One-dimensional Bose-Hubbard models are well known to obey a transition from regular to quantum-chaotic spectral statistics. We are extending this concept to relatively simple twodimensional many-body models. Also in two dimensions a transition from regular to chaotic spectral statistics is found and discussed. In particular, we analyze the dependence of the spectral properties on the bond number of the two-dimensional lattices and the applied boundary conditions. For maximal connectivity, the systems behave most regularly in agreement with the applicability of mean-field approaches in the limit of many nearest neighbor couplings at each site.

Background Tumor cell extensions called tumor microtubes (TMs) in glioma resemble neurites during neurodevelopment and connect glioma cells to a network that has considerable relevance for tumor progression and therapy resistance. The determination of interconnectivity in individual tumors has been challenging and the impact of tumor cell connectivity on patient survival remained unresolved so far. Material and Methods A connectivity signature was derived by single-cell RNA-sequencing (scRNA-Seq) of highly and lowly connected cells obtained from an SR101 dye transfer xenograft model of primary glioblastoma (GB). The signature was validated in different in vitro models of cell-to-cell connectivity and could be translated to GB clinical specimens. Results 34 of 40 connectivity genes were related to neurogenesis, neural tube development or glioma progression, including the TM-network-relevant GAP43 gene. Astrocytic-like and mesenchymal-like GB cells had the highest connectivity signature scores in scRNA-Seq data of patient-derived xenografts and patient samples. In 230 human GBs, high connectivity correlated with the mesenchymal expression subtype, TP53 wildtype, and with dismal patient survival. CHI3L1 was identified and validated as a robust molecular marker of connectivity with a functional relevance. Conclusion The connectivity signature gives novel insights into brain tumor biology, provides a proof-of-principle that tumor cell connectivity is relevant for patients’ prognosis, and serves as a robust biomarker that can be used for future clinical trials