Moo K Chung^{1}, Victoria Vilalta-Gil^{2}, Paul J Rathouz^{1}, Benjamin B Lahey^{3}, and David H Zald^{2}

We present a new integrative framework for analyzing paired images using hyper-networks. The method is applied to twin fMRI study in characterizing the amount of heritability in the functional network.

Sparse hyper-connectivity: Consider a collection of paired images $$$(x_1,y_1),(x_2, y_2),\cdots,(x_n, y_n).$$$ Let $$${\bf x}=(x_1,\cdots,x_n)'$$$ and $$${\bf y}=(y_1,\cdots,y_n)'$$$ be the vectors of images. We set up a hyper-network by relating the paired image vectors at different voxels $$$v_i$$$ and $$$v_j$$$: $${\bf y}(v_j)=\sum_{i{\neq}j}\beta_{ij}\;{\bf x}(v_i)+{\bf{e}}$$ for zero-mean noise vector **e **(Figure-1). The parameters $$$\beta=(\beta_{ij})$$$ are the weights of the hyper-edges. It is likely that we have significantly more number of voxels ($$$p$$$) than the number of images ($$$n$$$), so we estimate the parameters using sparse regression^{7}: $$\widehat{\beta}(\lambda)=\arg\min_{\beta}\frac{1}{2}\sum_{i,j=1}^p\parallel{\bf y}(v_j)-\sum_{i,j=1}^n\beta_{ij}\;{\bf x}({v_i})\parallel^{2}+\lambda\sum_{i,j =1}^p| \beta_{ij} |.$$

Sparse
cross-correlations: For this study, we consider a simpler
model
$$\widehat{\beta}(\lambda)=\arg\min_{\beta}\frac{1}{2}\sum_{i,j=1}^p\parallel{\bf y}(v_j)-\beta_{ij}\;{\bf x}({v_i})\parallel^{2}+\lambda\sum_{i,j =1}^p|\beta_{ij}|.$$ Without loss of generality, we can enter
and scale $$${\bf x}$$$ and $$${\bf y}$$$ such that
$$\sum_{k=1}^n
x_k(v_i) = \sum_{k=1}^ny_k(v_i)=0,\| {\bf x}(v_i)\|^2=\|{\bf y}(v_i)\|^2=1.$$ Then we have $$\widehat{\beta}_{ij}(\lambda)=\begin{cases}{\bf x}'(v_i) {\bf y}(v_j)-\lambda & \mbox{if }\;{\bf x}'(v_i) {\bf y}(v_j)>\lambda \\0& \mbox{if }\;|{\bf x}'(v_i) {\bf y}(v_j)|\leq \lambda \\{\bf x}'(v_i) {\bf y}(v_j)+\lambda & \mbox{if }\;{\bf x}'(v_i){\bf y}(v_j)<-\lambda \end{cases},$$ which is the sparse cross-correlation^{7} (Figure-2).

Topological
inference: The hyper-network is binarized by assigning value 1 to any nonzero hyper-edge weight and 0 otherwise. The resulting binary graph denoted as $$$G_{\lambda}$$$ induces a graph filtration^{2,7}, a collection of nested graphs:$$G_{\lambda_{(1)}}\supset G_{\lambda_{(2)}}\supset G_{\lambda_{(3)}}\supset\cdots\supset
G_{\lambda_{(q)}},$$
where
$$0 \leq \lambda_{(1)} \leq \lambda_{(2)} \leq \lambda_{(3)} \cdots \leq \lambda_{(q)}$$ are the sorted
hyper-edge weights.

Let $$$B$$$ be a monotonic graph function such as the number of connected components and the size of the largest component (Figure-3). Given
two hyper-networks $$$G_1,G_2$$$ and corresponding graph filtrations $$$G_{1\lambda},G_{2\lambda}$$$, we test the null hypothesis of the
equivalence of the two monotonic functions:
$$H_0:
B({G_1}_{\lambda}) = B({G_2}_{\lambda}) \; \mbox{ for all } \lambda\geq0.$$
The
test statistic^{7} $$D_p=\sup_{\lambda}|B({G_1}_{\lambda})-B({G_2}_{\lambda})|$$
is
used to compute the *p*-value using $$P\left( D_{p}/\sqrt{2(p-1)}\leq d\right)=1-2\sum_{i=1}^{\infty}(-1)^{i-1}e^{-2i^2d^2}.$$

Data:
We applied the method to the problem of determining the heritability of a fMRI network at
the network level. The study consists of 11 monozygotic (MZ) and 9 same-sex
dizygotic (DZ) twin pairs of 3T fMRI acquired in Intera-Achiava Phillips MRI
scanners with a state-of-the-art 32 channel SENSE head coil. BOLD functional
images were acquired with a gradient echoplanar sequence, with 38 axial-oblique
slices (3 mm thick), TR 2000ms, TE
25ms, flip angle=90. Subjects completed monetary incentive delay task^{8} involving $0, 1 and 5 rewards. The interest is in knowing the extent of the genetic influence on the functional network of these participants while anticipating the high reward as
measured by delays in hitting the response button. After fitting a GLM at each voxel, we obtained the contrast maps testing the significance of activity in the
delay period for $5 trials relative to $0 reward .
The paired contrast maps are $$${\bf x}$$$ and
$$${\bf y}$$$ in the model (Figure-1).

Heritability index (HI) determines the amount of variations due to genetic influence in a population and estimated using Falconer's formula. Extending Falcorner’s formula, HI matrix on the hyper-edges is given by $$\mbox{HI}=2\left(\widehat{\beta}_{MZ}-\widehat{\beta}_{DZ}\right),$$ the twice difference in the sparse cross-correlations in MZ- and DZ-twins. The diagonal entries are the node-level HI at each vertex (Figure-2). The statistical significance of HI is determined using statistic $$$D_p$$$.

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