Fig 1.
Distributions of the goodness-of-fit (GoF) values for all tested approaches and parameter spaces.
The considered optimization methods and the respective dimensions of the parameter spaces are indicated on the horizontal axes (BO 2D, CMAES 2D, BO 3D and CMAES 3D in the low-dimensional cases and BO 103D, CMAES 103D, BO 99D and CMAES 99D in the high-dimensional ones) along with the detected GoF values on the vertical axes. Violins show the distributions of the median GoF values obtained for all subjects across 30 algorithm executions with random initial data (option 2, see Methods). The medians (across subjects) of the relative increase between the results obtained in different parameter spaces for a given algorithm are indicated in the plots together with -values of the Wilcoxon signed-rank test and the considered atlas: (A) the Schaefer atlas (Sch100) and (B) the Harvard-Oxford atlas (HO0Thr). Statistically significant differences are marked with an asterisk. The significance level of 5%,
< 0.05, has been Bonferroni-corrected for multiple comparisons.
Fig 2.
Example of the results of the model fitting in the 103-dimensional parameter space.
Boxplots illustrate the distributions of the values for the optimal (A) delay , (B) coupling
, (C) noise intensity
, (D) goodness-of-fit (GoF) and (E,F) frequency parameters
found in 30 executions (with random initial data) of the CMAES algorithm for one subject. The considered modeling quantities are indicated in the titles of each plot, while their considered ranges are given on the vertical axes. Plot (E) shows the first half of the frequency parameters, belonging to the brain regions in the left hemisphere in the Schaefer atlas, and (F) shows the second half, which represents the right hemisphere. This figure was created with MATLAB R2021a (www.mathworks.com).
Fig 3.
Structure of optimized model parameters in the high-dimensional cases.
(A) Example of the correlations between optimized frequency parameter sets obtained at the model validation in the 103-dimensional parameter space via CMAES for the Schaefer atlas (Sch100). For a given subject, the correlations between any two sets of optimized frequency parameters obtained in different algorithm executions (Ex.) were calculated and then indicated in the table cells also highlighted in color. The algorithm executions were sorted, so that the cluster structure becomes observable. (B-I) Number of clusters and cluster size for all subjects and optimization approaches. (B-E) Subject-dependent number of the frequency clusters (# Clusters) for the atlases (Sch100 and HO0Thr) and methods (BO and CMAES) indicated in the titles. In each plot, the subjects were sorted according to the number of elements in the largest cluster as illustrated in the bottom plots. (F-I) Cluster size for all subjects, i.e., the number of algorithm executions in the largest frequency cluster. The number and size of the clusters were determined with help of the
-means clustering method. Both were set to zero if the absolute mean value of the off-diagonal elements in the correlation matrix (cf. (A)) was below 0.25 (corresponding to weak correlations between different frequency sets). The subjects considered in (A) and Supplementary Fig 6 in S1 Appendix are highlighted in the plots (F) and (G). This figure was created with MATLAB R2021a (www.mathworks.com).
Fig 4.
Reliability of the optimized model parameters of delay , coupling
and noise intensity
.
Colored bars illustrate the reliability of the modeling results as measured by the intraclass correlation coefficient (ICC). The considered optimization algorithms and parameter spaces are indicated on the horizontal axes along with the ICC scores on the vertical axes. Results are shown for (A-C) the Schaefer atlas (Sch100) and (D-F) the Harvard-Oxford atlas (HO0Thr). Dashed horizontal lines indicate the levels of poor, fair, good and excellent reliability in terms of the ICC as suggested in [72,73]. This figure was created with MATLAB R2021a (www.mathworks.com).
Fig 5.
Reliability of the optimized frequency parameters and sFC matrix edges in the high-dimensional cases as well as of the GoF.
(A-B,D-E) Colored, horizontal histograms visualize the distributions of the intraclass correlation coefficients (ICC) for the model parameters (gray) as well as for the above-diagonal edges of the matrices of sFC (orange and blue for BO and CMAES, respectively). The considered atlases, i.e., (A-B) the Schaefer atlas (Sch100) and (D-E) the Harvard-Oxford atlas (HO0Thr), are provided in the titles together with the utilized optimization algorithms and parameter spaces. (C,F) Colored bars illustrate the ICC scores of the GoF for all considered optimization algorithms and parameter spaces, which are indicated on the horizontal axes. The names of the atlases are again provided in the titles. Dashed horizontal lines indicate the levels of poor, fair, good and excellent ICC reliability as suggested in [72,73]. This figure was created with MATLAB R2021a (www.mathworks.com).
