2018/06/13 by Ann E. Sizemore, Sizemore, Ann E., Jennifer E. Phillips‐Cremins +5 · 7 citations
Computer Science · #55-01 #FOS: Biological sciences #Neurons and Cognition (q-bio.NC) #Quantitative Methods (q-bio.QM) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1806.05167
openalex publication_date 2018/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The application of network techniques to the analysis of neural data has\ngreatly improved our ability to quantify and describe these rich interacting\nsystems. Among many important contributions, networks have proven useful in\nidentifying sets of node pairs that are densely connected and that collectively\nsupport brain function. Yet the restriction to pairwise interactions prevents\nus from realizing intrinsic topological features such as cavities within the\ninterconnection structure that may be just as crucial for proper function. To\ndetect and quantify these topological features we must turn to methods from\nalgebraic topology that encode data as a simplicial complex built of sets of\ninteracting nodes called simplices. On this substrate, we can then use the\nrelations between simplices and higher-order connectivity to expose cavities\nwithin the complex, thereby summarizing its topological nature. Here we provide\nan introduction to persistent homology, a fundamental method from applied\ntopology that builds a global descriptor of system structure by chronicling the\nevolution of cavities as we move through a combinatorial object such as a\nweighted network. We detail the underlying mathematics and perform\ndemonstrative calculations on the mouse structural connectome, electrical and\nchemical synapses in \C. elegans, and genomic interaction data. Finally\nwe suggest avenues for future work and highlight new advances in mathematics\nthat appear ready for use in revealing the architecture and function of neural\nsystems.\n