The Graded–Spike Continuum Law: A Unified Framework Linking Discrete Spiking and Continuous Learning Dynamics

The Graded–Spike Continuum Law: A Unified Framework Linking Discrete Spiking and Continuous Learning 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 Research Article The Graded–Spike Continuum Law: A Unified Framework Linking Discrete Spiking and Continuous Learning Dynamics Atulya Thakur This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8010634/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 The investigation originates from an apparent contradiction between the discrete nature of neuronal spiking and the continuity required for gradient-based learning. Drawing an analogy to color synthesis—where a limited set of base channels generates a continuous visual spectrum—the study postulates that spike intensity, rather than spike count, can convey continuous information while preserving event-driven efficiency. From this analogy emerges the Graded–Spike Continuum Law (GSCL), which formalizes spiking activity as a smooth, amplitude-encoded surrogate of binary firing. The law demonstrates that introducing graded amplitudes transforms non-differentiable temporal dynamics into a mathematically continuous system, with provable convergence properties and an O(1/κ)bound on surrogate bias. Verification proceeds through symbolic derivation, and analytical proof, each confirming the theoretical predictions of stability and energy efficiency. The framework therefore establishes a reproducible bridge between biological realism and tractable optimization, extending the interpretive power of analog reasoning into formal neuromorphic theory. Spiking neural networks Surrogate Gradients Graded Activation Analog–Digital Continuum Energy Efficiency Neuromorphic Learning Human–AI Research Integration 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-8010634","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":538669507,"identity":"ab9b1297-b545-4ff6-9f11-bc4d5dc969e7","order_by":0,"name":"Atulya 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