The geometry of correlated variability leads to highly suboptimal discriminative sensory coding

Abstract

The brain represents the world through the activity of neural populations; however, whether the computational goal of sensory coding is to support discrimination of sensory stimuli or to generate an internal model of the sensory world is unclear. Correlated variability across a neural population (noise correlations) is commonly observed experimentally, and many studies demonstrate that correlated variability improves discriminative sensory coding compared to a null model with no correlations. However, such results do not address whether correlated variability is optimal for discriminative sensory coding. If the computational goal of sensory coding is discriminative, than correlated variability should be optimized to support that goal. We assessed optimality of noise correlations for discriminative sensory coding in diverse datasets by developing two novel null models, each with a biological interpretation. Across datasets, we found that correlated variability in neural populations leads to highly suboptimal discriminative sensory coding according to both null models. Furthermore, biological constraints prevent many subsets of the neural populations from achieving optimality, and subselecting based on biological criteria leaves red discriminative coding performance suboptimal. Finally, we show that optimal subpopulations are exponentially small as the population size grows. Together, these results demonstrate that the geometry of correlated variability leads to highly suboptimal discriminative sensory coding.NEW & NOTEWORTHY The brain represents the world through the activity of neural populations that exhibit correlated variability. We assessed optimality of correlated variability for discriminative sensory coding in diverse datasets by developing two novel null models. Across datasets, correlated variability in neural populations leads to highly suboptimal discriminative sensory coding according to both null models. Biological constraints prevent the neural populations from achieving optimality. Together, these results demonstrate that the geometry of correlated variability leads to highly suboptimal discriminative sensory coding.

Keywords: correlated variability; neurophysiology; null models; sensory coding.

Fig. 1. Correlated variability is a pervasive neural phenomenon.

a, b. Mean activity as a function of the angle of visually presented oriented bars (larger red circles) and trial-to-trial variability (small dots, with small x-axis offsets for visualization) for two RGC neurons (corresponding to Neuron 1 and Neuron 2, respectively, in e). c. Illustration of mean stimulus response curve (black line), less detrimental correlated variability (blue ellipse), and more detrimental correlated variability (orange ellipse) for two model neurons. The large black dot is the mean stimulus response corresponding to the covariances. The small black dots are the mean responses for neighboring stimuli. d-l. Each row refers to a different experimental dataset, while columns refer to an aspect of the dataset. d-f. Calcium imaging recordings from mouse retinal ganglion cells in response to drifting bars. g-i. Single-unit spike c ounts recorded from primary visual cortex of macaque monkey in response to drifting gratings. j-l. Micro-electrocorticography recordings (z-scored Hγ response) from rat primary auditory cortex in response to tone pips at varying frequencies. First column (d, g, j) depicts the recording region and stimulus for each dataset. Second column (e, h, k) shows the activity of two random RGCs/neurons/electrodes in the population to two neighboring stimuli. Individual points denote the unit activity on individual trials, while covariance ellipses denote the noise covariance ellipse at 2 standard deviations. Third column (f, i, l) plots the distribution of pairwise noise correlations, calculated for each pair of units across stimuli.

Fig. 2. Novel methods for assessing the optimality of neural codes.

