Down syndrome (DS) is caused by human chromosome 21 (HSA21) trisomy. It is characterized by a poorly understood intellectual disability (ID). We studied two mouse models of DS, one with an extra copy of the Dyrk1A gene (189N3) and the other with an extra copy of the mouse Chr16 syntenic region (Dp(16)1Yey). RNA-seq analysis of the transcripts deregulated in the embryonic hippocampus revealed an enrichment in genes associated with chromatin for the 189N3 model, and synapses for the Dp(16)1Yey model. A large-scale yeast two-hybrid screen (82 different screens, including 72 HSA21 baits and 10 rebounds) of a human brain library containing at least 107 independent fragments identified 1,949 novel protein–protein interactions. The direct interactors of HSA21 baits and rebounds were significantly enriched in ID-related genes (P-value < 2.29 × 10−8). Proximity ligation assays showed that some of the proteins encoded by HSA21 were located at the dendritic spine postsynaptic density, in a protein network at the dendritic spine postsynapse. We located HSA21 DYRK1A and DSCAM, mutations of which increase the risk of autism spectrum disorder (ASD) 20-fold, in this postsynaptic network. We found that an intracellular domain of DSCAM bound either DLGs, which are multimeric scaffolds comprising receptors, ion channels and associated signaling proteins, or DYRK1A. The DYRK1A-DSCAM interaction domain is conserved in Drosophila and humans. The postsynaptic network was found to be enriched in proteins associated with ARC-related synaptic plasticity, ASD, and late-onset Alzheimer’s disease. These results highlight links between DS and brain diseases with a complex genetic basis.