Ethics statement

All experiments on animals were conducted with approval of the Animal Care and Use Committee of the University of California, Berkeley.

Generative model—pairwise Ratio of Gaussians (RoG)

Here we describe in detail the RoG model and derive the Gaussian approximation to the ratio variable. Note that our RoG is entirely different from those used in [56] and in [48], despite the same acronym: [56] refers to the distribution obtained from the ratio between two Gaussian distributions, whereas we refer to the random variable that results from the ratio of two Gaussian variables. [48] does not refer to probability distributions at all, but rather to a surround suppression model in which the center and surround mechanisms are characterized by Gaussian integration over their respective receptive fields, while the RoG considered here is a model of neural covariability in general.

We build from the standard normalization model [35, 54] which computes the across-trial average neural response (e.g., firing rate) as the ratio between a driving input to a neuron (N) and the summed input from a pool of nearby neurons (D): (1) Where f_÷ is the division operator; this functional notation is convenient for later derivations in which we consider the derivative of division. Our goal is to model the joint activity of pairs of neurons, so we extend the normalization model by considering two model neurons R₁, R₂. Since we are interested in trial-to-trial variability, we assume that a pair of neural responses R_t = (R₁, R₂)_t on a single trial t can be written as the element-wise ratio of two Gaussian random vectors, N_t = (N₁, N₂)_t and D_t = (D₁, D₂)_t, with additive Gaussian random noise η_t = (η₁, η₂)_t to capture the residual (i.e., stimulus-independent) variability.

As detailed further below, the numerators of two neurons can be correlated, and similarly for the denominators. In general, there can be correlations between the numerators and denominators (e.g., (N₁, D₂) may be correlated), requiring us to consider the joint, four-dimensional Gaussian distribution for the vector (N_t, D_t). However, in this paper we consider the simpler model in which N_t and D_t are independent and are each distributed according to their respective two-dimensional Gaussian distributions. This assumption allows for simplified mathematical derivations and is supported by our previous work which found that including a parameter for the correlation between N and D caused over-fitting to single-neuron data [53]. However, we have also derived the equations for the case that numerators and denominators are correlated (see S1 Text), and implemented them in the associated code toolbox, so that interested researchers can test if their data warrant the inclusion of those additional free parameters.

We therefore write the generative model for the pairwise RoG as: (2) Where f_÷ is applied element-wise, μ_N, μ_D are the two-dimensional vectors of means of the numerator and denominator, respectively, Σ_N, Σ_D are the respective 2 × 2 covariance matrices, and (μ_η, Σ_η) is the mean and covariance matrix for the residual component of the model.

For the independent RoG, the ratio variable in general follows a Cauchy distribution whose moments are not well defined. To fit the model, we used the result that when the denominator has negligible probability mass at values less than or equal to zero, the ratio distribution can be approximated by a Gaussian distribution with mean and variance that can be derived from a Taylor expansion [57–60]. This assumption is justified since the denominator is the sum of the non-negative responses from a pool of neurons [36] and is therefore unlikely to attain values less than or equal to zero.

For the pairwise extension, we can use the multivariate delta method (an application of a Taylor expansion) to compute the mean and covariance for the joint distribution of ratio variables [61] under the assumption that μ_D > 0. We note that the true distribution of the ratio of bivariate or multivariate Gaussians vectors is unknown (although there is some work on ratios of complex Gaussian variables [62, 63]) and has higher-order statistics (e.g., skewness, kurtosis) that are not well approximated by an equivalent Gaussian. In this paper, we are interested in modeling the noise covariance as this is the most widely studied statistic in the field, and we show that the approximations we derive are very accurate (see Fig 1). Future work could extend the model to account for these statistics by using higher-order terms in the Taylor expansion or a non-Gaussian copula.

To derive equations for the mean and covariance of the pairwise RoG, we use a Taylor expansion around the point (μ_N, μ_D): (3)

Using only the first order terms, we derive expressions for the mean and covariance matrix of the RoG: (4) (5)

Note that the variance of the denominator influences the mean of the ratio variable through a second-order term, hence it does not appear in Eq (4) (see [58] for the second-order Taylor expansion for the mean of a ratio variable). From Eq (5), we can obtain expressions for the variance of each neuron in the pair and their covariance and correlation. First, we adopt the following notation to simplify the equations: let and let . Then: (6) (7)

Eq (7) is commonly referred to as the formula for “spurious” correlation of ratios found when comparing ratios of experimental variables [64], and we further generalize this in S1 Text. To the extent that tuning similarity between neurons reflects similarity in the driving inputs, and that those driving inputs are variable, neurons with more similar tuning would have larger ρ_N, which in turn implies larger noise correlations according to Eq (7). This is consistent with the widespread empirical observation that signal correlations and noise correlations are correlated [3].

