### 20. Characterizing neural dependencies with Poisson copula
models

Pietro Berkes^{} berkes@gatsby.ucl.ac.uk
Jonathan Pillow^{} pillow@gatsby.ucl.ac.uk
Frank Wood^{} fwood@gatsby.ucl.ac.uk
^{}Gatsby Computational Neuroscience Unit, UCL, London, UK

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 (e.g., 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 to continuous quantities and model them 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 can exhibit various kinds of
dependency.

Copulas are mathematical objects that specify a joint distribution’s dependency structure separately from
its marginal structure [1]. Copulas provide a principled way to quantify non-linear dependencies
that go beyond correlation coefficients, in a manner that is independent of rescaling of individual
variables.

Here we present some results on constructing joint distributions for the activity of pairs of neurons 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.

Acknowledgments

This work was supported by the Gatsby Charitable Foundation and the Royal Society USA/Canada Research
Fellowship.

References

[1] R.B. Nelson (1999) An introduction to copulas. Lecture notes in statistics 139, Springer Verlag, New York.