Docs:

• NIPS08 paper and poster: Berkes, P., Wood, F., and Pillow, J. (2009).
  Characterizing neural dependencies with copula models. To appear in Advances in Neural Information Processing Systems, 21.
  (paper.pdf, poster.pdf)
• Cosyne08 abstract and poster: Berkes, P., Pillow, J., and Wood, F. (2008).
  Characterizing neural dependencies with Poisson copula models. Cosyne 2008, Salt Lake City (abstract).
  (abstract.pdf, poster.pdf)

Demo code: copulas_demo.zip
The demo generates artificial data from a Clayton copula with Poisson marginals, and then fits a set a copulas to it. It then prints the cross-validation log likelihood for all copulas; the highest values should be the one for the Clayton copula, as it was the one used to generate the data. At the end the script displays the empirical copula and the Clayton fit, as in Fig. 5 of the paper. Execute demo.m with Matlab to start the demo.

Abstract:

The activities of individual neurons in cortex and many other areas of the brain are often well described by Poisson distributions. Unfortunately, there is no simple joint Poisson distribution that can incorporate statistical dependencies (i.e., correlations) between neurons. For this reason, neural population coding models often either assume that the individual neurons are independent, or transform the joint activity mathematically and model it using a multivariate distribution that naturally encodes dependency, such as the multivariate Gaussian. However, these solutions are sometimes poorly suited to describing neural population responses, failing to match either the marginal distributions of individual neurons or the detailed form of their dependencies. Here we develop a joint model for neural population responses using copulas, which allow Poisson marginal distributions to be combined into a joint distribution that captures dependencies between multiple neurons.

Copulas are mathematical objects that specify a joint distribution's dependency structure separately from its marginal structure [1]. More formally, copulas are joint probability distributions defined on the unit cube [0,1]^N. Copula models are constructed by projecting the original variables through their cumulative density functions onto the unit cube. This is a central result for copula models and is formalized in Sklar's theorem [2]. Copulas also provide a principled way to quantify non-linear dependencies that go beyond correlation coefficients (which are only appropriate for elliptical distributions), in a manner that is independent of rescaling of individual variables [1], and are applicable to the problem of estimating the mutual information between stimulus and response, as discussed in [3].

Here we present preliminary results on constructing a multivariate joint distribution for neural activity by choosing the marginals to be Poisson distributed, selecting an appropriate parametric family of copulas, and fitting the model parameters (of both the marginals and the copula) using Maximum Likelihood estimation. Different copula families are able to capture dependencies of different kinds (e.g., dependencies limited to the lower or upper tails of the distribution, or negative dependencies). The selection of an appropriate parametric family for the copula distribution can be addressed by cross-validation, and is the focus of current research.

References
[1] R.B. Nelson (1999) An introduction to copulas. Lecture notes in statistics 139, Springer Verlag, New York.
[2] A. Sklar (1959) Fonctions de repartition a n dimensions et leurs marges. Publications de l'Institut de Statistique de L'Universite de Paris 8, 229--231.
[3] R.L. Jenison and R.A. Reale (2004) The shape of neural dependence. Neural Comp., 16(4), 665--672.