Relating Divisive Normalization to Neuronal Response Variability

Abstract

Cortical responses to repeated presentations of a sensory stimulus are variable. This variability is sensitive to several stimulus dimensions, suggesting that it may carry useful information beyond the average firing rate. Many experimental manipulations that affect response variability are also known to engage divisive normalization, a widespread operation that describes neuronal activity as the ratio of a numerator (representing the excitatory stimulus drive) and denominator (the normalization signal). Although it has been suggested that normalization affects response variability, we lack a quantitative framework to determine the relation between the two. Here we extend the standard normalization model, by treating the numerator and the normalization signal as variable quantities. The resulting model predicts a general stabilizing effect of normalization on neuronal responses, and allows us to infer the single-trial normalization strength, a quantity that cannot be measured directly. We test the model on neuronal responses to stimuli of varying contrast, recorded in primary visual cortex of male macaques. We find that neurons that are more strongly normalized fire more reliably, and response variability and pairwise noise correlations are reduced during trials in which normalization is inferred to be strong. Our results thus suggest a novel functional role for normalization, namely, modulating response variability. Our framework could enable a direct quantification of the impact of single-trial normalization strength on the accuracy of perceptual judgments, and can be readily applied to other sensory and nonsensory factors.SIGNIFICANCE STATEMENT Divisive normalization is a widespread neural operation across sensory and nonsensory brain areas, which describes neuronal responses as the ratio between the excitatory drive to the neuron and a normalization signal. Normalization plays a key role in several important computations, including adjusting the neuron's dynamic range, reducing redundancy, and facilitating probabilistic inference. However, the relation between normalization and neuronal response variability (a fundamental aspect of neural coding) remains unclear. Here we develop a new model and test it on primary visual cortex responses. We show that normalization has a stabilizing effect on neuronal activity, beyond the known suppression of firing rate. This modulation of variability suggests a new functional role for normalization in neural coding and perception.

Keywords: divisive normalization; modeling; neuronal variability; single-trial inference; visual cortex.

Figure 1.

RoG model. a, The RoG model describes neuronal responses as the ratio of two stimulus-driven signals: the numerator N and denominator, or normalization signal, D. Both N and D are corrupted by additive Gaussian noise with Poisson-like variability (Eq. 1.9). b, Comparison of the approximation we derived for the RoG mean and variance (Eq. 1.4) and the true value (estimated from 1 million simulated trials), across 1000 experiments. Each experiment uses a different set of model parameters (i.e., corresponding to different neurons and stimuli), and each trial is a random draw from the corresponding distribution. Red triangles represent average percentage difference between true and approximated mean (left) or variance (right). c, RoG variance versus mean across 1000 simulated experiments. Model parameters for the simulations in b and c are drawn uniformly in the following intervals: μ_N ∈ [0,100]; μ_D ∈ [0.5, 1.5]; α_N = 1; α_D = 0.01; β_N = β_D ∈ [1,1.5]; ρ ∈ [0,0.5]; μ_η = 0; σ_η² = 0.1μ_N/μ_D.

Figure 2.

RoG model captures contrast modulation of firing rate and variability. a, b, Mean spike count (top), variance (middle), and Fano factor (bottom) as a function of contrast for two example neurons. Circles represent data. Black lines indicate 95% CI. Blue lines indicate RoG model fits. Green lines indicate modulated Poisson model fits. c, Histogram of percentage difference between Fano factors at high versus low contrast. d, Geometric mean (black lines) and 95% CI (shaded areas) of the Fano factor across neurons, as a function of contrast. Different colors correspond to different spike-count windows. e, Percentage difference in cross-validated goodness of fit between the RoG and the modulated Poisson models. For details on model fitting and evaluation, see Materials and Methods.

Figure 3.

