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.

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. The ever growing size of neural populations simultaneously recorded in electrophysiological experiments calls for urgent analytical progress in understanding how to compactly describe all sensory information present both in the spatial and temporal structure of single-trial neural population activity. Here we show the power of analytical methods, termed space-by-time tensor factorizations, which detect groups of simultaneously coactive neurons and the temporal profiles of their coactivation. By validation on simulated data and on retinal recordings, we show that the tensor decomposition performs competitively compared to other techniques both in terms of data robustness and ability to find informative patterns across diverse stimuli. We show that this method can determine the spatial and temporal resolution of neural population codes, and find which spatial or temporal components of neural responses carry information not available in other aspects of the population code. When applied to experimental data, the method demonstrates the importance of first-spike latencies in retinal population coding of visual images, particularly for decoding fine spatial details of natural images from population activity. This work shows that this methodology can improve our knowledge of population coding by allowing the discovery of informative spatial and temporal firing patterns in populations of simultaneously recorded neurons.

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. The brain has an enormous number of neurons that do not work alone but in an ensemble. Yet, mostly individual neurons were measured in the past and therefore models were restricted to independent neurons. With the advent of new multi-electrode techniques, however, it becomes possible to measure a great number of neurons simultaneously. As a result, models of how populations of neurons co-vary are becoming increasingly important. Here, we describe such a framework based on so-called copulas. Copulas allow to separate the neural variation structure of the population from the variability of the individual neurons. Contrary to standard models, versatile dependence structures can be described using this approach. We explore what additional information is provided by the detailed dependence. For simulated neurons, we show that the variation structure of the population allows inference of the underlying connectivity structure of the neurons. The power of the approach is demonstrated on a memory experiment in macaque monkey. We show that our framework describes the measurements better than the standard models and identify possible network connections of the measured neurons.

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 nonparametric copulas. This estimator, called the nonparametric 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 nonparametric copula information estimator will be a powerful tool in estimating mutual information in a broad range of data.

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.