Parametrization of the pairwise RoG for contrast responses

In the form described so far, the pairwise RoG has 10 stimulus-dependent parameters and 5 stimulus-independent parameters for the additive noise. For any stimulus condition, there are only five relevant measurements that can be derived from the neural data (the response means and variances for each neuron in a pair, and their correlation), so the model is over-parametrized. Therefore, to apply the RoG to neural data, we need to reduce the total number of parameters.

The generality of this model provides a procedure for converting a standard normalization model (i.e., a model for the mean response) into a RoG model that specifies both mean and (co)-variance. In this paper, we use the example of contrast-gain control, which has been widely used to model the response of single neurons and neural populations to visual input with varying contrast [36, 65–67]. By adapting such a model, we can reduce the stimulus dependence of the means of the numerator and denominator . In the contrast-gain control model, the neural response as a function of contrast c (0 − 100%) is computed as a “hyperbolic ratio” [36, 65]: (8) Where R^max is the maximum response rate, ϵ is the semi-saturation constant (the contrast at which R(ϵ) = R^max/2) to prevent division by 0, and R₀ is the spontaneous activity of the neuron (the response at 0% contrast). We can convert this standard model into an RoG by setting the mean of the numerator and denominator in the RoG to the numerator and denominator in this equation: (9)

By using this functional form, we can substitute the stimulus-dependent parameters of the RoG () with the stimulus-independent parameters . Another model simplification is to assume that individual neural variability and mean neural response are related by a power function as has been observed in the visual cortex [68–70]: (10)

This parametrization allows the Fano Factor (the ratio of the variance to the mean) to vary with stimulus input (as long as β ≠ 1) and for both over-dispersion (Fano factor >1) and under-dispersion (Fano factor <1). Moreover, as with the mean, the four stimulus-dependent variance parameters of the model () can be replaced with four pairs of stimulus-independent parameters (α_N, β_N, α_D, β_D). Lastly, in principle, the parameters controlling correlation (ρ_N, ρ_D) can vary with stimulus conditions but for computational simplicity we assume that (ρ_N, ρ_D) are stimulus-independent. However, even with this assumption, our model can capture stimulus-dependent noise correlations (see Pairwise Ratio of Gaussians model captures correlated variability in mouse V1) as often observed in vivo [41, 71, 72].

Fitting the RoG to data

We optimize the values of the parameters, given a dataset, by maximum likelihood estimation. In this paper, we validate various properties of the pairwise RoG using synthetic data produced from the generative model. We will demonstrate the applicability of this model to neural data analysis by fitting the pairwise RoG to calcium imaging data (see Data collection and processing).

Based on our previous discussion, we assume that the model parameters (collectively denoted Θ) are stimulus-independent. We consider our dataset {R_t(s)} where s is the stimulus and t indexes the trial. We assume that, for each stimulus, our data is independent and identically distributed according to , and that data is independent across stimuli. We can therefore compute the negative log-likelihood of the data using the following equation (see S2 Text for derivation): (11) Where and are the empirical mean and covariance across trials computed from the data.

In practice, we have found that it is computationally faster to first optimize the parameters for each neuron in the pair separately (which is equivalent to fitting the independent RoG model), and then optimize the correlation parameters (i.e., the ρ parameters) with the single-neuron model parameters fixed. This two-step optimization process is referred to as the inference functions for marginals method in the copula literature, and is known to be mathematically equivalent to maximum likelihood estimation for Gaussian copulas [73], which is the case we consider here. This points to an extension or alternative to the pairwise RoG that considers the bivariate distribution to be some non-Gaussian copula with Gaussian marginals, which we leave for future work. We assumed that the pairwise distribution is Gaussian for computational simplicity, but others have used non-Gaussian copulas to model neural populations [74].

Model comparison

We compared the pairwise RoG to a modified version of the modulated Poisson model [69], using Gaussian noise instead of Poisson [76]. We call this model the modulated Gaussian (MG). The original model is a compound Poisson-Gamma distribution, in which the Poisson rate parameter is the product of the mean tuning curve and a random gain variable that is Gamma distributed. The parameters of the Gamma distribution depend on the mean tuning curve (f(s)) and the variance of the gain variable (σ_G). Additionally, there are two sources contributing to (tuned) covariability: the correlation between the multiplicative gains (ρ_G), and the correlation between the Poisson processes (ρ_P). For the modulated Gaussian model, we use a bivariate Gaussian distribution whose moments (i.e., mean, variance, and covariance) are parametrized according to the moments of the modulated Poisson model. We made this modification to the modulated Poisson model for two main reasons. First, because we are examining continuously valued fluorescence traces as opposed to discrete spike count data, a continuous distribution is more appropriate for analysis. Second, the original modulated Poisson, while including a parametrization of the noise covariance between neurons, has no simple closed form for the bivariate distribution, which complicates the comparison of goodness of fit between the two models. By using a bivariate Gaussian distribution, we can more directly compare this model to our proposed pairwise RoG.

