Stable non-minimal fixed points of threshold-linear networks

TL;DR

This paper disproves a conjecture by providing a minimal counterexample of a stable non-minimal fixed point in a 3-neuron threshold-linear network, and explores the structure of fixed points in such networks.

Contribution

It presents the first explicit counterexample to the conjecture that all stable fixed points are minimal in TLNs and characterizes the possible configurations of nested fixed points.

Findings

Counterexample of a stable non-minimal fixed point in a 3-neuron TLN

No such fixed point exists in 2-neuron TLNs

Chains of nested stable fixed points can be arbitrarily long

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 $\tau \subsetneq \sigma$ for which $|\tau| = i$ and $|\sigma| = j$. 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.