Parametric Copula-GP model for analyzing multidimensional neuronal and behavioral relationships.

One of the main goals of current systems neuroscience is to understand how neuronal populations integrate sensory information to inform behavior. However, estimating stimulus or behavioral information that is encoded in high-dimensional neuronal populations is challenging. We propose a method based on parametric copulas which allows modeling joint distributions of neuronal and behavioral variables characterized by different statistics and timescales. To account for temporal or spatial changes in dependencies between variables, we model varying copula parameters by means of Gaussian Processes (GP). We validate the resulting Copula-GP framework on synthetic data and on neuronal and behavioral recordings obtained in awake mice. We show that the use of a parametric description of the high-dimensional dependence structure in our method provides better accuracy in mutual information estimation in higher dimensions compared to other non-parametric methods. Moreover, by quantifying the redundancy between neuronal and behavioral variables, our model exposed the location of the reward zone in an unsupervised manner (i.e., without using any explicit cues about the task structure). These results demonstrate that the Copula-GP framework is particularly useful for the analysis of complex multidimensional relationships between neuronal, sensory and behavioral variables.

Discovering Low-Dimensional Descriptions of Multineuronal Dependencies.

Coordinated activity in neural populations is crucial for information processing. Shedding light on the multivariate dependencies that shape multineuronal responses is important to understand neural codes. However, existing approaches based on pairwise linear correlations are inadequate at capturing complicated interaction patterns and miss features that shape aspects of the population function. Copula-based approaches address these shortcomings by extracting the dependence structures in the joint probability distribution of population responses. In this study, we aimed to dissect neural dependencies with a C-Vine copula approach coupled with normalizing flows for estimating copula densities. While this approach allows for more flexibility compared to fitting parametric copulas, drawing insights on the significance of these dependencies from large sets of copula densities is challenging. To alleviate this challenge, we used a weighted non-negative matrix factorization procedure to leverage shared latent features in neural population dependencies. We validated the method on simulated data and applied it on copulas we extracted from recordings of neurons in the mouse visual cortex as well as in the macaque motor cortex. Our findings reveal that neural dependencies occupy low-dimensional subspaces, but distinct modules are synergistically combined to give rise to diverse interaction patterns that may serve the population function.

Categorical encoding of decision variables in orbitofrontal cortex.

A fundamental and recurrent question in systems neuroscience is that of assessing what variables are encoded by a given population of neurons. Such assessments are often challenging because neurons in one brain area may encode multiple variables, and because neuronal representations might be categorical or non-categorical. These issues are particularly pertinent to the representation of decision variables in the orbitofrontal cortex (OFC)-an area implicated in economic choices. Here we present a new algorithm to assess whether a neuronal representation is categorical or non-categorical, and to identify the encoded variables if the representation is indeed categorical. The algorithm is based on two clustering procedures, one variable-independent and the other variable-based. The two partitions are then compared through adjusted mutual information. The present algorithm overcomes limitations of previous approaches and is widely applicable. We tested the algorithm on synthetic data and then used it to examine neuronal data recorded in the primate OFC during economic decisions. Confirming previous assessments, we found the neuronal representation in OFC to be categorical in nature. We also found that neurons in this area encode the value of individual offers, the binary choice outcome and the chosen value. In other words, during economic choice, neurons in the primate OFC encode decision variables in a categorical way.

Using Matrix and Tensor Factorizations for the Single-Trial Analysis of Population Spike Trains.

Advances in neuronal recording techniques are leading to ever larger numbers of simultaneously monitored neurons. This poses the important analytical challenge of how to capture compactly all sensory information that neural population codes carry in their spatial dimension (differences in stimulus tuning across neurons at different locations), in their temporal dimension (temporal neural response variations), or in their combination (temporally coordinated neural population firing). Here we investigate the utility of tensor factorizations of population spike trains along space and time. These factorizations decompose a dataset of single-trial population spike trains into spatial firing patterns (combinations of neurons firing together), temporal firing patterns (temporal activation of these groups of neurons) and trial-dependent activation coefficients (strength of recruitment of such neural patterns on each trial). We validated various factorization methods on simulated data and on populations of ganglion cells simultaneously recorded in the salamander retina. We found that single-trial tensor space-by-time decompositions provided low-dimensional data-robust representations of spike trains that capture efficiently both their spatial and temporal information about sensory stimuli. Tensor decompositions with orthogonality constraints were the most efficient in extracting sensory information, whereas non-negative tensor decompositions worked well even on non-independent and overlapping spike patterns, and retrieved informative firing patterns expressed by the same population in response to novel stimuli. Our method showed that populations of retinal ganglion cells carried information in their spike timing on the ten-milliseconds-scale about spatial details of natural images. This information could not be recovered from the spike counts of these cells. First-spike latencies carried the majority of information provided by the whole spike train about fine-scale image features, and supplied almost as much information about coarse natural image features as firing rates. Together, these results highlight the importance of spike timing, and particularly of first-spike latencies, in retinal coding.

