Modeling statistical dependencies in multi-region spike train data

Summary

Neural computations underlying cognition and behavior rely on the coordination of neural activity across multiple brain areas. Understanding how brain areas interact to process information or generate behavior is thus a central question in neuroscience. Here we provide an overview of statistical approaches for characterizing statistical dependencies in multi-region spike train recordings. We focus on two classes of models in particular: regression-based models and shared latent variable models. Regression-based models describe interactions in terms of a directed transformation of information from one region to another. Shared latent variable models, on the other hand, seek to describe interactions in terms of sources that capture common fluctuations in spiking activity across regions. We discuss the advantages and limitations of each of these approaches and future directions for the field. We intend this review to be an introduction to the statistical methods in multi-region models for computational neuroscientists and experimentalists alike.

Introduction

The idea that distinct areas of the brain support different cognitive and behavioral functions dates back to the early 19th century [1]. During this time, there was debate as whether various cortical regions had specific functionality or rather participated in all cognitive and psychological functions [2, 3]. These competing ideas remain a topic of study today [4], and despite significant progress in understanding brain function over the past two centuries, the extent to which different brain areas support distinct functions and convey different information to other brain areas remains largely unknown.

In the past decade, advances in neural recording technologies have provided exciting new opportunities to study how distinct brain regions interact. In particular, advances in wide field calcium imaging [5–7] and high-throughput neural probes [8] allow for simultaneous recording of neural activity from hundreds to thousands of neurons in multiple brain regions (Figure 1A). Recent work has made substantial progress in characterizing neural population codes within a single brain regions [9–15]. The challenge remains to extend these models to study how populations interact. This necessitates both adding to existing statistical tools and developing new tools to extract accurate and interpretable descriptions of interactions from multi-region neural recordings.

Figure 1:

Approaches to characterizing neural activity from simultaneously recorded brain regions. A. Schematic for simultaneous electrophysiological recording from two separate brain regions. B. Regression models are directional statistical descriptions that seek to characterize the firing of one brain region using the activity of the other region. C. Latent variable models characterize shared structure underlying response fluctuations in multiple brain regions.

A simple approach to analyzing multi-region datasets is to compute descriptive statistics that summarize the statistical relationships between regions. For example, an extensive literature has focused on using the cross-correlogram or cross-covariance between spike trains to make inferences about connectivity [16–19]. In recent work, covariance-based measures have been used to identify functional sub-networks across visual areas in rodent visual cortex [5, 15, 20]. However, moment-based methods are typically limited by the lack of an explicit statistical model for neural responses, and may lack statistical power for identifying weak interactions between areas. In this review, we therefore focus on model-based approaches, which provide explicit descriptions of the dependencies within and between brain areas.

Here we provide an overview of two modern approaches for characterizing dependencies in multi-region electrophysiological recordings. We begin by discussing regression-based approaches, which seek to build predictive models for neural activity in one region using neural activity in other regions (Figure 1B). We then discuss latent variable models, which characterize the structure of multi-region data in terms of shared low-dimensional time series (Figure 1C). We conclude by proposing future directions and ways of extending existing models to help further uncover the ways distinct brain regions interact.

Regression-based approaches

Regression models provide one framework for characterizing statistical dependencies between brain areas using data from multi-area recordings. The basic idea is to fit a model that predicts activity in one brain area using the activity from one or more other brain areas (Figure 1B). Models of this form contain a set of linear weights that specify how spikes from a group of input neurons modulate the firing rate of an output neuron, with Gaussian [21] or Poisson observation noise and time-lag between regions [22, 23]. The inferred regression weights in this class of models provide a measure of statistical dependency, often referred to as functional connectivity, between areas [22, 24]. When the regression model in question is a time-lagged linear-Gaussian “least-squares” regression, the resulting dependency measure is known as Granger causality [25, 26]. When the model consists of a linear-mapping following by a point-wise non-linearity and Poisson observations, the method is known as the Poisson generalized linear model (GLM). Recent work has extended this framework to allow for non-linear dependencies between neurons using deep neural networks [27]).

