Age-dynamic networks and functional correlation for early white matter myelination

Abstract

The maturation of the myelinated white matter throughout childhood is a critical developmental process that underlies emerging connectivity and brain function. In response to genetic influences and neuronal activities, myelination helps establish the mature neural networks that support cognitive and behavioral skills. The emergence and refinement of brain networks, traditionally investigated using functional imaging data, can also be interrogated using longitudinal structural imaging data. However, few studies of structural network development throughout infancy and early childhood have been presented, likely owing to the sparse and irregular nature of most longitudinal neuroimaging data, which complicates dynamic analysis. Here, we overcome this limitation and investigate through concurrent correlation the co-development of white matter myelination and volume, and structural network development of white matter myelination between brain regions as a function of age, using statistically well-supported methods. We show that the concurrent correlation of white matter myelination and volume is overall positive and reaches a peak at 580 days. Brain regions are found to differ in overall magnitudes and patterns of time-varying association throughout early childhood. We introduce time-dynamic developmental networks based on temporal similarity of association patterns in the levels of myelination across brain regions. These networks reflect groups of brain regions that share similar patterns of evolving intra-regional connectivity, as evidenced by levels of myelination, are biologically interpretable and provide novel visualizations of brain development. Comparing the constructed networks between different maternal education groups, we found that children with higher and lower maternal education differ significantly in the overall magnitude of the time-dynamic correlations.

Appendix

Estimation of correlation functions

The functional correlation ${\text{corr}}(X(t),Y(t))$ is estimated by the plug-in estimator

$$\widehat {{{\text{corr}}}}(X(t),Y(t))=\frac{{\widehat {{{\text{cov}}}}(X(t),Y(t))}}{{\sqrt {\widehat {{{\text{var}}}}(X(t)),\widehat {{{\text{var}}}}(Y(t))} }}.$$

(4)

We propose to estimate ${\text{cov}}(X(t),Y(t))$, ${\text{var}}(X(t))$, and ${\text{var}}(Y(t))$ separately by kernel local linear smoothing the pooled centered observations, which is detailed as follows. Assume we make observations $({t_{ij}},{X_{ij}},{Y_{ij}})$ at each time ${t_{ij}}$, for subject $i=1, \ldots ,n$ and visit $j=1, \ldots ,{n_i}$, where ${X_{ij}}=X({t_{ij}})$, ${Y_{ij}}=Y({t_{ij}})$, $n$ is the number of subjects and ${n_i}$ is the number of measurements per subject.

We first estimate ${\mu _X}(t):=E(X(t))$ and ${\mu _Y}(t):=E(Y(t))$ by kernel local linear smoothing. We define the local linear kernel smoother for ${\mu _X}(t)$ as ${\hat {\mu }_X}(t)={\hat {\beta }_0}$ by smoothing the pooled observation $\{ ({t_{ij}},{X_{ij}})\} _{{i=1}}^{n}{_{{j=1}}^{{{n_i}}}}$, where

$$({\hat {\beta }_0},{\hat {\beta }_1})=\mathop {{\text{arg min}}}\limits_{{{\beta _0},{\beta _1}}} \mathop \sum \limits_{{i=1}}^{n} \mathop \sum \limits_{{j=1}}^{{{n_i}}} K\left( {\frac{{{t_{ij}} - t}}{h}} \right){[{X_{ij}} - {\beta _0} - {\beta _1}(t - {t_{ij}})]^2},$$

(5)

$h>0$ is the bandwidth, and $K( \cdot )$ is a kernel function. The mean function ${\mu _Y}(t)$ of $Y(t)$ can be estimated similarly by smoothing $\{ ({t_{ij}},{Y_{ij}})\} _{{i=1}}^{n}{_{{j=1}}^{{{n_i}}}}$. Next we obtain the centered observations

$$\begin{aligned} {{\tilde {X}}_{ij}} & ={X_{ij}} - {{\hat {\mu }}_X}({t_{ij}}), \\ {{\tilde {Y}}_{ij}} & ={Y_{ij}} - {{\hat {\mu }}_Y}({t_{ij}}), \\ \end{aligned}$$

