{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T16:05:26Z","timestamp":1791475526523,"version":"4.3.4"},"reference-count":125,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T00:00:00Z","timestamp":1767830400000},"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:p>While the role of topology is established in active matter systems, its importance in cardiac electrophysiology, particularly concerning common arrhythmias, warrants further emphasis. Atrial fibrillation (AF), atrial flutter (AFL), and focal atrial tachycardia (FAT) are the most prevalent arrhythmias impacting human health. This article proposes a framework conceptualizing these atrial rhythm disturbances through the lens of topological states and phase transitions, drawing inspiration from the Kosterlitz-Thouless (KT) transition. Central to this framework is the hypothesis that distinct arrhythmia patterns emerge as discrete topological states constrained by the fundamental requirement that the net topological charge (associated with electrical phase singularities or vortices) must sum to zero across the atrial tissue. Within this constrained topological perspective, AF, characterised by disorganised activity, is likened to the KT unbound vortex state, dominated by disorder with repetitive vortex regeneration and an exponential decay in spatial correlation. In contrast, AFL, with its organized regularity, resembles the KT bound vortex state, where vortex-antivortex pairs result in ordered activity. Finally, FAT and Sinus Rhythm are characterized as topologically vortex-free states exhibiting ordered planar wave conduction. Importantly, while the resulting topological states show clear analogies, the specific biophysical mechanisms driving vortex defect formation, interaction, and unbinding in cardiac tissue likely differ significantly from the thermal free-energy considerations governing the classic KT transition. This viewpoint frames the transition between arrhythmias as a change in the topological organization of atrial electrical activity, governed by charge conservation principles and cardiac-specific dynamics. This perspective may offer novel diagnostic and therapeutic avenues applicable to human cardiac mapping procedures.<\/jats:p>","DOI":"10.3389\/fnetp.2025.1710567","type":"journal-article","created":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T06:35:10Z","timestamp":1767854110000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["A topological hypothesis for atrial fibrilllation, atrial flutter and focal atrial tachycardia: comparison and contrast with Kosterlitz-Thouless physics"],"prefix":"10.3389","volume":"5","author":[{"given":"Anand Narayan","family":"Ganesan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pawel","family":"Kuklik","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stanley","family":"Nattel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1965","published-online":{"date-parts":[[2026,1,8]]},"reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1142\/8572","article-title":"40 Years of Berezinskii?Kosterlitz?Thouless Theory","year":"2012","journal-title":"WORLD Sci"},{"key":"B2","doi-asserted-by":"publisher","first-page":"28962","DOI":"10.1038\/s41598-024-73544-z","article-title":"Analysis of complex excitation patterns using Feynman-like diagrams","volume":"14","author":"Arno","year":"","journal-title":"Sci. Rep."},{"key":"B3","doi-asserted-by":"publisher","first-page":"606","DOI":"10.1161\/CIRCEP.109.910125","article-title":"Electropathological substrate of long-standing persistent atrial fibrillation in patients with structural heart disease: longitudinal dissociation","volume":"3","author":"Allessie","year":"2010","journal-title":"Circ. Arrhythm. Electrophysiol."},{"key":"B4","doi-asserted-by":"publisher","first-page":"128402","DOI":"10.1103\/5pgp-1wj6","article-title":"Strings, branes and twistons: topological analysis of phase defects in excitable media such as the heart","volume":"135","author":"Arno","year":"","journal-title":"Phys. Rev. Lett"},{"key":"B5","doi-asserted-by":"publisher","first-page":"2924","DOI":"10.1111\/j.1540-8159.1997.tb05461.x","article-title":"The upper link of human common atrial flutter circuit: definition by multiple endocardial recordings during entrainment","volume":"20","author":"Arribas","year":"1997","journal-title":"Pacing Clin. Electrophysiol."},{"key":"B6","doi-asserted-by":"publisher","first-page":"euad348","DOI":"10.1093\/europace\/euad348","article-title":"Incidence and patterns of atrial fibrillation after catheter ablation of typical atrial flutter-the FLUTFIB study","volume":"26","author":"Attanasio","year":"2024","journal-title":"Europace"},{"key":"B7","doi-asserted-by":"publisher","first-page":"ehad655.378","DOI":"10.1093\/eurheartj\/ehad655.378","article-title":"Catheter ablation for atypical atrial flutter: characterization and recurrence predictors","volume":"44","author":"Barradas","year":"2023","journal-title":"Eur. Heart J."