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Efficient population coding depends on stimulus convergence and source of noise
Fig 6
Information landscape for the independent- and lumped-coding channels undergoes different phase transitions around critical noise levels.
A. Top: Optimal thresholds of the independent-coding channel for a population of two neurons as a function of output noise R. Bottom: Corresponding eigenvalues of the Hessian of the information landscape with respect to thresholds. At the critical noise value Rcrit ≈ 0.396 at which the threshold bifurcation occurs (vertical dashed line) one eigenvalue approaches zero. B. Information landscape Im(θ1, θ2) for the three output noise levels R indicated by arrows in A. Top: For R > Rcrit, there are two equal global maxima. Middle: At R = Rcrit, the eigenvectors of the Hessian are shown and scaled by the corresponding eigenvalue (the eigenvector with the smaller eigenvalue, 
, was artificially lengthened to show its direction). At the critical noise value the information landscape locally takes the form of a ridge. Bottom: For R < Rcrit, there is one global maximum, meaning that the optimal thresholds are equal (bottom). C. The mutual information as a function of the line x in (θ1, θ2) space connecting the two maxima in B. Top: For R > Rcrit (low noise), there are two inflection points (dashed vertical lines) with zero curvature along the line x. The point with equal thresholds corresponds to a local minimum. Middle: At R = Rcrit, the two maxima, the minimum, and the two inflection points merge into one point, thus the curvature is zero. Bottom: For R < Rcrit, there is a single global maximum with a negative curvature. D. As in A but for a population with N = 3 neurons. E. As in A but for the lumped-coding channel. Both the optimal thresholds and the eigenvalues show a discontinuity at the critical noise level. F. Information landscape as in B for the lumped-coding channel and noise values indicated by arrows in E. Local optima are shown in cyan, global ones in red. G. Similar to C for the lumped-coding channel. Here the abscissa denotes the (non-straight) path connecting the three optima in F. H. Illustration of discontinuous threshold bifurcations, where the global maximum at θ1 ≠ θ2 at low noise (red, solid) becomes a local maximum for high noise (cyan, solid), while θ1 = θ2 (dashed) becomes global. As their respective derivatives are different, there is a discontinuity in the first derivative when only taking the global maximum into account (red lines), corresponding to a first-order phase transition.
doi: https://doi.org/10.1371/journal.pcbi.1008897.g006