Discrete Morse Graph Construction for High-Dimensional Transcriptomic Data

Single-cell transcriptomic atlases resolve thousands of putative brain cell types, yet common analytical workflows often rely on low-dimensional embeddings that can distort neighborhood relationships and obscure continuous variation between closely related populations. We introduce single-cell Discrete Morse Graph Construction (scDMGC), a deterministic framework based on topological data analysis and discrete Morse theory that extracts a compact graph from neighborhood structure in high-dimensional gene-expression space. Because graph edges are derived from the original k-nearest-neighbor complex, scDMGC provides an interpretable multiscale representation of transcriptomic organization without requiring a low-dimensional embedding. Local maxima represent coherent transcriptional states, saddles quantify their separation, and gradient paths capture intermediate states and continuous transitions. Applied to cortical and hippocampal datasets, scDMGC recovers established inhibitory-neuron classes and elucidates transcriptional gradients. In mouse whole-cortex data, Morse graph structure evaluates cell-type hierarchy and type robustness. Persistence across scales provides a quantitative measure of cell-type identity and discrete versus continuous relationships. Finally, in Alzheimers disease single-nucleus data, scDMGC distinguishes changes in cell-type abundance from disease-associated shifts in transcriptional state. scDMGC therefore provides a multiscale framework for quantifying whether transcriptional populations form persistent cell types, how strongly external annotations are supported by intrinsic data structure, how cell types relate to one another, and where discrete organization transitions into continuous variation.