},{"key":"B8","first-page":"493","article-title":"Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems","volume":"32","author":"Berezinskii","year":"1971","journal-title":"Sov. Phys. JETP"},{"key":"B9","doi-asserted-by":"publisher","first-page":"677","DOI":"10.1142\/s0218127498000474","article-title":"A three-dimensional autowave turbulence","volume":"8","author":"Biktashev","year":"1998","journal-title":"Int. J. Bifurcation Chaos"},{"key":"B10","doi-asserted-by":"publisher","first-page":"709","DOI":"10.1146\/annurev.fluid.32.1.709","article-title":"Recent developments in rayleigh-b\u00e9nard convection","volume":"32","author":"Bodenschatz","year":"2000","journal-title":"Annu. Rev. fluid Mech."},{"key":"B11","doi-asserted-by":"publisher","first-page":"1167","DOI":"10.1016\/0002-9149(80)90475-0","article-title":"Natural and evoked atrial flutter due to circus movement in dogs. Role of abnormal atrial pathways, slow conduction, nonuniform refractory period distribution and premature beats","volume":"45","author":"Boineau","year":"1980","journal-title":"Am. J. Cardiol."},{"key":"B12","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1109\/10.387197","article-title":"A technique for measurement of the extent of spatial organization of atrial activation during atrial fibrillation in the intact human heart","volume":"42","author":"Botteron","year":"1995","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"B13","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1161\/01.cir.93.3.513","article-title":"Quantitative assessment of the spatial organization of atrial fibrillation in the intact human heart","volume":"93","author":"Botteron","year":"1996","journal-title":"Circulation"},{"key":"B14","doi-asserted-by":"publisher","first-page":"010501","DOI":"10.1103\/PhysRevX.12.010501","article-title":"Symmetry, thermodynamics, and topology in active matter","volume":"12","author":"Bowick","year":"2022","journal-title":"Phys. Rev. X"},{"key":"B15","doi-asserted-by":"publisher","first-page":"051902","DOI":"10.1103\/PhysRevE.65.051902","article-title":"Considerations in phase plane analysis for nonstationary reentrant cardiac behavior","volume":"65","author":"Bray","year":"2002","journal-title":"Phys. Rev. E Stat. Nonlin Soft Matter Phys."},{"key":"B16","volume-title":"Josephson's clinical cardiac electrophysiology: techniques and interpretations","author":"Callans","year":"2020"},{"key":"B17","doi-asserted-by":"publisher","first-page":"1530","DOI":"10.1111\/jce.16700","article-title":"Superwide left atrial circumferential ablation for persistent atrial fibrillation: a randomized controlled trial","volume":"36","author":"Cao","year":"2025","journal-title":"J. Cardiovasc Electrophysiol."},{"key":"B18","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1016\/0167-2789(88)90065-6","article-title":"Spatio-temporal intermittency in coupled map lattices","volume":"32","author":"Chat\u00e9","year":"1988","journal-title":"Phys. D. Nonlinear Phenom."},{"key":"B19","first-page":"111","article-title":"Spatiotemporal intermittency","volume-title":"Turbulence: a tentative dictionary","author":"Chat\u00e9","year":"1994"},{"key":"B20","doi-asserted-by":"publisher","first-page":"1700","DOI":"10.1161\/01.cir.99.13.1700","article-title":"Right atrial flutter due to lower loop reentry: mechanism and anatomic substrates","volume":"99","author":"Cheng","year":"1999","journal-title":"Circulation"},{"key":"B21","volume-title":"Phase transitions and collective phenomena","author":"Cheung","year":"2011"},{"key":"B22","doi-asserted-by":"crossref","DOI":"10.5948\/UPO9780883859339","volume-title":"First concepts of topology: the geometry of mappings of segments, curves, circles, and disks","author":"Chinn","year":"1966"},{"key":"B23","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1038\/nature26001","article-title":"Electromechanical vortex filaments during cardiac fibrillation","volume":"555","author":"Christoph","year":"2018","journal-title":"Nature"},{"key":"B24","doi-asserted-by":"publisher","first-page":"286","DOI":"10.1103\/PhysRevLett.60.286","article-title":"Spatiotemporal intermittency in rayleigh-b\u00e9nard convection","volume":"60","author":"Ciliberto","year":"1988","journal-title":"Phys. Rev. Lett."},{"key":"B25","doi-asserted-by":"publisher","first-page":"043127","DOI":"10.1063\/1.3043805","article-title":"Vortex filament dynamics in computational models of ventricular fibrillation in the heart","volume":"18","author":"Clayton","year":"2008","journal-title":"Chaos"},{"key":"B26","doi-asserted-by":"publisher","first-page":"55","DOI":"10.15420\/aer.2017.5.2","article-title":"Atrial flutter, typical and atypical: a review","volume":"6","author":"Cosio","year":"2017","journal-title":"Arrhythm. Electrophysiol. Rev."