Stable non-minimal fixed points of threshold-linear networks
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918191700115456 |
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| author | Geneson, Jesse |
| author_facet | Geneson, Jesse |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05517 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable non-minimal fixed points of threshold-linear networks Geneson, Jesse Neurons and Cognition Discrete Mathematics Combinatorics 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. |
| title | Stable non-minimal fixed points of threshold-linear networks |
| topic | Neurons and Cognition Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2511.05517 |