{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,4]],"date-time":"2026-08-04T11:46:57Z","timestamp":1785844017287,"version":"3.56.0"},"reference-count":24,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T00:00:00Z","timestamp":1783555200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Netw. Physiol."],"abstract":"<jats:sec>\n                    <jats:title>Background<\/jats:title>\n                    <jats:p>\n                      This companion empirical study established fundamental heterogeneity in pre-arrhythmic dynamical pathways across nine PhysioNet cohorts (N &amp;gt; 1,500 records): 93%\u201398% of events do not follow the canonical suppress-to-late-rise trajectory, Phase V (scaling manifold collapse, R\n                      <jats:sup>2<\/jats:sup>\n                      &amp;lt; 0.93) is absent in healthy subjects and increases monotonically with disease severity, and healthy dynamics are predominantly supercritical (CHI = 2 (\u03b1\n                      <jats:sub>1<\/jats:sub>\n                      \u2212 \u03b1\n                      <jats:sub>2<\/jats:sub>\n                      ) = +0.334). Although these empirical markers characterize the dynamical state, they lacked formal theoretical grounding. The present study addresses this gap by constructing and validating a minimal stochastic dynamical model of cardiovascular criticality.\n                    <\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Methods<\/jats:title>\n                    <jats:p>\n                      A coupled Langevin system, grounded in Haken\u2019s synergetic order-parameter formalism, is reduced to d\u03c8\/dt = A (R\n                      <jats:sup>2<\/jats:sup>\n                      )\u03c8 \u2212 B\u03c8 + \u03b5(\u03c8)\u03b7(t) via adiabatic elimination. Parameters estimated by Fokker\u2013Planck KL-divergence minimization, cross-cohort regression, and Kramers calibration. Euler\u2013Maruyama and Heun\/Stratonovich simulations (N = 50,000; SEED = 42) provide large-noise ground-truth benchmarks.\n                    <\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Results<\/jats:title>\n                    <jats:p>\n                      Rc = 0.991, a\n                      <jats:sub>0<\/jats:sub>\n                      = 20.5, D = 0.06 \u00b1 0.02. A five-condition P_self benchmark (Kramers additive\/EM sim\/Kramers multiplicative\/Heun sim\/empirical: 0.42|0.49|0.65|0.50|0.70 for Suppress; 0.50|0.41|0.63|0.43|0.58 for Late-rise) establishes that multiplicative Kramers overestimates simulation in the large-noise regime, quantifying the boundary conditions of Kramers-based approximations. Individual-level estimation (78 patients and 135 mr records) yields ICC = 0.37\u20130.41, demonstrating real between-patient heterogeneity with dominant within-patient non-stationarity. This ICC range is comparable to that of published values for DFA \u03b1\n                      <jats:sub>1<\/jats:sub>\n                      (ICC \u22480.40\u20130.60; Penttil\u00e4 et al., 2001; Aubert et al., 2003) and reflects the expected property of a dynamical rather than trait marker: CHI measures the current phase of the regulatory system, which varies within individuals across states, activity levels, and time of day. Low ICC is, therefore, not a limitation of CHI as a dynamic biomarker but a consequence of measuring a quantity that changes with the physiological context\u2014analogous to blood pressure ICC being lower than body height ICC. High test\u2013retest reliability would indicate that CHI is insensitive to physiological dynamics, which would contradict its role as an order parameter. CHI follows Student\u2019s-t (\u03bd = 12.4; \u0394AIC = 74.4 vs. Gaussian). Cross-cohort validation supports P1 (Spearman \u03c1 = 0.867, p = 0.001, N = 10; extended to N = 13 with nsrdb and Fantasia external cohorts, \u03c1 = 0.989; p &amp;lt; 0.0001) and P3 (\u03b2 = 1.255 [BCa 95%CI: 0.69\u20132.89], permutation p = 0.014, directionally consistent with pitchfork prediction; exploratory at N = 6). Within-patient temporal prediction (N = 78; 7,555 rolling windows; Cox model) yields HR_CHI = 0.61 (p &amp;lt; 0.001) and HR_PV = 2.31 (p = 0.001); temporal receiver operating characteristic (ROC) area under the curve (AUC) increases from 0.71 to 0.86 as VT\/VF onset approaches; Harrell\u2019s C = 0.78 with CHI trend \u0394CHI included.\n                    <\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Conclusion<\/jats:title>\n                    <jats:p>This study provides the first formal dynamical framework for the ECSoC empirical markers, deriving the pitchfork bifurcation structure of cardiovascular criticality from the first principles approach and validating it against cross-cohort data and stochastic simulations. The model is structurally validated across analytical, simulation, and empirical domains, with quantitative constraints in the large-noise regime delineating the boundary conditions for Kramers-based approximations. Six falsifiable predictions (P1\u2013P6) define the roadmap for prospective quantitative validation.<\/jats:p>\n                  <\/jats:sec>","DOI":"10.3389\/fnetp.2026.1865256","type":"journal-article","created":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T04:21:10Z","timestamp":1783570870000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Empirically constrained order parameter dynamics in cardiovascular criticality: a synergetic Langevin framework for arrhythmic transitions with cross-cohort parameter estimation and Kramers escape-time validation"],"prefix":"10.3389","volume":"6","author":[{"given":"Hiroyuki","family":"Okabe","sequence":"first","affiliation":[{"name":"Independent Researcher","place":["Tokyo, Japan"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1965","published-online":{"date-parts":[[2026,7,9]]},"reference":[{"key":"B21","doi-asserted-by":"publisher","first-page":"1674","DOI":"10.1016\/S0140-6736(06)68735-7","article-title":"Deceleration capacity of heart rate as a predictor of mortality after myocardial infarction: cohort study","volume":"367","author":"Bauer","year":"2006","journal-title":"Lancet"},{"key":"B23","doi-asserted-by":"publisher","first-page":"456","DOI":"10.1016\/j.jacc.2005.11.026","article-title":"Microvolt T-wave alternans and the risk of death or sustained ventricular arrhythmias in patients with left ventricular dysfunction","volume":"47","author":"Bloomfield","year":"2006","journal-title":"J. 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