For this review, we focus on the Poisson GLM, which provides a simple yet powerful regression modeling framework for spike train data. The single-neuron Poisson GLM describes how the spike rate of a single neuron depends on external covariates such as visual stimuli, spatial position, or theta oscillation phase [28–32]. So called “auto-regressive” Poisson GLMs include weights on spike history from one or more neurons, which allow them to characterize dependencies between neurons in multi-region recordings [22–24, 33–39].

Formally, a Poisson GLM describes each neuron by an instantaneous firing rate, also known as the conditional intensity, denoted λ_t, which defines the probability of observing a spike in a small time bin at time t. This firing rate is given by the projection of an input vector x_t onto the linear weights w, transformed by a rectifying nonlinear function f:

$${\lambda }_t=f\left(w^{\top }{x}_t+b\right),$$ (1)

where b is a constant offset or bias term. Common choices of nonlinearity f include exponential and the soft-rectified or “softplus” function, f(x) = log(1 + e^x), both of which ensure that firing rates be non-negative. The spike count y_t in a time bin of size Δ has a conditionally Poisson distribution:

$$y_t\mid x_t~\text{Poisson}\left(\Delta {\lambda }_t\right),$$ (2)

although other count distributions (e.g., Bernoulli, binomial, negative binomial) have also be considered [40–43].

To model statistical dependencies between neurons, the spike history of other neurons can be included as a regressor. In this case the firing rate of neuron j can be re-written as:

$${\lambda }_t^{(j)}=f\left(w_j^{\top }{x}_t+h_j^{\top }{y}_{{\text{hist}}_t}^{(j)}+\sum _{i\ne j}{c}_{ij}^{\top }{y}_{{\text{hist}}_t}^{(i)}+b_j\right)$$ (3)

where w_j and b_j are stimulus weights and offset for neuron j, h_j denotes the “self-coupling” spike-history filter for neuron j, c_ij denotes a “coupling” filter from neuron i to neuron j, and $y_{{\text{hist }}_t}^{(i)}$ denotes a vector representing the spike history of neuron i at time t. This is typically defined to be a vector of binned spike counts from time bins (t – τ, … , t − 1), for some number of time lags τ before the current time bin.

When defined this way, it can be seen that the coupled Poisson GLM is actually a collection of single-neuron models, one for each neuron in the population (schematized for a single neuron in Fig. 2). Each neuron is characterized by the set of filters that determine its inputs. Fitting the coupled Poisson GLM to multi-neuron data therefore involves fitting a separate GLM to each neuron in the population, which can be carried out in parallel.

Figure 2:

The coupled Poisson GLM is a regression model for capturing dependencies in multi-region spike train recordings. To predict spike trains in region B using spike trains from region A, the model contains “coupling” filters c_ij that capture statistical dependencies between previous spikes from neuron i and the spike rate of neuron j at time t. It may also contain filters capturing dependencies on external stimuli, and on the spike history of other neurons in region B (not shown). The full Poisson GLM for a single brain region contains an independent set of filters for every spike train, meaning that it is really a collection of independent models, one for every neuron in downstream population B. One can similarly construct a Poisson GLM to predict spike trains in region A from the spikes in region B.

One way to characterize statistical dependencies with the Poisson GLM is to analyze the coupling filters between neurons in different brain regions. The strength of dependencies between regions can be assessed by computing the predictive performance of models fit with and without coupling filters between regions [22]. A simple metric is the difference in cross-validated log-likelihood, given by (log P(Y_test|θ_coupled) − log P(Y_test|θ_uncoupled), where Y_test represents held-out test data not used for fitting, and θ_coupled and θ_uncoupled represent fitted parameters of the coupled and uncoupled models, respectively. This quantity has units of bits when taking log base 2, and can be divided by the number of seconds or number of samples to get units of bits/s or bits/sample [31, 42]. Another way to use GLMs to characterize dependencies between brain regions is to directly analyze the shape and amplitude of fitted coupling filters. The peak amplitude of each coupling filter provides a summary of coupling strength, while the sign of the peak indicates whether the effective dependencies are excitatory or suppressive [38].