for $i=1, \ldots ,n$ and $j=1, \ldots ,{n_i}$. Finally, $\widehat {{\operatorname{cov} }}(X(t),Y(t))$ (resp. $\widehat {{\operatorname{var} }}(X(t))$ and $\widehat {{\operatorname{var} }}(Y(t))$) is obtained by smoothing $\{ ({t_{ij}},{\tilde {X}_{ij}}{\tilde {Y}_{ij}})\} _{{i=1}}^{n}{_{{j=1}}^{{{n_i}}}}$ (resp. $\{ ({t_{ij}},{\tilde {X}_{ij}}^{2})\} _{{i=1}}^{n}{_{{j=1}}^{{{n_i}}}}$ and $\{ ({t_{ij}},{\tilde {Y}_{ij}}^{2})\} _{{i=1}}^{n}{_{{j=1}}^{{{n_i}}}}$) as in (5). For all kernel local smoothing we used Gaussian kernel for $K( \cdot )$ with bandwidth $h$ equal to 150 days.

Note that one can write $\operatorname{var} (X(t))=E({X^2}(t)) - {\mu _X}{(t)^2}$ and thus construct another plug-in estimate $\widehat {{\operatorname{var} }}(Y(t))$ from $\hat {E}(X{(t)^2}) - {\hat {\mu }_X}{(t)^2}$ by smoothing $\{ ({t_{ij}},X_{{ij}}^{2})\} _{{i=1}}^{n}{_{{j=1}}^{{{n_i}}}}$ for $\hat {E}(X{(t)^2})$ and $\{ {t_{ij}},{X_{ij}}\} _{{i=1}}^{n}{_{{j=1}}^{{{n_i}}}}$ for ${\hat {\mu }_X}(t)$. This alternative procedure is known to have larger bias than the proposed procedure (see for example Fan and Yao 1998; Zhang and Wang 2016) and thus is not used here. An alternative approach is Frechet regression (Petersen et al. 2018).

Modes of variation

The modes of variation for functional data was discussed by Castro et al. (1986) and Jones and Rice (1992). Given a random function $X(t)$, we target to summarize its important variability using a few basis functions. Denoting ${X^C}(t)=X(t) - \mu (t)$ as the centered process, our goal is to approximate ${X^C}(t)$ by $X_{J}^{C}(t)=\sum\nolimits_{{j=1}}^{J} {{\xi _j}{\psi _j}(t)}$ using a few basis functions, where $\{ {\psi _j}(t)\} _{{j=1}}^{J}$ is an orthonormal basis of ${L^2}$and the ${\xi _j}$ are the $j$th Fourier coefficients of ${X^C}$ projected onto ${\psi _j}$. Using a suitably defined notion of total variation for functional data, the best J-dimensional approximation $X_{J}^{C}(t)$ to ${X^C}(t)$ in terms of total variation explained is given by the orthonormal basis that solves

$$\mathop {\hbox{min} }\limits_{{\begin{array}{*{20}{c}} {{\psi _1}, \ldots ,{\psi _J}} \\ {||{\psi _j}||=1,} \\ {\langle {\psi _j},{\psi _l}\rangle =0{\text{ for }}1 \leq j \ne l \leq J} \end{array}}} E\left( {\int {{{\left( {{X^C}(t) - \sum\limits_{{j=1}}^{J} {{\xi _j}{\psi _j}(t)} } \right)}^2}} \,{\text{d}}t} \right).$$

(6)

An explicit solution to (6) is given by the eigenfunctions of $G(s,t)=\,{\text{cov}}(X(s),X(t))$. Covariance function $G$ has spectral decomposition

$$G(s,t)=\mathop \sum \limits_{{j=1}}^{\infty } {\lambda _j}{\phi _j}(s){\phi _j}(t),$$

where the ${\lambda _1} \geq {\lambda _2} \geq \cdots \geq 0$ are the eigenvalues and the ${\phi _{\text{j}}}({\text{t}})$ are the corresponding orthonormal eigenfunction. It is then well-known the first $J$ eigenfunctions ${\phi _1}, \ldots ,{\phi _J}$ of the covariance operator $G$ is a solution to (6), corresponding to the principal modes of variation, and the eigenvalue ${\lambda _j}$ associated with ${\phi _j}$ quantifies how much variation is explained by the $j$th eigenfunction. The fraction of total variation explained by the $j$th eigenfunction is ${\lambda _j}/\sum\nolimits_{{j=1}}^{\infty } {{\lambda _j}}$.