# Functional Connectome of the Human Brain with Total Correlation
## 5 Experiment 3: Brain Functional Connectivity Analysis Using Total Correlation
### First Total-Correlation-Based Clustering Example from fMRI Data  
The data were taken from a resting-state fMRI experiment in which a subject was watching and maintaining alert wakefulness, but not performing any other behavioral task. Meanwhile, the BOLD signal was recorded. These data were downloaded from Nitime ([[https://nipy.org/nitime/index.html](https://nipy.org/nitime/index.html)]([https://nipy.org/nitime/index.html](https://nipy.org/nitime/index.html)) accessed on 12 October 2022). The data were preprocessed, and time series were extracted from different Regions Of Interest (ROIs)  
Figure 5: A flowchart for the construction of a functional brain network by fMRI. 1 Time series extraction from fMRI data within each anatomical unit (i.e., network node). 2 Estimation of functional connectivity with CC, I, and TC (CorEx), respectively. 3 Visualization of functional connectivity as tree and circle graphs (i.e., network edges and network nodes).  
in the brain. The ROIs' abbreviations and related full names are listed as follows: Cau, caudate; Pau, Paudate; Thal, Thalamus; Fpol, Frontal pole; Ang, Angular gyrus; SupraM, Supramarginal gyrus; MTG, Middle Temporal Gyrus; Hip, Hippocamp; PostPHG, Posterior Parahippocamapl Gyrus; APHG, Anterior Parahippocamapl Gyrus; Amy, Amygdala; ParaCing, Paracingulate gyrus; PCC, Posterior Cingulate Cortex; Prec, Precuneus; R, Right hemisphere; L, Left hemisphere. First, we estimated the pairwise functional connectivity metrics with Pearson correlation, mutual information, and the corresponding functional connectivity, a circle-weighted graph used to visualize the outcome of pairwise functional connectivity. In Figure 6, top row (left and right), Pearson correlation and mutual information estimate the same pairwise dependencies, but later approaches capture stronger weights between ROIs, such as LPCC and RPCC, LThal and RThal, and LAmy and RAmy.  
Meanwhile, we also used weighted graph theory to cluster dependence among ROIs, and we thresholded edges with a weight of less than 0.16 for legibility with the CorEx approach. As we mentioned above, mutual information only estimates a more robust relationship between ROIs compared to correlation. However, when we go beyond pairwise  
Figure 6: Functional connectivity representation with graph-based networks. The functional connectivity is represented in the cycle (**a**) and tree (**b**) graphs. Top row: the left and right figures correspond to Pearson correlation with a threshold of 0.14 and mutual information with a threshold of 0.02, respectively. Bottom row: the figures show the Total Correlation with a threshold of 0.16 that was estimated by CorEx. To more directly display the statistical dependencies of brain regions, we here converted the circle graph to a tree graph. The weights are shown by the thickness of the edges, which shows how strongly information is coupled between or among brain regions.  
ROIs, CorEx captures richer information among all ROIs (see Figure 6 (bottom row)). Here, we selected \(m_{1}=10\), \(m_{2}=3\), \(m_{3}=1\) as the latent dimension for each layer in our estimate of TC with CorEx, and their corresponding convergent curves are plotted in Figure 7; it shows the Total Correlation lower bound stops increasing. Figure 6 (bottom row) shows the overall structure of the learned hierarchical model. Edge thickness is determined by \(\alpha_{i,j}\)\(\left(X_{i}:Y_{j}\right)\). The size of each node is proportional to the Total Correlation that a latent factor explains about its children. The discovered structure captures several significant relationships among ROIs that are consistent with correlation and mutual information results, e.g., LPCC and RPCC, LThal and RThal, LParaCing and RParaCing, and LPut and RPut. Furthermore, TC discovered some beyond pairwise unknown relationships; for example, LCau, RCau, LFpol, and RFpol are clustered under Node 0, which explains why they have dense dependency during this cognitive task compared to other ROIs in the brain.