2.3.4. Competitive connectional topography mapping approaches

In addition to the proposed co‐clustering based group‐wise connectional topography mapping method, there are three other prior connectivity, ICA and clustering‐based mapping approaches that have been presented to construct the connectional topography of the brain at the group level. In the first approach, a substructure (i.e., thalamus here) considered for mapping its connectional topography is parcellated into subregions by assigning each voxel of the substructure to one of the prior defined brain targets with the strongest group‐level functional connectivity (Hwang et al., 2017; Jiang et al., 2019; Zhang et al., 2008, 2010). In our experiments, five cortical regions were selected as brain targets in the prior connectivity based mapping approach, which are prefrontal zone, temporal zone, primary motor/premotor zone, somatosensory zone, and posterior parietal/occipital zone (Zhang et al., 2008, 2010). In addition, the group‐level functional connectivity strength between the substructure's voxels and the brain targets was measured by using t value calculated following the Subsection 2.3.2.

In the second approach, the ICA is applied to the functional similarity matrix between two brain ROIs of a certain neural circuit (i.e., thalamocortical system here) considered for connectional topography mapping (Wu et al., 2018). The group‐level functional similarity matrix M_g for the thalamocortical system calculated as described in the Subsection 2.1.2 is firstly decomposed into mutually linked independent components (ICs) by means of ICA, and the number of pairs of decomposed mutually linked ICs is set to the optimal clustering number k^* determined in our proposed co‐clustering based mapping method. Then, each voxel within the thalamocortical system is assigned to one IC based on its maximum z score among the decomposed k^* ICs by means of the winner‐take‐all (WTA) rule. The resulting WTA map is the constructed connectional topography of the thalamocortical system at the group level with the ICA based mapping approach.

In the third approach, many clustering algorithms are adopted to construct the connectional topography of the brain, and the normalized cut is one of the excellent clustering algorithms (Heuvel & Pol, 2010; Lee et al., 2012; Salehi et al., 2018; Toro‐Serey et al., 2020; Van Den Heuvel et al., 2008; Yeo et al., 2011). So, the normalized cut based mapping method is chosen as a representative approach for comparison, which is a two‐level mapping approach. The two‐level normalized cut based mapping method can construct more robust and accurate group‐level connectional topography than the connectional topography identified by applying a clustering approach (such as a normalized cut) to group‐level functional connectivity (Van Den Heuvel et al., 2008). The normalized cut based mapping method firstly constructs the connectional topography at the individual level. Then, this method constructs the group‐level connectional topography through clustering the consistency of resulting individual connectional topographies across all subjects in a group, and the constructed group‐level connectional topography is chosen for comparison. Particularly, the normalized cut is applied to each subject's Pearson correlation matrix of the brain region (i.e., thalamocortical system here) considered for parcellation after thresholding (cutoff threshold 0.4), and the matrix element (i.e., the Pearson correlation coefficient between functional signals) is calculated according to Equation (1). The partitions from each subject are utilized to calculate a group‐level weight matrix as follows. For each element of the weight matrix, the number of subjects is counted when two voxels from the thalamocortical system have the same partitioning label. This number is divided by the total number of subjects in a group, and the ratio is adopted as the value for the element of the weight matrix. Mathematically, the value of the element in the weight matrix is defined as

where S_i is one of the N subjects in a data set, countSiu,v is equal to 1 if voxels u and v have the same partitioning label in subject S_i and 0 otherwise. Then, the normalized cut is applied to the group‐level weight matrix for parcellation of the thalamocortical system. The final partition is the constructed connectional topography of the thalamocortical system at the group level with the normalized cut based mapping approach. For both individual and group partitions, the clustering number is set to the optimal clustering number k* determined in our proposed co‐clustering based mapping method.

To evaluate the proposed method's quality, the proposed method is compared quantitatively to the three alternative methods with their aforementioned parameter settings. In particular, the proposed method and the three alternative methods firstly construct the connectional topography at the group level. Then, the modified SI index defined in Equation (5) is used to measure the functional homogeneity of the mapped connectional topographies obtained by these methods, respectively. Finally, the proposed method and the three alternative methods are compared with respect to the functional homogeneity measured by the modified SI index. The SI value is in the range (−∞, +∞). When the mapped connectional topography is an incorrect partition, the SI value is negative, and if the mapped connectional topography is a good partition, the SI value is near 1.