1 Copula-GP model

Many questions in neuroscience require accurate statistical models of neuronal responses to a certain stimulus. For instance, such models can be used for Bayesian decoding of the stimulus [20] or for information-theoretic analysis [33] in order to assess the coding precision of a neuronal population. Here we describe a general framework for constructing such statistical models, called Copula-GP; the technical details related to model implementation and fitting can be found in Methods.

Our statistical model is based on copulas: multivariate distributions with uniform marginals. Sklar’s theorem [34] states that any multivariate joint distribution can be written in terms of univariate marginal distribution functions p_i(y_i) and a unique copula which characterizes the dependence structure: (1)

Here, F_i(⋅) are the marginal cumulative distribution functions (CDF) corresponding to the probability density functions (PDF): .

The model in (1) has been previously applied to neuronal data [18, 23, 35]. In this paper, we extend that model by conditioning it on a continuous variable x: (2) where x, depending on the design of the experimental task, can represent any continuous task-related variable such as time, phase, coordinate in space, velocity, direction or orientation. In the further sections we will show that if this variable x also uniquely determines the stimulus, then the components of the model (2) can also be interpreted in terms of ‘noise correlations’ (i.e. joint trial-to-trial variability of responses to a given stimulus).

We are using Gaussian Processes (GP) to model the conditional dependencies of copula parameters (i.e. c(.|x)), following the approach by Hernández-Lobato et al. [31] which was originally developed for analysis of financial time-series. GPs are ideally suited for copula parametrization, as they make the dependencies explicit and provide an estimate of the uncertainty in model parameters, which we utilize in our model selection process (see Sec. 6 in Methods). Similarly to previous computational studies [23, 28, 30, 36, 37], we use vine copula constructions to scale the method to higher dimensions (Sec. 7). Here, we extend the Copula-GP framework with more flexible copula mixture models (see Fig 1 and Methods), suitable for inter-neuronal dependencies, and utilise the repeated trial structure in order to estimate conditional marginal distributions p_i(y_i|x) (e.g. distributions of single neuron responses at every moment x in the trial). We develop model selection algorithms, based on the fully-Bayesian Watanabe–Akaike information criterion (WAIC), that construct the best suited mixture models for a given data set. In the following section we show the utility of our extended Copula-GP model for analysis of neuronal and behavioral data.

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Fig 1. Copula families used in the mixture models in our framework.

Each panel shows a scatter plot of samples drawn from a parametric copula family (named in the title of each plot) with a fixed parameter θ (shown in the bottom right corner). In total, we used 10 different copula elements—Gaussian + Frank + 4 × Clayton + 4 × Gumbel—to construct copula mixtures. The rightmost figure shows a mixture of two copulas from different copula families taken with equal mixing weights (0.5). Blue points here correspond to the samples drawn from a Clayton copula, orange points—to a 90°-rotated Gumbel copula. Note, that a mixture of copulas is a copula itself.

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2 Copula-GP separates noise correlations from stimulus statistics

We first examine a synthetic ‘toy’ model of calcium imaging data in order to illustrate and interpret the components of our Copula-GP model. We use a Generalized linear model (GLM) [38] and a model of the kinetics of the calcium indicator [39] in order to generate synthetic calcium imaging data of the activity of two neurons (see Methods). GLMs are widely used to model population spike trains based on spike history filters and coupling filters. While these models can be readily fit to small neuronal populations, they have parameter identifiability issues that lead to degraded performance in predicting neuronal activity patterns [40]. GLMs are also not directly applicable to calcium imaging data and require some additional spike inference algorithm (deconvolution, e.g. Deneux et al. 2016 [39]) to obtain spikes. So, instead of fitting GLMs, we propose to explicitly model the dynamic neuronal activity patterns (including the calcium dynamics) with Copula-GP. Our approach is complementary to GLMs: while GLMs consider time-lagged interactions via temporal spike history filters and cannot model instantaneous coactivation [41], our model focuses exclusively on the instantaneous correlations between calcium signals.

Calcium imaging in neurons is used as a proxy for monitoring their spiking activity since action potentials are tightly coupled to large increases in intracellular free calcium concentration [42]. Two-photon imaging is used to monitor fluorescence changes of calcium indicators in large population of neurons, with single cell resolution. When dense populations of neurons are imaged, the signals imaged in a given focal plane can be contaminated by signals from a volume around this plane. Different methods for correcting such out-of-focus contamination have been developed [43]. We assumed that each decontaminated recording y_i(t) of calcium dynamics for each neuron is a function of its spiking activity n_i(t), which is independent of the activity of other neurons (i.e. y_i(n₁…n_N, t) = y_i(n_i, t), given time). This assumption is also implicitly made in deconvolution algorithms, which incorporate inductive biases for calcium dynamics and are also applied to individual neurons independently. We will consider the decontaminated recordings y_i(t) to be a surrogate of neural spike trains n_i(t), which can be approximated by blurring (convolving) the spikes with an exponential kernel (Deneux et al. 2016 [39]). The time constant in this kernel corresponds to calcium decay and also sets the time scale of interactions between spiking neurons, which would be captured by an instantaneous correlation of their calcium traces. Keeping these assumptions in mind, we aim to build instantaneous dependence models of calcium recordings for statistical and information-theoretic analysis.

We first consider a pair of neurons, which have the same tuning (receive the same input x(t) in Fig 2A), but are not coupled. Since the spiking activity of these neurons in response to a given stimulus is completely independent, we expect that their calcium traces, which are a temporally blurred versions of spike trains, to also be independent given the stimulus. A few simulated calcium transients (fluorescence across time) for one of these identical neurons are shown in Fig 2B.