More explicitly, the pairwise neural responses are distributed as: (13) where we assume the mean tuning cure is the contrast-response curve (Eq (8)), is the standard deviation of the multiplicative gain for neuron i, and ρ_G is the correlation between the multiplicative gains. ρ_P is no longer interpreted as the point process correlation; instead, ρ_P controls the portion of the tuned covariability that is independent of the shared gain. As with the RoG, we also model the untuned variability η as additive bivariate Gaussian noise. We then fit the model parameters to data by minimizing the negative log-likelihood (Eq (11), with and defined in Eq (13)). As with the pairwise RoG, we use cross-validation to account for model complexity and compute the goodness of fit scores using Eq (12). An extension to this model was recently proposed that incorporates normalization by assuming the rate parameter is a ratio term in which the denominator is a Gaussian random variable, then deriving moments of the distribution for optimization [77]. However, this model does not currently account for noise correlations, so we chose to instead adapt the Poisson-Gamma model.

The most relevant difference between RoG and MG is in how each model accounts for the effect of normalization on (co)variability. In the RoG, normalization directly influences variability by division operating on random variables. This creates flexible dependencies between the mean firing rate, individual neuron variability and shared covariability. In the MG, normalization influences the gain of neurons through the interaction between the mean firing rate (i.e., the standard normalization model) and the gain parameter σ_G, which is assumed be a slowly fluctuating source of variability that scales how the mean firing rate effects variability. In this way, the normalization signal for the MG is a deterministic factor. The MG is therefore a simpler model that can only account for overdispersion, whereas the RoG allows for both overdispersion and underdispersion, and for diverse patterns of covariability (see Pairwise Ratio of Gaussians model captures correlated variability in mouse V1) albeit at the cost of additional parameters.

Inference of single trial normalization from measured neural activity

Because of the probabilistic formulation of the RoG, we can use Bayes theorem to compute the posterior probability of the normalization variable D_t in a single trial, given the observed neural responses R_t: (14) Where multiplication by |D₁D₂| occurs due to the change of variables formula for probability density functions. From this distribution, we can find the maximum a posteriori (MAP) estimates of the normalization strength in a single trial by differentiating the posterior distribution with respect the denominator variables and finding the maxima by setting the partial derivatives to 0. For ease of computation, we solve the equivalent problem of finding the zeros of the partial derivatives of the negative logarithm of the posterior distribution. In our previous work [53], we found that, when subtracting the mean additive noise from the simulated activity, the MAP estimate remained unbiased. Thus, for simplicity, we assume that we can subtract off the mean spontaneous activity and consider instead the posterior p(D|R − μ_η). To obtain an estimate for the denominator strength D_t we look at the partial derivatives of the negative log posterior with respect to D₁, D₂ and solve to obtain the MAP estimate. This procedure leads to a two-dimensional system of bivariate quadratic equations: (15) Where the coefficients A₁, A₂, B₁, B₂, C are functions of the parameters of the model (see S3 Text for the derivation of Eq (15) and the full expressions of these coefficients).

A basic result from the algebra of polynomial systems (Bézout’s theorem) tells us that this system has four pairs of possibly complex valued solutions [78]. In fact, as solving this system amounts to solving a quartic equation in one variable, there exists an algebraic formula (with radicals) for solutions to this system as a function of the coefficients. This solution is too long to include here and uninformative but was found using the Symbolic Math toolbox from MATLAB and is included in our toolbox (Code and Data Availability).

Because all the variables involved are real-valued, we are only interested in the existence of real solutions to this two-dimensional system. However, there is no theoretical guarantee that there will be any real solutions. In practice we take the real part of the algebraic solution to this system and find which pair of solutions minimize the negative log posterior. Alternatively, we can consider finding the MAP by directly minimizing Eq (14) (see S3 Text) using numerical optimization. We have verified that, when real-valued solutions exist to Eq (15), these coincide with numerically minimizing Eq (14). However, as optimization of Eq (14) must be computed on a per-trial basis, it is far too time consuming to perform when there are many experimental trials, so we utilize the algebraic solution to Eq (15).

Generating realistic pairwise neural activity from the model

To constrain our simulations to realistic parameter values for the contrast response function (Eq (8)), we took the single-neuron best-fit parameters to macaque V1 data analyzed in our previous work (for details see [53]) and created parameter pairs by considering all combinations (N = 11628 pairs) of these parameters. Using the generative model for the pairwise RoG (Eq (2)) and the contrast-response parametrization (Eq (8)), we can simulate single-trial neural activity from these parameter pairs and specific values for (ρ_N, ρ_D). These synthetic data allow us to explore properties of the pairwise model without having to exhaustively explore the full parameter space.