One general advantage of regression-based approaches is the ability to characterize directed dependencies between brain regions. Because we regress the activity in each brain area against the time-lagged activity from other areas, we can identify asymmetric relationships between areas. For example, we might identify that spikes in area A predict increased spiking area B, whereas spikes in area B predict decreased firing in area A. However, it is important to emphasize that the statistical dependencies identified by GLMs are statistical and not causal, and are susceptible to omitted variable bias [44]. Thus, for example, the dependencies between two brain regions identified by GLM analysis may in fact reflect shared input with different time lags from a third, unobserved brain region [45, 46].

Another advantage of GLMs is the ability to parallelize over neurons when fitting, since the full model is simply a collection of single-neuron models. This confers computational advantages over latent variable models (see next section), which must be fit jointly to all neurons at once. However, the total number of coupling filters in a GLM grows quadratically with the number of neurons, which can lead to overfitting and problems of interpretation in large neural populations. Sparsity-inducing priors, which drive some coupling filters to zero, can help mitigate both problems; these priors reduce the number of model parameters while preserving the most significant connections between neurons [35, 38, 47].

Latent variable modeling approaches

An alternative to regression-based approaches is to model dependencies between brain regions using shared latent variables. Here, the standard approach is to model the response fluctuations in a large population of neurons as arising from a small number of hidden or ‘latent’ sources. These models are typically not directed, meaning that they they seek to describe dependencies between brain regions in terms of shared modulation by a small number of hidden or unobserved sources, as opposed to seeking to describe dependencies in terms of separate feedforward and feedback components.

In recent years, latent variable models have become a popular approach for analyzing structure in single-region datasets. Typically, the time-course of the latent variable is either assumed to follow a linear dynamical system (LDS) [45, 48–53], or is simply assumed to be smooth in time using a Gaussian process (GP) prior [43, 54–60]. Both LDS and GP latent variable models have been augmented and modified for different purposes, and here we will highlight how each form can be extended to understand multi-region data.

The key distinction between LDS and GP latent variable models lies in the assumed prior over the latent variable. In LDS models, the latent vector x_t evolves according to linear dynamics with Gaussian noise:

$$x_t=A{x}_{t-1}+{ϵ}_t,{ϵ}_t~\mathcal{N}(0,Q)$$ (4)

where A describes the discrete-time linear dynamics. Although the assumption of linear dynamics is almost certainly too restrictive for real neural circuits, the noise component ϵ_t, often referred to as “innovations” noise, can be viewed as accounting for both the true noise as well model mismatch. One notable extension of this model, the switching Linear Dynamical System (sLDS), allows for discrete switching between different linear dynamics matrices, which can approximate nonlinear dynamics in the same way that a series of line segments can approximate a curve [61–65].

An alternate approach to LDS models is to describe the latent variable with a Gaussian process (GP), which makes no assumption about the dynamics beyond smoothness in time. A GP defines a joint Gaussian distribution over the latent x_t (an element of the latent vector x_t) at all time points t ∈ [0, T]. The correlations of this latent process are specified by a kernel function:

$$\text{cov}\left(x_t,x_{t^{\prime }}\right)=k\left(t,t^{\prime }\right),$$ (5)

which determines the a priori correlation between the latent variable at any pair of times t and t′. A common choice of kernel function is the ‘Gaussian’ or ‘radial basis function’ (RBF) covariance:

$$k\left(t,t^{\prime }\right)=e^{-\frac{{\left(t-t^{\prime }\right)}^2}{2{\mathcal{l}}^2}},$$ (6)

which is governed by a single hyperparameter ℓ known as the “length scale”, which determines degree of the smoothness. In a classic model known as Gaussian process factor analysis (GPFA), each element of the latent vector x_t is assumed to have an independent zero-mean GP prior, each with its own length scale [54].

