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Analyzing Short-Term Noise Dependencies of Spike-Counts in Macaque Prefrontal Cortex Using Copulas and the Flashlight Transformation

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  • Arno Onken
  • Steffen Grünewälder
  • Matthias H J Munk
  • Klaus Obermayer

Abstract

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.Author Summary: 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.

Suggested Citation

  • Arno Onken & Steffen Grünewälder & Matthias H J Munk & Klaus Obermayer, 2009. "Analyzing Short-Term Noise Dependencies of Spike-Counts in Macaque Prefrontal Cortex Using Copulas and the Flashlight Transformation," PLOS Computational Biology, Public Library of Science, vol. 5(11), pages 1-13, November.
  • Handle: RePEc:plo:pcbi00:1000577
    DOI: 10.1371/journal.pcbi.1000577
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    References listed on IDEAS

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    1. Ines Fortin & Christoph Kuzmics, 2002. "Tail‐dependence in stock‐return pairs," Intelligent Systems in Accounting, Finance and Management, John Wiley & Sons, Ltd., vol. 11(2), pages 89-107, April.
    2. Elad Schneidman & Michael J. Berry & Ronen Segev & William Bialek, 2006. "Weak pairwise correlations imply strongly correlated network states in a neural population," Nature, Nature, vol. 440(7087), pages 1007-1012, April.
    3. Ling Hu, 2006. "Dependence patterns across financial markets: a mixed copula approach," Applied Financial Economics, Taylor & Francis Journals, vol. 16(10), pages 717-729.
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    Cited by:

    1. Marcel Wollschlager & Rudi Schafer, 2015. "Impact of non-stationarity on estimating and modeling empirical copulas of daily stock returns," Papers 1506.08054, arXiv.org.
    2. Hideaki Shimazaki & Shun-ichi Amari & Emery N Brown & Sonja Grün, 2012. "State-Space Analysis of Time-Varying Higher-Order Spike Correlation for Multiple Neural Spike Train Data," PLOS Computational Biology, Public Library of Science, vol. 8(3), pages 1-27, March.
    3. Arno Onken & Valentin Dragoi & Klaus Obermayer, 2012. "A Maximum Entropy Test for Evaluating Higher-Order Correlations in Spike Counts," PLOS Computational Biology, Public Library of Science, vol. 8(6), pages 1-12, June.
    4. Enrico Bibbona & Laura Sacerdote & Emiliano Torre, 2016. "A Copula-Based Method to Build Diffusion Models with Prescribed Marginal and Serial Dependence," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 765-783, September.
    5. Savita Jain & Suresh K. Sharma & Kanchan Jain, 2022. "Using Copulas for Bayesian Meta-analysis," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 14(1), pages 23-41, April.
    6. Kim, Daeyoung & Kim, Jong-Min & Liao, Shu-Min & Jung, Yoon-Sung, 2013. "Mixture of D-vine copulas for modeling dependence," Computational Statistics & Data Analysis, Elsevier, vol. 64(C), pages 1-19.

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