Data collection and processing

Modulations of correlated variability depend on sharing of normalization

Within the RoG modeling framework, there are two main sources of response (co)-variability: the numerator (excitatory drive) and the denominator (normalization). Depending on the value of the corresponding ρ parameters, each of these sources of variability can be independent (ρ = 0) or shared (ρ ≠ 0), and therefore contribute differently to noise correlations. Consequently, understanding modulations of noise correlations in the pairwise RoG requires understanding how normalization and external stimuli affect the relative amplitude of these sources, and how the effects depend on whether those sources are shared.

First, we studied the relationship between normalization and noise correlations for the lowest contrast stimuli (Fig 2, yellow symbols). We define normalization strength for a given neuron as the mean of the denominator; for a fixed stimulus contrast, this is determined by the semi-saturation constant ϵ in Eq (8). When the normalization signals are positively correlated (shared normalization, ρ_D = 0.5) but the excitatory drive is independent (ρ_N = 0), increasing normalization strength tends to decrease the magnitude of noise correlations (Fig 2A). Conversely, when the normalization signals are independent (ρ_D = 0) but there is shared driving input (ρ_N = 0.5), the magnitude of noise correlations tends to increase with increasing normalization (Fig 2B). Intuitively, this is due to how the model partitions neuronal covariability into two sources and how normalization separately effects variability of these sources. As mentioned at the beginning of this section, these terms describe the correlations among the numerators and among the denominators. The correlations arising from the numerator are unaffected by normalization strength, while the correlations arising from the denominator tend to decrease with normalization strength. So, when ρ_N = 0, the noise correlations are solely due to the denominator cofluctuations and thus tend to decrease with normalization. However, when ρ_D = 0, the numerator covariability drives the response covariability, which is unchanged by normalization strength. The reason why we see increased noise correlations in this scenario is because normalization decreases individual neuronal response variability [53] so the proportional contribution of the numerator term increases. This is derived more completely later in this section. Therefore, increasing normalization strength reduces a source of noise correlations when normalization is shared, whereas it reduces a source of independent noise when normalization is not shared.

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Fig 2. Relationship between noise correlations, normalization strength and contrast depends on the source of variability.

Each panel shows, for a combination of ρ_N, ρ_D specified in the panel title, the median noise correlation of all generated neural pairs binned according to ϵ₁ × ϵ₂, a contrast independent measure of the common normalization strength. Bins with less than 100 pairs were discarded. Neural responses were generated from the contrast-response parametrization (Eq (8)). Noise correlation strength was computed across 1e3 simulated trials drawn from the pairwise RoG model. For each contrast level and combination of ρ_N, ρ_D, 1e5 simulated experiments were created. See S1 Fig for a more systematic exploration of the factors influencing modulation of noise correlations for two simulated pairs that matches the large scale experiment considered here. Model parameters were drawn uniformly from the following intervals:

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We found similar effects of normalization on noise correlations at higher contrast levels (Fig 2, orange and red symbols) although the slope of the relationship became shallower as contrast increased. This is due to the saturating effect of the contrast response function (Eq (8)): at high contrast, (co)fluctuations of the normalization signal across trials have relatively little effect on the responses of a neural pair, so the correlation in neural responses will be relatively unaffected by normalization strength. We also observed that the magnitude of noise correlations generally increased with contrast when the denominators were correlated and the numerators independent (Fig 2A and S2 Fig), whereas it decreased when the numerators were correlated (Fig 2B and 2C and S2 Fig). Importantly, for the analysis of normalization strength at fixed contrast, we used the contrast semi-saturation constants (Eq (8); i.e., a pure change in the denominator) as a measure of normalization strength. Conversely, increasing stimulus contrast increases both the numerator and denominator (Eq (8)). This explains our observation that, even though normalization is stronger at higher contrast, noise correlations can be modulated in different ways by contrast and by normalization, because changing stimulus contrast also affects the numerator term.

Indeed, these results can be derived from examining the equation for the correlation in the pairwise RoG (Eq (7)) rearranged as follows (see S5 Text for more details): (16) and by recognizing that the coefficient of variation of D_i () is a decreasing function of normalization strength (μ_D or ϵ), whereas the ratio is often an increasing function of contrast.