},{"key":"B27","doi-asserted-by":"publisher","first-page":"965","DOI":"10.1111\/j.1540-8159.1996.tb03394.x","article-title":"Atrial flutter mapping and ablation II. Radiofrequency ablation of atrial flutter circuits","volume":"19","author":"Cosio","year":"","journal-title":"Pacing Clin. Electrophysiol."},{"key":"B28","doi-asserted-by":"publisher","first-page":"841","DOI":"10.1111\/j.1540-8159.1996.tb03368.x","article-title":"Atrial flutter mapping and ablation. I. Studying atrial flutter mechanisms by mapping and entrainment","volume":"19","author":"Cosio","year":"","journal-title":"Pacing Clin. Electrophysiol."},{"key":"B29","doi-asserted-by":"publisher","first-page":"1619","DOI":"10.1103\/PhysRevLett.62.1619","article-title":"Defect-mediated turbulence","volume":"62","author":"Coullet","year":"1989","journal-title":"Phys. Rev. Lett."},{"key":"B30","doi-asserted-by":"publisher","first-page":"3388","DOI":"10.1103\/PhysRevA.42.3388","article-title":"Transition to turbulence via spatiotemporal intermittency in one-dimensional rayleigh-B\u00e9nard convection","volume":"42","author":"Daviaud","year":"1990","journal-title":"Phys. Rev. A"},{"key":"B31","doi-asserted-by":"publisher","first-page":"208302","DOI":"10.1103\/PhysRevLett.101.208302","article-title":"Filament-induced surface spiral turbulence in three-dimensional excitable media","volume":"101","author":"Davidsen","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"B32","doi-asserted-by":"publisher","first-page":"056203","DOI":"10.1103\/PhysRevE.70.056203","article-title":"Topological constraints on spiral wave dynamics in spherical geometries with inhomogeneous excitability","volume":"70","author":"Davidsen","year":"2004","journal-title":"Phys. Rev. E"},{"key":"B33","doi-asserted-by":"publisher","first-page":"e003648","DOI":"10.1161\/CIRCEP.115.003648","article-title":"Direct proof of endo-epicardial asynchrony of the atrial wall during atrial fibrillation in humans","volume":"9","author":"de Groot","year":"2016","journal-title":"Circulation Arrhythmia Electrophysiol."},{"key":"B34","doi-asserted-by":"publisher","first-page":"1286","DOI":"10.1038\/s41591-025-03517-w","article-title":"Artificial intelligence for individualized treatment of persistent atrial fibrillation: a randomized controlled trial","volume":"31","author":"Deisenhofer","year":"2025","journal-title":"Nat. Med."},{"key":"B35","doi-asserted-by":"publisher","first-page":"e2117568119","DOI":"10.1073\/pnas.2117568119","article-title":"Terminating spiral waves with a single designed stimulus: teleportation as the mechanism for defibrillation","volume":"119","author":"DeTal","year":"2022","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"B36","doi-asserted-by":"publisher","first-page":"1786","DOI":"10.1016\/j.hrthm.2021.10.008","article-title":"Ganesan AN. M\/M\/Infinity birth-death processes \u2013 a quantitative representational framework to summarize and explain phase singularity and wavelet dynamics in atrial fibrillation","volume":"11","author":"Dharmaprani","year":"2021","journal-title":"Front. physiology"},{"key":"B37","doi-asserted-by":"publisher","first-page":"e007569","DOI":"10.1161\/CIRCEP.119.007569","article-title":"Renewal theory as a universal quantitative framework to characterize phase singularity regeneration in mammalian cardiac fibrillation","volume":"12","author":"Dharmaprani","year":"2019","journal-title":"Circ. Arrhythm. Electrophysiol."},{"key":"B38","doi-asserted-by":"publisher","first-page":"616866","DOI":"10.3389\/fphys.2020.616866","article-title":"M\/M\/Infinity birth-death processes \u2013 a quantitative representational framework to summarize and explain phase singularity and wavelet dynamics in atrial fibrillation","volume":"11","author":"Dharmaprani","year":"2021","journal-title":"Front. physiology"},{"key":"B39","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1016\/j.hrthm.2021.10.008","article-title":"A governing equation for rotor and wavelet number in human clinical ventricular fibrillation: implications for sudden cardiac death","volume":"19","author":"Dharmaprani","year":"2022","journal-title":"Heart rhythm."},{"key":"B40","doi-asserted-by":"publisher","first-page":"e012684","DOI":"10.1161\/circep.123.012684","article-title":"Observable atrial and ventricular fibrillation episode durations are conformant with a power law based on system size and spatial synchronization","volume":"17","author":"Dharmaprani","year":"2024","journal-title":"Circ. Arrhythm. Electrophysiol."},{"key":"B41","doi-asserted-by":"publisher","first-page":"044207","DOI":"10.1103\/PhysRevE.110.044207","article-title":"Influence of conduction heterogeneities on transient spatiotemporal chaos in cardiac excitable media","volume":"110","author":"Dix","year":"2024","journal-title":"Phys. Rev. E"},{"key":"B42","doi-asserted-by":"publisher","first-page":"e013102","DOI":"10.1161\/CIRCEP.124.013102","article-title":"Atrial topology for a unified understanding of typical and atypical flutter","volume":"17","author":"Duytschaever","year":"2024","journal-title":"Circulation Arrhythmia Electrophysiol."