Fig 1.
Bridging between brain structure and function by calculating commute time.
Connectivity matrices derived from diffusion MRI and functional MRI data are illustrated from an arbitrarily chosen UK Biobank individual (subject id: 1000366). Brighter colors reflect larger values (denser white matter tracts, larger commute times, or larger correlation coefficients), while structural matrix elements equal to 0 (no white matter tracts) before log transformation are colored white. Rows and columns refer to 84 brain regions from the Desikan-Killiany atlas [29,30]. Brain regions are labeled such that the first half corresponds to the left hemisphere; second half, right hemisphere. The structural connectivity matrix (left map) yields the commute time matrix in the middle (Methods), which is then compared with correlations calculated from the fMRI time-series (right map). Figure created with biorender.com.
Fig 2.
Commute time captures functional connectivity in a mean-field Ising simulation of brain signaling.
The Ising simulation with a coupling strength λ = 6.0 is run on a dMRI-derived structure from an arbitrary UK Biobank individual (subject id: 1000366) under the Desikan-Killiany atlas (84 brain regions). A. Data points correspond to all possible pairs of brain regions (n = 84x83/2) and are colored to capture the density of points, with brighter colors corresponding to greater density. The ordinate represents the elements of the functional connectivity (FC) matrix obtained by an Ising simulation, and the abscissa shows the corresponding commute time based on graph theoretical analysis on dMRI data (Eq 6). B. Same as A, using only the top two principal components of the FC matrix. The symbol ρ corresponds to the Spearman correlation coefficient with its p-value following in parenthesis. C. Commute time exhibits a stronger correlation with simulated functional connectivity compared to three other metrics, connectivity, search information, and communicability. 100 replicate Ising runs are performed. For visualization purposes, the additive inverse (–) of search information and commute time are taken such that all structure-based metrics are positively correlated with FC. Each metric is described by two distributions: orange, based on the top two principal components of the FC matrix and blue, all components. Dashed lines within distributions correspond to quartiles: 25th percentile, 50th percentile (median) and 75% percentile. Corresponding Kolmogorov-Smirnov pairwise tests between metrics are summarized in S2 Table in SI.
Fig 3.
Commute time marginally captures functional connectivity from functional MRI across the UK Biobank.
fMRI and dMRI data are processed according to the Desikan-Killiany atlas (84 brain regions). A-B. Calculated commute times (abscissa) are plotted against FCs indicated by fMRI data for one individual. See caption for Fig 2 for more details. C. Comparison of different metrics derived from structural connectivity. The ordinate shows their correlation to fMRI data. Corresponding Kolmogorov-Smirnov pairwise tests between metrics are summarized in S3 Table in SI. N corresponds to the number of human subjects considered from the UK Biobank. D. Values from the top mode of commute time (abscissa) are plotted against those from the top mode FC.
Fig 4.
The strength of the commute time-functional connectivity relationship demonstrates weak pathological significance and weak dependency on age.
fMRI and dMRI data are processed according to the Desikan-Killiany atlas throughout (84 brain regions). A. Small differences are seen between individuals with mental and nerve disorders compared to healthy individuals in the UK Biobank. The corresponding Kolmogorov-Smirnov pairwise tests are not significant (S5 Table in SI). B. Older brains have slightly stronger correlations between commute time and FC. Two distributions are plotted, where the orange distribution only includes the top mode of the FC matrix. C-D. Corresponding plots of average commute time (green) and average FC (yellow) across all possible pairs distributed over individuals. C. Individuals with mental health disorders have slightly higher average commute times and much lower average FCs. D. Strengthening commute time-FC correlations shown in B seem to be driven by a decrease in average commute time across age. Note that all reported correlations are calculated by considering all data points, i.e., they are not calculated for the binned data.
Fig 5.
Schematic of the mean-field Ising simulation evolving in time.
We start with a fixed structure whose spins are randomly initialized either up (blue) or down (red) at time 0. We then randomly choose a node and attempt to flip its spin. At time 0, we attempt to flip node a). It is accepted according to the Metropolis-Hastings algorithm because the energy becomes stabilized by λ units. At time 1, we attempt to flip node f). Because the energy is destabilized by 3*λ, we accept it with a probability of e-3λ according to the Metropolis-Hastings algorithm. In the figure, this particular realization of f)’s spin flip is rejected. At time 2, we attempt to flip node c). The change in energy is 0, so we accept the spin flip. This schematic is simplified such that all edges have the same weight. The simulation also accounts for the number of tracts for each edge derived from the diffusion MRI of an arbitrarily chosen individual. Figure created with biorender.com.