We further analyze this equation, first in the case of changing normalization strength while keeping contrast fixed. The term proportional to ρ_N is an increasing function of the mean normalization strength since the denominator is a decreasing function of μ_D. Conversely, the term proportional to ρ_D decreases with normalization since the denominator increases with μ_D. In these two cases, the monotonic dependence of noise correlation on normalization strength is guaranteed by Eq (16) regardless of the specific parameter values. Similar patterns emerge when ρ_N, ρ_D < 0, except the signs of the noise correlations are reversed (S2 Fig). When the correlations of input and normalization signals are both different from zero, the relationship between noise correlation and normalization strength resembles a combination of the two previously described scenarios, and the specific parameter values determine which of the two terms in Eq (16) dominates. For instance, in our simulations with (ρ_N = 0.5, ρ_D = 0.5) (Fig 2C), the magnitude of noise correlations increased with normalization strength on average similar to (Fig 2B), indicating that the magnitude of the term proportional to ρ_N is usually larger than the magnitude of the ρ_D term, but this trend is not consistent, as evidenced by the increased spread of the scatter. When the input strength and normalization signal have opposite correlations (e.g., ρ_N = 0.5, ρ_D = −0.5), we obtained similar results; however, the magnitude of noise correlations was on average closer to 0 due to cancellation between the two sources of covariability (S2 Fig).

A similar analysis of Eq (16) shows that the effects of stimulus contrast are opposite to those of a pure change of normalization strength, because is often an increasing function of contrast but a decreasing function of normalization strength. Notice that we assumed there is no residual noise component (η ≡ 0), but all the analyses above remain valid when the amplitude of noise variance σ_η is relatively small compared to (δ_N, δ_D) (see S5 Text).

In summary, our analysis shows that normalization and stimulus contrast can have diverse effects on noise correlations, depending on whether neurons share their normalization signals and on the interplay between multiple sources of variability.

Inference of correlation parameters

The above analysis demonstrates that the relationship between noise correlations and normalization depends on how this correlated variability arises: either through cofluctuations in the excitatory drive or normalization signal (determined by ρ_N, ρ_D respectively). To employ these insights when fitting to data, we need to know how well we can infer these parameters from data. To do so, we generated synthetic neural data using realistic values for the single-neuron parameters (see Generating realistic pairwise neural activity from the model) and uniformly randomly sampled ρ_N, ρ_D parameters in the range [-0.9,0.9]. We then assessed the quality of the maximum likelihood estimate of the parameters by calculating bootstrapped confidence intervals (with N = 1000 bootstrap samples) and comparing the estimator and true values (see Quantifying the accuracy of the estimated correlation parameters).

First, we assessed the validity of the confidence interval by examining how well the empirical coverage probability matches the confidence level as constructed. To do so, we constructed 90% confidence intervals for the (ρ_N, ρ_D) parameters via bootstrap resampling. We then grouped by the ground truth ρ values using a sliding window and counted the proportion of cases in that bin for which the bootstrap confidence interval contains the true value. We found that for both ρ_N and ρ_D the coverage is near the nominal level: for ρ_N it is slightly lower (Fig 3A left) while for ρ_D it is nearly equivalent (Fig 3A right). This indicates that the confidence intervals constructed via the bootstrap are valid and can be used for further analysis.

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Fig 3. Accuracy of inference of ρ parameters.

Plots were generated with 11628 synthetic parameter pairs with uniformly randomly generated ρ_N, ρ_D ∈ [−0.9, 0.9], contrast levels {6.25, 12.5, 25, 50, 100}, 1000 synthetic trials and 1000 bootstrap resamples. The left column are the results of the analysis for ρ_N, the right column for ρ_D. (A) Empirical coverage probability for the 90% confidence intervals as a function of the ground truth ρ values. The dotted line indicates the nominal confidence level. Coverage probability was computed as the proportion of cases with a specified range of ρ values for which the 90% confidence intervals contained the true value. We used a moving window with width 0.4 and a step size of 0.2. (B) Direct comparison between the true ρ value and the maximum likelihood estimator for the ρ value. The darker colors are the pairs for which the ρ parameters are significantly different from 0 (i.e., the 90% confidence interval excludes 0), whereas the lighter colors are not significant (i.e., the 90% confidence interval includes 0).

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Next, we directly compared the the true and inferred ρ values (Fig 3B). In general, the maximum likelihood estimators are largely unable to recover the true generating values (the overall Pearson correlation between the fit and true (ρ_N, ρ_D) = (0.48, 0.18), the mean squared error across pairs is (0.32, 0.57)), indicating that these parameters are not identifiable with the parametrization we considered (contrast tuning). Further, this analysis indicates that inference of ρ_D is in general more difficult than it is for ρ_N. The lack of identifiability of these parameters is likely due to the numerous multiplicative interactions between the parameters. For instance, by looking closer at Eq (7), we can see that the contribution of the ρ parameters to the noise correlation is multiplied by the respective standard deviations for the numerator or denominator variables. Such interactions may make it difficult to infer the exact value of the ρ parameters (see S6 Text for further discussion).

