Emergent topological structure in spontaneous brain-organoid activity

Neural activity is widely held to organize on low-dimensional structure embedded in a high-dimensional state space. Persistent homology reads such structure directly from the pattern of pairwise correlations, without assuming in advance which variables are relevant. We apply persistent homology to microelectrode-array (MEA) recordings of spontaneous activity from human (Lancaster) and mouse (Pa\c{s}ca) cortical organoids, spanning $26$--$234$ simultaneously sorted units, and ask whether topological data analysis resolves structure at the node counts that neural recordings actually deliver. Building weighted networks in correlation space and characterizing them by Vietoris--Rips filtration, we find that the first homology ($H_1$, loops) rises significantly above a rate- and population-preserving null in $14$ of $18$ datasets. This loop structure occupies a non-redundant core: it is robust to random removal of units yet disrupted by targeted removal of the units that carry it. Topological richness grows with network size, and second homology ($H_2$) emerges significantly above the null only in the larger networks. These results show that persistent homology resolves structured topology in neural recordings at the scale experiments actually deliver.