted by noise at some point in the circuit, and thus relatively little information can propagate.