The Sign Rule and Beyond: Boundary Effects, Flexibility, and Noise Correlations in Neural Population Codes
For fixed neuronal responses variances and tuning curves, we compute coding performance – quantified by information values – for different values of the pair-wise noise correlations. To be physically realizable, the correlation coefficients must form a positive semi-definite matrix. This constraint defines a spectrahedron, or a swelled tetrahedron, for the cells used. The colors of the points represent information values. With different parameters and (see values in Methods Section “Details for numerical examples and simulations”), the optimal configuration can appear at different locations, either unique (A) or attained at multiple disjoint places (B), but always on the boundary of the spectrahedron. In both panels, plot titles give the maximum value of attained over the allowed space of noise correlations, and the value of that would obtained with the given tuning curves, and perfectly deterministic neural responses. This provides an upper bound on the attainable (see text Section “Noise cancellation”). Interestingly, in panel (A), the noisy population achieves this upper bound on performance, but this is not the case in (B). Details of parameters used are in Methods Section “Details for numerical examples and simulations”.