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Fig 2. Copula-GP finds that uncoupled neurons are independent given the stimulus.

A GLM model of two identical uncoupled neurons that receive the same time-dependent input x(t); B simulated calcium transients (fluorescence across time) showing dynamic responses to the stimulus x(t) for one of the neurons; C calcium transients of two neurons (y₁(t), y₂(t)) projected onto a unit cube by the probability integral transform based on unconditional marginals; colored points show transformed samples (u₁, u₂) corresponding to times t (color-coded). The clusters of similarly colored points (e.g. green) illustrate that the copula c(u) depends on time t; the particular shape and the location of the clusters depends on the function x(t); only 10% of data-points are shown (selected randomly). D same as C, but based on conditional marginals F_i(y_i|t). The resulting empirical copula describes ‘noise correlations’ between two neurons. The colored data-points (,) are uniformly distributed on the unit square, which suggests that there is no noise correlation between these neurons, the copula c(u^t) is independent of time t, and the neurons are independent given the time-dependent stimulus.

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By design, the activity of these neurons is independent given the stimulus (Fig 2A), but they do respond similarly to the stimulus. As a result of a naive analysis of their joint statistics, such as measuring correlation between recorded activities over time, their activity would appear dependent. The structure of this dependence can be visualized with a static copula model, as in (1).

To fit a copula-based model to a given dataset, one typically starts with the marginals (i.e. single variable distributions). Here, we model the single neuron marginals with their empirical distributions, estimated directly from the data in the form of empirical cumulative distribution functions (eCDFs). We then use these eCDFs to project the simulated neuronal recordings onto a unit cube using the probability integral transform: F(y_i)→u_i ∼ U_[0,1], such that each u_i becomes uniformly distributed. Note, that both the neural responses y_i and their transformed versions u_i still depend on both the stimulus x(t) and time t. We can confirm that by visualizing the empirical copula c(u₁…u_N) = c(F₁(y₁)…F_N(y_N)) (Fig 2C). The colors of the data points here indicate time t. Similarly colored clusters of points in copula space (e.g. green in Fig 2C) indicate that both the dependence and the marginals in such an unconditional model depend on time t and on the presented stimulus x(t). Hence, such dependence characterizes the joint statistics of neural responses to stimulus x(t), but not the interaction between neurons.

Spurious correlations between neuronal recordings are a well known problem when modeling responses to complex stimuli (e.g. time-dependent sequences of stimuli x(t)), which can be solved by distinguishing between noise and stimulus correlations [40, 44]. In our Copula-GP model, this separation is achieved by using conditional marginals and copulas:

Here, we only assume that the stimulus x(t) is fully determined by time t. In general, our framework (2) is applicable to any complex stimuli determined by any continuous variable (e.g. time, phase, coordinate in space, velocity, direction, orientation, etc.). The conditional marginals p_i(y_i|t) in this model account for the within-trial variability (e.g. dynamics of responses), while the copula model accounts for the trial-to-trial variability in neural responses.

The conditional marginals p_i(y_i|t) are estimated using the non-parametric fastKDE [45] algorithm. This approach is suitable for relatively large datasets, which have enough data points for direct estimation of the eCDF (see Sec D in S3 Text). If the number of datapoints is insufficient, one might consider using parametric marginal models instead. Note that the fastKDE also assumes smooth changes in the marginal distributions with respect to the conditioning variable. The corresponding conditional marginal CDFs F_i(y_i|t) are then used to map the data onto a unit hypercube using: , such that is uniformly distributed for any t. The resulting empirical copula correctly demonstrates the absence of dependence between y₁(t) and y₂(t), as the points are uniformly distributed on a unit square. The density of points in Fig 2D illustrates that for every given value of the transformed neural response , the conditional distribution of the other neural responses is the same (). This means that the variables and are independent , and their copula is the Independence copula. This result demonstrates that once we made the distinction between noise correlations and stimulus correlations, we could correctly identify that these two neurons are uncoupled (Fig 2A).

The noise correlations were previously linked to anatomical and functional relationships between cortical neurons [46, 47]. However, these relationships should be interpreted with caution, since correlation between neurons does not necessarily imply the mechanistic coupling between them [48]. Despite lack of mechanistic interpretability, accurate modelling of noise correlations is useful for understanding neural code from the information-theoretic perspective. Generally, noise correlations themselves can depend on the stimulus [49], and taking their stimulus dependence into account can improve the decoding accuracy [44, 50, 51]. In order to model such tuning of the noise correlations, not only the marginals but also the corresponding copula c(…|t) itself must be conditioned on t, as in (2).

We next consider two coupled neurons: one excitatory and one inhibitory (Fig 3A.i). They again receive the same input x(t) as in Fig 2A, in all trials. We added two time-dependent filters h₁₂ and h₂₁, that couple the spike train history of each neuron to the other (Fig 3A.ii) [38]. The synthetic calcium traces (Fig 3B and 3C) demonstrate some non-trivial activity in both neurons after the stimulus presentation window, where all of the recurrent circuit dynamics unfolds. We expect that these dynamics will be reflected in some non-independent trial-to-trial co-variability of neural responses. Since one neuron inhibits the other, we also expect that their responses will be overall negatively correlated. While the unconditional copula analysis again demonstrates strong stimulus correlations (Fig 3D), the conditional copula (conditioned on the stimulus) reveals some structure in noise correlations (Fig 3E, see more points are concentrated in the upper-left corner).

