Low-rank gluing rules enable predictable neural network compositionality

Brains generate complex behaviors from stable structures using compositionality—decomposing tasks into reusable primitives. But a rigorous mathematical link between structural modularity and functional compositionality in nonlinear networks has been lacking. This work from Juliana Londono Alvarez formally bridges that gap by studying inhibition-dominated threshold-linear networks (TLNs). The author introduces a novel class of modular network assemblies called low-rank gluings, where component subnetworks with arbitrary internal connectivity are connected via specific low-rank couplings. The key result: global fixed points of these glued networks are constrained to be combinations of the local fixed points of their constituent modules. For a more structured subclass called rank-1 gluings, the paper provides a complete characterization that determines exactly which combinations of local fixed points yield global ones.

These theoretical results are applied to graph-based networks, extending fixed point decomposition rules from combinatorial threshold-linear networks (CTLNs) to the more flexible family of generalized CTLNs (gCTLNs)—proving the rules are more robust than previously thought. The work also demonstrates that these gluing rules provide a mathematically tractable recipe for engineering compositional dynamics. This enables the construction of networks with a combinatorially large repertoire of predictable attractors that can be understood from simpler component motifs, from compositions of fixed points to compositional limit cycles. The implications span both theoretical neuroscience and AI: a mathematical foundation for building modular neural networks with predictable, scalable emergent behaviors from simple building blocks.