Evidence for Hill-function of Oxyhemoglobin Saturation to quantify Stevens Law of Subjective Cognitive Load

Evidence for Hill-function of Oxyhemoglobin Saturation to quantify Stevens Law of Subjective Cognitive Load | 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 Article Evidence for Hill-function of Oxyhemoglobin Saturation to quantify Stevens Law of Subjective Cognitive Load Norbert Fürstenau This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8895239/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 6 You are reading this latest preprint version Abstract I investigate here the biophysical energetic foundation of a logistic resource limitation model of subjective mental workload (MWL) and behavioral objective task load. It is based on a previously published initial model version with experimental evidence by means of human-in-the-loop air traffic control simulation with domain experts. Subjective “instantaneous self-assessment” (ISA) data of cognitive load as function of objective task load had been formalized by a two parametric power law through transformation of variables. I show that it is formally equivalent to the classical Hill function of oxyhemoglobin (HbO 2 ) saturation as dependent on O 2 -partial pressure, represented by an inverse hyperbola. For quantitative comparison of variables and parameter estimates the cognitive Hill function is transformed into a psychophysical Stevens power law with exponent of the order of 1. Assuming oxygen partial pressure of ca.100 mmHg, sufficient for high arterial HbO 2 saturation, Hill’s physiological parameter estimates turn out to agree with the cognitive ones within order of magnitude, and even better for Stevens’ exponent. A derived generalized linear relationship between cognitive and physiological Stevens laws suggests the derivation of direct evidence for the hypothesized energetic resource limitation of MWL by means of functional near-infrared spectroscopy. Physical sciences/Engineering Physical sciences/Mathematics and computing Biological sciences/Neuroscience Physical sciences/Physics Biological sciences/Physiology Biological sciences/Psychology Social science/Psychology subjective workload cognitive resource limit energy resources logistic resource limitation Hill function Stevens law Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 15 May, 2026 Reviewers invited by journal 06 May, 2026 Editor assigned by journal 05 May, 2026 Editor invited by journal 02 Mar, 2026 Submission checks completed at journal 28 Feb, 2026 First submitted to journal 28 Feb, 2026 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-8895239","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":640224732,"identity":"bfb9bcbe-7f5b-4926-bc1b-36f7a1405b7e","order_by":0,"name":"Norbert Fürstenau","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCElEQVRIie2PsUoDQRCG5zi4awZsFyR5hg1bJMKhr7LLwlUbLXyBq6wiaU8EfYWIYBVwQMg1eYAtT4StPWxOCGhyYgrxFuws9iuG4Wc+fgYgEPifJN1Mi30QE6sBouLX673CAWm3yy6QTP5RQe5Vxumlq5vlBpDpVfPeZqcHpWmsWsJg3qMczarxqHTbFpbra5T5ObPT+4l0IK56arjNk0MkDifMiBjkkyrs9IFJArUgj7KhXcvZW9TKD3VrjeuUR58CnWJiQElqYU3y1dLzC1+v0tGMBCI6EWOu1d3aiYkkJsq+luoiqVsaDjHVL1GbHaubSj/bV8oG8573v8GfAfPfBwKBQMDLJ36bXO0h3TqWAAAAAElFTkSuQmCC","orcid":"","institution":"German Aerospace Center (DLR), Institute of Flight Guidance","correspondingAuthor":true,"prefix":"","firstName":"Norbert","middleName":"","lastName":"Fürstenau","suffix":""}],"badges":[],"createdAt":"2026-02-16 17:38:56","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8895239/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8895239/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109296971,"identity":"a14370e6-a463-496a-8a34-cb57deb2194c","added_by":"auto","created_at":"2026-05-15 08:52:20","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":892685,"visible":true,"origin":"","legend":"","description":"","filename":"BloodoxygenationCognitiveHillFunctionsubjectivementalloadpaperv53.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8895239/v1_covered_894696f5-6bda-4bdc-8e87-fc44e23ee367.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Evidence for Hill-function of Oxyhemoglobin Saturation to quantify Stevens Law of Subjective Cognitive Load","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"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":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"subjective workload, cognitive resource limit, energy resources, logistic resource limitation, Hill function, Stevens law","lastPublishedDoi":"10.21203/rs.3.rs-8895239/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8895239/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eI investigate here the biophysical energetic foundation of a logistic resource limitation model of subjective mental workload (MWL) and behavioral objective task load. It is based on a previously published initial model version with experimental evidence by means of human-in-the-loop air traffic control simulation with domain experts. Subjective \u0026ldquo;instantaneous self-assessment\u0026rdquo; (ISA) data of cognitive load as function of objective task load had been formalized by a two parametric power law through transformation of variables. I show that it is formally equivalent to the classical Hill function of oxyhemoglobin (HbO\u003csub\u003e2\u003c/sub\u003e) saturation as dependent on O\u003csub\u003e2\u003c/sub\u003e-partial pressure, represented by an inverse hyperbola. For quantitative comparison of variables and parameter estimates the cognitive Hill function is transformed into a psychophysical Stevens power law with exponent of the order of 1. Assuming oxygen partial pressure of ca.100 mmHg, sufficient for high arterial HbO\u003csub\u003e2\u003c/sub\u003e saturation, Hill\u0026rsquo;s physiological parameter estimates turn out to agree with the cognitive ones within order of magnitude, and even better for Stevens\u0026rsquo; exponent. A derived generalized linear relationship between cognitive and physiological Stevens laws suggests the derivation of direct evidence for the hypothesized energetic resource limitation of MWL by means of functional near-infrared spectroscopy.\u003c/p\u003e","manuscriptTitle":"Evidence for Hill-function of Oxyhemoglobin Saturation to quantify Stevens Law of Subjective Cognitive Load","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-15 03:32:30","doi":"10.21203/rs.3.rs-8895239/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"87904895883118442071979644367408794688","date":"2026-05-15T12:59:10+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-05-06T07:28:51+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-05-05T10:53:31+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2026-03-02T12:27:46+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-02-28T13:45:23+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2026-02-28T12:56:05+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"4aa44752-807b-4c51-9639-240049771483","owner":[],"postedDate":"May 15th, 2026","published":true,"recentEditorialEvents":[{"type":"reviewerAgreed","content":"87904895883118442071979644367408794688","date":"2026-05-15T12:59:10+00:00","index":92,"fulltext":""},{"type":"reviewersInvited","content":"11","date":"2026-05-06T07:28:51+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-05-05T10:53:31+00:00","index":"","fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":68136637,"name":"Physical sciences/Engineering"},{"id":68136638,"name":"Physical sciences/Mathematics and computing"},{"id":68136639,"name":"Biological sciences/Neuroscience"},{"id":68136640,"name":"Physical sciences/Physics"},{"id":68136641,"name":"Biological sciences/Physiology"},{"id":68136642,"name":"Biological sciences/Psychology"},{"id":68136643,"name":"Social science/Psychology"}],"tags":[],"updatedAt":"2026-05-15T03:32:30+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-15 03:32:30","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8895239","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8895239","identity":"rs-8895239","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}