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Fig 3. Copula-GP describes the noise correlations between dynamically coupled neurons with a Clayton copula.

A i. GLM model of two coupled neurons (excitatory and inhibitory) that receive the same time-dependent input x(t); ii. the spike history coupling filters h₁₂ and h₂₁; B-C simulated calcium transients (fluorescence across time) showing dynamic responses to the stimulus x(t) for excitatory and inhibitory neurons, respectively; D calcium transients of two neurons (y₁(t), y₂(t)) projected onto a unit cube by the probability integral transform based on unconditional marginals; colored points show transformed samples (u₁, u₂) corresponding to times t (color-coded). The clusters of similarly colored points (e.g. green) illustrate that the copula c(u) depends on time t; the particular shape and the location of the clusters depends on the function x(t). E same as D, but based on conditional marginals F_i(y_i|t). The resulting copula describes ‘noise correlations’ between two neurons. The colored data-points (,) are not uniformly distributed on the unit square, which suggests that the noise correlation between these neurons and the copula c(u^t) itself depends on time t. F Clayton copula parameter (θ) that characterizes the strength of the non-linear noise correlation between neurons (see Methods for details); G probability density plots illustrating the stimulus-dependent shape of noise correlations. The empirical dependence estimated from data samples is shown with black outlines, while the predictions of the Clayton copula model are shown in shades of blue. The proportion of the variance explained is indicated in the upper-right corner for each time interval. Orange circles indicate the heavy tail of the distribution, which can be best seen in the range t ∈ [0.6, 1.0] where the variables are stronger correlated.

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In order to analyse the structure of the noise correlations in this example, we apply our Copula-GP model. For simplicity, here we use a single 90°-rotated Clayton copula, parameterized by a Gaussian Process (GP); for details regarding the inference scheme, see Methods. The inferred GP parameter reconstructs the stimulus-dependent changes in noise correlations (Fig 3F), which are most pronounced after the stimulation window. The corresponding Clayton copula model can accurately describe the shape of the conditional dependence, which we quantify with the proportion of variance explained in Fig 3G (see Sec. 10 in Methods).

This noise correlation model in Fig 3G can be interpreted as follows. The red point in the middle of each plot in Fig 3G corresponds to the median response for both neurons: 50% of recorded responses from a given neuron were higher, while 50% were lower. More generally, the values of the transformed neural responses u^t correspond to the percentile scores of each response y in a marginal distribution of neural responses. The shades in Fig 3G correspond to the probability density, i.e. how likely it is to jointly observe a pair of responses and . Typically, most of the density is concentrated along one of the diagonals of the unit square: around if neurons are positively correlated, and around if neurons are negatively correlated. In this case, the density is mostly concentrated around the negative diagonal, suggesting that the responses of these two neurons are generally negatively correlated (as expected, given that one of the neurons inhibits the activity of the other). However, this dependence also has a heavy tail: a high degree of association between some of the extreme values of the variables. We can observe this association as a probability mass concentrated in the corner of the unit square (see orange circles in Fig 3G). It indicates that there is a high probability to observe a strongly activated inhibitory neuron together with the inactivated excitatory neuron. Such asymmetry in the joint trial-to-trial variability of two neurons suggests that their dependence is non-linear: their dependence is not well characterized by a single linear correlation coefficient as there is additional structure in the dependence beyond linear association. In our GLM model, this non-linearity stems from the different timescales of spike history filters: fast inhibitory and slow excitatory neurons.

The confidence intervals (95% CI, two standard deviations) shown in Fig 3F correspond to the uncertainty in model parameters, captured by the Gaussian process. The uncertainty in model parameters is not essential for this example, but necessary when modeling high-dimensional datasets with relatively low sample numbers, which is often the case in neuronal data with hundreds to thousands of recorded neurons and only few hundreds of trials. These uncertainties can be propagated through the model and produce the uncertainty measures for the decoded stimulus (when using Copula-GP for Bayesian decoding) or in estimated mutual information.

As we demonstrated in these synthetic examples, our Copula-GP model can separate the statistics corresponding to the network from the single unit statistics, or, in other words, distinguish (stimulus-dependent) noise correlations (i.e. shared neuronal trial-to-trial variability) from stimulus correlations. Moreover, it provides a visualization of the shape of noise correlations (Fig 3G), and reveals that the noise correlations are tuned to a particular time point (t ≈ 0.7) in a trial (Fig 3F). The main advantage of using copula models [18] is their expressive power in modeling the shape of the relationship between variables, which, in this case, is asymmetric due to the dynamic connections in the neuronal circuit being asymmetric (Fig 3). It was also shown previously that shared excitatory or inhibitory inputs can result in similar non-linear dependencies between neurons [24]. While copula models of the noise correlations do not provide any mechanistic interpretation of these dependencies (e.g. whether neurons are recurrently coupled or share an input), their expressive power allows one to build statistical models for information-theoretic analysis or Bayesian decoding. An additional explicit fully-Bayesian parameterization provided by a GP (Fig 3F) extends the copula model to time-dependent, location-dependent or otherwise continuously parameterized behavioral tasks.