},{"key":"B43","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1016\/S0378-4371(97)00026-5","article-title":"Spiral defect chaos in Rayleigh-Be\u00b4nard convection: defect population statistics","volume":"239","author":"Ecke","year":"1997","journal-title":"Phys. A Stat. Mech. its Appl."},{"key":"B44","doi-asserted-by":"publisher","first-page":"1704","DOI":"10.1126\/science.269.5231.1704","article-title":"Excitation of spirals and chiral symmetry breaking in rayleigh-benard convection","volume":"269","author":"Ecke","year":"1995","journal-title":"Science"},{"key":"B45","doi-asserted-by":"publisher","first-page":"733","DOI":"10.1038\/35008013","article-title":"Mechanisms of extensive spatiotemporal chaos in rayleigh-benard convection","volume":"404","author":"Egolf","year":"2000","journal-title":"Nature"},{"key":"B46","doi-asserted-by":"publisher","first-page":"852","DOI":"10.1063\/1.1504242","article-title":"Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity","volume":"12","author":"Fenton","year":"2002","journal-title":"Chaos"},{"key":"B47","doi-asserted-by":"publisher","first-page":"1138","DOI":"10.1103\/PhysRevA.41.1138","article-title":"Statistical properties of defect-mediated turbulence","volume":"41","author":"Gil","year":"1990","journal-title":"Phys. Rev. A"},{"key":"B48","doi-asserted-by":"publisher","first-page":"174502","DOI":"10.1103\/PhysRevLett.87.174502","article-title":"Nonequilibrium defect-unbinding transition: defect trajectories and loop statistics","volume":"87","author":"Granzow","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"B49","doi-asserted-by":"publisher","first-page":"1222","DOI":"10.1126\/science.270.5239.1222","article-title":"Mechanisms of cardiac fibrillation","volume":"270","author":"Gray","year":"1995","journal-title":"Science"},{"key":"B50","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1038\/32164","article-title":"Spatial and temporal organization during cardiac fibrillation","volume":"392","author":"Gray","year":"1998","journal-title":"Nature"},{"key":"B51","doi-asserted-by":"publisher","first-page":"364","DOI":"10.1136\/heartjnl-2022-321729","article-title":"Cryoballoon pulmonary vein isolation as first-line treatment for typical atrial flutter","volume":"109","author":"Gupta","year":"2023","journal-title":"Heart"},{"key":"B52","doi-asserted-by":"publisher","first-page":"659","DOI":"10.1056\/NEJM199809033391003","article-title":"Spontaneous initiation of atrial fibrillation by ectopic beats originating in the pulmonary veins","volume":"339","author":"Haissaguerre","year":"1998","journal-title":"N. Engl. J. Med."},{"key":"B53","doi-asserted-by":"publisher","first-page":"1078","DOI":"10.1093\/cvr\/cvaa141","article-title":"Ventricular fibrillation mechanism and global fibrillatory organization are determined by gap junction coupling and fibrosis pattern","volume":"117","author":"Handa","year":"2020","journal-title":"Cardiovasc Res."},{"key":"B54","doi-asserted-by":"publisher","first-page":"1698","DOI":"10.1093\/europace\/euab078","article-title":"Size matters in atrial fibrillation: the underestimated importance of reduction of contiguous electrical mass underlying the effectiveness of catheter ablation","volume":"23","author":"Hartley","year":"2021","journal-title":"EP Eur."},{"key":"B55","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1146\/annurev-physiol-031720-085307","article-title":"Dynamics of atrial fibrillation mechanisms and comorbidities","volume":"83","author":"Heijman","year":"2021","journal-title":"Annu. Rev. Physiol."},{"key":"B56","volume-title":"Clinical arrhythmology and electrophysiology: a companion to Braunwald's heart disease","author":"Issa","year":"2009"},{"key":"B57","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1114\/1.1335538","article-title":"An experimentalist's approach to accurate localization of phase singularities during reentry","volume":"29","author":"Iyer","year":"2001","journal-title":"Ann. Biomed. Eng."},{"key":"B58","doi-asserted-by":"publisher","first-page":"2928","DOI":"10.1161\/01.CIR.101.25.2928","article-title":"Mapping and ablation of left atrial flutters","volume":"101","author":"Ja\u00efs","year":"2000","journal-title":"Circulation"},{"key":"B59","doi-asserted-by":"publisher","first-page":"480","DOI":"10.1111\/j.1540-8167.2008.01373.x","article-title":"A deductive mapping strategy for atrial tachycardia following atrial fibrillation ablation: importance of localized reentry","volume":"20","author":"Jais","year":"2009","journal-title":"J. Cardiovasc Electrophysiol."