### Project Description

The twin study design offers a
very effective way of determining the heritability
of the human brain. The difference in variability
between monozygotic and same-sex dizygotic twins
can be used to determine the heritability of an
imaging-based phenotype at each voxel. Except for
few well known neural circuits, the extent to
which heritability influences the brain network is
not well established. Compared to many existing
studies on univariate phenotypes in brain imaging,
there are not many studies on the heritability of
brain networks. Measures of network topology and
features may be worth investigating as
intermediate phenotypes or endophenotypes.
However, the existing brain network analysis has
not yet been adapted for this purpose. Determining
the extent of heritability of brain networks with
large number of nodes is the first necessary
prerequisite for identifying network-based
endophenotypes. This requires constructing the
large-scale brain networks by taking every voxel
in the brain as network nodes with at least a
billion connections, which is a serious
computational bottleneck. We propose to map the
heritability of large-scale brain networks by
taking every voxel in images as network nodes
using persistent homology
and sparse network models
(Figure 1,2). By exploiting the topological
structure of data, we can show that it is possible
to bypass the bottleneck and learn networks with
billion connections without much computational
resource such as cluster computing.

### Matlab Codes

The exact topological inference procedure is packaged into a single zipped file: matlab.2017.IPMI.zip. The package will run two simulation studies given in Chung et al. (2017). It contains codes for computing probability given in Theorem 2 and 3 as well as codes for computing Kolmogorove-Smirnov (KS) like test statistic and Gromov-Housdorff (GH) distance for networks.The sparse hyper-network construction is done via soft-thresholding as explained in Chung et al. (2017). Soft-thresholding based sparse network construction is first explained in Chung et al. (2015).

The procedure is packaged into a single zipped file: hyper-network.zip, where a toy example is used as an illustration for how the method works. Heritability index of the constructed sparse hyper-network at the network level is given as heritability-network.zip.

### Publication

[1] Chung,
M.K., Vilalta, V.G., Rathouz, P.J., Lahey, B.B.,
Zald, D.H. 2016 Mapping
heritability of large-scale brain networks
with a billion connections via persistent
homology. arXiv:1509.04771 [2] Chung, M.K., Vilalta, V.G., Rathouz, P.J., Lahey, B.B., Zald, D.H. 2016 Heritability of large-scale functional brain networks. Organization for Human Brain Mapping (OHBM). Lecture slides, Poster

[3] Chung, M.K., Vilalta, V.G., Rathouz, P.J., Lahey, B.B., Zald, D.H. 2017 Hyper network analysis on paired images. International Society of Magnetic Resonance in Medicine (ISMRM) lecture slides

[4] Chung, M.K., Vilalta, V.G., Lee, H., Rathouz, P.J., Lahey, B.B., Zald, D.H. 2017 Exact topological inference for paired brain networks via persistent homology. Information Processing in Medical Imaging (IPMI). lecture slides

[5] Chung, M.K., Hanson, J.L., Ye, J., Davidson, R.J. Pollak, S.D. 2015 Persistent homology in sparse regression and its application to brain morphometry. IEEE Transactions on Medical Imaging, 34:1928-1939