Moo K Chung^{1}, Jamie L Hanson^{2}, Nagesh Adluru^{1}, Andrew L Alexander^{1}, Richard J Davidson^{3}, and Seth D Pollak^{3}

We present evidence that the structural brain network follows the exponential decay law; these data are inconsistent with the view the brain follows the more complicated truncated exponential power law. Our model is then used in application, to show that children exposed to high levels of early life stress have more sparsely connected hub nodes. These data are consistent with rodent models of the effects of stress on synaptogenesis.

Data:
The study consisted of 23 children who
experienced maltreatment early in their lives, and 31 age-matched normal
controls. The maltreated children were raised in institutional
settings, where the quality of care has been below the standard necessary for
healthy development. The age distribution is 11.44±1.64 years. The average amount of time spent in
institutional care range from 3 months to 5.4 years. Children were on average
3.2 years when they were adopted, with a range of 3 months to 7.7 years. DTI
were collected on a 3T GE SIGNA scanner. Diffusion tensor encoding was achieved
using 12 optimum non-collinear encoding directions with a diffusion weighting
of 1114 s/mm^{2}. The details on acquisition parameters are given
in Hanson^{4}.
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Processing:
DTI processing follows the pipeline established
in the previous studies^{4,5,6,7}. Spatial
normalization of DTI data was done using diffeomorphic registration^{5}.
Tractography was done using the TEND algorithm^{7}. Anatomical Automatic
Labeling (AAL) with 116 parcellations were used to establish network nodes^{8}.
The tracts counts were used in
building adjacency matrices.

Degree distributions: The node degree $$$k$$$, which counts the number of connections at the node, was measured. The two-sample *t*-test on the node degree did not yield any group difference (Figure-1). It was necessary to use a more sophisticated approach. The previous studies reported the probability distribution $$$P(k)$$$ to follow one of three laws^{1,2,3}:

Power law: $$ P_{p}(k){\sim}k^{-\gamma},$$

Exponential decay: $$P_{e}(k){\sim}e^{-\lambda{k}},$$

Exponentially truncated power law: $$P_{etp}(k){\sim}k^{-\gamma}e^{-\lambda k}.$$ Estimating the parameters from the empirical distribution is challenging due to small sample size in the tail region. The parameters were estimated from the cumulative distribution functions (CDF) that accumulate the probability from low to high degrees to reduces noise in the tail region. We further combined all the degrees across the subjects to increase the sample size further. The parameters were estimated by minimizing the sum of squared errors (SSE) between the theoretical and empirical CDFs. For instance, in the exponentially truncated power law $$(\widehat\gamma,\widehat\lambda)=\arg\min_{\gamma\geq0,\lambda\geq0} \int_{0}^{\infty}\big| F_{etp}(k)-\widehat{F}_{etp}(k)\big|_2^2\;dk.$$

Exponential decay: The best model was for the power law was when γ=0.2794 with SSE=1.01. For the exponential decay law, the best model was when λ=0.1500 with SSE=0.046. For the truncated power law, the best model was when γ=0,λ=0.1500 with SSE=0.046. The best truncated power law became the exponential decay when γ=0. Thus, our data supports the exponential decay model (Figure-2).

The maltreated subjects had higher concentration in low degree nodes. The controls had much higher concentration of hub nodes. This pattern is also observed in the cumulative distribution functions (CDF) for each group. The maltreated subjects clearly show higher cumulative probability at lower degree.

Group
comparisons: Once we determined the degree distribution follows the
exponential decay model for all 54 subjects, we fitted the exponential decay
model for each subject separately to see if there is any group difference in
the estimated parameter. The estimated parameters were 0.1673±0.0341 for the
maltreated children and 0.1458±0.0314 for the controls. The controls showed a slightly
heavier tail compared to that of the maltreated children. The two-sample *t*-test was performed on
the parameter difference obtaining significant result (*t*-stat.=2.40, *p*-value<0.02). The controls had a higher concentration of hub nodes.

Hub nodes: We determined the 13 most connected nodes (Figure-3), all of which showed higher numbers of connections. The probability of this outcome occurring by a chance is 2^{-13}=0.00012. Thus, the controls are more highly connected in the hub nodes compared to the maltreated children.

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