},{"key":"B60","doi-asserted-by":"publisher","first-page":"766","DOI":"10.1093\/cvr\/cvq364","article-title":"D\u00e9j\u00e0 vu in the theories of atrial fibrillation dynamics","volume":"89","author":"Jalife","year":"2011","journal-title":"Cardiovasc Res."},{"key":"B61","doi-asserted-by":"publisher","first-page":"032101","DOI":"10.1063\/5.0062095","article-title":"Understanding the origins of the basic equations of statistical fibrillatory dynamics","volume":"32","author":"Jenkins","year":"2022","journal-title":"Chaos"},{"key":"B62","doi-asserted-by":"publisher","first-page":"063136","DOI":"10.1063\/5.0141890","article-title":"Markov modeling of phase singularity interaction effects in human atrial and ventricular fibrillation","volume":"33","author":"Jenkins","year":"2023","journal-title":"Chaos"},{"key":"B63","doi-asserted-by":"publisher","first-page":"2692","DOI":"10.1111\/jce.16543","article-title":"Atypical atrial flutter: electrophysiological characterization and effective catheter ablation","volume":"36","author":"Johner","year":"2025","journal-title":"J. Cardiovasc Electrophysiol."},{"key":"B64","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1111\/j.1540-8167.1997.tb00775.x","article-title":"Electrocardiographic and electrophysiologic characterization of atypical atrial flutter in man: use of activation and entrainment mapping and implications for catheter ablation","volume":"8","author":"Kalman","year":"1997","journal-title":"J. Cardiovasc Electrophysiol."},{"key":"B65","doi-asserted-by":"publisher","first-page":"1033","DOI":"10.1143\/ptp.74.1033","article-title":"Spatiotemporal intermittency in coupled map lattices","volume":"74","author":"Kaneko","year":"1985","journal-title":"Prog. Theor. Phys."},{"key":"B66","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1016\/0375-9601(90)90534-U","article-title":"Supertransients, spatiotemporal intermittency and stability of fully developed spatiotemporal chaos","volume":"149","author":"Kaneko","year":"1990","journal-title":"Phys. Lett. A"},{"key":"B67","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1063\/1.166024","article-title":"Electrical alternans and spiral wave breakup in cardiac tissue","volume":"4","author":"Karma","year":"1994","journal-title":"Chaos"},{"key":"B68","doi-asserted-by":"publisher","first-page":"2486","DOI":"10.1172\/jci119791","article-title":"Spatiotemporal complexity of ventricular fibrillation revealed by tissue mass reduction in isolated swine right ventricle. Further evidence for the quasiperiodic route to chaos hypothesis","volume":"100","author":"Kim","year":"1997","journal-title":"J. Clin. Invest"},{"key":"B69","doi-asserted-by":"publisher","first-page":"1857","DOI":"10.1111\/jce.14298","article-title":"Persistent atrial fibrillation in the setting of pulmonary vein isolation-where to next?","volume":"31","author":"Kistler","year":"2019","journal-title":"J. Cardiovasc Electrophysiol."},{"key":"B70","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1001\/jama.2022.23722","article-title":"Effect of catheter ablation using pulmonary vein isolation with vs without posterior left atrial wall isolation on atrial arrhythmia recurrence in patients with persistent atrial fibrillation: the CAPLA randomized clinical trial","volume":"329","author":"Kistler","year":"2023","journal-title":"JAMA"},{"key":"B71","doi-asserted-by":"publisher","first-page":"e35","DOI":"10.1161\/01.RES.0000160709.49633.2b","article-title":"Mechanisms of atrial fibrillation termination by pure sodium channel blockade in an ionically-realistic mathematical model","volume":"96","author":"Kneller","year":"2005","journal-title":"Circ. Res."},{"key":"B72","doi-asserted-by":"publisher","first-page":"026001","DOI":"10.1088\/0034-4885\/79\/2\/026001","article-title":"Kosterlitz-thouless physics: a review of key issues","volume":"79","author":"Kosterlitz","year":"2016","journal-title":"Rep. Prog. Phys."},{"key":"B73","doi-asserted-by":"publisher","first-page":"040501","DOI":"10.1103\/revmodphys.89.040501","article-title":"Nobel lecture: topological defects and phase transitions","volume":"89","author":"Kosterlitz","year":"2017","journal-title":"Rev. Mod. Phys."},{"key":"B74","doi-asserted-by":"publisher","first-page":"1181","DOI":"10.1088\/0022-3719\/6\/7\/010","article-title":"Ordering, metastability and phase transitions in two-dimensional systems","volume":"6","author":"Kosterlitz","year":"1973","journal-title":"J. Phys. C Solid State Phys."},{"key":"B75","doi-asserted-by":"publisher","first-page":"775","DOI":"10.1016\/j.compbiomed.2010.07.005","article-title":"Spiral wave breakup in excitable media with an inhomogeneity of conduction anisotropy","volume":"40","author":"Kuklik","year":"2010","journal-title":"Comput. Biol. Med."},{"key":"B76","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1016\/0045-7825(91)90051-7","article-title":"Defect-mediated turbulence","volume":"89","author":"Lega","year":"1991","journal-title":"Comput. methods Appl. Mech. Eng."