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From a neuronal recurrence equation of memory size $h$ which describes a cycle of length $\\rho(m) \\times lcm(p_0, p_1,..., p_{-1+\\rho(m)})$, we construct a set of $\\rho(m)$ neuronal recurrence equations whose dynamics describe respectively the transient of length $O(\\rho(m) \\times lcm(p_0, ..., p_{d}))$ and the cycle of length $O(\\rho(m) \\times lcm(p_{d+1}, ..., p_{-1+\\rho(m)}))$ if $0 \\leq d \\leq -2+\\rho(m)$ and 1 if $d=\\rho(m)-1$.\n  This result shows the expone"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1110.3586","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.NE","submitted_at":"2011-10-17T07:10:26Z","cross_cats_sorted":["math.DS","nlin.CD"],"title_canon_sha256":"90cae1c5ed988421851e6acf42c324f0c9ca93af93a9bbe3ea2585bbbfcf3863","abstract_canon_sha256":"00e2e745471425770af65004f189035a2c36b3c54e8cdc0002157074237c75c9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:58:24.513091Z","signature_b64":"1QgdBld5mSBEM7H64KxpbBmW9ep7xS9bHWsTIQStt3Ruk0hnlU3PteBZEXHrTHkie24KqdALitCtU72SZBWZAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b596a910866c6fad025951f0d28151d8528bc675127768e0b44d7b2b639ef94","last_reissued_at":"2026-05-18T03:58:24.512222Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:58:24.512222Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Period-halving Bifurcation of a Neuronal Recurrence Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","nlin.CD"],"primary_cat":"cs.NE","authors_text":"Ren\\'e Ndoundam","submitted_at":"2011-10-17T07:10:26Z","abstract_excerpt":"We study the sequences generated by neuronal recurrence equations of the form $x(n) = {\\bf 1}[\\sum_{j=1}^{h} a_{j} x(n-j)- \\theta]$. From a neuronal recurrence equation of memory size $h$ which describes a cycle of length $\\rho(m) \\times lcm(p_0, p_1,..., p_{-1+\\rho(m)})$, we construct a set of $\\rho(m)$ neuronal recurrence equations whose dynamics describe respectively the transient of length $O(\\rho(m) \\times lcm(p_0, ..., p_{d}))$ and the cycle of length $O(\\rho(m) \\times lcm(p_{d+1}, ..., p_{-1+\\rho(m)}))$ if $0 \\leq d \\leq -2+\\rho(m)$ and 1 if $d=\\rho(m)-1$.\n  This result shows the expone"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.3586","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1110.3586","created_at":"2026-05-18T03:58:24.512387+00:00"},{"alias_kind":"arxiv_version","alias_value":"1110.3586v3","created_at":"2026-05-18T03:58:24.512387+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.3586","created_at":"2026-05-18T03:58:24.512387+00:00"},{"alias_kind":"pith_short_12","alias_value":"HNMWVEIIM3DP","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_16","alias_value":"HNMWVEIIM3DPVUBF","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_8","alias_value":"HNMWVEII","created_at":"2026-05-18T12:26:30.835961+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HNMWVEIIM3DPVUBFSUPQ2KAVDW","json":"https://pith.science/pith/HNMWVEIIM3DPVUBFSUPQ2KAVDW.json","graph_json":"https://pith.science/api/pith-number/HNMWVEIIM3DPVUBFSUPQ2KAVDW/graph.json","events_json":"https://pith.science/api/pith-number/HNMWVEIIM3DPVUBFSUPQ2KAVDW/events.json","paper":"https://pith.science/paper/HNMWVEII"},"agent_actions":{"view_html":"https://pith.science/pith/HNMWVEIIM3DPVUBFSUPQ2KAVDW","download_json":"https://pith.science/pith/HNMWVEIIM3DPVUBFSUPQ2KAVDW.json","view_paper":"https://pith.science/paper/HNMWVEII","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1110.3586&json=true","fetch_graph":"https://pith.science/api/pith-number/HNMWVEIIM3DPVUBFSUPQ2KAVDW/graph.json","fetch_events":"https://pith.science/api/pith-number/HNMWVEIIM3DPVUBFSUPQ2KAVDW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HNMWVEIIM3DPVUBFSUPQ2KAVDW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HNMWVEIIM3DPVUBFSUPQ2KAVDW/action/storage_attestation","attest_author":"https://pith.science/pith/HNMWVEIIM3DPVUBFSUPQ2KAVDW/action/author_attestation","sign_citation":"https://pith.science/pith/HNMWVEIIM3DPVUBFSUPQ2KAVDW/action/citation_signature","submit_replication":"https://pith.science/pith/HNMWVEIIM3DPVUBFSUPQ2KAVDW/action/replication_record"}},"created_at":"2026-05-18T03:58:24.512387+00:00","updated_at":"2026-05-18T03:58:24.512387+00:00"}