In both LDS and GP latent variable models, the probability distribution over neural data is defined in terms of a noisy projection of the low-dimensional latent variable to the space of high-dimensional neural activity. The simplest case is to assume a linear mapping, as found in a class of so-called “factor models”. If we assume Gaussian additive noise with covariance R, we obtain [54]:

$$y_t=W{x}_t+{ϵ}_t,ϵ~\mathcal{N}(0,R),$$ (7)

where y_t is the set of n neurons’ spike counts at time t, x_t is the p-element latent vector at time t, and W is a (n × p) matrix of loading weights. Each column of W describes how a single latent affects the population spiking activity, and each row describes how a single neuron’s activity depends on the different latents contained in x_t.

A better model for spike count data is to assume Poisson noise after transformation by a fixed nonlinearity. In this case, the factor model can be written:

$$y_t\mid x_t~\text{Poiss}(\Delta f(Wx)),$$ (8)

where as before Δ denotes time bin width and f(·) is fixed non-negative output function [43, 57, 64]. Gaussian models are tractable and simple to fit because the log-likelihood can be evaluated in closed form. Poisson models, by contrast, my be more accurate for spike count data, particularly when spike rates are low, but require approximate inference methods to fit [43, 57, 64, 66, 67]. Recent work has introduced a latent variable models with flexible nonlinear mappings between latent variables and observations [60, 66–69], but for the purposes of this review we focus on models with linear mappings.

To extend the two latent variable modeling approaches described above to multi-region data, we will first consider recording from two brain regions, denoted A and B. We can perform inference on an LDS or GP factor model by concatenating data from two regions, and then examine the inferred loadings matrix, W. In the standard inference case, the model will uncover latent structure agnostic to the multi-region nature of the data. The latent factors will reflect the dimensions of shared variability, across time, for the entire population. A way to isolate per-region latent structure is to constrain the loadings matrix, W, to have block structure, with a subset of latents mapping only to region A, and another subset mapping only to region B. The model, shown for the Poisson observation case, takes the following form:

$$\left[\begin{array}{l}{y}_A\hfill \\ y_B\hfill \end{array}\right]=\text{Poiss}\left(f\left(\left[\begin{array}{cc}{W}_A& 0\\ 0& W_B\end{array}\right]\left[\begin{array}{l}{x}_A\hfill \\ x_B\hfill \end{array}\right]\right)\right),$$ (9)

where x_A and x_B reflect the latents corresponding to regions A and B and W_A,B the respective loadings for each region, and for simplicity we have dropped indexing on time.

This model is equivalent to fitting two entirely separate models, one for region A and another for region B, with no shared structure between them. We wish to contrast the above model with one that includes both shared and independent latent variables:

$$\left[\begin{array}{l}{y}_A\hfill \\ y_B\hfill \end{array}\right]=\text{Poiss}\left(f\left(\left[\begin{array}{ccc}{W}_A& W_{S1}& 0\\ 0& W_{S2}& W_B\end{array}\right]\left[\begin{array}{c}{x}_A\\ x_S\\ x_B\end{array}\right]\right)\right).$$ (10)

Here x_S refers to latent variables that are shared between regions, while x_A and x_B denote region-specific latent variables. This block structure in the loadings matrix W allows us to extract shared and independent sources of variability using either GP [70, 71] or LDS [65] priors over the latent variables. We show a schematic for this model in Figure 3. Here, a loadings matrix with independent components and shared components map independent and shared latents to a set of firing rates. These firing rates describe Poisson spiking from populations of neurons in each brain region.

Figure 3:

The multi-region Poisson GPFA (Gaussian-Process Factor Analysis) model contains a block-structured loadings matrix, with latent variables local to each region, and “shared” latents that capture shared variability across regions. Latent firing rates are passed through a rectifying nonlinearity f to obtain firing rates, which drive spiking via a Poisson process.

