Bayesian Inference for Generalized Linear Models for Spiking Neurons

Generalized Linear Models (GLMs) are commonly used statistical methods for modelling the relationship between neural population activity and presented stimuli. When the dimension of the parameter space is large, strong regularization has to be used in order to fit GLMs to datasets of realistic size without overfitting. By imposing properly chosen priors over parameters, Bayesian inference provides an effective and principled approach for achieving regularization. Here we show how the posterior distribution over model parameters of GLMs can be approximated by a Gaussian using the Expectation Propagation algorithm. In this way, we obtain an estimate of the posterior mean and posterior covariance, allowing us to calculate Bayesian confidence intervals that characterize the uncertainty about the optimal solution. From the posterior we also obtain a different point estimate, namely the posterior mean as opposed to the commonly used maximum a posteriori estimate. We systematically compare the different inference techniques on simulated as well as on multi-electrode recordings of retinal ganglion cells, and explore the effects of the chosen prior and the performance measure used. We find that good performance can be achieved by choosing an Laplace prior together with the posterior mean estimate

How biased are maximum entropy models?

Maximum entropy models have become popular statistical models in neuroscience and other areas in biology, and can be useful tools for obtaining estimates of mutual information in biological systems. However, maximum entropy models fit to small data sets can be subject to sampling bias; i.e. the true entropy of the data can be severely underestimated. Here we study the sampling properties of estimates of the entropy obtained from maximum entropy models. We show that if the data is generated by a distribution that lies in the model class, the bias is equal to the number of parameters divided by twice the number of observations. However, in practice, the true distribution is usually outside the model class, and we show here that this misspecification can lead to much larger bias. We provide a perturbative approximation of the maximally expected bias when the true model is out of model class, and we illustrate our results using numerical simulations of an Ising model; i.e. the second-order maximum entropy distribution on binary data.

Neural population coding: combining insights from microscopic and mass signals

Behavior relies on the distributed and coordinated activity of neural populations. Population activity can be measured using multi-neuron recordings and neuroimaging. Neural recordings reveal how the heterogeneity, sparseness, timing, and correlation of population activity shape information processing in local networks, whereas neuroimaging shows how long-range coupling and brain states impact on local activity and perception. To obtain an integrated perspective on neural information processing we need to combine knowledge from both levels of investigation. We review recent progress of how neural recordings, neuroimaging, and computational approaches begin to elucidate how interactions between local neural population activity and large-scale dynamics shape the structure and coding capacity of local information representations, make them state-dependent, and control distributed populations that collectively shape behavior

Inferring decoding strategy from choice probabilities in the presence of noise correlations

The activity of cortical neurons in sensory areas covaries with perceptual decisions, a relationship often quantified by choice probabilities. While choice probabilities have been measured extensively, their interpretation has remained fraught with difficulty. Here, we derive the mathematical relationship between choice probabilities, read-out weights and noise correlations within the standard neural decision making model. Our solution allows us to prove and generalize earlier observations based on numerical simulations, and to derive novel predictions. Importantly, we show how the read-out weight profile, or decoding strategy, can be inferred from experimentally measurable quantities. Furthermore, we present a test to decide whether the decoding weights of individual neurons are optimal, even without knowing the underlying noise correlations. We confirm the practical feasibility of our approach using simulated data from a realistic population model. Our work thus provides the theoretical foundation for a growing body of experimental results on choice probabilities and correlations

Simultaneous identification of models and parameters of scientific simulators

Many scientific models are composed of multiple discrete components, and scien tists often make heuristic decisions about which components to include. Bayesian inference provides a mathematical framework for systematically selecting model components, but defining prior distributions over model components and developing associated inference schemes has been challenging. We approach this problem in an amortized simulation-based inference framework: We define implicit model priors over a fixed set of candidate components and train neural networks to infer joint probability distributions over both, model components and associated parameters from simulations. To represent distributions over model components, we introduce a conditional mixture of multivariate binary distributions in the Grassmann formalism. Our approach can be applied to any compositional stochastic simulator without requiring access to likelihood evaluations. We first illustrate our method on a simple time series model with redundant components and show that it can retrieve joint posterior distribution over a set of symbolic expressions and their parameters while accurately capturing redundancy with strongly correlated posteriors. We then apply our approach to drift-diffusion models, a commonly used model class in cognitive neuroscience. After validating the method on synthetic data, we show that our approach explains experimental data as well as previous methods, but that our fully probabilistic approach can help to discover multiple data-consistent model configurations, as well as reveal non-identifiable model components and parameters. Our method provides a powerful tool for data-driven scientific inquiry which will allow scientists to systematically identify essential model components and make uncertainty-informed modelling decisions

Signatures of criticality arise in simple neural population models with correlations

Large-scale recordings of neuronal activity make it possible to gain insights into the collective activity of neural ensembles. It has been hypothesized that neural populations might be optimized to operate at a 'thermodynamic critical point', and that this property has implications for information processing. Support for this notion has come from a series of studies which identified statistical signatures of criticality in the ensemble activity of retinal ganglion cells. What are the underlying mechanisms that give rise to these observations? Here we show that signatures of criticality arise even in simple feed-forward models of retinal population activity. In particular, they occur whenever neural population data exhibits correlations, and is randomly sub-sampled during data analysis. These results show that signatures of criticality are not necessarily indicative of an optimized coding strategy, and challenge the utility of analysis approaches based on equilibrium thermodynamics for understanding partially observed biological systems.Comment: 36 pages, LaTeX; added journal reference on page 1, added link to code repositor

Statistical Analysis of Multi-Cell Recordings: Linking Population Coding Models to Experimental Data