},{"key":"B77","doi-asserted-by":"publisher","first-page":"022215","DOI":"10.1103\/PhysRevE.98.022215","article-title":"Scaling behavior of the terminal transient phase","volume":"98","author":"Lilienkamp","year":"","journal-title":"Phys. Rev. E"},{"key":"B78","doi-asserted-by":"publisher","first-page":"094101","DOI":"10.1103\/PhysRevLett.120.094101","article-title":"Terminal transient phase of chaotic transients","volume":"120","author":"Lilienkamp","year":"","journal-title":"Phys. Rev. Lett."},{"key":"B79","doi-asserted-by":"publisher","first-page":"054101","DOI":"10.1103\/PhysRevLett.119.054101","article-title":"Features of chaotic transients in excitable media governed by spiral and scroll waves","volume":"119","author":"Lilienkamp","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"B80","doi-asserted-by":"publisher","first-page":"093936","DOI":"10.1063\/1.5003259","article-title":"Dynamical mechanism of atrial fibrillation: a topological approach","volume":"27","author":"Marcotte","year":"2017","journal-title":"Chaos An Interdiscip. J. Nonlinear Sci."},{"key":"B81","first-page":"183","article-title":"On the multiple wavelet hypothesis of atrial fibrillation","volume":"140","author":"Moe","year":"1962","journal-title":"Arch. Int. Pharmacodyn. Ther."},{"key":"B82","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1016\/0002-8703(59)90274-1","article-title":"Atrial fibrillation as a self-sustaining arrhythmia independent of focal discharge","volume":"58","author":"Moe","year":"1959","journal-title":"Am. Heart J."},{"key":"B83","doi-asserted-by":"publisher","first-page":"200","DOI":"10.1016\/0002-8703(64)90371-0","article-title":"A computer model of atrial fibrillation","volume":"67","author":"Moe","year":"1964","journal-title":"Am. Heart J."},{"key":"B84","doi-asserted-by":"publisher","first-page":"358","DOI":"10.1016\/S0167-2789(00)00087-7","article-title":"Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients","volume":"144","author":"Mormann","year":"2000","journal-title":"Phys. D. Nonlinear Phenom."},{"key":"B85","doi-asserted-by":"publisher","first-page":"2026","DOI":"10.1103\/PhysRevLett.71.2026","article-title":"Spiral defect chaos in large aspect ratio rayleigh-b\u00e9nard convection","volume":"71","author":"Morris","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"B86","first-page":"249","article-title":"Drowning by numbers: topology and physics in fluid dynamics","volume-title":"Imagine math 6: between culture and mathematics","author":"Mouchet","year":"2018"},{"key":"B87","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1038\/415219a","article-title":"New ideas about atrial fibrillation 50 years on","volume":"415","author":"Nattel","year":"2002","journal-title":"Nature"},{"key":"B88","doi-asserted-by":"publisher","first-page":"1396","DOI":"10.1161\/CIRCRESAHA.116.310489","article-title":"Controversies about atrial fibrillation mechanisms: aiming for order in chaos and whether it matters","volume":"120","author":"Nattel","year":"2017","journal-title":"Circ. Res."},{"key":"B89","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1038\/nrcardio.2017.37","article-title":"Demystifying rotors and their place in clinical translation of atrial fibrillation mechanisms","volume":"14","author":"Nattel","year":"2017","journal-title":"Nat. Rev. Cardiol."},{"key":"B90","volume-title":"Elements of phase transitions and critical phenomena","author":"Nishimori","year":"2011"},{"key":"B91","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1063\/1.166287","article-title":"Spiral breakup as a model of ventricular fibrillation","volume":"8","author":"Panfilov","year":"1998","journal-title":"Chaos"},{"key":"B92","doi-asserted-by":"publisher","first-page":"1315","DOI":"10.1098\/rsta.2001.0833","article-title":"Evolution of the multiple-wavelet hypothesis","volume":"359","author":"Panfilov","year":"2001","journal-title":"Philosophical Trans. Math. Phys. Eng. Sci."},{"key":"B93","doi-asserted-by":"publisher","first-page":"2738","DOI":"10.1103\/PhysRevLett.84.2738","article-title":"Topological constraint on scroll wave pinning","volume":"84","author":"Pertsov","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"B94","doi-asserted-by":"publisher","first-page":"101018","DOI":"10.1016\/j.ijcha.2022.101018","article-title":"Substrate-based ablation of atypical atrial flutter in patients with atrial cardiomyopathy","volume":"40","author":"Pott","year":"2022","journal-title":"Int. J. Cardiol. Heart Vasc."},{"key":"B95","doi-asserted-by":"publisher","first-page":"016212","DOI":"10.1103\/PhysRevE.79.016212","article-title":"Defect-mediated turbulence in the belousov-zhabotinsky reaction","volume":"79","author":"Qiao","year":"2009","journal-title":"Phys. Rev. E Stat. Nonlin Soft Matter Phys."