3 Validation of mutual information estimation on artificial data

Apart from conditioning on time or on some complex continuous stimuli, Copula-GP models can also provide accurate information estimates. Static copula models are generally well suited for information estimation, as the mutual information between the variables corresponds to the negative entropy of their copula [18, 34]. This also applies to the conditional entropy of our conditional copula models H(y|x), as it factorizes into marginal and copula components (see Eq (11) in Methods). However, calculating the mutual information between the variable y and the conditioning variable x is more complicated. We propose two methods for estimating this mutual information I(y, x): ‘integrated’ (12) and ‘estimated’(13). The first one assumes no conditional dependence in marginals (i.e. p_i(y_i) = p_i(y_i|x) for every y_i and every value of x) and only requires a conditional copula model for p(y|x), which is then integrated using the Monte-Carlo (MC) algorithm (over both x and y) in order to obtain p(y). The second method has no limitations on the conditional dependencies in marginals, but requires an additional copula model for the unconditional distribution p(y) apart for a conditional copula model of p(y|x) (see Sec. 11 in Methods for details).

We expect our information estimates to be unbiased for every parametric copula model in our framework, due to the MC estimator being unbiased [52]. The only source of bias in our information estimates comes from the accuracy of the model fit obtained via variational inference, which is inevitably biased. This bias, however, is expected to affect ‘integrated’ and ‘estimated’ Copula-GP methods differently, since different dependencies are being modeled with the parametric copulas.

In this section, we evaluate the performance of our Copula-GP method and compare it with the other commonly used non-parametric algorithms for mutual information estimation: Kraskov-Stögbauer-Grassberger (KSG [53]), Bias-Improved-KSG by Gao et al. (BI-KSG [54]) and the Mutual Information Neural Estimator (MINE [55]). We construct 3 datasets, which best illustrate the strengths and the limitations of our semi-parametric and other non-parametric methods. These datasets comprise the synthetic samples from relatively low dimensional distributions (≤10D), for which we can still directly calculate the true mutual information. These distributions have uniform marginals, which allows us to compare ‘integrated’ and ‘estimated’ approaches.

Baseline scenario: Gaussian dataset.

First, we consider a dataset sampled from a multivariate Gaussian distribution, with cov(y_i, y_j) = ρ + (1 − ρ)δ_ij, where δ_ij is Kronecker’s delta and ρ = −0.1 + 1.1 x, x ∈ [0, 1]. Our method is expected to perform well on this dataset because the ground truth model is a special case of our parametric copula model. On the other hand, the dataset is expected to expose the limited scalability of non-parametric methods that do not separate the dependence structure from the marginals and do not make any assumptions on the shape of the dependence.

Our Bayesian model selection algorithm (Sec. 6), based on the fully-Bayesian Watanabe–Akaike information criterion (WAIC, see Sec. 6), selected a Gaussian copula on these data, which perfectly matches the true distribution. As a result, we confirm that Copula-GP measures both entropy and mutual information without bias for any number of dimensions in this dataset (within integration tolerance, see Fig 4A; also tested up to ρ = 0.999 and up to 20 dimensions in Sec A in S3 Text). The same exact estimation applies to any linear mixture of copulas in our framework as well (Fig 1; see validation on Clayton copula in Sec B in S3 Text), and is covered by the automated tests on simulated data (see S2 Text).

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Fig 4.

Comparison of the Copula-GP model against the non-parametric information estimators, performed on three benchmarking datasets A Multivariate Gaussian. B Multivariate Student T. C Multivariate Gaussian y (same as A), morphed into another distribution with a tail dependence, while . In each row, the plots show: i. the probability density plots from each dataset: the unconditional dependency structure p(u) (left) and conditional dependency structures at the beginning and the end of the parameter domain dom x = [0, 1] (middle and right, respectively). ii. conditional entropy H(y|x); the black line shows the true values, the red line—Copula-GP, the orange line—BI-KSG; Note, that MINE is not included in this comparison, as it does not produce estimates of H(y|x). iii. mutual information I(x, y); black line—true value; red—Copula-GP (solid: MC integration (12); dashed: estimated MI (13)); orange—BI-KSG; green—KSG; blue—MINE (dotted: 100 HU, dashed: 200 HU, solid: 500 HU). Gray intervals show either standard error of mean (SE, 6 repetitions), or for integrated variables. Note, that MINE estimates are sensitive to the choice of hyper-parameters (e.g. number of hidden units, shown in different line styles).

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The performance of the non-parametric methods on this dataset is lower: KSG/BI-KSG captured only around 35%-40% of the true mutual information in 10 dimensions, and MINE captured from 70% to 90% depending on the number of hidden units. This agrees with the previous studies, in which KSG and MINE both severely underestimate the MI for high-dimensional Gaussians with high correlation (e.g. see Fig 1 in Belghazi et al. [55]).

4 Validation of Copula-GP method on neuronal population activity from the visual cortex of behaving mice

In this section, we investigate the dependencies observed in neuronal and behavioral data and showcase possible applications of the Copula-GP framework. We used two-photon calcium imaging data of neuronal population activity in the primary visual cortex of mice engaged in a visuospatial navigation task in virtual reality (data from Henschke et al. [16], decontaminated with FISSA algorithm [43]). The visual stimulus in this experiment was uniquely determined by the position in the virtual reality, similarly to the examples in Figs 2 and 3 where the shared input x was determined by time t. Briefly, the mice learned to run through a virtual corridor with vertical gratings on the walls (Fig 5A, 0–120 cm) until they reached black walls in the reward zone (Fig 5A, 120–140 cm), where they could get a reward by licking a reward spout as defined in the original publication of this dataset [5].

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Fig 5. Validation of Copula-GP method on neuronal population activity and behavioral variables from awake mice.