As we established the validity of the bootstrapped confidence intervals (Fig 3A), one could select pairs for which the parameter inference is accurate to the desired precision (i.e., selecting those pairs whose confidence intervals are a certain width). Additionally, one can also perform population-level analyses on the ρ parameters, as we demonstrate for experimental data (see Pairwise Ratio of Gaussians model captures correlated variability in mouse V1). Here, we introduce and validate another kind of analysis one can perform with the bootstrapped confidence intervals. Rather than the exact magnitude of the ρ parameters, we are often only interested in the sign of these parameters, as in deriving the relationship between normalization strength and noise correlations (Modulations of correlated variability depend on sharing of normalization). Moreover, because all the parameters in Eq (7) are a priori positive besides the ρ parameters, we reasoned that the accuracy of sign inference might be higher. Therefore, we considered the subset of cases where the parameters are significantly different from 0, which we define as cases where the 90% confidence interval does not include 0 and the pairwise goodness of fit is greater than the independent goodness of fit (see Cross-validated goodness of fit). First, for those pairs (for ρ_N, 4598/11628 pairs were significant, for ρ_D, 1625/11628), we see a much stronger relationship between the fit and true ρ values (Fig 3B, darker colors; Pearson correlation between significant fit and true (ρ_N, ρ_D) = (0.89, 0.57); mean squared error across pairs = (0.09,0.43)). Importantly, the plots also show that, for pairs with ρ parameters significantly different from 0, the sign of the inferred ρ parameter is very frequently equivalent to the sign of the true ρ parameter, in a similar proportion of cases for ρ_N and ρ_D, although ρ_N has a much higher proportion of cases (for ρ_N, 4509/4598 of pairs significantly different from 0 has the same sign; for ρ_D, 1309/1625). From this, we conclude that, for these significant ρ parameters, the sign of the inferred ρ parameter is accurate.

In summary, our analysis indicates that it is difficult to estimate the precise value of the ρ parameters in general, for the stimulus parametrization considered here (i.e., the classical normalization model for contrast tuning). However, we have provided a method to calculate bootstrapped confidence intervals for the maximum likelihood estimators of the ρ parameters and have shown that these confidence intervals accurately represent the uncertainty around those estimates. We then demonstrated one possible use-case for these confidence intervals: for ρ estimates that are significantly different from 0, the ρ estimators are accurately able to recover the sign of the ground truth ρ parameters.

Pairwise model improves single-trial inference of normalization strength even when noise correlations are small

In past work that connected normalization to modulation of noise correlations, stimulus and experimental manipulations (e.g., contrast, attention) are used as proxies for normalization strength [51, 52, 55] because normalization strength cannot be measured directly. However, these manipulations also affect other factors that drive neural responses (as we have illustrated in Modulations of correlated variability depend on sharing of normalization, Fig 2), which could confound these as measures of normalization signal. Therefore, quantitatively testing the relationship between noise correlations and normalization requires estimating the single-trial normalization strength for a pair of neurons. One of the advantages of our probabilistic, generative formulation of the pairwise RoG model (Eq (2)) is that it allows us to infer the single-trial normalization strength from measured neural activity (see Inference of single trial normalization from measured neural activity). The independent RoG model also provides an estimate for the single-trial normalization, which is known to be a valid estimator for the ground-truth normalization strength for data generated from the independent RoG [53]. We found similar results for the pairwise RoG, so we examined how the pairwise estimates for the single-trial normalization strength compares to estimates based on the independent model.

One possibility is that the estimate derived from the pairwise model would outperform the independent model as the magnitude of noise correlations increase. However, this is not necessarily the case. Fig 4A and 4B demonstrates this with two example synthetic neuron pairs. Because the single-trial normalization inference depends on the single-trial neural activity, correlations between neurons will induce correlations between the inferred normalization signals even for the independent RoG (Fig 4A). However, when noise correlations are small due to cancellation between ρ_N and ρ_D, the independent model will infer minimal correlation between normalization signals while the pairwise model will correctly infer that the single-trial normalization is correlated (Fig 4B).

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Fig 4. Inference of Single-trial normalization depends on ρ parameters and contrast level.