a-c. Null models of correlated variability. Solid, purple ellipses denote idealized trial-to-trial variability observed about the mean stimulus activity (solid point). Samples from the null models are depicted by dashed ellipses. a. The shuffle null model maintains per-neuron variance and samples correlations near zero. b. The uniform correlation null model maintains per-neuron variance and samples uniform correlations. c. The factor analysis null model combines a fixed private variability (estimated from the experimental data, left gray inset) with shared variability (right teal inset) that can be rotated to form null samples (dash styles are consistent between the teal shared variabilities in the inset and the purple null samples in the main panel). d. For a synthetic 2d dataset, the LFI for the uniform correlation (UC) parameterization as a function of the pairwise correlation, $\rho$, is shown at the top, the bottom plots are the covariance and samples as a function of $\rho$. e. For a synthetic 2d dataset, the LFI for the factor analysis (FA) parameterization as a function of the rotation angle, θ, is shown at the top, the bottom plots are the covariance and samples as a function of θ. f. To calculate an observed LFI or percentile under a null model, d units were randomly subsampled from the population. Then, two neighboring stimuli, $s_1$ and $s_2$, were chosen. The subpopulation and stimulus pairing together constitute a pair of design matrices $\left[{\mathbf{X}}_{s_1}^d,{\mathbf{X}}_{s_2}^d\right]$. These matrices are the inputs into a LFI calculation or null model analysis and form the basis for the distributions of calculated quantities. g. Responses in the retinal data for the depicted stimulus pairing (colors) from f.

Fig. 3. The geometry of correlated variability leads to suboptimal discriminative sensory coding.

Each column corresponds to one of the datasets. Color legend is preserved across all panels. a-c. The median LFI is plotted (solid lines, log-scale y-axis) as a function of the subpopulation size (x-axis) for the observed correlated variability and null model samples (colors in legend). Shaded regions indicate the 95% CI of the median LFI (note that CIs are often comparable to the median line width). d-f. Histograms of null LFIs are shown for the shuffle, uniform correlation, and factor analysis null models for one subpopulation and stimulus for each dataset. The observed LFI is denoted by the black vertical line in each plot. Null percentiles for each null model are reported. g-i. Median observed subpopulation and stimulus null percentiles are shown (solid lines) as a function of subpopulation size, for each dataset and null model. S haded regions indicate the 95% CI of the median observed null percentile (note that CIs are often comparable to the median line width). Black dashed lines divide optimal (Opt), near-chance (NC), and suboptimal (Sub) regions.

Fig. 4. Optimal correlated variability for sensory discrimination is typically biologically inaccessible.

Each column corresponds to a separate dataset. 2d histograms are plotted with a log-density color scale with shared colorbar. Color legend in i is shared across panels. a-c. 2d-histogram across subpopulations and stimuli of the observed null percentile under the FA null model versus the absolute log-ratio of the observed and FA-optimal covariance Fano factors for d = 3. Blue line is the median binned null percentile as a function of the absolute log-ratio of observed and FA-optimal covariance Fano factors. d-f. 2d-histogram across subpopulations and stimuli of the null percentile under the FA null model versus the absolute difference of negative densities (ND) of the observed and FA-optimal covariance Fano factors for d = 3. Red line is the median binned null percentile as a function of the absolute difference in NDs. g-i. The Spearman correlation coefficient between the null percentile and absolute log-FF ratio or absolute difference of NDs, respectively is shown as a function of subpopulation size. Dashed black line indicates zero correlation.

Fig. 5. Optimal subpopulations are exponentially small.

Color legend in a is shared across panels. a-c. For the uniform correlation and FA null model, subpopulations and stimuli were subselected to maximize the units’ tuning (solid lines, highest 10% subselected). Additionally, for the retina and PAC datasets, subpopulations and stimuli were subselected to minimize the average pairwise distance between the RGC RoIs in a subpopulation and stimulus (dashed lines, lowest 10% subselected). The median percentiles are shown as a function of dimension. Black dashed lines indicate the 1/3 and 2/3 null percentile range. Shaded regions indicate the 95% CI of the median percentiles. d-f. For each subpopulation size, the largest possible fraction of subpopulation and stimulus percentiles such that their median is ≥ 2/3 is plotted. Shaded regions indicate 95% CI. For the uniform correlation null model, subpopulation sizes where no samples e xceeded the 2/3 threshold are not plotted. Black dashed line indicates the optimal fraction if null percentiles were drawn from a uniform distribution. Gray dotted line indicates the minimum non-zero optimal fraction that can be estimated due to finite sampling.