{
  "abstract": "Temporal integration gives continuous-time recurrent networks memory, but in deep stacks it also delays bottom-up signals and attenuates top-down errors. We develop Recursive Quadrature Filters (RQFs), biologically motivated complex-valued temporal filters that are a special case of diagonal state-space models (SSMs), and ask whether this failure mode can be addressed by making each layer's bottom-up input prospective. Starting from an energy model, we derive the RQF dynamics and show that each RQF is a band-pass filter whose learnable parameters control its tuning frequency and bandwidth. We then make each layer's bottom-up input prospective using a parameter-free two-tap update that leaves the recurrent transition and parallel scan unchanged. We extend this correction to general diagonal SSMs and show that it mitigates depth-dependent gradient attenuation when temporal gradients are truncated, i.e., spatial-only backpropagation. We evaluate the intervention in RQFs, S5, and ORGaNICs (a nonlinear gated RNN) trained using full backpropagation through time (BPTT) and spatial-only backpropagation. Under full BPTT, prospective variants match or outperform their non-prospective controls in every model and configuration. A non-residual width-32 six-layer RQF reaches 96.09% accuracy on raw-audio Speech Commands with 31.9k parameters; a width-64 six-layer RQF reaches 83.56% on the 16,384-step Path-X task. These results identify RQFs as a parameter-efficient recurrent substrate and prospective-input coding as an input-side correction for deep continuous-time recurrent networks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04134",
  "author_total": 4,
  "authors": [
    {
      "name": "Shivang Rawat"
    },
    {
      "name": "Mirko Morello"
    },
    {
      "name": "Flaviano Morone"
    },
    {
      "name": "David J. Heeger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f2trbz44sgdhn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04134",
  "fulltext_url": null,
  "id": "f2trbz44sgdhn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04134",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NE"
    ],
    [
      "arxiv",
      "q-bio.NC"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7208790002",
  "pdf_url": "https://arxiv.org/pdf/2609.04134",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f2trbz44sgdhn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:37Z",
      "native_id": "2609.04134",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04134"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.NC",
      "subfield": "neuroscience"
    }
  ],
  "title": "Prospective Coding Improves Learning in Deep Continuous-Time Recurrent Networks",
  "title_text": "Prospective Coding Improves Learning in Deep Continuous-Time Recurrent Networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:37Z",
  "url": "https://research.draconitic.com/paper/f2trbz44sgdhn",
  "venue": "arXiv",
  "versions": []
}