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Modelling Large- and Small-Scale Brain Networks

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SNAW01 - Graph limits and statistics

Investigations of the human brain with neuroimaging have recently seen a dramatic shift in focus, from “brain mapping”, identifying brain regions related to particular functions, to connectivity or “connectomics”, identifying networks of coordinated brain regions, and how these networks behave at rest and during tasks. In this presentation I will discuss two quite different approaches to modeling brain connectivity. In the first work, we use Bayesian time series methods to allow for time-varying connectivity. Non-stationarity connectivity methods typically use a moving-window approach, while this method poses a single generative model for all nodes, all time points. Known as a “Multiregression Dynamic Model” (MDM), it comprises an extension of a traditional Bayesian Network (or Graphical Model), by posing latent time-varying coefficients that implement a regression a given node on its parent nodes. Intended for a modest number of nodes (up to about 12), a MDM allows inference of the structure of the graph using closed form Bayes factors (conditional on a single estimated “discount factor”, reflecting the balance of observation and latent variance. While originally developed for directed acyclic graphs, it can also accommodate directed (possibly cyclic) graphs as well. In the second work, we use mixtures of simple binary random graph models to account for complex structure in brain networks. In this approach, the network is reduced to a binary adjacency matrix. While this is invariably represents a loss of information, it avoids a Gaussianity assumption and allows the use of much larger graphs, e.g. with 100's of nodes. Daudin et al. (2008) proposed a “Erdos-Reyni Mixture Model”, which assumes that, after an unknown number of latent node classes have been estimated, that connections arise as Bernoulli counts, homogeneously for each pair of classes. We extend this work to account for multisubject data (where edge data are now Binomially distributed), allowing

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