2025/10/26 by Jesse Geneson, Geneson, Jesse
Computer Science · Neuroscience · #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Biological sciences #FOS: Computer and information sciences #FOS: Mathematics #Neural Networks Stability and Synchronization #Neural dynamics and brain function #Neurons and Cognition (q-bio.NC) #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2511.05517
openalex publication_date 2025/10/26 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28
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.