Normalization reduces response variability. a, Relation between the normalization strength (abscissa) and mean (ordinate, top), variance (middle), or Fano factor (bottom) in the RoG model. Columns represent different values of the exponent of the relation between mean and variance of the denominator. Different colors represent different values of the proportionality factor between mean and variance of the denominator. Other parameters are set to μ_N = 35; μ_D ∈ [0.1,10]; α_N = 1; β_N = 1; ρ = 0; μ_η = 0; σ_η² = 0.1μ_N/μ_D. b–d, Relation between the z-scored normalization strength and the z-scored response mean (b), variance (c), or Fano factor (d) in the data. z scoring was performed across all neurons, but separately within each contrast condition. Gray symbols represent individual neurons and contrasts. Blue symbols represent average across neurons and contrasts. Shaded areas represent 95% CI. e–g, Similar to b–d, but with z scoring performed across all neurons and contrast levels with similar firing rate. Firing rate bins were logarithmically spaced between the minimum and maximum firing rates measured. Black symbols represent average across neurons and contrasts. Shaded areas represent 95% CI

Figure 4.

Inference of single-trial normalization strength in the RoG model. a, b, Comparison of the true and inferred single-trial normalization strength, across 100 trials of two simulated experiments. Error bars: standard deviation of the inferred values. c, Histogram of Spearman correlation coefficients between true and inferred values, across 10,000 simulated experiments each with 100 trials. The model was formulated as in Methods eq. (1.16) for the contrast response function, with parameters drawn from the uniform distribution in the following intervals: R_max ∈ [10,100]; ϵ ∈ [15,25]; β_N = β_D ∈ [1.5,2]; ρ = 0; μ_η = 0; σ_η² = 0, with contrast between 20 and 50%, and with α_N, α_D set to enforce a Fano factor of 1 at a 75% contrast. d, Correlation between true and inferred values (ordinate) as a function of the ratio between the variances of the denominator and numerator in the RoG model (abscissa). Correlation coefficients smaller than the vertical line (c) or horizontal line (d) are not statistically significant (p > 0.05). e, Difference between true and inferred values of the denominator, expressed as a fraction of the true value.

Figure 5.

Reduced response variability during epochs with strong normalization. a, Each symbol denotes, for one neuron and one contrast condition, the mean spike count across trials with inferred strong (ordinate) versus trials with inferred weak (abscissa) normalization signal. Only neurons with large across-trials variance of the normalization signal (at least 1% of the variance of the numerator) and trials with at least one spike are included. b, C, Same as a, but for response variance (b) and Fano factor (c). d, e, f, Same as a, b, c, but using as a proxy of single-trial normalization strength the total activity of a population of simultaneously recorded neurons.

Figure 6.

Reduced variability during epochs with strong inferred normalization is robust to choice of threshold. a, Average spike count (ordinate) across neurons and contrast conditions for which the ratio between denominator variance and numerator variance was larger than the threshold (abscissa). Averages are computed over the subset of trials with strong (blue) or weak (red) inferred normalization signal. Shaded areas: 95% CI Black vertical line: the threshold used for Fig. 5. b, c, Same as (a) but for the spike count variance (b) and Fano factor (c). d–f, Same as a–c, but including neurons and contrast conditions for which the ratio between denominator variance and numerator variance was smaller than the threshold (abscissa). The largest threshold in d–f and the smallest threshold in a–c correspond to including all neurons and contrast conditions.

Figure 7.

Normalization is shared between neurons and reduces noise correlation. a, Histogram of correlation coefficients of the inferred single-trial denominator for pairs of simultaneously recorded neurons with similar tuning. Gray bars: pairs with non-significant correlation; black bars: pairs with significant correlation. Triangle: average correlation across all pairs. b, Black symbols and gray shaded area: Median noise correlation and 95% CI, as a function of the average normalization strength of each pair. For each pair, the inferred values of D of each neuron were z-scored across trials; then trials were sorted by the average z-scored D of the pair. Each point in the plot represents averages computed in windows of 12 trials. Yellow area: 95% CI for a control with trial order shuffled independently for each neuron, to remove noise correlations. c, Histogram of noise correlation coefficients during trials with large (blue) or small (red) inferred normalization signal. These estimates of correlations are based on few trials (as little as 5 in some cases, see related text), hence the large range of values. Triangles denote group medians.