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Combinatorial threshold-linear networks (CTLNs) are a special class ofinhibition-dominated TLNs defined from directed graphs. Like more general TLNs,they display a wide variety of nonlinear dynamics including multistability,limit cycles, quasiperiodic attractors, and chaos. In prior work, we havedeveloped a detailed mathematical theory relating stable and unstable fixedpoints of CTLNs to graph-theoretic properties of the underlying network. Herewe find that a special type of fixed points, corresponding to core motifs, arepredictive of both static and dynamic attractors. Moreover, the attractors canbe found by choosing initial conditions that are small perturbations of thesefixed points. This motivates us to hypothesize that dynamic attractors of anetwork correspond to unstable fixed points supported on core motifs. We testedthis hypothesis on a large family of directed graphs of size $n=5$, and foundremarkable agreement. Furthermore, we discovered that core motifs with similarembeddings give rise to nearly identical attractors. This allowed us toclassify attractors based on structurally-defined graph families. Our resultssuggest that graphical properties of the connectivity can be used to predict anetwork's complex repertoire of nonlinear dynamics.

Core motifs predict dynamic attractors in combinatorial threshold-linear networks