},{"key":"B96","doi-asserted-by":"publisher","first-page":"755","DOI":"10.1114\/1.1289474","article-title":"Origins of spiral wave meander and breakup in a two-dimensional cardiac tissue model","volume":"28","author":"Qu","year":"2000","journal-title":"Ann. Biomed. Eng."},{"key":"B97","doi-asserted-by":"publisher","first-page":"660","DOI":"10.1002\/joa3.12363","article-title":"Prospective cross-sectional study using poisson renewal theory to study phase singularity formation and destruction rates in atrial fibrillation (RENEWAL-AF): study design","volume":"36","author":"Quah","year":"2020","journal-title":"J. Arrhythm."},{"key":"B98","doi-asserted-by":"publisher","first-page":"77","DOI":"10.15420\/aer.2020.42","article-title":"Reconceptualising atrial fibrillation using renewal theory: a novel approach to the assessment of atrial fibrillation dynamics","volume":"10","author":"Quah","year":"2021","journal-title":"Arrhythmia Electrophysiol. Res. Rev."},{"key":"B99","doi-asserted-by":"publisher","first-page":"809532","DOI":"10.3389\/fnetp.2022.809532","article-title":"Stochastic termination of spiral wave dynamics in cardiac tissue","volume":"2","author":"Rappel","year":"2022","journal-title":"Front. Netw. Physiology"},{"key":"B100","doi-asserted-by":"publisher","first-page":"e012041","DOI":"10.1161\/CIRCEP.123.012041","article-title":"Spatially conserved spiral wave activity during human atrial fibrillation","volume":"17","author":"Rappel","year":"2024","journal-title":"Circulation Arrhythmia Electrophysiol."},{"key":"B101","doi-asserted-by":"publisher","first-page":"1807","DOI":"10.1007\/s10840-023-01475-2","article-title":"Atypical atrial flutter catheter ablation in the era of high-density mapping","volume":"66","author":"Raymond-Paquin","year":"2023","journal-title":"J. Interventional Cardiac Electrophysiol."},{"key":"B102","doi-asserted-by":"publisher","first-page":"643","DOI":"10.1111\/j.1540-8159.2006.00413.x","article-title":"Focal atrial tachycardia I: clinical features, diagnosis, mechanisms, and anatomic location","volume":"29","author":"Roberts-Thomson","year":"2006","journal-title":"Pacing Clin. Electrophysiol."},{"key":"B103","doi-asserted-by":"publisher","first-page":"3484","DOI":"10.1161\/01.cir.96.10.3484","article-title":"Relationship between atrial fibrillation and typical atrial flutter in humans: activation sequence changes during spontaneous conversion","volume":"96","author":"Roithinger","year":"1997","journal-title":"Circulation"},{"key":"B104","doi-asserted-by":"publisher","first-page":"273","DOI":"10.15420\/aer.2019.08","article-title":"Challenges associated with interpreting mechanisms of AF","volume":"8","author":"Roney","year":"2020","journal-title":"Arrhythmia & Electrophysiol. Rev."},{"key":"B105","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1016\/j.jacep.2023.09.024","article-title":"Identification of 2 distinct boundaries distinguishes critical from noncritical isthmuses in ablating atypical atrial flutter","volume":"10","author":"Santucci","year":"2024","journal-title":"JACC Clin. Electrophysiol."},{"key":"B106","doi-asserted-by":"publisher","first-page":"1162","DOI":"10.1053\/euhj.2001.2658","article-title":"A classification of atrial flutter and regular atrial tachycardia according to electrophysiological mechanisms and anatomical bases; a statement from a joint expert group from the working Group of arrhythmias of the european society of Cardiology and the north American society of pacing and electrophysiology","volume":"22","author":"Saoudi","year":"2001","journal-title":"Eur. Heart J."},{"key":"B107","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1152\/physrev.00031.2009","article-title":"Pathophysiological mechanisms of atrial fibrillation: a translational appraisal","volume":"91","author":"Schotten","year":"2011","journal-title":"Physiol. Rev."},{"key":"B108","doi-asserted-by":"publisher","first-page":"011929","DOI":"10.1103\/PhysRevE.75.011929","article-title":"Spiral-wave dynamics depend sensitively on inhomogeneities in mathematical models of ventricular tissue","volume":"75","author":"Shajahan","year":"2007","journal-title":"Phys. Rev. E Stat. Nonlin Soft Matter Phys."},{"key":"B109","doi-asserted-by":"publisher","first-page":"e4738","DOI":"10.1371\/journal.pone.0004738","article-title":"Spiral-wave turbulence and its control in the presence of inhomogeneities in four mathematical models of cardiac tissue","volume":"4","author":"Shajahan","year":"2009","journal-title":"PLoS One"},{"key":"B110","doi-asserted-by":"publisher","first-page":"108002","DOI":"10.1103\/PhysRevLett.121.108002","article-title":"Defect unbinding in active nematics","volume":"121","author":"Shankar","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"B111","doi-asserted-by":"publisher","first-page":"380","DOI":"10.1038\/s42254-022-00445-3","article-title":"Topological active matter","volume":"4","author":"Shankar","year":"2022","journal-title":"Nat. Rev. Phys."