Core motif–attractor correspondence in CTLNs

Determine whether dynamic attractors in combinatorial threshold-linear networks (CTLNs) correspond exactly to minimal fixed point supports (core motifs) and, if so, establish precise conditions under which the correspondence holds for graphs G with parameters in the legal range (θ>0, δ>0, 0<ε<δ/(δ+1)). Characterize any exceptions and prove the bidirectional implication between core motifs and dynamic attractors.

Background

CTLNs are inhibition-dominated threshold-linear networks whose connectivity is prescribed by a simple directed graph G. The set FP(G) collects fixed point supports and has been used to predict network dynamics. Minimal fixed point supports known as core motifs are widely used heuristically to anticipate attractors such as limit cycles.

The dissertation reports a conjectured correspondence between core motifs and dynamic attractors and illustrates it through examples, but a general proof or characterization is not provided. Establishing this correspondence would bridge combinatorial structure and dynamics, enabling reliable attractor prediction from FP(G).