Space-by-time decomposition for single-trial decoding of M/EEG activity.

We develop a novel methodology for the single-trial analysis of multichannel time-varying neuroimaging signals. We introduce the space-by-time M/EEG decomposition, based on Non-negative Matrix Factorization (NMF), which describes single-trial M/EEG signals using a set of non-negative spatial and temporal components that are linearly combined with signed scalar activation coefficients. We illustrate the effectiveness of the proposed approach on an EEG dataset recorded during the performance of a visual categorization task. Our method extracts three temporal and two spatial functional components achieving a compact yet full representation of the underlying structure, which validates and summarizes succinctly results from previous studies. Furthermore, we introduce a decoding analysis that allows determining the distinct functional role of each component and relating them to experimental conditions and task parameters. In particular, we demonstrate that the presented stimulus and the task difficulty of each trial can be reliably decoded using specific combinations of components from the identified space-by-time representation. When comparing with a sliding-window linear discriminant algorithm, we show that our approach yields more robust decoding performance across participants. Overall, our findings suggest that the proposed space-by-time decomposition is a meaningful low-dimensional representation that carries the relevant information of single-trial M/EEG signals.

Retrospective for the Dynamic Sensorium Competition for predicting large-scale mouse primary visual cortex activity from videos.

Understanding how biological visual systems process information is challenging because of the nonlinear relationship between visual input and neuronal responses. Artificial neural networks allow computational neuroscientists to create predictive models that connect biological and machine vision. Machine learning has benefited tremendously from benchmarks that compare different model on the same task under standardized conditions. However, there was no standardized benchmark to identify state-of-the-art dynamic models of the mouse visual system. To address this gap, we established the SENSORIUM 2023 Benchmark Competition with dynamic input, featuring a new large-scale dataset from the primary visual cortex of ten mice. This dataset includes responses from 78,853 neurons to 2 hours of dynamic stimuli per neuron, together with the behavioral measurements such as running speed, pupil dilation, and eye movements. The competition ranked models in two tracks based on predictive performance for neuronal responses on a held-out test set: one focusing on predicting in-domain natural stimuli and another on out-of-distribution (OOD) stimuli to assess model generalization. As part of the NeurIPS 2023 competition track, we received more than 160 model submissions from 22 teams. Several new architectures for predictive models were proposed, and the winning teams improved the previous state-of-the-art model by 50%. Access to the dataset as well as the benchmarking infrastructure will remain online at www.sensorium-competition.net.

A maximum entropy test for evaluating higher-order correlations in spike counts.

Evaluating the importance of higher-order correlations of neural spike counts has been notoriously hard. A large number of samples are typically required in order to estimate higher-order correlations and resulting information theoretic quantities. In typical electrophysiology data sets with many experimental conditions, however, the number of samples in each condition is rather small. Here we describe a method that allows to quantify evidence for higher-order correlations in exactly these cases. We construct a family of reference distributions: maximum entropy distributions, which are constrained only by marginals and by linear correlations as quantified by the Pearson correlation coefficient. We devise a Monte Carlo goodness-of-fit test, which tests--for a given divergence measure of interest--whether the experimental data lead to the rejection of the null hypothesis that it was generated by one of the reference distributions. Applying our test to artificial data shows that the effects of higher-order correlations on these divergence measures can be detected even when the number of samples is small. Subsequently, we apply our method to spike count data which were recorded with multielectrode arrays from the primary visual cortex of anesthetized cat during an adaptation experiment. Using mutual information as a divergence measure we find that there are spike count bin sizes at which the maximum entropy hypothesis can be rejected for a substantial number of neuronal pairs. These results demonstrate that higher-order correlations can matter when estimating information theoretic quantities in V1. They also show that our test is able to detect their presence in typical in-vivo data sets, where the number of samples is too small to estimate higher-order correlations directly.

Mixed vine copula flows for flexible modeling of neural dependencies.

