There has been significant amount of work^4,12-16 on parcellating the human brain with resting-state fMRI (rs-fMRI). Given the large inter-subject variability in brain organization^2,3,10, estimating individual-specific brain parcellation is an important step for biomarker development^5-7,9. We propose a hidden Markov Random Field (hMRF) model to parcellate the cerebral cortex of individual subjects with rs-fMRI. The MRF likelihood captures inter-subject variability in functional connectivity profiles, while the MRF prior consists of a smoothness prior and a prior that captures inter-subject variability in the spatial distribution of brain networks. We evaluated the individual-level parcellations on a test-retest dataset¹⁷ to assess whether the parcellations were able to capture stable properties of individual subjects’ intrinsic cortical organization.

We considered rs-fMRI data from 744 GSP subjects (https://thedata.harvard.edu/dvn/dv/GSP) and 30 HNU subjects each scanned on ten different days (http://dx.doi.org/10.15387/fcp_indi.corr.hnu1). The GSP and HNU data underwent rs-fMRI preprocessing previously reported in Holmes et al.⁸ and Zuo et al.¹⁷ respectively, and was projected to the FreeSurfer fsaverage5 surface space. Following the approach of Yeo et al.¹⁶, for each subject, we computed the connectivity profile of each vertex by correlating the vertex’s fMRI timecourse with 1175 uniformly-distributed cortical ROIs. Each 1175-length connectivity profile was normalized to unit length. Let $X^s_n$ denote the normalized functional connectivity profile of subject $s$ and vertex $n$.

The normalized connectivity profiles were averaged across the GSP subjects resulting in a group-level connectivity profiles $X^g_{1:N}$ at vertices $1$ to $N$. The profiles $X^g_{1:N}$ were modeled as a von Mises-Fisher (vMF) mixture model: $$p(X^g_n|l^g_n=l,\mu^g_{1:L},\kappa^g)=p(X^g_n|\mu^g_l,\kappa^g)=z(\kappa^g)e^{\kappa^g{X^g_n}^T\mu^g_l}$$ where $l_n^g$ was the parcellation label at vertex $n$, $\mu^g_{1:L}$ were the mean directions for cluster $1$ to $L$, and $\kappa^g$ was the concentration parameter. Expectation-Maximization (EM) was employed to estimate the group-level vMF parameters $\{\mu^g_{1:L}, \kappa^g\}$ and parcellation labels $L^g=\{l^g_1,…,l^g_N\}$. Note that the resulting parcellation corresponded to that of Yeo et al.¹⁶.

To estimate the cerebral cortex parcellation of subject $s$, we considered a hMRF model. The MRF likelihood followed a vMF mixture model:

$$p(X^s_n|l^s_n=l,\mu^s_{1:L},\kappa^s)=p(X^s_n|\mu^s_l,\kappa^s)=z(\kappa^s)e^{\kappa^s{X^s_n}^T\mu^s_l},$$ where $\mu^s_{1:L}$ were the mean directions and $\kappa^s$ was the concentration parameters of the subject-specific vMF mixture model. The conjugate prior on $\mu^s_l$ was a vMF distribution whose mean direction corresponded to the group-level mean direction $\mu^g_l$:

$$p(\mu^s_l|\mu^g_l,\epsilon)=z(\epsilon)e^{\epsilon{\mu^s_l}^T\mu^g_l},$$ where $\epsilon$ controls how much the subject-specific mean direction $\mu^s_l$ can deviate from the group-level mean direction $\mu^g_l$. The MRF prior on the parcellation labels $l^s_{1:N}$ was given by

$$p(l^s_{1:N})=\frac{1}{\cal{Z}}\exp(\sum^N_{n=1}U(l^s_n|L^g)-\sum_{n=1}^N\sum_{m\in \cal{N}_n}V(l^s_n,l^s_m)),$$ where the unary potential $U$ encoded the likelihood of a label occurring at a vertex and was obtained by the LogOdds representation¹¹ of the group-level parcellation $L^g$. The pairwise potential encouraged neighboring vertices to have the same parcellation labels: $V(l^s_n,l^s_m)$ is equal to $c$ if $l^s_n=l^s_m$ and equal to 1 otherwise.

Conditioned on the group-level vMF parameters $\{\mu^g_{1:L}, \kappa^g\}$ and parcellation labels $L^g$ from the GSP dataset, we performed leave-one-out cross-validation where we estimated $\epsilon$ and $c$ from the first session of 29 HNU subjects and utilized variational EM to estimate the vMF parameters $\{\mu^s_{1:L}, \kappa^s\}$ and parcellation labels $L^s=\{l^s_1,…,l^s_N\}$ in the first session of the remaining subject.

We compared our approach (K-MRF) with three alternative approaches: (1) group-level parcellation (GP) computed from the GSP subjects, and (2&3) vMF mixture model¹⁶ applied to individual subjects by initialization with the GP and running E-step once (GP-E) or running EM till convergence (GP-EM). All four approaches are evaluated by computing the parcellation homogeneity (defined as pairwise correlations among vertices of the same cluster) in the remaining nine sessions of the leave-out subject.

Figure 1 shows the group-level parcellation (GP) and parcellations of three individual subjects with our approach (K-MRF). Figure 2 shows the parcellation homogeneity of the four approaches in the unseen sessions of the leave-out subjects. Our approach achieves the highest homogeneity (p < 1e-9). The improvements are modest but highly consistent across subjects, which is why the p-values are very small.

Inter-subject functional connectivity differences arise from both intra-subject (inter-session) and true inter-subject variability¹⁰. It is worth noting that GP-EM achieved the highest homogeneity in the first session of the leave-out subjects (used to estimate the parcellation), while K-MRF achieved the highest homogeneity in unseen sessions of the leave-out subjects. This suggests that K-MRF’s group-level priors were effective in removing intra-subject (inter-session), rather than true inter-subject variability.

We proposed a hMRF model to parcellate the cerebral cortex of individual subjects with rs-fMRI. We demonstrated improved parcellation homogeneity in new unseen sessions of the individual subjects suggesting that the individual-specific parcellations are capturing stable properties of individual subjects’ intrinsic brain organization, instead of transient noise or session-dependent variations.