Dynamical mechanisms and transitions of mixed-mode bursting in a closed-loop respiratory control model

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

Abnormal respiratory rhythms, a hallmark of many respiratory diseases, largely arise from transitional firing dynamics in the pre-Bötzinger complex (pre-BötC) neurons. Mixed-mode bursting (MMB), characterized by alternating depolarization block (DB) and square-wave (SW) bursts, represents a particularly complex firing pattern of significant research interest. Based on Diekman’s closed-loop respiratory control model, this study applies a fast-slow dynamical analysis and the bifurcation theory to investigate how variations in hemoglobin concentration ([Hb]) govern the emergence and disappearance of MMB. Numerical simulations show that [Hb] variations induce clear transitions between MMB and single SW bursting. A dynamical analysis reveals that the intrinsic slow variable h determines the type of individual bursting, while the slow feedback variable g_tonic, which is regulated by the arterial blood oxygen partial pressure, evolves on a comparable timescale and influences the global trajectories by shifting the critical bifurcation structures. The interaction between these slow variables enables the emergence of MMB. When the slow feedback pathway is fixed such that g_tonic remains constant, MMB can no longer be sustained. These results indicate that MMB in closed-loop systems relies on the dynamic coupling of multiple slow variables rather than on the static value of a single parameter, thus providing a dynamical mechanism for complex rhythm generation.

Keywords

pre-Bötzinger complex / closed-loop control model / mixed-mode bursting / bifurcation / depolarization block

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] M. Gorini, A. Spinelli, R. Ginanni, R. Duranti, F. Gigliotti, G. Scano, Neural respiratory drive and neuromuscular coupling in patients with chronic obstructive pulmonary disease (COPD), Chest, 98 (1990), 1179-1186. https://doi.org/10.1378/chest.98.5.1179

[2] M. Gorini, G. Misuri, A. Corrado, R. Duranti, I. Iandelli, E. De Paola, et al., Breathing pattern and carbon dioxide retention in severe chronic obstructive pulmonary disease, Thorax, 51 (1996), 677-683. https://doi.org/10.1136/thx.51.7.677

[3] R. B. Gorman, D. K. McKenzie, N. B. Pride, J. F. Tolman, S. C. Gandevia, Diaphragm length during tidal breathing in patients with chronic obstructive pulmonary disease, Am. J. Respir. Crit. Care Med., 166 (2002), 1461-1469. https://doi.org/10.1164/rccm.200111-087OC

[4] R. L. Horner, Pathophysiology of obstructive sleep apnea, J. Cardiopulmonary Rehabil. Prev., 28 (2008), 289-298. https://doi.org/10.1097/01.HCR.0000336138.71569.a2

[5] J. L. Feldman, C. A. Del Negro, Looking for inspiration: new perspectives on respiratory rhythm, Nat. Rev. Neurosci., 7 (2006), 232-241. https://doi.org/10.1038/nrn1871

[6] S. P. Lieske, M. Thoby-Brisson, P. Telgkamp, J. M. Ramirez, Reconfiguration of the neural network controlling multiple breathing patterns: eupnea, sighs and gasps, Nat. Neurosci., 3 (2000), 600-607. https://doi.org/10.1038/75776

[7] P. Banovcin, J. Seidenberg, H. von der Hardt, Assessment of tidal breathing patterns for monitoring of bronchial obstruction in infants, Pediatr. Res., 38 (1995), 218-220. https://doi.org/10.1203/00006450-199508000-00014

[8] A. K. Tryba, F. Peña, S. P. Lieske, J. C. Viemari, M. Thoby-Brisson, J. M. Ramirez, Differential modulation of neural network and pacemaker activity underlying eupnea and sigh-breathing activities, J. Neurophysiol., 99 (2008), 2114-2125. https://doi.org/10.1152/jn.01192.2007

[9] A. L. Bianchi, M. Denavit-Saubié, J. Champagnat, Central control of breathing in mammals: neuronal circuitry, membrane properties, and neurotransmitters, Physiol. Rev., 75 (1995), 1-45. https://doi.org/10.1152/physrev.1995.75.1.1