These shared latent variable models for multi-region data are structurally related to canonical correlation analysis (CCA) [72], which have been used in neuroscience settings for multi-region data before [4]. In particular, the LVMs outlined above relate closely to statistical variants of CCA known in the machine learning community as probabilistic CCA [73] and Bayesian CCA (BCCA) [74]. Though probabilistic CCA and BCCA do not include neuroscience-specific model features such as point-wise nonlinearities, Poisson observations, and GP priors, they are factor analytic models which identify shared low-dimensional subspaces that map to distinct upstream observations [73]. In the case of BCCA, these statistical models, like the LVMs described above, assert region-specific and shared factors through a block-structured loadings matrix [74, 75]. The ongoing work in the machine learning community on probabilistic CCA and Bayesian CCA may have useful implications to the neuroscience community. In particular, recent developments of incorporating sparsity priors over the block loadings matrix may be well suited to neuroscience contexts [75]

Implementing LVMs with block loadings structure is an exciting future for multi-region analysis. Some preliminary results suggest that models of this sort can incorporate time-delays to identify latents that represent feed-forward and feedback communication between brain regions [71]. Other work has isolated trial-varying latent structure that is shared across regions during repeated presentations of a visual stimulus [70]. However, a technical challenge here remains in properly identifying the dimensionality of the shared an independent subspaces, as a exhaustive search over all possibilities in computationally cumbersome. Greedy approaches, that add dimensions one-by-one if it improves cross-validation performance, is a good initial start [70]. An exciting possibility would be to impose an explicit regularization over the loadings matrix, similar to the approaches used in [75], such as an automatic relevance determination prior, or nuclear norm regularization. These more sophisticated statistical methods would isolate within and across region sub-spaces without the need directly specify the nature of the block structured loadings structure.

Hybrid approaches: low-dimensional regression models

So far we have considered regression models, which seek to explain activity in one brain area as a function of activity in another, and latent variable models, which seek to explain activity in multiple brain areas using a small number of components or factors. It is of course natural to seek to combine the strengths of these two approaches: models that can explain activity in one brain area in terms of a small number of components arising from another brain area. Dimensionality-reduction methods have had great success in extracting interpretable features from high-dimensional neural recordings [9]. Combining dimensionality reduction with regression thus provides a powerful approach for extracting more interpretable statistical interactions between brain regions. Although we have emphasized models with Poisson noise up to now, this section will focus primarily on models with Gaussian noise.

A simple approach to low-dimensional regression for multi-region data is to work in two stages: first perform dimensionality reduction on the upstream region, and then use regression on this low dimensional representation to predict downstream activity. When using principal components analysis (PCA) for dimensionality reduction followed by ordinary least squares regression, this technique is known as principal components regression (PCR). Kaufman et al. [76] used PCR to analyze the relationship between neural activity in pre-motor (PMd) and primary motor cortex (M1) during movement preparation, arguing that PMd activity falls primarily in the null space of the linear mapping to M1. Semedo et al. [21] used a similar approach, substituting factor analysis for PCA, to analyze dependencies in neural activity between V1 and V2, concluding that principal axes of V1 variability were not the dimensions that best predicted activity in downstream area V2.

An alternate approach to low-dimensional regression modeling is reduced rank regression (RRR), which unifies dimensonality reduction and regression into a single model. The RRR model explicitly imposes a rank constraint on the parameters of a regression model that govern the mapping from one brain region to another. For linear regression with Gaussian noise, the RRR objective is

$$\underset{W\in {ℝ}^{m\times n}}{\text{min}}{\Vert Y_B-Y_{A}W\Vert }_2^2,\text{\hspace{1em} where \hspace{1em}rank}(W)\le r$$ (11)

where Y_A denotes an T × m matrix of responses from m area-A neurons over T time bins, Y_B denotes a T × n matrix of responses from n area-B downstream neurons, and W is a m × n matrix of regression weights of rank at most r.

Remarkably, there is a closed-form solution to this optimization problem. This solution can be obtained using the top r right singular vectors of $Y_A{\widehat{W}}_{OLS}$, where ${\widehat{W}}_{OLS}$ is the matrix of ordinary least squares regression weights, ${\widehat{W}}_{OLS}={\left(Y_A^{\top }{Y}_A\right)}^{-1}{Y}_A^{\top }{Y}_B$. If we assemble these singular vectors into a matrix V, the RRR solution is simply ${\widehat{W}}_{RRR}={\widehat{W}}_{OLS}V{V}^{\top }$. The reduced-rank regression model can of course be extended to the setting of Poisson GLMs, although in this case fitting requires numerical optimization of the log-likelihood, as there is no longer an eigenvector solution.

