A framework to quantify controlled directed interactions in network physiology applied to cognitive function assessment

Figures

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  Figure 1. Original and modified TDS, as well as the new CTDS are compared in this simulation study. The strength of the TDS and CTDS links are colour-coded where 10% to 95% range is scaled between 0 and 1 in colour-codes. In the figure, (a) shows the original network generated through Eq. (4), with colour-coded coefficients (b) illustrates the network determined by the original TDS method proposed in2. Although TDS can uncover the couplings between the nodes, it cannot show the direction and whether the couplings are uni- or bi-directional. The network in (c) resulted from modified TDS to show the coupling directions only based on the delays and (d) shows the network based on the new CTDS method. CTDS not only shows the directions, but also refines the couplings by controlling for indirect relationships and highlighting the primary links. It can better distinguish between primary and indirect links, e.g. the modified TDS found the indirect link from 3 to 1 to be stronger than the primary link from 2 to 4, while in the network derived by the proposed CTDS method, all primary links were stronger than indirect links.

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  Figure 2. Average strength of the links for the simulated network in Fig. 1a is illustrated for the links derived by (a) modified TDS which indicates coupling directions, and (b) the new CTDS framework. The original TDS is not presented as it does not provide any information about the coupling direction. The primary links (as shown in Fig. 1a) are in bold and marked with ‘*’. The figure indicates that the proposed CTDS method can distinguish better between primary and indirect links e.g. using a threshold of 40%, while the modified TDS, resulted in a stronger indirect link (3 to 1) than the primary one (2 to 4).

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  Figure 3. The link strength obtained by (a) the modified TDS and (b) CTDS, are shown for various coefficients of links for randomly generated networks with 6 nodes and a range of noise amplitude (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {k}=0.01$$\end{document}k=0.01 to 100), excluding the outliers. Comparison of link strength found by the modified TDS and CTDS shows that both methods fail to effectively differentiate weak links (coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<0.25$$\end{document}<0.25 for this example) from non-existent/non-primary links (with zero coefficients). For larger coefficients (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}…

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  Figure 4. Group-averaged CTDS coloured matrices and corresponding network physiology of left- and right-side brain rhythms, heart and breathing rates, interactions during initial rest; WAFA alertness test; transition (rest); WAFF attention test and final recovery (rest). Each matrix element colour-maps the average strength of the CTDS (the fraction of time with stable delay over the total phase duration) for each pair of physiological systems. The coloured matrices show the strength averaged over all subjects during each phase. The number in each matrix element is the percentage of cases with significant links (CTDS strength > threshold) out of the total number of cases (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=110$$\end{document}N=110). The threshold was determined by surrogate analysis (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}…

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  Figure 5. Surrogate test results indicated that a threshold set at 19% for link strength (shown with the vertical dashed line) allows only significant links with relevant physiological information to be reliably included in the analysis.

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  Figure 6. (a) The boxplots show the distribution of the CTDS strength of significant links over 110 subjects. A larger variability range for the link strength for subjects at rest, and lower link strengths during the WAFF test were observed. Overall, from Fig. 6b and this figure, there are fewer, but stronger links for rest, compared to WAFA and WAFF. According to the MWW test results, the strength of the links significantly differs between WAFA and all other states (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}<0001$$\end{document}p<0001), and between WAFF and all other states (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}<0.0001$$\end{document}p<0.0001) for both sides of the brain. No significant difference was found between the left and right sides of the brain during the same cognitive state (WAFA and WAFF). The…

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  Figure 7. (a) The boxplots show the distribution of the TDS strength of significant links over 110 subjects. Generally TDS strength were higher than that of CTDS, for most links, with less variability between states, compared to CTDS. TDS also found similar patterns for changes in the link strength between states. It also found a larger range variability during rest, similar to CTDS. Consistent with CTDS results but with less significance, there are fewer, but stronger links for rest, compared to WAFA and WAFF. According to the MWW test results, the strength of the links significantly differs between WAFA and all other states (p < 0001), except for the rest (left-side), and between WAFF and all other states (p < 0.0001) for both sides of the brain. No significant difference was found between the left and right sides of the brain during the same cognitive state, except for recovery (p = 0.04). (b) The boxplots show the distribution of the number of significant links over all subjects (N =110) found by the modified TDS. Generally higher number of links was found by TDS than CTDS. However, the trend of changes in the number of links across states is similar to that of CTDS results. It shows…