Copula-GP accurately models the neuronal and behavioral heavy tailed dependencies in the data from the visual cortex of awake mice, and quantifies more mutual information between various combinations of variables than the alternative methods. A Schematic of the navigational experimental task [5, 16] in virtual reality; B Example traces from ten example trials: x is a position in virtual reality, y is a vector of neuronal (blue) and behavioral (red) variables; these traces show that variables have different timescales and different signal-to-noise ratios, which result in different distributions of single variables y_i (i.e. different marginal statistics). C-D Copula probability density plots for: the noise correlation between two neurons (number 3 and 63) (C) and for the correlation between one neuronal activity (60) vs. one behavioral variable (licks) (D); Black outlines show empirical copula, shades of blue—the best fitting Copula-GP model: a mixture of Gaussian + 90°-rotated Clayton copula in (C) and a mixture of Frank + 0°-rotated Clayton + 270°-rotated Gumbel copula in (D) (see S4 Text for model parameters). Similarly to the example with two dynamically coupled neurons (Fig 3G), these copulas are heavy-tailed. The goodness-of-fit for these models is measured with the proportion of the variance explained , which is indicated in the upper-right corner of each plot corresponding to a range of positions in virtual reality; E-G Conditional entropy for the bivariate examples (E-F) and the population-wide statistics (G) all peak in the reward zone; this entropy is equivalent to the mutual information between variables, given the position x, which means that the variables carry the most information about each other when the animal is in the reward zone. H Comparison of Copula-GP method (“integrated”) vs. non-parametric MI estimators (MINE [55] and KSG [53]) on estimating the amount of information about the location x from the subsets of variables u^x. While the true I(x, u^x) is unknown, the validation on synthetic data (Fig 4) suggest that Copula-GP “integrated” does not overestimate the amount of mutual information. Yet, Copula-GP “integrated” quantifies more information about the position x from the large subsets of data u^x than MINE and KSG methods.

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For demonstration of the suitability of Copula-GP models for neuronal data, we selected one example dataset from an animal in which 102 neurons were imaged in visual cortex, during the first day of the animal’s training to the task. We conditioned our Copula-GP model on the position in the virtual environment x and studied the joint distribution of the behavioral () and neuronal () variables (dim y = 109, Table 1). Fig 5B shows examples of neuronal responses from the selected mouse running along the virtual corridor. The traces show changes in the position x of the mouse as well as the activity of 3 selected neurons and the licking rate.

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Table 1. A list of variables in the navigational task dataset from Pakan et al.; for a detailed description of the task and the reward, see the original paper [5].

The variables are grouped according to their type; their order does not correspond to the order of variables y₁…y₁₀₉ in vine models (see text).

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The goal of our analysis is to build a statistical model of the joint distribution of these variables p(y|x) and measure how much information these variables carry about each other (i.e. the redundancy) and about the location in the virtual environment. These variables have different patterns of activity depending on x and different signal-to-noise ratios (Fig 5B), which results in drastically different distributions of individual variables (i.e. marginal statistics). For such data, copulas are an uniquely suited tool for ‘gluing’ these variables together and constructing a statistical model of their joint distribution.

Comparison with alternative methods

Throughout the paper, we compare our model with the state-of-the-art general-purpose information estimators, which were also recently applied to the neuronal data [58, 59]. We show, that both of the state-of-the-art information estimators—KSG and MINE—have poor performance on high dimensional neuronal data. Intuitively, the difference in performance between our parametric method and these non-parametric methods can be explained as follows. The estimators based on k-nearest neighbor distances, such as KSG [53] / BI-KSG [54] rely on the volumes of the nearest neighborhoods of the samples for density estimation, which expand with the increase in the number of dimensions, resulting in poor information estimates. The neural network approaches, such as MINE [55], provide better scalability, but have many parameters and require high sample numbers in order to achieve a given accuracy with a given confidence. As a result, both classes have poor performance when the number of samples (trials) is low and the dimensionality (number of neurons and behavioral variables) is high. To overcome this ‘curse of dimensionality’, some assumptions about the dependency between variables are required [60].

Our Copula-GP method follows this approach in that it assumes certain shapes of the dependence between variables by specifying parametric copula distributions (see Fig 1). These constraints on the shape of the dependence, however, are less restrictive than the assumptions in other commonly used models, such as maximum entropy models [61]. Some of these maximum entropy models utilized conditioning on the overall population activity in order to take some of the higher-order correlations into account [62–64]. These models resemble the conditioning used in the first tree of our vine copulas, especially when we select the global neuropil activity as the first conditioning variable (see Sec. 7). However, all these population tracking maximum entropy models assumed linear coupling of each individual neuron with the population rate, characterized by a single parameter (thus, ∼N² parameters in total). We relax this constraint by introducing mixtures of copulas from different copula families, which model complex nonlinear pairwise dependencies between variables. The complexity of the copula mixture ‘adapts’ to the complexity of the data, since we select the least complex model that can capture the dependence via the information criterion (WAIC). As a result, our semi-parametric Copula-GP model takes the best of both worlds, performing well given a limited number of high-dimensional samples, and, at the same time, being more flexible in representing the conditional dependencies than maximum entropy models.

