The Systemic Coherence Function: Gradient-Flow Modeling of Coherence v2

The Systemic Coherence Function: Gradient-Flow Modeling of Coherence v2 | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 8 December 2025 V3 Latest version Share on The Systemic Coherence Function: Gradient-Flow Modeling of Coherence v2 Author : Peter Brunzelle 0009-0005-7109-6745 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176463655.51111306/v3 773 views 261 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract Coherence—the emergence of stable, aligned, and integrative structure—appears across physical, biological, neural, cognitive, and social systems. Yet most disciplines treat it indirectly through domain-specific constructs such as order parameters, synchronization indices, pattern-forming instabilities, or consensus variables. This work introduces the Systemic Coherence Function (SCF) as a phenomenological gradient-flow framework that treats coherence itself as the modeled field. We define a scalar coherence field Φs(r,t) governed by a gradient flow on a coherence-tension functional F[Φs]. The resulting SCF equation is a driven nonlinear parabolic partial differential equation that combines diffusion, local restoring forces, external drive fields, and polarization-like transport in a substrate-generalizable form. We provide operational definitions of Φs across domains, sketch a generic microscopic-to-macroscopic route to an SCF-type equation, and then develop a concrete neural case study: starting from a spatially extended Kuramoto model, we apply an Ott–Antonsen reduction, write z = ΦseiΨ for the local complex order parameter, and explicitly obtain an approximate evolution equation for ∂tΦs. Near the threshold for synchrony, this yields an SCF equation with a quartic potential whose quadratic coefficient α(K,∆) recovers the classical Kuramoto synchrony threshold, with Kc = 2∆/J0 in SCF parameters. We further present numerical simulations of one-dimensional driven SCF systems with double-well potentials. A well-mixed model exhibits multistability and hysteresis in Φs as a function of the drive amplitude. A spatially extended model with localized Gaussian drive shows coherence-front propagation from a seeded region, and the numerically computed coherence-tension functional F[Φs(t)] decays monotonically in time, confirming its Lyapunov character. We outline how SCF predictions can be compared directly to spatial Kuramoto simulations by matching front speeds and coherence profiles. Throughout, SCF is presented as a phenomenological modeling program for coherence dynamics—a canonical parametrization of coherence-seeking flows that is mathematically standard but conceptually unifying. We discuss empirical pathways for testing SCF in neural and other systems, outline multi-field and cross-scale extensions, and highlight limitations and open questions for future work in coherence measurement and coherence engineering. Supplementary Material File (the_systemic_coherence_function__gradient_flow_modeling_of_coherence (2).pdf) Download 543.69 KB Information & Authors Information Version history V1 Version 1 02 December 2025 V2 Version 2 03 December 2025 V3 Version 3 08 December 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords coherence dynamics cross-scale dynamics gradient flow systems multi-agent systems neural-field theory nonlinear partial differential equations (pdes) pattern formation synchronization and alignment systemic coherence function (scf) Authors Affiliations Peter Brunzelle 0009-0005-7109-6745 [email protected] Convergence Sciences Initiative View all articles by this author Metrics & Citations Metrics Article Usage 773 views 261 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Peter Brunzelle. The Systemic Coherence Function: Gradient-Flow Modeling of Coherence v2. Authorea . 08 December 2025. DOI: https://doi.org/10.22541/au.176463655.51111306/v3 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . Format Please select one from the list RIS (ProCite, Reference Manager) EndNote BibTex Medlars RefWorks Direct import Tips for downloading citations document.getElementById('citMgrHelpLink').addEventListener('click', function() { popupHelp(this.href); return false; }); $(".js__slcInclude").on("change", function(e){ if ($(this).val() == 'refworks') $('#direct').prop("checked", false); $('#direct').prop("disabled", ($(this).val() == 'refworks')); }); View Options View options PDF View PDF Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. 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