The rank constraint in the reduced-rank regression model can be seen as imposing a bottleneck for information transmission between brain areas. For example, if we use a rank-2 RRR model, the interactions between brain areas are confined to a 2-dimensional subspace, regardless of the total number of latent dimensions in each area. Recent work from Semedo et al [21] coined the term ‘communication subspace’ to refer to this phenomenon, and showed that the dependencies between neural activity in visual areas V1 and V2 are indeed low-dimensional relative to the dimensionality of activity in each brain region.

The rank constraint in RRR also provides a form of regularization by reducing the number of parameters and allowing for parameter-sharing across neurons. The RRR model may therefore achieve better generalization performance than a (full-rank) GLM with all-to-all coupling due reduced overfitting. More importantly, RRR allows for direct estimation of the communication subspace from one population to another, without requiring separate steps for dimensionality-reduction and regression. In addition to being more accurate, the RRR model is also be more interpretable due to the fact that it provides a concise summary of the statistical dependencies between areas.

Challenges and opportunities

Recent advances in technology have ushered in a new era for multi-region optical and electrophysiological recording. These techniques hold great potential for deciphering the complex the flow of information within neural circuits. However, the challenges posed by analyzing such datasets highlight the need for innovative new statistical tools and approaches. As we move forward with the development of such tools, we propose that future work be guided by the principles of intepretable structure, scalability, and statistical efficiency.

Thinking carefully about interpretable statistical structure is important due to the large number of possible modelling options for multi-region data. When developing a model for multi-region data, scientists should consider what the model fits might tell us in the context of a particular hypothesis or set of hypotheses of interest. Successful examples include multi-region models built to understand signal and noise representations [70], alignment of activity subspaces [21], feedforward and feedback signals [71], or functional coupling properties across and within regions [23]. In each of these cases, a particular neuroscience question or network property motivates the development of the model, and the resultant fits provide insight into the network property of interest. Developing or using a multi-region model without first thinking about how to interpret it may be a useful mathematical exercise, but a good fit to neural data will not on its own provide any insight into multi-region neural activity.

Scalability of statistical models for multi-region data is vital for future work. High-throughput imaging and electrophysiology preparations will soon be routinely recording from thousands of neurons simultaneously. The estimation procedures used to fit the models described here must be able to contend not only with an expanding volume of data, but with the exploding degrees of freedom that accompany it. Fitting regression models in the large-scale setting, including parameters characterizing the coupling between cells, results in geometric growth in the number of parameters with the number of cells. One promising solution is to rely on approximate modeling methods [38]. Another approach that applies more generally to population-level modeling is to explicitly reduce the number of degrees freedom of the model, in which combinations of regression and latent variable techniques are applied synergistically.

We are especially excited about possibilities for combining latent variable models and regression models, and the potential for such combined models to reveal new insights into the flow of information between and within brain regions. Model-based targeted dimensionality reduction [77, 78] represents one recent method that combines a regression model with explicit low-dimensional constraints on population activity. Bayesian latent structure discovery [79] is another approach that merges regression and latent variable models. Both of these approaches model population-level activity by sharing degrees of freedom among cells but leave enough flexibility to identify functionally significant, low dimensional latent structure. These approaches have yet to be used in the multi-region setting but we expect their applicability to the multi-region setting to be straight forward.

Conclusion

The development of statistical analysis tools for multi-region data is in its infancy. We believe that a conceptual understanding of information processing in the brain depends on appropriate characterization of population-level activity [80]. As multi-region datasets become increasingly common, there will be greater need for analytical tools and appropriate statistical summaries for each region as well as inter-region interactions. Here, we have described two general methodological approaches for understanding multi-region data that are newly active areas of research and make suggestions for future directions.