A unified theoretical framework for the geometry of cognitive dynamics

A unified theoretical framework for the geometry of cognitive dynamics | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Biological Sciences - Article A unified theoretical framework for the geometry of cognitive dynamics Huayifu Lv This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7926249/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract How the brain flexibly switches between different cognitive contexts and deploys internal representational models of vastly different geometries is a core puzzle in neuroscience. Large-scale neural recordings have recently revealed a contradictory phenomenon: during different tasks, the low-dimensional manifold of neural activity sometimes manifests as zerodimensional points (fixed points), sometimes as one-dimensional lines, and at other times as two-dimensional or even higher-dimensional continuous spaces [1–3]. However, a unified theoretical framework capable of explaining why and how these attractors of disparate dimensions can coexist within the same biological system has been missing. This paper proposes a dynamical paradigm based on first principles that provides a concise and powerful solution to this problem. This paradigm reveals a two-layer, decoupled topological design principle: a top-level "camp" competition network determines the number of discrete, stable contexts (multistability) [4, 5], while a bottom-level "citadel" topology precisely endows each context with its local geometric form, such as points, lines, or planes [6, 7]. Through a series of numerical experiments, we not only demonstrate that this system can stably emerge coexisting attractors of different dimensions but also, by simulating targeted interventions, causally prove the switching mechanisms of multistability and the construction rules for attractor dimensions. This theoretical framework unifies a seemingly chaotic experimental landscape into a simple, programmable topological rule, offering a fundamental theoretical insight into the neural basis of cognitive flexibility and context-dependent computation. Biological sciences/Computational biology and bioinformatics/Computational neuroscience/Dynamical systems Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Complex networks Neural Manifolds Cognitive Flexibility Dynamical Systems Multistability Continuous Attractors Unified Theory Full Text Additional Declarations There is NO Competing Interest. Supplementary Files Supplementaryinformation.pdf Supplementary Information Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7926249","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Biological Sciences - Article","associatedPublications":[],"authors":[{"id":538272088,"identity":"6755ba30-0066-4036-8257-1bcf809a21f9","order_by":0,"name":"Huayifu Lv","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0ElEQVRIiWNgGAWjYBACPgkGhg8JBv/k2NgbGx9+IEYLmwQD44wHFQeM+XgONxtLEKtl5oMzBxLlJNLbBHiI0iLdY9iQ2HYngU3yYRuDBIOdnG4DIS0yZ0BanuWxSSe2PShgSDY2O0DQYTnmDxLbmIuBWtoNJBgOJG4jQgvIFubENsmDbRI8RGtJOHM4sU2CkWgtaYUNCRVpxmw8icBANiDCL/wSyRsbfxjYyMm3H3/48EOFnRxBLQwMHAZIHAOcypAB+wOilI2CUTAKRsEIBgBi/0LLL/rPJgAAAABJRU5ErkJggg==","orcid":"","institution":"Beijing Normal University Zhuhai Campus","correspondingAuthor":true,"prefix":"","firstName":"Huayifu","middleName":"","lastName":"Lv","suffix":""}],"badges":[],"createdAt":"2025-10-22 19:20:27","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7926249/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7926249/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":95223482,"identity":"63e05297-cf78-4172-9091-d6427f00977e","added_by":"auto","created_at":"2025-11-05 16:22:21","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":6126266,"visible":true,"origin":"","legend":"Article File","description":"","filename":"Aunifiedtheoreticalframeworkforthegeometryofcognitivedynamics.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7926249/v1_covered_f8254bc3-8fdd-46bf-8839-13e39b494247.pdf"},{"id":95071961,"identity":"29cfe2e5-bd4d-4bd2-87fa-3b8ff169e0ac","added_by":"auto","created_at":"2025-11-04 03:26:38","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":606212,"visible":true,"origin":"","legend":"Supplementary Information","description":"","filename":"Supplementaryinformation.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7926249/v1/319cc004c081601818dcf773.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"A unified theoretical framework for the geometry of cognitive dynamics","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Neural Manifolds, Cognitive Flexibility, Dynamical Systems, Multistability, Continuous Attractors, Unified Theory","lastPublishedDoi":"10.21203/rs.3.rs-7926249/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7926249/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eHow the brain flexibly switches between different cognitive contexts and deploys internal representational models of vastly different geometries is a core puzzle in neuroscience. 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