{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T04:20:48Z","timestamp":1762057248939,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,6,23]],"date-time":"2022-06-23T00:00:00Z","timestamp":1655942400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Chiesi Pharmaceutical S.p.A., Parma, Italy"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Biomolecules"],"abstract":"<jats:p>Cardiac EC coupling is triggered by rhythmic depolarizing current fronts originating from the sino-atrial node, and the way variability in rhythm is associated with variability in action potential duration (APD) and, in turn, in the variability of calcium transient amplitude (CTA) and contraction is a key determinant of beating stability. Sinusoidal-varying pacing rate is adopted here in order to establish whether APD and CTA oscillations, elicited in a human ventricular AP model (OR) under oscillatory pacing, are consistent with the dynamics of two coupled harmonic oscillators, e.g., a two-degree-of-freedom system of mass and springs (MS model). I show evidence that this is the case, and that the MS model, preliminarily fitted to OR behavior, retains key features of the physiological system, such as the dependence of APD and CTA oscillation amplitudes from average value and from beat-to-beat changes in pacing rate, and the phase relationship between them. The bi-directionality of coupling between APD and CTA makes it difficult to discriminate which one leads EC coupling dynamics under variable pacing. The MS model suggests that the calcium cycling, with its greater inertia chiefly determined by the SR calcium release, is the leading mechanism. I propose the present approach to also be relevant at the whole organ level, where the need of compact representations of electromechanical interaction, particularly in clinical practice, remains urgent.<\/jats:p>","DOI":"10.3390\/biom12070873","type":"journal-article","created":{"date-parts":[[2022,6,23]],"date-time":"2022-06-23T21:25:00Z","timestamp":1656019500000},"page":"873","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Ventricular Repolarization and Calcium Transient Show Resonant Behavior under Oscillatory Pacing Rate"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8808-2239","authenticated-orcid":false,"given":"Massimiliano","family":"Zaniboni","sequence":"first","affiliation":[{"name":"Department of Chemistry, Life Sciences and Environmental Sustainability, University of Parma, Parco Area delle Scienze 11\/A, 43124 Parma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Murray, J.D. (1993). Biological oscillators and switches. Mathematical Biology, Springer. [2nd ed.].","DOI":"10.1007\/978-3-662-08542-4_6"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1242\/jeb.81.1.281","article-title":"An atlas of cellular oscillators","volume":"81","author":"Rapp","year":"1979","journal-title":"J. Exp. Biol."},{"key":"ref_3","first-page":"198","article-title":"Cardiac excitation-contraction coupling","volume":"415","author":"Bers","year":"2002","journal-title":"Rev. Nat."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Zaniboni, M., Cacciani, F., and Lux, R.L. (2014). Beat-to-beat cycle length variability of spontaneously beating guinea pig sinoatrial cells: Relative contributions of the membrane and calcium clocks. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0100242"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1210","DOI":"10.1016\/j.hrthm.2014.03.049","article-title":"Synchronization of sinoatrial node pacemaker cell clocks and its autonomic modulation impart complexity to heart beating intervals","volume":"11","author":"Yaniv","year":"2014","journal-title":"Heart Rhythm"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"S2","DOI":"10.1111\/j.1540-8167.2006.00378.x","article-title":"Action potential duration, rate of stimulation, and intracellular sodium","volume":"17","author":"Carmeliet","year":"2006","journal-title":"J. Cardiovasc. Electrophysiol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1507","DOI":"10.1529\/biophysj.104.043299","article-title":"A Computational Model of the Human Left-Ventricular Epicardial Myocyte","volume":"87","author":"Iyer","year":"2004","journal-title":"Biophys. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1113\/jphysiol.1978.sp012576","article-title":"A study of the factors responsible for rate-dependent shortening of the action potential in mammalian ventricular muscle","volume":"285","author":"Boyett","year":"1978","journal-title":"J. Physiol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"S140","DOI":"10.1046\/j.1540.8167.90303.x","article-title":"The electrical restitution curve revisited: Steep or flat slope-which is better?","volume":"14","author":"Franz","year":"2003","journal-title":"J. Cardiovasc. Electrophysiol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1113\/expphysiol.2007.037986","article-title":"Temporal variability of repolarization in rat ventricular myocytes paced with time-varying frequencies","volume":"92","author":"Zaniboni","year":"2007","journal-title":"Exp. Physiol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"L33","DOI":"10.1529\/biophysj.106.100982","article-title":"Inferring the cellular origin of voltage and calcium alternans from the spatial scales of phase reversal during discordant alternans","volume":"92","author":"Sato","year":"2007","journal-title":"Biophys. J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1016\/j.pbiomolbio.2008.02.014","article-title":"Cellular mechanisms of arrhythmogenic cardiac alternans","volume":"97","author":"Laurita","year":"2008","journal-title":"Prog. Biophys. Mol. Biol."