Alternatively to the information-theoretic approaches, some popular non-copula methods could model the neuronal responses in the aforementioned navigational task dataset. These methods include GPFA or GLMs on the deconvolved spikes [65]. Also, for the multi-photon calcium recordings (ΔF/F), the newly developed ValPACa method could, perhaps, be an alternative [66]. These models, however, require certain assumptions on the marginal statistics (e.g. Poisson or Gaussian), and only model the joint distribution of the neuronal activity (without behavioral variables). Contrary to these approaches, copula methods allow us to model the dependencies between neuronal and behavioral variables, combining the elements with utterly different statistics (e.g. licks or velocity with neuronal activity). In addition, Copula-GP explicitly represents the dependencies as a function of time or position. As a result, the conditional entropy calculated with our model (Fig 5E–5G) revealed insightful information about the task structure, highlighting the location of the reward zone. To the best of our knowledge, there are currently no other methods with this combination of properties that include: explicit conditioning on stimuli, flexibility in probability density estimation, scalability and accuracy in information estimation.

5 Parametric copula mixtures with Gaussian process priors

In this paper, we use semi-parametric approach: non-parametric marginals and parametric copulas. We used the two-stage inference for margins (IFM) training scheme, which is typically used high dimensional datasets [72]. First, univariate marginals were estimated and used to map the data onto a multidimensional unit cube. Second, the parameters of the copula model were inferred.

Bivariate copula families.

We use 4 copula families as the building blocks for our copula models: Gaussian, Frank, Clayton and Gumbel copulas (Fig 1). All of these families have a single parameter, corresponding to the rank correlation (Table 2). We also use rotated variants (90°, 180°, 270°) of Clayton and Gumbel copulas in order to express upper tail dependencies and negative correlation.

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Table 2. Bivariate copula families and their GPLink functions.

https://doi.org/10.1371/journal.pcbi.1009799.t002

Since we are primarily focused on the analysis of neuronal data, we have first visualized the dependencies in calcium signal recordings after a probability integral transform, yielding empirical conditional copulas. As a distinct feature in neuronal datasets, we observed changes in tail dependencies with regard to the conditioning variable. Since none of the aforementioned families alone could describe such conditional dependency, we combined multiple copulas into a linear mixture model: (3) where M is the number of elements, ϕ_j(x) is the concentration of the jth copula in a mixture, c_j is the pdf of the jth copula, and θ_j is its parameter.

Each of the copula families includes the Independence copula as a special case. To resolve this overcompleteness, we add the Independence copula as a separate model with zero parameters (Table 2). For independent variables y_ind, the Independence model will be preferred over the other models in our model selection algorithm (Sec. 6), since it has the smallest number of parameters.

Approximate inference.

Since our model has latent variables with GP priors and intractable posterior distribution, direct maximum likelihood Type-II estimation is not possible and an approximate inference is needed. Such an inference problem with copula models has previously been solved with the expectation propagation algorithm (Hernández-Lobato et al. [31], see a direct comparison with our method in Sec C in S3 Text), which was not suitable for large scale data. Recently, a number of scalable approximate inference methods were developed, including stochastic variational inference (SVI) [73, 74], scalable expectation propagation (SEP) [75], and MCMC based algorithms [76], as well as a scalable exact GP [77]. We chose to use SVI due to availability of the well-established GPU-accelerated libraries: PyTorch [67] and GPyTorch [68].

In particular, we used stochastic variational inference (SVI) with a single evidence lower bound [78]: (4) implemented as VariationalELBO in GPyTorch [68]. Here N is the number of data samples, u are the inducing points, q(u) is the variational distribution and q(f) = ∫p(f|u)q(u)d u.

Following the Wilson et al. [79] approach (KISS-GP), we then constrain the inducing points to a regular grid, which applies a deterministic relationship between f and u. As a result, we only need to infer the variational distribution q(u), but not the positions of u. The number of grid points is one of the model hyper-parameters: grid_size. Note that while approximate inference with KISS-GP uses a discrete set of inducing points (pseudo-inputs), it does not imply discretization of the input variable. The Gaussian process still uses a continuous kernel and assumes a continuous input.

Eq (4) enables joint optimization of the GP hyperparameters (constant mean μ and two kernel parameters: scale and bandwidth) and parameters of the variational distribution q (mean and covariance at the inducing points: ) [78]. We have empirically discovered by studying the convergence on synthetic data, that the best results are achieved when the learning rate for the GP hyperparameters (base_lr) is much greater than the learning rate for the variational distribution parameters (var_lr, see Table 3).

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Table 3. Hyper-parameters of the bivariate Copula-GP model.

https://doi.org/10.1371/journal.pcbi.1009799.t003

Priors. For both the neuronal and the synthetic data, we use a standard normal prior for a variational distribution. Note, that the parametrization for mixture models was chosen such that the aforementioned choice of the variational distribution prior with zero mean corresponds to a priori equal mixing coefficients ϕ_j = 1/M for j = 1…M. In our experiments with the simulated and real neuronal data, we observed that the GP hyper-parameter optimisation problem often had 2 minima (which is a common situation, see Fig 5.5 on page 116 in [80]). One of those corresponds to a short kernel lengthscale (λ) and low noise (min_f σ²), which we interpret as overfitting. To prevent overfitting, we used prior on RBF kernel lengthscale parameter.

Optimization. We use the Adam optimizer with two learning rates for GP hyper-parameters (base_lr) and variational distribution parameters (var_lr). We monitor the loss (averaged over 50 steps) and its changes in the last 50 steps: Δ loss = mean(loss[-100:-50])—mean(loss[-50:]). If the change becomes smaller than check_waic, then we evaluate the model WAIC and check if it is lower than −WAIC_tol. If it is higher, we consider that either the variables are independent, or the model does not match the data. Either way, this indicates that further optimisation is counterproductive. If the WAIC < −WAIC_tol, we proceed with the optimisation until the change of loss in 50 steps Δloss becomes smaller than loss_tol (see Table 3).