},{"key":"B112","doi-asserted-by":"publisher","first-page":"056211","DOI":"10.1103\/PhysRevE.80.056211","article-title":"Transient spatiotemporal chaos is extensive in three reaction-diffusion networks","volume":"80","author":"Stahlke","year":"2009","journal-title":"Phys. Rev. E Stat. Nonlin Soft Matter Phys."},{"key":"B113","doi-asserted-by":"publisher","first-page":"062902","DOI":"10.1103\/PhysRevE.68.062902","article-title":"Influence of nonexcitable cells on spiral breakup in two-dimensional and three-dimensional excitable media","volume":"68","author":"Ten Tusscher","year":"2003","journal-title":"Phys. Rev. E"},{"key":"B114","doi-asserted-by":"publisher","first-page":"H1088","DOI":"10.1152\/ajpheart.00109.2006","article-title":"Alternans and spiral breakup in a human ventricular tissue model","volume":"291","author":"Ten Tusscher","year":"2006","journal-title":"Am. J. physiology Heart circulatory physiology"},{"key":"B115","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1016\/0167-2789(88)90062-0","article-title":"Singular perturbation theory of traveling waves in excitable media (a review)","volume":"32","author":"Tyson","year":"1988","journal-title":"Phys. D. Nonlinear Phenom."},{"key":"B116","doi-asserted-by":"publisher","first-page":"e84595","DOI":"10.1371\/journal.pone.0084595","article-title":"A study of early afterdepolarizations in a model for human ventricular tissue","volume":"9","author":"Vandersickel","year":"2014","journal-title":"PLoS One"},{"key":"B117","doi-asserted-by":"publisher","first-page":"euad122.078","DOI":"10.1093\/europace\/euad122.078","article-title":"Unique topological classification of complex reentrant atrial tachycardias enables optimal ablation strategy","volume":"25","author":"Vandersickel","year":"2023","journal-title":"EP Eur."},{"key":"B118","doi-asserted-by":"publisher","first-page":"1586","DOI":"10.1093\/eurheartj\/ehae032","article-title":"Impact of topology on the number of loops during macro-re-entrant atrial tachycardia","volume":"45","author":"Vandersickel","year":"2024","journal-title":"Eur. Heart J."},{"key":"B119","doi-asserted-by":"publisher","first-page":"737","DOI":"10.1161\/01.cir.56.5.737","article-title":"Entrainment and interruption of atrial flutter with atrial pacing: studies in man following open heart surgery","volume":"56","author":"Waldo","year":"1977","journal-title":"Circulation"},{"key":"B120","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1016\/S0022-5193(89)80200-0","article-title":"Electrical instability in cardiac muscle: phase singularities and rotors","volume":"138","author":"Winfree","year":"1989","journal-title":"J. Theor. Biol."},{"key":"B121","doi-asserted-by":"publisher","first-page":"1003","DOI":"10.1126\/science.7973648","article-title":"Electrical turbulence in three-dimensional heart muscle","volume":"266","author":"Winfree","year":"1994","journal-title":"Science"},{"key":"B122","doi-asserted-by":"publisher","first-page":"148302","DOI":"10.1103\/PhysRevLett.91.148302","article-title":"Mechanism for spiral wave breakup in excitable and oscillatory media","volume":"91","author":"Yang","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"B123","doi-asserted-by":"publisher","first-page":"e010081","DOI":"10.1161\/CIRCEP.121.010081","article-title":"Propagation vectors facilitate differentiation between conduction block, slow conduction, and wavefront collision","volume":"14","author":"Yavin","year":"2021","journal-title":"Circulation Arrhythmia Electrophysiol."},{"key":"B124","doi-asserted-by":"publisher","first-page":"168302","DOI":"10.1103\/PhysRevLett.92.168302","article-title":"Intermittent self-organization of scroll wave turbulence in three-dimensional excitable media","volume":"92","author":"Zaritski","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"B125","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/0002-9149(75)90865-6","article-title":"Termination of ventricular fibrillation in dogs by depolarizing a critical amount of myocardium","volume":"36","author":"Zipes","year":"1975","journal-title":"Am. J. Cardiol."}],"container-title":["Frontiers in Network Physiology"],"original-title":[],"link":[{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fnetp.2025.1710567\/full","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T06:35:13Z","timestamp":1767854113000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fnetp.2025.1710567\/full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,8]]},"references-count":125,"alternative-id":["10.3389\/fnetp.2025.1710567"],"URL":"https:\/\/doi.org\/10.3389\/fnetp.2025.1710567","relation":{},"ISSN":["2674-0109"],"issn-type":[{"value":"2674-0109","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,8]]},"article-number":"1710567"}}