Recordings of complex neural population responses provide a unique opportunity for advancing our understanding of neural information processing at multiple scales and improving performance of brain computer interfaces. However, most existing analytical techniques fall short of capturing the complexity of interactions within the concerted population activity. Vine copula-based approaches have shown to be successful at addressing complex high-order dependencies within the population, disentangled from the single-neuron statistics. However, most applications have focused on parametric copulas which bear the risk of misspecifying dependence structures. In order to avoid this risk, we adopted a fully non-parametric approach for the single-neuron margins and copulas by using Neural Spline Flows (NSF). We validated the NSF framework on simulated data of continuous and discrete types with various forms of dependency structures and with different dimensionality. Overall, NSFs performed similarly to existing non-parametric estimators, while allowing for considerably faster and more flexible sampling which also enables faster Monte Carlo estimation of copula entropy. Moreover, our framework was able to capture low and higher order heavy tail dependencies in neuronal responses recorded in the mouse primary visual cortex during a visual learning task while the animal was navigating a virtual reality environment. These findings highlight an often ignored aspect of complexity in coordinated neuronal activity which can be important for understanding and deciphering collective neural dynamics for neurotechnological applications.

Neural System Identification With Spike-Triggered Non-Negative Matrix Factorization.

Neuronal circuits formed in the brain are complex with intricate connection patterns. Such complexity is also observed in the retina with a relatively simple neuronal circuit. A retinal ganglion cell (GC) receives excitatory inputs from neurons in previous layers as driving forces to fire spikes. Analytical methods are required to decipher these components in a systematic manner. Recently a method called spike-triggered non-negative matrix factorization (STNMF) has been proposed for this purpose. In this study, we extend the scope of the STNMF method. By using retinal GCs as a model system, we show that STNMF can detect various computational properties of upstream bipolar cells (BCs), including spatial receptive field, temporal filter, and transfer nonlinearity. In addition, we recover synaptic connection strengths from the weight matrix of STNMF. Furthermore, we show that STNMF can separate spikes of a GC into a few subsets of spikes, where each subset is contributed by one presynaptic BC. Taken together, these results corroborate that STNMF is a useful method for deciphering the structure of neuronal circuits.

Building population models for large-scale neural recordings: Opportunities and pitfalls.

Modern recording technologies now enable simultaneous recording from large numbers of neurons. This has driven the development of new statistical models for analyzing and interpreting neural population activity. Here, we provide a broad overview of recent developments in this area. We compare and contrast different approaches, highlight strengths and limitations, and discuss biological and mechanistic insights that these methods provide.

Analyzing short-term noise dependencies of spike-counts in macaque prefrontal cortex using copulas and the flashlight transformation.

Simultaneous spike-counts of neural populations are typically modeled by a Gaussian distribution. On short time scales, however, this distribution is too restrictive to describe and analyze multivariate distributions of discrete spike-counts. We present an alternative that is based on copulas and can account for arbitrary marginal distributions, including Poisson and negative binomial distributions as well as second and higher-order interactions. We describe maximum likelihood-based procedures for fitting copula-based models to spike-count data, and we derive a so-called flashlight transformation which makes it possible to move the tail dependence of an arbitrary copula into an arbitrary orthant of the multivariate probability distribution. Mixtures of copulas that combine different dependence structures and thereby model different driving processes simultaneously are also introduced. First, we apply copula-based models to populations of integrate-and-fire neurons receiving partially correlated input and show that the best fitting copulas provide information about the functional connectivity of coupled neurons which can be extracted using the flashlight transformation. We then apply the new method to data which were recorded from macaque prefrontal cortex using a multi-tetrode array. We find that copula-based distributions with negative binomial marginals provide an appropriate stochastic model for the multivariate spike-count distributions rather than the multivariate Poisson latent variables distribution and the often used multivariate normal distribution. The dependence structure of these distributions provides evidence for common inhibitory input to all recorded stimulus encoding neurons. Finally, we show that copula-based models can be successfully used to evaluate neural codes, e.g., to characterize stimulus-dependent spike-count distributions with information measures. This demonstrates that copula-based models are not only a versatile class of models for multivariate distributions of spike-counts, but that those models can be exploited to understand functional dependencies.

State-dependent brainstem ensemble dynamics and their interactions with hippocampus across sleep states.

