Hierarchical vector auto-regressive models and their applications to multi-subject effective connectivity

Abstract

Vector auto-regressive (VAR) models typically form the basis for constructing directed graphical models for investigating connectivity in a brain network with brain regions of interest (ROIs) as nodes. There are limitations in the standard VAR models. The number of parameters in the VAR model increases quadratically with the number of ROIs and linearly with the order of the model and thus due to the large number of parameters, the model could pose serious estimation problems. Moreover, when applied to imaging data, the standard VAR model does not account for variability in the connectivity structure across all subjects. In this paper, we develop a novel generalization of the VAR model that overcomes these limitations. To deal with the high dimensionality of the parameter space, we propose a Bayesian hierarchical framework for the VAR model that will account for both temporal correlation within a subject and between subject variation. Our approach uses prior distributions that give rise to estimates that correspond to penalized least squares criterion with the elastic net penalty. We apply the proposed model to investigate differences in effective connectivity during a hand grasp experiment between healthy controls and patients with residual motor deficit following a stroke.

Understanding the Impact of Stroke on Brain Motor Function: A Hierarchical Bayesian Approach

“…To our knowledge, there is a dearth of literature describing how activity within a region predicts subsequent activation in the region. A recent study from our group did identify that this intraregional activation prediction is present for LM1, LPMd, SMA, and RPMd (Gorrostieta et al, 2013), largely confirming what we observed in the current analyses. Together the data suggest possible positive feedforward connectivity in these regions.…”

“…Following Gorrostieta et al (2013), the noise u ( s,r ) ( t ) ∈ ℝ P is assumed to follow a Gaussian vector autoregressive model (VAR) with pre-determined order L and condition-dependent VAR coefficients, i.e.,…”

“…Methods for brain activation are proposed in Friston et al (1994); Worsley and Friston (1995); Smith and Fahrmeir (2007); David et al (2008); Guo and Pagnoni (2008); Xu et al (2009); Degras and Lindquist (2014). Methods for brain connectivity are developed in Friston et al (2003); Harrison et al (2003); Bowman et al (2008); Cribben et al (2012); Kang et al (2012); Gorrostieta et al (2013); Luo (2014).…”

“…Harrison et al (2003) estimated connectivity using vector-autoregressive (VAR) models, while Cribben et al (2012) developed dynamic connectivity regression which models changes in connectivity, but none of these approaches take into account the systematic changes in the BOLD response induced by external stimuli. Gorrostieta et al (2013) considered the BOLD response evoked by stimuli and also the differential connectivity across different experimental conditions. However, the BOLD response shape was based on a fixed HRF shape for all brain regions and subjects, which may lead to erroneous conclusions when inferring activation and connectivity (Yu et al, 2015).…”

Is First-Order Vector Autoregressive Model Optimal for fMRI Data?

“…Vector autoregressive (VAR) modeling of fMRI time series data has been widely used to infer directed connectivity networks of distinct brain regions (Harrison, Penny, & Friston, 2003;Valdés-Sosa, 2004;Deshpande, LaConte, James, Peltier, & Hu, 2009;Valdés-Sosa et al, 2005;Gorrostieta, Ombao, Bedard, & Sanes, 2012;Gorrostieta, Fiecas, Ombao, Burke, & Cramer, 2013;Seghouane & Amari, 2012;Ting, Seghouane, Salleh, & Noor, 2014;Ting, Seghouane, Salleh, & Noor, 2015). Determining the optimal model order p for fMRI data modeling is an important practical as well as a theoretical issue when using VAR models.…”