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The Sign Rule and Beyond: Boundary Effects, Flexibility, and Noise Correlations in Neural Population Codes

Figure 4

Heterogeneous neural population and violations of the sign rule with increasing correlation strength.

We consider signal encoding in a population of 20 neurons, each of which has a different dependence of its mean response on the stimulus (heterogeneous tuning curves shown in A). We optimize the coding performance of this population with respect to the noise correlations, under several different constraints on the magnitude of the allowed noise correlations. Panel (B) shows the resultant – optimal given the constraint – values of OLE information , with different noise correlation strengths (blue circles). The strength of correlations is quantified by the Euclidean norm (Eq. (18)). For comparison, the red crosses show information obtained for correlations that obey the sign rule (in particular, pointing along the gradient giving greatest information for weak correlations); this information is always less than or equal to the optimum, as it must be. Note that correlations that follow the sign rule fail to exist for large correlation strengths, as the defining vector points outside of the allowed region (spectrahedron) beyond a critical length (labeled (ii)). For correlation strengths beyond this point, distinct optimized noise correlations continue to exist; the information values they obtain eventually saturate at noise-free levels (see text), which is for the example shown here. This occurs for a wide range of correlation strengths. Panel (C) shows how well these optimized noise correlations are predicted from the corresponding signal correlations (by the sign rule), as quantified by the statistic (between 0 and 1, see Fig. 5). For small magnitudes of correlations, the values are high, but these decline when the noise correlations are larger.

Figure 4