The brainstem plays a crucial role in sleep-wake regulation. However, the ensemble dynamics underlying sleep regulation remain poorly understood. Here, we show slow, state-predictive brainstem ensemble dynamics and state-dependent interactions between the brainstem and the hippocampus in mice. On a timescale of seconds to minutes, brainstem populations can predict pupil dilation and vigilance states and exhibit longer prediction power than hippocampal CA1 neurons. On a timescale of sub-seconds, pontine waves (P-waves) are accompanied by synchronous firing of brainstem neurons during both rapid eye movement (REM) and non-REM (NREM) sleep. Crucially, P-waves functionally interact with CA1 activity in a state-dependent manner: during NREM sleep, hippocampal sharp wave-ripples (SWRs) precede P-waves. On the other hand, P-waves during REM sleep are phase-locked with ongoing theta oscillations and are followed by burst firing of CA1 neurons. This state-dependent global coordination between the brainstem and hippocampus implicates distinct functional roles of sleep.

Information estimation using nonparametric copulas.

Estimation of mutual information between random variables has become crucial in a range of fields, from physics to neuroscience to finance. Estimating information accurately over a wide range of conditions relies on the development of flexible methods to describe statistical dependencies among variables, without imposing potentially invalid assumptions on the data. Such methods are needed in cases that lack prior knowledge of their statistical properties and that have limited sample numbers. Here we propose a powerful and generally applicable information estimator based on non-parametric copulas. This estimator, called the non-parametric copula-based estimator (NPC), is tailored to take into account detailed stochastic relationships in the data independently of the data's marginal distributions. The NPC estimator can be used both for continuous and discrete numerical variables and thus provides a single framework for the mutual information estimation of both continuous and discrete data. By extensive validation on artificial samples drawn from various statistical distributions, we found that the NPC estimator compares well against commonly used alternatives. Unlike methods not based on copulas, it allows an estimation of information that is robust to changes of the details of the marginal distributions. Unlike parametric copula methods, it remains accurate regardless of the precise form of the interactions between the variables. In addition, the NPC estimator had accurate information estimates even at low sample numbers, in comparison to alternative estimators. The NPC estimator therefore provides a good balance between general applicability to arbitrarily shaped statistical dependencies in the data and shows accurate and robust performance when working with small sample sizes. We anticipate that the non-parametric copula information estimator will be a powerful tool in estimating mutual information between a broad range of data.

State-Dependent Decoding Algorithms Improve the Performance of a Bidirectional BMI in Anesthetized Rats.

Brain-machine interfaces (BMIs) promise to improve the quality of life of patients suffering from sensory and motor disabilities by creating a direct communication channel between the brain and the external world. Yet, their performance is currently limited by the relatively small amount of information that can be decoded from neural activity recorded form the brain. We have recently proposed that such decoding performance may be improved when using state-dependent decoding algorithms that predict and discount the large component of the trial-to-trial variability of neural activity which is due to the dependence of neural responses on the network's current internal state. Here we tested this idea by using a bidirectional BMI to investigate the gain in performance arising from using a state-dependent decoding algorithm. This BMI, implemented in anesthetized rats, controlled the movement of a dynamical system using neural activity decoded from motor cortex and fed back to the brain the dynamical system's position by electrically microstimulating somatosensory cortex. We found that using state-dependent algorithms that tracked the dynamics of ongoing activity led to an increase in the amount of information extracted form neural activity by 22%, with a consequently increase in all of the indices measuring the BMI's performance in controlling the dynamical system. This suggests that state-dependent decoding algorithms may be used to enhance BMIs at moderate computational cost.

Inference of neuronal functional circuitry with spike-triggered non-negative matrix factorization.

Neurons in sensory systems often pool inputs over arrays of presynaptic cells, giving rise to functional subunits inside a neuron's receptive field. The organization of these subunits provides a signature of the neuron's presynaptic functional connectivity and determines how the neuron integrates sensory stimuli. Here we introduce the method of spike-triggered non-negative matrix factorization for detecting the layout of subunits within a neuron's receptive field. The method only requires the neuron's spiking responses under finely structured sensory stimulation and is therefore applicable to large populations of simultaneously recorded neurons. Applied to recordings from ganglion cells in the salamander retina, the method retrieves the receptive fields of presynaptic bipolar cells, as verified by simultaneous bipolar and ganglion cell recordings. The identified subunit layouts allow improved predictions of ganglion cell responses to natural stimuli and reveal shared bipolar cell input into distinct types of ganglion cells.How a neuron integrates sensory information requires knowledge about its functional presynaptic connections. Here the authors report a new method using non-negative matrix factorization to identify the layout of presynaptic bipolar cell inputs onto retinal ganglion cells and predict their responses to natural stimuli.