Structure of basins in competitive threshold-linear networks (CTLNs)

Characterize the structure and boundary properties of the basins of attraction in competitive threshold‑linear networks defined by dx_i/dt = −x_i + [∑_{j=1}^n W_{ij} x_j + b_i]^+ with weights W_{ij} constructed from a directed graph (W_{ii}=0, W_{ij}=−1+ε if i←j, W_{ij}=−1−δ otherwise) and uniform inputs b_i=1, including the specific multistable 9‑node network example, by determining whether the basin boundaries are smooth or fractal and how multiple attractors partition the state space.

Background

The paper studies a class of competitive threshold‑linear networks (CTLNs) built from directed graphs, showing multistability and complex basin patterns in a 9‑node example. While multistability is documented, the basins’ detailed geometry and boundary regularity have not been analyzed.

The authors explicitly note that the basins of this system have not been studied in detail and highlight an open question regarding their structure, pointing to the broader paucity of research on basin boundaries in neural networks and potential relevance for learning dynamics.