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Information-limiting correlations

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Abstract

Computational strategies used by the brain strongly depend on the amount of information that can be stored in population activity, which in turn strongly depends on the pattern of noise correlations. In vivo, noise correlations tend to be positive and proportional to the similarity in tuning properties. Such correlations are thought to limit information, which has led to the suggestion that decorrelation increases information. In contrast, we found, analytically and numerically, that decorrelation does not imply an increase in information. Instead, the only information-limiting correlations are what we refer to as differential correlations: correlations proportional to the product of the derivatives of the tuning curves. Unfortunately, differential correlations are likely to be very small and buried under correlations that do not limit information, making them particularly difficult to detect. We found, however, that the effect of differential correlations on information can be detected with relatively simple decoders.

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Figure 1: The effect of correlations on a population code with translation invariant tuning curves.
Figure 2: Decorrelation does not necessarily increase information.
Figure 3: Differential correlations induced by a shifting hill.
Figure 4: Shared connections do not necessarily induce information limiting correlations.
Figure 5: Differential correlations.
Figure 6: Small differential correlations can have a large effect on information.
Figure 7: Estimating information from neuronal populations with differential correlations (Supplementary Modeling).

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Acknowledgements

R.M.-B. was supported by the Ramón y Cajal Spanish Award RYC-2010-05952 and by the Marie Curie FP7-PEOPLE-2010-IRG grant PIRG08-GA-2010-276795. X.P. was supported in part by US National Institutes of Health grant T32DC009974. P.L. was supported by the Gatsby Charitable Foundation. A.P. was supported by a grant from the Swiss National Science Foundation (#31003A_143707), and a grant from the Human Frontier Science Program.

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Contributions

R.M.-B., J.B., P.L. and A.P. conceived the project. R.M.-B., J.B., I.K., X.P., P.L. and A.P. developed the theory and wrote the manuscript.

Corresponding author

Correspondence to Alexandre Pouget.

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The authors declare no competing financial interests.

Integrated supplementary information

Supplementary Figure 1 Effect of observation time.

Output information for a non-leaky integrate and fire network as a function of network size, N for several values of the observation time, T :100 ms (pink), 500 ms (purple), 2 seconds (red), and 10 seconds (green). The black solid line is the information in the input, Eq. (25).Parameters correspond to those of the network with high correlations (Table 1 with JIE = 5.4 and JII = 8.9); this is the same network we used to make Fig. 2d of the main text. The open green circles here correspond to the ones in Fig. 2d of the main text; slight differences between the open green circles in this figure and in Fig. 2d are because we used a different seed in the random number generator in the two sets of simulations; those differences are within error bars. Small observation times reduce information, but the effect diminishes as the number of neurons increases.

Supplementary Figure 2 Visualizing information limiting correlations.

All figures lie in firing rate space, here shown as two (out of N) dimensional. Red indicates the noise; in panels a, c and d, it denotes the noise covariance ellipse. We are assuming that one of the eigenvalues of the noise covariance ellipse is O(N) and the rest are O(1).

a. When the angle between f′(s) and the long (i.e, O(N)) direction of the noise covariance ellipse is O(1/N1/2), the correlations are information limiting.

b. Realistic information limiting noise tracks f(s). The thin black curve is the covariance ellipse from panel a.

c. When there are information limiting correlations, a suboptimal linear decoder, w, predicts that information saturates with N even if the angle between w and the long direction of the noise covariance ellipse is O(1).

d. When there are no information limiting correlations, so that information is O(N), the same linear decoder still predicts that information saturates with N. To observe the O(N) scaling, w would have to be nearly perpendicular to the long direction of the covariance ellipse.

Supplementary Figure 3 Empirically estimated information using ridge regularization.

The black lines on the right show the upper bound on information given by 1/ò (see Eq. (64)); the dashed blue lines are the true information. Different colors indicate different number of samples (M) used to estimate information: from top to bottom, M is 400, 800, 1600, 2400, 3200, and 4000. On the left, differential correlations are not present and information grows with the number of neurons. On the right, differential correlations are present and information saturates at the black line.

First row: The ridge parameter, λ, is fixed at 0.1.

Second row: The ridge parameter is chosen so that the effective number of degrees of freedom of the resulting covariance matrix is 90\% of the maximum number of degrees of freedom possible (see Eq. (107)); typically λ = 0.25.

Third row: The inverse of the variance of a decoder regularized using ridge regression and optimized by minimizing cross-validation error.

Supplementary information

Supplementary Text and Figures

Supplementary Figures 1–3 and Supplementary Modeling (PDF 2065 kb)

Supplementary Methods Checklist (PDF 346 kb)

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Moreno-Bote, R., Beck, J., Kanitscheider, I. et al. Information-limiting correlations. Nat Neurosci 17, 1410–1417 (2014). https://doi.org/10.1038/nn.3807

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