(A and B) Two simulated experiments drawn from the pairwise RoG with different noise correlations arising from different underlying ρ_N, ρ_D values to compare the pairwise and independent estimators of single-trial normalization with the ground truth normalization signal. (A) has overall noise correlation of 0.21 across contrasts generated with ρ_N = 0, ρ_D = 0.3, (B) has overall noise correlation of -0.05 across contrasts generated with ρ_N = −0.3, ρ_D = 0.3. Z-scoring performed across trials. Random parameters drawn from R^max ∈ [10, 100], ϵ ∈ [15, 25], α_N, α_D ∈ [0.1, 1], β_N, β_D ∈ [1, 1.5], η ≡ 0. Contrasts levels were {6.25, 12.5, 25, 50, 100}. (C and D) Comparison of the single-trial normalization inference in the pairwise and independent RoG models for the mean-squared error (C) and the correlation between the estimate and the true value (D), as it depends on ρ_N, ρ_D and the contrast levels. Left: lowest contrast level (6.25); middle: intermediate contrast level (25); right: full contrast (100). Each bin corresponds to the median difference between the pairwise and independent models across simulated experiments. (C and D) used 11628 synthetic pairs (see Generating realistic pairwise neural activity from the model).

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To demonstrate this principle, we computed the difference of the mean squared error (Fig 4C) and correlations (Fig 4D) between the pairwise RoG and independent RoG normalization inference (with respect to the ground truth values) for many simulated pairs (11628) with systematically varying ρ_N, ρ_D values. First, we found that the distinction between the two models depended on the contrast level: as contrast levels increase, the quality of the pairwise and independent estimators of normalization signal become more distinct. Second, the improvement of the pairwise RoG estimate over the independent estimate increased when the magnitude of the ρ_D parameter increased. Consistent with the intuition provided by Fig 4B, the largest improvement occurred when the ρ_N parameter had a large value that was the opposite sign of ρ_D. The dependence on the ρ_D parameter reflects that the estimator for the pairwise model incorporates knowledge about correlation between the normalization signals.

In summary, this analysis shows that the pairwise model estimates of the single trial normalization can improve upon the independent model even when noise correlations are small. As the single-trial normalization estimator from the independent model was previously shown to be accurate [53], our results imply that the pairwise model estimate is also able to recover the ground-truth normalization strength. Additionally, we have outlined the conditions in which those estimates are preferable to those obtained with the independent model.

Pairwise Ratio of Gaussians model captures correlated variability in mouse V1

To test how well the model captures experimental data, we applied it to calcium imaging data recorded in V1 of mice responding to sinusoidal gratings of varying contrast levels (see Data collection and processing). We analyzed neurons that were strongly driven by the visual stimuli (N = 295 neurons, 5528 simultaneously recorded pairs; see Data collection and processing for inclusion criteria). We focused on stimulus contrast tuning (Eq (8)) because the formulation of the corresponding standard normalization model captures firing rate data well [36], and visual contrast affects both normalization strength and the strength of noise correlations [41].

Because the RoG framework has not been validated before on mouse V1 fluorescence data, we first applied the independent RoG and found that it provided excellent fits (average cross-validated goodness of fit 0.85, 95% c.i [0.846,0.858], 5163/5528 pairs with goodness of fit >0.5) on par with that found in macaque V1 data recorded with electrode arrays [53], thus demonstrating that the RoG framework is flexible enough to capture datasets with different statistics. In both cases, the analysis was performed on visually responsive neurons, that therefore exhibited strong contrast tuning of the firing rate, partly explaining the high performance. However, the independent RoG could not capture correlated variability, which was prominent in the data (median noise correlation across all pairs and contrasts 0.117, c.i. [0.115, 0.120], 2781/5528 pairs had noise correlations significantly different from 0).

Therefore, we tested if the pairwise RoG could capture correlated variability in the data. Fig 5A and 5B demonstrates that the model can capture contrast-dependent noise correlations, both for pairs with positive (example in Fig 5A; 4327/5528 pairs) and negative median noise correlations (example in Fig 5B; 1201/5528 pairs). Importantly, even though the ρ_N, ρ_D parameters were stimulus-independent, the pairwise model captured substantial changes in noise correlations with contrast for many of the pairs analyzed (2991/5528 had greater than 0.5 correlation between the observed noise correlations and model fit, across contrast levels). However, this ability to capture correlations comes at the cost of larger model complexity. To account for this, we next compared quantitatively the pairwise and independent models, using a cross-validated goodness of fit score (Eq (12)). The pairwise model slightly outperformed the independent model on average and for most pairs (median difference in goodness of fit = 0.0121, p < 0.001, 4123/5528 pairs with pairwise goodness of fit greater than independent), denoting that the additional free parameters are warranted. Furthermore, because the independent model is a special case of the pairwise with noise correlations fixed at zero, we found as expected that the performance difference between the two models increased for pairs of neurons with larger noise correlations (see S7 Text).

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Fig 5. Pairwise RoG captures contrast-dependent noise correlations in mouse V1.

