Extended Data Fig. 8: PLS-based decoding methods are robust to multiplicative gain modulation and common mode fluctuations in the neural ensemble dynamics and yield identical conclusions to regularized regression.

a, b, To test whether PLS analysis and dimensionality reduction might lead to underestimates of d′, we compared d′ values determined using an L2-regularized regression (L2RR) performed in the full space of neural responses (a) to those found by PLS analysis (b). The two methods yielded similar estimates of d′, which both saturated with increasing numbers of neurons. Plots show d′ values (mean ± s.d.) for neural responses within [0.83 s, 1.11 s] after stimulus onset, computed across 100 different randomly chosen subsets of neurons and visual stimulation trials (Extended Data Fig. 5b). For PLS analyses, we used half of the trials in each subset for decoder training and the other half for testing. For L2RR we used 90% of the trials in each subset to determine the regression vector and the other 10% to determine d′. We varied the regularization parameter, k, within [1, 10⁵] and used the maximum d′ value so obtained, as determined independently for each mouse, subset of neurons, and subset of trials (217–332 trials per stimulus condition in each of 5 mice). c–h, The conclusions of our study depend on comparisons of decoding performance between real and trial-shuffled datasets. Thus, we checked whether our PLS-based decoding methods would robustly detect information-limiting correlations in models in which such correlations were present but weak; avoid reporting information-limiting correlations in models lacking such correlations; and be robust to the potential presence of other strong sources of neural trial-to-trial variability—such as common mode fluctuations and multiplicative gain modulation—even when they make an order-of-magnitude greater contribution to neural variability than the information-limiting noise fluctuations. We studied these issues using two different computational models (Methods). For both models we plotted empirically determined (d′)² values as a function of the number of neurons in the ensemble. We compared determinations of (d′)² using PLS-based decoding and those made using L2RR to the actual ground truth values of (d′)² in each model. In each panel, the top and bottom plots show results for unshuffled and trial-shuffled datasets, respectively. Data points and error bars denote mean ± s.d. values across 30 different simulations. To examine the combined effects of information-limiting noise correlations and common mode fluctuations (c–f) we studied a model of neural ensemble responses in which the noise covariance matrix exhibited information-limiting noise correlations via a single eigenvector f, the eigenvalue of which grew linearly with the number of cells in the ensemble. In addition to this rank 1 component, we included a noise term that was uncorrelated between different cells, as well as a common mode fluctuation, yielding a noise covariance matrix with the form Σ* = σ²I + ε_commonJ + ε f^T f, where σ² = 1 is the amplitude of uncorrelated noise, I is the identity matrix, J is a rank 1 matrix of all ones, reflecting a common mode fluctuation, and f is the information-limiting direction, a vector that we chose randomly in each individual simulation from a multi-dimensional Gaussian distribution with unity variance in each dimension. The amplitude of information-limiting correlations was ε = 0.002, approximately matching the level observed in the experimental data. We chose the difference in the means of the two stimulus response distributions, Δμ, to be aligned with f (Fig. 3a) and to have a magnitude of 0.2 so that the asymptotic value of d′ for large numbers of cells approximately matched that of the data. We compared decoding results attained with and without the presence of the common mode fluctuations in the neural responses. In the version of the model without common mode fluctuations, we set ε_common to zero. In this case (c) both PLS- and L2RR-based decoders correctly detected the saturation of information in the real data but not in trial-shuffled datasets. (See Extended Data Fig. 10h, k for theoretical results showing how the accuracy of d′ estimates from PLS analysis depends on the numbers of neurons and experimental trials in this particular model.) To verify that our methods would not incorrectly report an information saturation when it was in fact absent, we next set ε = 0 and confirmed that in the absence of information-limiting noise correlations (d), neither decoder detected a saturation of information in the real or shuffled data. In the version of the model with common mode fluctuations, we set ε_common = 0.02, ten times the value of ε = 0.002. In this case (e), both PLS- and L2RR-based decoders correctly detected the information saturation in the real but not in the shuffled data. To verify that common mode fluctuations alone cannot induce an illusory saturation of information (f), we set ε = 0 while maintaining ε_common = 0.02 and confirmed that neither PLS- nor L2RR-based decoders reported an illusory information saturation. Overall, these results indicate that our methods accurately detect the presence of weak information-limiting correlations buried within common mode noise that can be an order of magnitude larger, without falsely detecting information-limiting correlations when they are absent. To study the possible effects of multiplicative gain modulation (g, h), we compared two versions of a model in which the responses of the V1 neural population either were or were not subject to a multiplicative stochastic gain modulation but were otherwise statistically equivalent. We modelled the V1 cell population as a set of Gabor filters (see Appendix section 5). In the model version with gain modulation, on each visual stimulation trial we multiplied the output of each Gabor filter by a randomly chosen factor, uniformly distributed between 50%–150%, the value of which was the same for all cells but varied from trial to trial. In the model version without gain modulation (g) both PLS- and L2RR-based decoders detected the information saturation in the real but not in the trial-shuffled datasets. When we added global gain modulation to the model (h) both decoders correctly found the information saturation in the real but not in the shuffled datasets.