Fig 6.
GoF values for males and females together with the corresponding effect sizes (ES) of the observed differences. (A-B) Colored boxplots show the distributions of the median model fits obtained for males (brown) and females (pink) in 30 algorithm executions with random initial data (option 2, see Methods). The considered dimensions of the parameter spaces combined with the group abbreviations of males (M) or females (F) are indicated on the horizontal axes along with the detected GoF values on the vertical axes. Solid vertical lines separate the results for the Schaefer atlas (Sch100) and the Harvard-Oxford atlas (HO0Thr). The names of the utilized optimization algorithms are given in the titles. Statistically significant sex differences as assessed by the Wilcoxon rank-sum test are marked with an asterisk. The significance level of 5%, < 0.05, has been Bonferroni-corrected for multiple comparisons. The corrected
-values of the sex differences are indicated in the plots. (C-D) Boxplots illustrate the distributions of the ES for the sex differences observed during a random selection of one of the 30 available algorithm executions for every subject and a subsequent comparison of the corresponding GoF values across both groups (1000 repetitions). The ES of the changes in ES from the low- (2D) to the high-dimensional (99D, 103D) cases are indicated in the plots. The Wilcoxon rank-sum test was applied. Statistically significant differences are marked with an asterisk. The significance level of 5%,
< 0.05, has been Bonferroni-corrected for multiple comparisons. (E-F) Same as (C-D), but the ES of the sex differences were computed for the GoF residuals obtained by regressing out the intracranial volume of the individual subjects from the GoF values. This figure was created with MATLAB R2021a (www.mathworks.com).
Fig 7.
Distributions of optimized coupling parameters and median FC strengths for males and females.
Colored boxplots together with gray circles in the background show the distributions of (A,C) the values of approximated optimal coupling parameters and (B,D) the median connectivity strength in the FC matrices for males (brown) and females (pink). Empirical (eFC, Emp) as well as simulated (sFC) data were considered. For every subject, the eFC from the concatenated sessions as well as the sFC matrix and coupling parameters yielding the highest GoF across algorithm executions were selected (option 1, see Methods). We refrained from including the 3D cases in the figures as we found no remarkable differences to the 2D cases. The names of the utilized optimization algorithms and considered dimensions of the parameter spaces combined with the group abbreviations of males (M) or females (F) are indicated on the horizontal axes along with the median FC and optimal C values on the vertical axes. The names of the considered atlases are given in the titles, i.e., (A,B) the Schaefer atlas (Sch100) and (C,D) the Harvard-Oxford atlas (HO0Thr). Statistically significant sex differences as assessed by the Wilcoxon rank-sum test are marked with an asterisk. The significance level of 5%,
< 0.05, has been Bonferroni-corrected for multiple comparisons. The effect sizes (ES) and corrected
-values of the sex differences are indicated in the plots. This figure was created with MATLAB R2021a (www.mathworks.com).
Fig 8.
Model-based sex classification accuracies.
(A-D) Boxplots together with gray circles in the background visualize the balanced accuracies of a logistic regressor trained for sex classification based on (A-B) the GoF or (C-D) the optimized values of the coupling parameter . For every subject, the GoF and parameter values derived from the algorithm execution yielding the highest GoF were selected (option 1, see Methods). We refrained from including the 3D cases in the figures as we found no remarkable differences to the 2D cases. The names of the utilized parameter optimization algorithms and parameter spaces are provided on the horizontal axes along with the balanced accuracy values on the vertical axes. Dashed horizontal lines indicate the chance level of 50%. Indications of which property (GoF or coupling parameter) was used for classification (classific.) are provided in the titles. The names of the considered atlases (Sch100 for the Schaefer atlas and HO0Thr for the Harvard-Oxford atlas) are given in the plots. Statistically significant performance gains between low- and high-dimensional parameter spaces as assessed by the Wilcoxon rank-sum test are marked with an asterisk. The significance level of 5%,
< 0.05, has been Bonferroni-corrected for multiple comparisons. The effect sizes (ES) and corrected
-values of the increases are indicated in the plots. This figure was created with MATLAB R2021a (www.mathworks.com).