Copula directional dependence for inference and statistical analysis of whole-brain connectivity from fMRI data

Abstract

Introduction: Inferring connectivity between brain regions has been raising a lot of attention in recent decades. Copula directional dependence (CDD) is a statistical measure of directed connectivity, which does not require strict assumptions on probability distributions and linearity.

Methods: In this work, CDDs between pairs of local brain areas were estimated based on the fMRI responses of human participants watching a Pixar animation movie. A directed connectivity map of fourteen predefined local areas was obtained for each participant, where the network structure was determined by the strengths of the CDDs. A resampling technique was further applied to determine the statistical significance of the connectivity directions in the networks. In order to demonstrate the effectiveness of the suggested method using CDDs, statistical group analysis was conducted based on graph theoretic measures of the inferred directed networks and CDD intensities. When the 129 fMRI participants were grouped by their age (3-5 year-old, 7-12 year-old, adult) and gender (F, M), nonparametric two-way analysis of variance (ANOVA) results could identify which cortical regions and connectivity structures correlated with the two physiological factors.

Results: Especially, we could identify that (a) graph centrality measures of the frontal eye fields (FEF), the inferior temporal gyrus (ITG), and the temporopolar area (TP) were significantly affected by aging, (b) CDD intensities between FEF and the primary motor cortex (M1) and between ITG and TP were highly significantly affected by aging, and (c) CDDs between M1 and the anterior prefrontal cortex (aPFC) were highly significantly affected by gender.

Software: The R source code for fMRI data preprocessing, estimation of directional dependences, network visualization, and statistical analyses are available at https://github.com/namgillee/CDDforFMRI.

Keywords: Brodmann area; connectivity; cortex; directional dependence; functional magnetic resonance imaging (fMRI); group analysis.

Figure 1

Illustration of the data generation procedure for the simulated fMRI data. (a) The original bivariate data generated by the predefined asymmetric copula distribution on the square region ${\left[0,1\right]}^2$ with β = 0.1 and T = 15, that is, n = 60T/a = 9000. (b) The transformed data to the standard normal distribution. (c) The simulated bivariate BOLD time series data after the convolution with the Gaussian HRF with the FWHM of 4 s. (d) The simulated fMRI data subsampled at every TR = 2 s

Figure 2

Scatter plots of the simulated fMRI data for $\beta =0.1,0.3,0.5$ and T = 15, together with the regression curves fitted by the proposed method

Figure 3

The root‐mean‐squared error (RMSE) values for various values of the model parameter β and the data length T for the simulated fMRI data in the simulated experiments

Figure 4

Boxplots of the differences, $\mathrm{\Delta }{\rho }_{y_1,y_2}^2$, obtained by the bootstrap resampling procedure for the three types of noise for the three‐dimensional VAR models in the simulated experiments. The negative sign $\mathrm{\Delta }{\rho }_{y_1,y_2}^2<0$ implies that the determined direction is y ₂ → y ₁

Figure 5

Each location of the selected seven voxels on the cortex is indicated by a circle with a label and color and illustrated in the lateral view. The selected voxels belong to a certain area among the Brodmann areas of the brain. The color of each circle in the figure indicates that it belongs to either the frontal lobe (yellow), temporal lobe (orange), occipital lobe (cyan), or limbic lobe (green)

Figure 6

An empirical distribution of estimated CDDs for a participant selected from the group of Adult and F (black). The CDDs have been transformed to normal distributions via Fisher's z‐transformation. The null and alternative distributions estimated by the FDR procedure of Strimmer (2008) are shown by the red dotted curve and blue straight line, respectively. On the top of the figure, the standard deviation (σ = 0.19) and proportion (η = 0.707) of the null distribution are displayed

Figure 7

A preprocessed sample fMRI time series for a participant selected from the group of Adult and F (left panel), and a normal Q‐Q plot for a sample fMRI time series from the BA18 V2 in the left hemisphere (right panel)

Figure 8

Sample scatter plots for the fMRI time series from the regions BA18 V2 in both hemispheres for a participant selected from the group of Adult and F (female). The red line in each plot represents the regression line by beta regression

Figure 9

A sample connectivity network of brain regions for a participant selected from the group of Adult and F (female). Each of the edges in the directed network was pruned if the local FDR score was greater than or equal to 0.2, that is, $\mathrm{fdr}\left({\rho }_{U\to V}\right)\ge 0.2$. The “R” and “L” in the node labels represent the right and left hemispheres, and the letters from “A” to “G” represent the brain regions described in Table 4. An arrow from brain regions U to V represents the difference of the estimated CDDs, $\mathrm{\Delta }{\rho }_{U,V}^2$, for ${\rho }_{U\to V}^2>{\rho }_{V\to U}^2$, and it is colored in red if $\mathrm{\Delta }{\rho }_{U,V}^2$ was significantly larger than zero, that is, $LB\left(\mathrm{\Delta }{\rho }^2\right)>0$

Figure 10

Connectivity networks of brain regions for participants selected from each of the six groups classified by three age levels and two gender levels. The connectivity networks for every participant are available at http://github.com/namgillee/ CDDforFMRI. The “R” and “L” in the node labels represent the right and left hemispheres, and the letters from “A” to “G” represent the brain regions described in Table 4. An arrow from brain regions U to V represents the difference of the estimated CDDs, $\mathrm{\Delta }{\rho }_{U,V}^2$, for ${\rho }_{U\to V}^2>{\rho }_{V\to U}^2$, and it is colored in red if $\mathrm{\Delta }{\rho }_{U,V}^2$ was significantly larger than zero, that is, $LB\left(\mathrm{\Delta }{\rho }^2\right)>0$

Figure 11

Boxplots for comparing distributions of total‐degree between groups

Figure 12

Boxplots for comparing distributions of ${\rho }_{U\to V}^2$ with U < V between groups