Modern recording techniques such as multi-electrode arrays and two-photon imaging methods are capable of simultaneously monitoring the activity of large neuronal ensembles at single cell resolution. These methods finally give us the means to address some of the most crucial questions in systems neuroscience: what are the dynamics of neural population activity? How do populations of neurons perform computations? What is the functional organization of neural ensembles? While the wealth of new experimental data generated by these techniques provides exciting opportunities to test ideas about how neural ensembles operate, it also provides major challenges: multi-cell recordings necessarily yield data which is high-dimensional in nature. Understanding this kind of data requires powerful statistical techniques for capturing the structure of the neural population responses, as well as their relationship with external stimuli or behavioral observations. Furthermore, linking recorded neural population activity to the predictions of theoretical models of population coding has turned out not to be straightforward. These challenges motivated us to organize a workshop at the 2009 Computational Neuroscience Meeting in Berlin to discuss these issues. In order to collect some of the recent progress in this field, and to foster discussion on the most important directions and most pressing questions, we issued a call for papers for this Research Topic. We asked authors to address the following four questions: 1. What classes of statistical methods are most useful for modeling population activity? 2. What are the main limitations of current approaches, and what can be done to overcome them? 3. How can statistical methods be used to empirically test existing models of (probabilistic) population coding? 4. What role can statistical methods play in formulating novel hypotheses about the principles of information processing in neural populations? A total of 15 papers addressing questions related to these themes are now collected in this Research Topic. Three of these articles have resulted in “Focused reviews” in Frontiers in Neuroscience (Crumiller et al., 2011; Rosenbaum et al., 2011; Tchumatchenko et al., 2011), illustrating the great interest in the topic. Many of the articles are devoted to a better understanding of how correlations arise in neural circuits, and how they can be detected, modeled, and interpreted. For example, by modeling how pairwise correlations are transformed by spiking non-linearities in simple neural circuits, Tchumatchenko et al. (2010) show that pairwise correlation coefficients have to be interpreted with care, since their magnitude can depend strongly on the temporal statistics of their input-correlations. In a similar spirit, Rosenbaum et al. (2010) study how correlations can arise and accumulate in feed-forward circuits as a result of pooling of correlated inputs. Lyamzin et al. (2010) and Krumin et al. (2010) present methods for simulating correlated population activity and extend previous work to more general settings. The method of Lyamzin et al. (2010) allows one to generate synthetic spike trains which match commonly reported statistical properties, such as time varying firing rates as well signal and noise correlations. The Hawkes framework presented by Krumin et al. (2010) allows one to fit models of recurrent population activity to the correlation-structure of experimental data. Louis et al. (2010) present a novel method for generating surrogate spike trains which can be useful when trying to assess the significance and time-scale of correlations in neural spike trains. Finally, Pipa and Munk (2011) study spike synchronization in prefrontal cortex during working memory. A number of studies are also devoted to advancing our methodological toolkit for analyzing various aspects of population activity (Gerwinn et al., 2010; Machens, 2010; Staude et al., 2010; Yu et al., 2010). For example, Gerwinn et al. (2010) explain how full probabilistic inference can be performed in the popular model class of generalized linear models (GLMs), and study the effect of using prior distributions on the parameters of the stimulus and coupling filters. Staude et al. (2010) extend a method for detecting higher-order correlations between neurons via population spike counts to non-stationary settings. Yu et al. (2010) describe a new technique for estimating the information rate of a population of neurons using frequency-domain methods. Machens (2010) introduces a novel extension of principal component analysis for separating the variability of a neural response into different sources. Focusing less on the spike responses of neural populations but on aggregate signals of population activity, Boatman-Reich et al. (2010) and Hoerzer et al. (2010) describe methods for a quantitative analysis of field potential recordings. While Boatman-Reich et al. (2010) discuss a number of existing techniques in a unified framework and highlight the potential pitfalls associated with such approaches, Hoerzer et al. (2010) demonstrate how multivariate autoregressive models and the concept of Granger causality can be used to infer local functional connectivity in area V4 of behaving macaques. A final group of studies is devoted to understanding experimental data in light of computational models (Galán et al., 2010; Pandarinath et al., 2010; Shteingart et al., 2010). Pandarinath et al. (2010) present a novel mechanism that may explain how neural networks in the retina switch from one state to another by a change in gap junction coupling, and conjecture that this mechanism might also be found in other neural circuits. Galán et al. (2010) present a model of how hypoxia may change the network structure in the respiratory networks in the brainstem, and analyze neural correlations in multi-electrode recordings in light of this model. Finally, Shteingart et al. (2010) show that the spontaneous activation sequences they find in cultured networks cannot be explained by Zipf’s law, but rather require a wrestling model. The papers of this Research Topic thus span a wide range of topics in the statistical modeling of multi-cell recordings. Together with other recent advances, they provide us with a useful toolkit to tackle the challenges presented by the vast amount of data collected with modern recording techniques. The impact of novel statistical methods on the field and their potential to generate scientific progress, however, depends critically on how readily they can be adopted and applied by laboratories and researchers working with experimental data. An important step toward this goal is to also publish computer code along with the articles (Barnes, 2010) as a successful implementation of advanced methods also relies on many details which are hard to communicate in the article itself. In this way it becomes much more likely that other researchers can actually use the methods, and unnecessary re-implementations can be avoided. Some of the papers in this Research Topic already follow this goal (Gerwinn et al., 2010; Louis et al., 2010; Lyamzin et al., 2010). We hope that this practice becomes more and more common in the future and encourage authors and editors of Research Topics to make as much code available as possible, ideally in a format that can be easily integrated with existing software sharing initiatives (Herz et al., 2008; Goldberg et al., 2009)