(A and B) Pairwise neural responses for two example pairs of neurons in mouse V1 with positive (A) and negative (B) median noise correlations. From left to right: 1) empirical mean and covariance ellipses (∼1 standard deviation from the empirical mean) for pairwise responses at each contrast level; 2) the RoG model predicted means and covariance ellipses, the panel title includes the cross-validated goodness of fit score; 3) the modulated Gaussian (MG) predicted means and covariance ellipses; 4) compares the two model fit noise correlation values (continuous lines), with the empirical values as a function of contrast (error bars are 68% bootstrapped confidence interval). Neuronal pair in (A) had 93 repeats of each stimulus contrast, pair in (B) had 68 repeats. (C) Scatter plot across all pairs of the goodness of fit score for modulated Gaussian vs. the goodness of fit for the RoG. (D) Histogram of the difference between the scores in (C). Contrast levels are {2,8,16,32,64,80,100}.

https://doi.org/10.1371/journal.pcbi.1011667.g005

To benchmark the RoG against a widely adopted alternative model, we considered the modulated Poisson model that was previously shown to capture noise correlations in macaque V1 [69]. For application to our imaging dataset and for a fair comparison with the RoG, we used Gaussian noise instead of Poisson [76] and termed this the modulated Gaussian (MG) model (see Model comparison). The example pairs demonstrate that, while in some pairs the MG can capture the modulation of noise correlations with contrast as well as the RoG (Fig 5A), it is not able to capture it in other pairs while the RoG can (Fig 5B). Across the dataset, for the majority of pairs (5238/5528), the pairwise RoG had a higher goodness of fit score (Eq (12)) than the MG (Fig 5C and 5D, median difference between goodness of fit for RoG and MG = 0.238, 95% c.i. [0.232,0.245]). These results were also largely independent of the specific preprocessing method applied to the calcium imaging data (see S4 Text). Moreover, although both models capture the tuning of noise correlations with contrast level by using stimulus-independent correlation parameters, the RoG model better predicts the trend in noise correlations with contrast than the MG (median difference in correlations between model fit noise correlations and empirical noise correlations = 0.0771, 95% c.i. [0.071, 0.084]; 1450/5528 pairs had statistically significant correlation between the pairwise RoG predictions and empirical noise correlations compared to 1008/5528 for the MG). In principle the MG model’s ability to capture the modulation of noise correlations with contrast could be improved by including contrast dependence in the correlation parameters explicitly, although this would increase model complexity.

These results demonstrate that the pairwise RoG captures a range of effects of stimulus contrast on noise correlations observed in experimental data and performs competitively against a popular alternative model that does not account for normalization explicitly.

Next, we analyzed the correlation parameters (ρ_N, ρ_D) in the model fit (Fig 6). We first only selected those pairs of neurons whose pairwise goodness of fit exceeded 0.5 and the independent goodness of fit measure (3920/5528 total), and we computed the bootstrapped 90% confidence interval for the (ρ_N, ρ_D) parameters (see Quantifying the accuracy of the estimated correlation parameters).

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Fig 6. Inferred ρ_N, ρ_D are positive in Mouse V1.

Histograms comparing the inferred ρ_N (A) and ρ_D (B) values for all neuronal pairs meeting our goodness of fit criteria (outlined) and the subset of those pairs significantly different from zero with 90% confidence (filled). The histograms for all pairs and pairs significantly different from 0 are normalized separately.

https://doi.org/10.1371/journal.pcbi.1011667.g006

Examining the correlation parameters for all of the pairs meeting the goodness of fit criteria (Fig 6A and Fig 6B outlined), we see a significant bias towards positive values (median ρ_N, ρ_D parameter values = 0.84, 1). This is partially due to the large number of cases in which the fit parameters were exactly equal to ±1 (for ρ_N, ρ_D, 1442 and 2700 fit values were ±1). However, even when excluding these pairs, the trend within the population is still towards positive fit ρ values (median fit ρ_N, ρ_D values excluding extreme pairs is 0.29, 0.22). This analysis suggests that these signals are, on aggregate, shared among the population recorded; in particular, this suggests that normalization is typically shared between the pairs recorded.

As a complementary analysis, we then focused on the cases where the parameters were assessed to be significantly different from zero (1270/3920 for ρ_N, 192/3920 for ρ_D) (Fig 6A and 6B filled). The proportion of pairs for which the estimated ρ parameters are significantly different from 0 is similar to the synthetic data (see Inference of correlation parameters). For these pairs that were significant, we found that nearly all inferred ρ_N (1239) and ρ_D (191) parameters were positive (see Fig 6), suggesting that normalization signals are generally shared for these pairs of neurons.

In summary, our results demonstrate a new approach to quantify how strongly normalization signals are shared between neurons, and to explain the diverse effects of normalization on noise correlations.