The cerebral cortex and hippocampus are important for the control of cognitive functions and social behaviors, many of which are sexually dimorphic and tightly regulated by gonadal steroid hormones via activation of their respective nuclear receptors. As different levels of sex steroid hormones are present between the sexes during early development and their receptors act as transcription factors to regulate gene expression, we hypothesize that sexually dimorphic gene expression in the developing mouse cortex and hippocampus might result in sex differences in brain structures and neural circuits governing distinct behaviors between the sexes as adults. To test our hypothesis, we used gene expression microarrays to identify 90 candidate genes differentially expressed in the neonatal cortex/hippocampus between male and female mice, including 55 male-biased and 35 female-biased genes. Among these genes, sexually dimorphic expression of eight sex chromosome genes was confirmed by reverse transcription with quantitative PCR (RT-qPCR), including three located on the X chromosome (Xist, Eif2s3x, and Kdm6a), three on the Y chromosome (Ddx3y, Eif2s3y, and Kdm5d), and two in the pseudoautosomal region of the X and Y chromosomes (Erdr1 and Mid1). In addition, five autosomal genes (Cd151, Dab2, Klk8, Meg3, and Prkdc) were also validated for their sexually dimorphic expression in the neonatal mouse cortex/hippocampus. Gene Ontology annotation analysis suggests that many of these sexually dimorphic genes are involved in histone modifications, cell proliferation/death, androgen/estrogen signaling pathways, and synaptic organization, and these biological processes have been implicated in differential neural development, cognitive function, and neurological diseases between the sexes.

This paper presents a nonlinear model reduction method for systems of equations using a structured neural network. The neural network takes the form of a "three-layer" network with the first layer constrained to lie on the Grassmann manifold and the first activation function set to identity, while the remaining network is a standard two-layer ReLU neural network. The Grassmann layer determines the reduced basis for the input space, while the remaining layers approximate the nonlinear input-output system. The training alternates between learning the reduced basis and the nonlinear approximation, and is shown to be more effective than fixing the reduced basis and training the network only. An additional benefit of this approach is, for data that lie on low-dimensional subspaces, that the number of parameters in the network does not need to be large. We show that our method can be applied to scientific problems in the data-scarce regime, which is typically not well-suited for neural network approximations. Examples include reduced order modeling for nonlinear dynamical systems and several aerospace engineering problems.