Hyper-parameter selection. The hyper-parameters of our model (Table 3) were manually tuned, often considering the trade off between model accuracy and evaluation time. A more systematic hyper-parameter search might yield improved results and better determine the limits of model accuracy.

6 Bayesian model selection

We use the Watanabe–Akaike information criterion (WAIC [81]) for model selection. WAIC is a fully Bayesian approach to estimating the Akaike information criterion (AIC) (see (31) in the original paper by Watanabe (2013) [81]). The main advantage of the method is that it avoids the empirical estimation of the effective number of parameters, which is often used for approximation of the out-of-sample bias. It starts with the estimation of the log pointwise posterior predictive density (lppd) [82]: where {θ^s}_S is a draw from a posterior distribution, which must be large enough to represent the posterior. Next, the p_WAIC approximates the bias correction, where represents sample variance. Therefore, the bias-corrected estimate of the log pointwise posterior predictive density is given by:

In the model selection process, we aim to choose the model with the lowest WAIC. Since our copula probability densities are continuous, their values can exceed 1 and the resulting WAIC is typically negative. Zero WAIC corresponds to the Independence model (pdf = 1 on the whole unit square).

Since the total number of combinations of 10 copula elements (Fig 1, considering rotations) is large, exhaustive search for the optimal model is not feasible. In our framework, we propose two model algorithms for constructing close-to-optimal copula mixtures: greedy and heuristic (see S1 Text for details). The greedy algorithm is universal and can be used with any other copula families without adjustment, while the heuristic algorithm is fine-tuned to the specific copula families used in this paper (Fig 1). Both model selection algorithms were able to select the correct 1- and 2-component model on simulated data and at least find a close approximation (within WAIC_tol = 0.005) for more complex models (see validation of model selection in S2 Text).

7 Copula vine constructions

High-dimensional copulas can be constructed from bivariate copulas by organizing them into hierarchical structures called copula vines [21]. There are many possible decompositions based on different assumptions about conditional independence of specific elements in a model, which can be classified using graphical models called regular vines [83, 84]. A regular vine can be represented using a hierarchical set of trees, where each node corresponds to a conditional distribution function (e.g. F(u₂|u₁)) and each edge corresponds to a bivariate copula (e.g. c(u₂, u₃|u₁)). The copula models from the lower trees are used to obtain new conditional distributions (new nodes) with additional conditional dependencies for the higher trees, e.g. a ccdf of a copula c(u₂, u₃|u₁) and a marginal conditional distribution F(u₂|u₁) from the 1st tree provide a new conditional distribution F(u₃|u₁, u₂) for a 2nd tree. Therefore, bivariate copula parameters are estimated sequentially, starting from the lowest tree and moving up the hierarchy. The total number of edges in all trees (= the number of bivariate copula models) for an m-dimensional regular vine equals m(m − 1)/2.

The regular vines often assume that the conditional copulas c(u_i, u_j|{u_k}) themselves are independent of their conditioning variables {u_k}, but depend on the them indirectly through the conditional distribution functions (nodes) [85]. This is known as the simplifying assumption for vine copulas [86], which, if applicable, allows to escape the curse of dimensionality in high-dimensional copula construction.

In this study, we focus on the canonical vine or C-vine, which has a unique node in each tree, connected to all of the edges in that tree. It factorizes the high-dimensional copula probability density function as follows: (5) where {k}_k<i = 1, …, i − 1 and F(.|.) is a conditional CDF. For graphical illustration, see, for example, Fig 2 in Aas et al. [21]. Note, that all of the copulas in (5) can also be conditioned on x via Copula-GP model.

The C-vine was shown to be a good choice for neuronal datasets [23], as they often include some proxy of neuronal population activity as an outstanding variable, strongly correlated with the rest. This variable provides a natural choice for the first conditioning variable in the lowest tree. In the neuronal datasets from [16], this outstanding variable is the global fluorescence signal in the imaged field of view (global neuropil, variable in Table 1).

To construct a C-vine for describing the neuronal and behavioral data from [16], we used a heuristic element ordering based on the sum of absolute values of Kendall’s τ of a given element with all of the other elements. It was shown by Czado et al. [36] that this ordering facilitates C-vine modeling. For all of the animals and most of the recordings (14 out of 16) the first variable after such ordering was the global neuropil activity. This again confirms, that a C-vine with the global neuropil activity as a first variable is an appropriate model for the dependencies in neuronal datasets.

8 Algorithmic complexity

In this section, we discuss the algorithmic complexity of the parameter inference for a C-vine copula model.

The parameter inference for each of the bivariate Copula-GP models scales as , where n is the number of samples, since we use a scalable kernel interpolation KISS-GP [79]. As we mentioned in Sec. 7, a full m-dimensional C-vine model requires m(m − 1)/2 bivariate copulas, trained sequentially. As a result, the GP parameter inference has to be repeated m(m − 1)/2 times, which yields complexity.

In practice, the computational cost (in terms of time) of the parameter inference for each bivariate model varies from tens of seconds to tens of minutes. The heuristic model selection is designed in such a way, that it discards independent variables in just around 20 seconds (line 3 in Alg. 2). As a result, most of the models are quickly skipped and further considered as Independence models. When the model is evaluated, the Independence components are also efficiently ‘skipped’ during sampling, as ppcf function is not called for them. The Independence models also add zero to C-vine log probability, so they are also ‘skipped’ during log probability calculation. They also reduce the total memory storage, as no GP parameters, which predominate the memory requirements, are stored for these models.

