Stable non-minimal fixed points of threshold-linear networks
Jesse Geneson
TL;DR
The paper disproves the conjecture that every stable fixed point of a threshold-linear network must have a minimal support. It provides a concrete 3-neuron competitive TLN with a stable full-support fixed point containing a smaller stable fixed-point support, and shows this phenomenon cannot occur in 2-neuron networks. Building on this, it introduces constructions that realize nested stable fixed-point supports with arbitrary sizes and unbounded chain lengths, and proves robustness under perturbations. It also distinguishes general competitive TLNs from combinatorial CTLNs by showing that CTLNs preserve minimality for clique-supported fixed points, while broader TLNs admit richer hierarchical attractor structures. Collectively, the results reveal a richer, non-minimal attractor landscape in TLNs and open questions about the combinatorial motifs governing these hierarchies.
Abstract
In threshold-linear networks (TLNs), a fixed point is called minimal if no proper subset of its support is also a fixed point. Curto et al (Advances in Applied Mathematics, 2024) conjectured that every stable fixed point of any TLN must be a minimal fixed point. We provide a counterexample to this conjecture: an explicit competitive TLN on 3 neurons that exhibits a stable fixed point whose support is not minimal (it contains the support of another stable fixed point). We prove that there is no competitive TLN on 2 neurons which contains a stable non-minimal fixed point, so our 3-neuron construction is the smallest such example. By expanding our base example, we show for any positive integers $i, j$ with $i < j-1$ that there exists a competitive TLN with stable fixed point supports $τ\subsetneq σ$ for which $|τ| = i$ and $|σ| = j$. Using a different expansion of our base example, we also show that chains of nested stable fixed points in competitive TLNs can be made arbitrarily long.