{
  "abstract": "How is computation organized and reused across tasks and time in a trained recurrent network? Most analyses emphasize the geometry of neural activity, dynamical motifs, or local perturbation growth. We instead study the network's global first-order perturbation response. The finite-horizon Green's operator maps perturbations at each source along a trajectory to their downstream state-space responses and therefore directly represents perturbation routing. Simple reductions of this operator provide task-to-task and time-to-time views of the same computation, while matrix-free products make these views accessible without constructing the full operator. In a flexible multitask recurrent network, task reductions reveal structured reuse of known computational motifs, while temporal reductions reveal causal pathways and how they emerge during training. Our main point is simple: the Green's operator provides a global response geometry for mapping the organization of learned dynamical computation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40292",
  "author_total": 1,
  "authors": [
    {
      "name": "James Hazelden"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ek70xktz2emdt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40292",
  "fulltext_url": null,
  "id": "ek70xktz2emdt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40292",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "q-bio.NC"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7215093063",
  "pdf_url": "https://arxiv.org/pdf/2609.40292",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ek70xktz2emdt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:52Z",
      "native_id": "2609.40292",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40292"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.NC",
      "subfield": "neuroscience"
    }
  ],
  "title": "Disentangling Computation in Multi-Task Neural Networks with the Green's Operator",
  "title_text": "Disentangling Computation in Multi-Task Neural Networks with the Green's Operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:52Z",
  "url": "https://research.draconitic.com/paper/ek70xktz2emdt",
  "venue": "arXiv",
  "versions": []
}