},{"key":"ref_13","first-page":"1","article-title":"Impaired Sarcoplasmic Reticulum Calcium Uptake and Release Promote Electromechanically and Spatially Discordant Alternans: A Computational Study","volume":"10","author":"Weinberg","year":"2016","journal-title":"Clin. Med. Insights Cardiol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2735","DOI":"10.1529\/biophysj.106.083865","article-title":"Action potential duration restitution portraits of mammalian ventricular myocytes: Role of calcium current","volume":"91","author":"Tolkacheva","year":"2006","journal-title":"Biophys. J."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1016\/j.bpj.2008.09.025","article-title":"The transfer functions of cardiac tissue during stochastic pacing","volume":"96","author":"Kucera","year":"2009","journal-title":"Biophys. J."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"O\u2019Hara, T.O., Virag, L., Varro, A., and Rudy, Y. (2011). Simulation of the Undiseased Human Cardiac Ventricular Action Potential: Model Formulation and Experimental Validation. PLoS Comput. Biol., 7.","DOI":"10.1371\/journal.pcbi.1002061"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Weinberg, S.H., and Tung, L. (2012). Oscillation in cycle length induces transient discordant and steady-state concordant alternans in the heart. PLoS ONE, 7.","DOI":"10.1371\/journal.pone.0040477"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1136\/adc.54.6.432","article-title":"Sinusoidal heart rate rhythms in severe neonatal hypoxia","volume":"54","author":"Reid","year":"1979","journal-title":"Arch. Dis. Child."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1161\/CIRCHEARTFAILURE.108.806588","article-title":"Novel use of cardiac pacemakers in heart failure to dynamically manipulate the respiratory system through algorithmic changes in cardiac output","volume":"2","author":"Baruah","year":"2009","journal-title":"Circ. Heart Fail."},{"key":"ref_20","first-page":"3017","article-title":"CellML and associated tools and techniques","volume":"366","author":"Garny","year":"2008","journal-title":"Philos. Trans. A Math. Phys. Eng. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"H1088","DOI":"10.1152\/ajpheart.00109.2006","article-title":"Alternans and spiral breakup in a human ventricular tissue model","volume":"291","author":"Panfilov","year":"2006","journal-title":"Am. J. Physiol.-Heart Circ. Physiol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2382","DOI":"10.1016\/j.bpj.2019.08.020","article-title":"Restitution and Stability of Human Ventricular Action Potential at High and Variable Pacing Rate","volume":"117","author":"Zaniboni","year":"2019","journal-title":"Biophys. J."},{"key":"ref_23","unstructured":"Rao, S.S. (2017). Two-degree-of-freedom systems. Mechanical Vibrations, Pearson Editions. [6th ed.]."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Zaniboni, M. (2018). Short-term action potential memory and electrical restitution: A cellular computational study on the stability of cardiac repolarization under dynamic pacing. PLoS ONE, 13.","DOI":"10.1371\/journal.pone.0193416"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1594","DOI":"10.1093\/europace\/euz205","article-title":"Restitution and adaptation measurements for the estimate of short-term cardiac action potential memory: Comparison of five human ventricular models","volume":"21","author":"Zaniboni","year":"2019","journal-title":"Europace"},{"key":"ref_26","first-page":"H782","article-title":"Cycle length effect on restitution of action potential duration in dog cardiac fibers","volume":"244","author":"Elharrar","year":"1983","journal-title":"Am. J. Physiol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1426","DOI":"10.1038\/sj.bjp.0702443","article-title":"Kinetics of rate-dependent shortening of action potential duration in guinea-pig ventricle; effects of IK1 and IKr blockade","volume":"126","author":"Williams","year":"1999","journal-title":"Br. J. Pharmacol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1046\/j.1540-8167.2004.03550.x","article-title":"The restitution portrait: A new method for investigating rate-dependent restitution","volume":"15","author":"Kalb","year":"2004","journal-title":"J. Cardiovasc. Electrophysiol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1152\/jappl.1968.25.2.191","article-title":"A graphic method for the study of alternation in cardiac action potentials","volume":"25","author":"Nolasco","year":"1968","journal-title":"J. Appl. Physiol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6061","DOI":"10.1073\/pnas.090492697","article-title":"Preventing ventricular fibrillation by flattening cardiac restitution","volume":"97","author":"Garfinkel","year":"2000","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"74","DOI":"10.2174\/1573403X16999201231203854","article-title":"Analysis of Heart rate Variability and Implication of Different Factors on Heart Rate Variability","volume":"17","author":"Tiwari","year":"2021","journal-title":"Curr. Cardiol. Rev."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1150","DOI":"10.1046\/j.1540-8167.2002.01150.x","article-title":"Electrical restitution and ventricular fibrillation: Negotiating a slippery slope","volume":"13","author":"Gilmour","year":"2002","journal-title":"J. Cardiovasc. Electrophysiol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1046\/j.1540-8167.2004.03330.x","article-title":"Restitution dynamics during pacing and arrhythmias in isolated pig hearts","volume":"15","author":"Banville","year":"2004","journal-title":"J. Cardiovasc. Electrophysiol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"S64","DOI":"10.1111\/j.1540-8167.2006.00385.x","article-title":"Restitution of Ca(2+) release and vulnerability to arrhythmias","volume":"17","author":"Sobie","year":"2006","journal-title":"J. Cardiovasc. Electrophysiol."