An analitically tractable model for studying beyond pairwise common input correlations

FERNANDO MONTANI; LISANDRO MONTANGIE

Juan-les-Pins

Congreso; 2nd International Conference on Mathematical Neuroscience; 2016

Recent results from experiments involving a relatively large population of neurons have shown a significant amount of higher-order correlations, as synchronous activity that cannot be reduced to pairwise statistics. Finding suitable mathematical models for capturing the statistical structure of firing patterns distributed across several neurons provides a challenge and a prerequisite for understanding population codes. The Dichotomized Gaussian model \cite{firstref}, in which binary patterns are thought of being generated by thresholding a multivariate Gaussian random variable, provides an appropiate statistical model to study neural correlations. In this model, correlations between neurons arise from pairwise correlations in the underlying Gaussian inputs, an analitically tractable framework for generating population spike trains with specified mean and pairwise statistics \cite{secondref}. Although inputs in the model are Gaussian distributed and therefore have no interactions beyond second order, the nonlinear threshold spiking may give rise to statistical interactions of all orders and can be used to construct quantitative predictions on how departures from pairwise models depend on common Gaussian like neuronal inputs \cite{thirdref,fourthref}. This approach has been developed within the Central Limit Theorem context, which ensures that the probability distribution function of any measurable quantity is a normal distribution, provided that a sufficiently large number of independent random variables with exactly the same mean and variance are being considered. However, the Central Limit Theorem does not hold if correlations between random variables cannot be neglected and, perhaps undetectable, higher-order input correlations may well have an important effect at population level.Little is known of how correlations affect the integration and firing behavior of a population of neurons beyond the second order statistics. To investigate how higher-order inputs correlations can shape information coding in the brain, we developed a model which constitutes the natural extension of the Dichotomized Gaussian model, where the inputs are distributed according to deformed Gaussians (i.e. q-Gaussians) and therefore can exhibit more complex input interactions \cite{fifthref}. Furthermore, q-Gaussians are often favored for their heavy tails in comparison to Gaussians, allowing for better fitting of deviations in amplitude distributions of local field potentials. This Dichotomized q-Gaussian model arises due to the Extended Central Limit Theorem \cite{sixthref}, in which the independence constraint for the independent and identically distributed variables is relaxed to an extent defined by the q parameter, and converges to the Dichotomized Gaussian model when we consider the limit of the Central Limit Theorem framework (that is, $q\rightarrow 1$). This approach allows for generating binary spike trains with beyond second order input correlations, and so it provides a means for further studying higher-order correlations in neuronal populations.