In practice, this means that the algorithmic complexity of the model is much better than the naïve theoretical prediction , based on the structure of the graphical model. Suppose that the actual number of the non-Independence models N_nI in a vine model is much smaller than m(m − 1)/2 and can be characterized by an effective number of dimensions . In this case, instead of the scaling with the number of variables, the complexity of parameter inference highly depends on the sparsity of the dependencies in the graphical model and scales with as .

Therefore, the our method is especially efficient on the datasets with a low effective dimensionality m_eff, such as the neuronal data. The dimensionality m itself has little effect on the computational cost and memory requirements, as the total complexity scales as , where c is a small constant (≈ 0.05) equal to the ratio between independency test time to the extra time required for model selection to complete. However, the second term can become significant for extremely large datasets, where many bivariate models have to be tested for dependence vs. independence. If this becomes an issue, some additional independence assumptions (e.g. for distant neurons) can be incorporated into the model.

9 Generation of the synthetic calcium recording from a GLM model

We generated the synthetic spike counts using a Generalized linear model (GLM) with an exponential non-linearity and Poisson emission model [38]: (6) (7) where T = 20 is the length of the filters, x^−T:0 are the values of the stimuli in the preceding T steps (up to, but excluding 0) of size (T × 1), y^−T:0 is a matrix (T × 2) of neural responses, and h is a tensor (2 × 2 × T) of coupling filters: (8) for coupled neurons in Fig 3. For independent neurons in Fig 2, the off-diagonal elements were equal to 0.

We simulate 100 identical trials with fixed duration (τ = 100 steps), in which the same sequence of stimuli x(t) was applied (see, for example, Fig 2A). In order to generate continuous data mimicking the calcium optical recordings, we convolve the simulated spike count data y(t) with an exponential temporal kernel (as in Deneux et al. [39] model). A characteristic time of the exponent in the synthetic calcium dynamics was 4 time steps (whole interval t ∈ [0, τ], τ = 100 steps). The resulting calcium traces are shown in Figs 2B, 3B and 3C.

10 Goodness-of-fit

We measure the accuracy of the density estimation with the proportion of variance explained R². We compare the empirical conditional CDF ecdf(u₂|u₁ = y) vs. estimated conditional CDF ccdf(u₂|u₁ = y) and calculate: (9) where R²(y) quantifies the portion of the total variance of u₂ that our copula model can explain given u₁ = y, and . The sum was calculated for u₂ = 0.05 n, n = 0…20.

Next, we select all of the samples from a certain interval of the task (x ∈ [x₁, x₂]) matching one of those shown in Fig 3 in the paper. We split these samples u₁ ∈ [0, 1] into 20 equally sized bins: {I_i}₂₀. For each bin I_i, we calculate (9). We evaluate ccdf(u₂|u₁ = y_i) ≈ ccdf(u₂|u₁ ∈ I_i) using a copula model from the center of mass of the considered interval of x: x_μ = mean(x) for samples x ∈ [x₁, x₂]. We use the average measure: (10) to characterize the goodness of fit for a bivariate copula model. Since u₁ is uniformly distributed on [0, 1], the probabilities for each bin p(u₁ ∈ I_i) are equal to 1/20, and the resulting measure is just an average R² from all bins. The results were largely insensitive to the number of bins (e.g. 20 vs. 100).

11 Entropy and mutual information

Our framework provides tools for efficient sampling from the conditional distribution and for calculating the probability density p(y|x). Therefore, for each x = t the entropy H(y|x = t) can be estimated using Monte Carlo (MC) integration: The probability p(y|x = t) factorizes into the conditional copula density and marginal densities (2), hence the entropy also factorizes [18] as H(y|x = t) = ∑H(y_i|x = t) + H_c(u^x|x = t), where u^x = F(y|x). The conditional entropy can be integrated as (11) separating the entropy of the marginals {y_i}_N from the copula entropy.

Now, I(x, y) = I(x, G(y)) if G(y) is 1) a homeomorphism, 2) independent of x [53]. If marginal statistics are independent of x, then the probability integral transform u = F(y) satisfies both requirements, and I(x, y) = I(x, u). Then, in order to calculate the mutual information I(x, u) ≔ H(u) − H(u|x), we must also rewrite it using only the conditional distribution p(u|x), which is modelled with our Copula-GP model. This can be done as follows: (12)

The last term in (12) involves nested integration, which is computationally difficult and does not scale well with N = dim u. Therefore, we propose an alternative way of estimating I(x, y), which avoids double integration and allows us to use the marginals conditioned on x (u^x = F(y|x)). We can use two separate copula models, one for estimating p(y) and calculating H(y), and another one for estimating p(y|x) and calculating H(y|x): (13) where both entropy terms are estimated with MC integration. Here we only integrate over the unit cube [0, 1]^N and then parameter domain domx, whereas (12) required integration over [0, 1]^N × dom x.

The performance of both (12) and (13) critically depends on the approximation of the dependence structure, i.e. how well the parametric copula approximates the true copula probability density. If the joint distribution p(y₁…y_N) has a complex dependence structure, as we saw in synthetic examples (Sec. 3), then the mixture of parametric copulas may provide a poor approximation of p(y) and overestimate H_c(u₁, …, u_N), thereby overestimating I(x, y). The direct integration (12), on the other hand, typically underestimates the I(x, y) due to imperfect approximation of p(y|x), and is only valid under assumption that the marginals can be considered independent of x.

We refer to the direct integration approach (12) as “Copula-GP integrated” and to the alternative approach (13) as “Copula-GP estimated” and assess both of them on synthetic and real data.