A Discrete-time Continuous-space Neural Model for Shell Patterns in Mollusks

A Discrete-time Continuous-space Neural Model for Shell Patterns in Mollusks | 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 Research Article A Discrete-time Continuous-space Neural Model for Shell Patterns in Mollusks Rahnuma Islam, Bard Ermentrout, Sabrina Streipert This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7992040/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 We introduce a discrete-time continuous-space neural model to produce diverse shell structures and pigmentation patterns observed in aquatic mollusks. The model builds on an earlier neural model for shell patterns by incorporating the inhibition as a separate population and thus eliminates the need for a ''refractory" substance, yet is still able to produce many varieties of molluscan pigmentation patterns. The model utilizes a system of neural excitation and inhibition to conduct secretory activity and successfully replicates various natural shell patterns found in these organisms. Through an analysis of local stability around equilibria and an analysis of bifurcation, we establish the critical role of parameters involved in our system on the bifurcations in governing the emergence of spatial, temporal, and spatio-temporal patterns. Mathematical and Theoretical Biology Shell pattern formation Neural excitation-inhibition model Spatio-temporal dynamics Bifurcation analysis Molluscan pigmentation Full Text Additional Declarations The authors declare no competing interests. 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-7992040","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":537664344,"identity":"144055a7-3404-4e7d-900e-e53853355131","order_by":0,"name":"Rahnuma Islam","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAr0lEQVRIiWNgGAWjYFACxuYHCRUMPGA2D5Fa2gwenGHg4WEjXgsDg+TDNqBqorWYtx9uMEicd0fGXr6B8cHbNiK0yJxJbHiQuO0ZyGHMhnOJ0SLBkAi0ZdthkBY2aV6itPA/bJBInAPWwv6bOC0SiUAtDRBbmInU8rDNIOEY0C/HEpsl55wjymHpjx/+qLljz958+OCHN2VEaIGCA0DM2EC8eqiWUTAKRsEoGAU4AAAFkDK+QNPOWwAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-9420-5896","institution":"University of Pittsburgh","correspondingAuthor":true,"prefix":"","firstName":"Rahnuma","middleName":"","lastName":"Islam","suffix":""},{"id":537666427,"identity":"330bf8b4-f6fe-4c84-a1ca-169d9dd2bdad","order_by":1,"name":"Bard Ermentrout","email":"","orcid":"","institution":"University of Pittsburgh","correspondingAuthor":false,"prefix":"","firstName":"Bard","middleName":"","lastName":"Ermentrout","suffix":""},{"id":537666455,"identity":"94bf8267-b3bc-4c98-9cac-919d6dc68dec","order_by":2,"name":"Sabrina Streipert","email":"","orcid":"","institution":"University of Pittsburgh","correspondingAuthor":false,"prefix":"","firstName":"Sabrina","middleName":"","lastName":"Streipert","suffix":""}],"badges":[],"createdAt":"2025-10-30 18:35:05","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7992040/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7992040/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":94987996,"identity":"bafd2084-24b8-4d72-bdf8-5b091d95653d","added_by":"auto","created_at":"2025-11-03 07:02:43","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4025765,"visible":true,"origin":"","legend":"","description":"","filename":"MolluskShellPattern5.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7992040/v1_covered_6a60ff3e-031c-43da-8387-3cf640cbe3d7.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eA Discrete-time Continuous-space Neural Model\u003c/p\u003e\n\u003cp\u003efor Shell Patterns in Mollusks\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"University of Pittsburgh","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"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":"Shell pattern formation, Neural excitation-inhibition model, Spatio-temporal dynamics, Bifurcation analysis, Molluscan pigmentation","lastPublishedDoi":"10.21203/rs.3.rs-7992040/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7992040/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe introduce a discrete-time continuous-space neural model to produce diverse shell structures and pigmentation patterns observed in aquatic mollusks. 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