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1161\/01.RES.0000240542.03986.e7","article-title":"Spatially discordant alternans in cardiac tissue: Role of calcium cycling","volume":"99","author":"Sato","year":"2006","journal-title":"Circ. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"103119","DOI":"10.1063\/1.4932961","article-title":"Spatial discordance and phase reversals during alternate pacing in discrete-time kinematic and cardiomyocyte ionic models","volume":"25","author":"Weinberg","year":"2015","journal-title":"Chaos"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2930","DOI":"10.1016\/S0006-3495(99)77126-2","article-title":"Intracellular Ca(2+) dynamics and the stability of ventricular tachycardia","volume":"77","author":"Chudin","year":"1999","journal-title":"Biophys. J."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1244","DOI":"10.1161\/01.RES.0000224540.97431.f0","article-title":"From pulsus to pulseless: The saga of cardiac alternans","volume":"98","author":"Weiss","year":"2006","journal-title":"Circ. Res."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"108567","DOI":"10.1016\/j.mbs.2021.108567","article-title":"Evolution of mathematical models of cardiomyocyte electrophysiology","volume":"334","author":"Amuzescu","year":"2021","journal-title":"Math. Biosci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1038\/s41567-019-0458-z","article-title":"Going into resonance","volume":"105","author":"Buchanan","year":"2019","journal-title":"Nat. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Lemay, M., De Lange, E., and Kucera, J.P. (2012). Uncovering the dynamics of cardiac systems using stochastic pacing and frequency domain analyses. PLoS Comput. Biol., 8.","DOI":"10.1371\/journal.pcbi.1002399"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1113\/expphysiol.1993.sp003672","article-title":"Phase-dependent properties of the cardiac sarcoplasmic reticulum oscillator in cat right atrium: A mechanism contributing to dysrhythmias induced by Ca2+ overload","volume":"78","author":"Rubenstein","year":"1993","journal-title":"Exp. Physiol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"844","DOI":"10.1161\/01.RES.57.6.844","article-title":"Frequency, amplitude, and propagation velocity of spontaneous Ca++-dependent contractile waves in intact adult rat cardiac muscle and isolated myocytes","volume":"57","author":"Kort","year":"1985","journal-title":"Circ. Res."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"3074","DOI":"10.1096\/fasebj.2.15.2847949","article-title":"Cytosolic calcium oscillators","volume":"2","author":"Berridge","year":"1988","journal-title":"FASEB J."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1007\/s10484-006-9009-3","article-title":"Characteristics of resonance in heart rate variability stimulated by biofeedback","volume":"31","author":"Vaschillo","year":"2006","journal-title":"Appl. Psychophysiol. Biofeedback"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"570400","DOI":"10.3389\/fnins.2020.570400","article-title":"A Practical Guide to Resonance Frequency Assessment for Heart Rate Variability Biofeedback","volume":"14","author":"Shaffer","year":"2020","journal-title":"Front. Neurosci."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"756","DOI":"10.3389\/fpsyg.2014.00756","article-title":"Heart rate variability biofeedback: How and why does it work?","volume":"5","author":"Lehrer","year":"2014","journal-title":"Front. Psychol."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.autneu.2009.08.020","article-title":"Frequency of slow oscillations in arterial pressure and R\u2013R intervals during muscle metaboreflex activation","volume":"152","author":"Kiviniemi","year":"2010","journal-title":"Auton. Neurosci. Basic Clin."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/978-1-4615-1893-8_17","article-title":"Metabolic oscillations in heart cells","volume":"382","author":"Ramza","year":"1995","journal-title":"Adv. Exp. Med. Biol."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"2285","DOI":"10.1113\/JP273626","article-title":"Dynamical effects of calcium-sensitive potassium currents on voltage and calcium alternans","volume":"595","author":"Kennedy","year":"2017","journal-title":"J. Physiol."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"023204","DOI":"10.1103\/PhysRevE.90.023204","article-title":"Dynamics of periodic mechanical structures containing bistable elastic elements: From elastic to solitary wave propagation","volume":"90","author":"Nadkarni","year":"2014","journal-title":"Phys. Rev. E Stat. Nonlin. Soft Matter Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1111\/j.1469-7793.2000.00795.x","article-title":"Functional coupling between glycolysis and excitation-contraction coupling underlies alternans in cat heart cells","volume":"524","author":"Huser","year":"2000","journal-title":"J. Physiol."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1161\/CIRCRESAHA.107.160929","article-title":"Gain and cardiac E-C coupling, revisited and revised","volume":"101","author":"Wier","year":"2007","journal-title":"Circ. Res."}],"container-title":["Biomolecules"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2218-273X\/12\/7\/873\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:38:24Z","timestamp":1760139504000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2218-273X\/12\/7\/873"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,23]]},"references-count":53,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["biom12070873"],"URL":"https:\/\/doi.org\/10.3390\/biom12070873","relation":{},"ISSN":["2218-273X"],"issn-type":[{"type":"electronic","value":"2218-273X"}],"subject":[],"published":{"date-parts":[[2022,6,23]]}}}