[10] M. I. Cohen, Neurogenesis of respiratory rhythm in the mammal, Physiol. Rev., 59 (1979), 1105-1173. https://doi.org/10.1152/physrev.1979.59.4.1105

[11] D. W. Richter, K. Ballanyi, S. Schwarzacher, Mechanisms of respiratory rhythm generation, Curr. Opin. Neurobiol., 2 (1992), 788-793. https://doi.org/10.1016/0959-4388(92)90135-8

[12] I. A. Rybak, A. P. L. Abdala, S. N. Markin, J. F. R. Paton, J. C. Smith, Spatial organization and state-dependent mechanisms for respiratory rhythm and pattern generation, Prog. Brain Res., 165 (2007), 201-220. https://doi.org/10.1016/S0079-6123(06)65013-9

[13] J. C. Smith, H. H. Ellenberger, K. Ballanyi, D. W. Richter, J. L. Feldman, Pre-Bötzinger complex: a brainstem region that may generate respiratory rhythm in mammals, Science, 254 (1991), 726-729. https://doi.org/10.1126/science.1683005

[14] S. M. Johnson, J. C. Smith, G. D. Funk, J. L. Feldman, Pacemaker behavior of respiratory neurons in medullary slices from neonatal rat, J. Neurophysiol., 72 (1994), 2598-2608. https://doi.org/10.1152/jn.1994.72.6.2598

[15] J. C. Rekling, J. L. Feldman, PreBötzinger complex and pacemaker neurons: hypothesized site and kernel for respiratory rhythm generation, Annu. Rev. Physiol., 60 (1998), 385-405. https://doi.org/10.1146/annurev.physiol.60.1.385

[16] N. Koshiya, J. C. Smith, Neuronal pacemaker for breathing visualized in vitro, Nature, 400 (1999), 360-363. https://doi.org/10.1038/22540

[17] C. A. Del Negro, S. M. Johnson, R. J. Butera, J. C. Smith, Models of respiratory rhythm generation in the pre-Bötzinger complex. III. Experimental tests of model predictions, J. Neurophysiol., 86 (2001), 59-74. https://doi.org/10.1152/jn.2001.86.1.59

[18] R. J. Butera, J. Rinzel, J. C. Smith, Models of respiratory rhythm generation in the pre- Bötzinger complex. I. Bursting pacemaker neurons, J. Neurophysiol., 82 (1999), 382-397. https://doi.org/10.1152/jn.1999.82.1.382

[19] R. J. Butera, J. Rinzel, J. C. Smith, Models of respiratory rhythm generation in the pre-Bötzinger complex. II. Populations of coupled pacemaker neurons, J. Neurophysiol., 82 (1999), 398-415. https://doi.org/10.1152/jn.1999.82.1.398

[20] J. Best, A. Borisyuk, J. Rubin, D. Terman, M. Wechselberger, The dynamic range of bursting in a model respiratory pacemaker network, SIAM J. Appl. Dyn. Syst., 4 (2005), 1107-1139. https://doi.org/10.1137/050625540

[21] R. H. Lee, C. J. Heckman, Essential role of a fast persistent inward current in action potential initiation and control of rhythmic firing, J. Neurophysiol., 85 (2001), 472-475. https://doi.org/10.1152/jn.2001.85.1.472

[22] J. F. R. Paton, A. P. L. Abdala, H. Koizumi, J. C. Smith, W. M. St John, Respiratory rhythm generation during gasping depends on persistent sodium current, Nat. Neurosci., 9 (2006), 311-313. https://doi.org/10.1038/nn1650

[23] R. W. Pace, D. D. Mackay, J. L. Feldman, C. A. Del Negro, Role of persistent sodium current in mouse preBötzinger complex neurons and respiratory rhythm generation, J. Physiol., 580 (2007), 485-496. https://doi.org/10.1113/jphysiol.2006.124602

[24] R. W. Pace, D. D. Mackay, J. L. Feldman, C. A. Del Negro, Inspiratory bursts in the preBötzinger complex depend on a calcium-activated nonspecific cation current linked to glutamate receptors in neonatal mice, J. Physiol., 582 (2007), 113-125. https://doi.org/10.1113/jphysiol.2007.133660

[25] E. A. Crowder, M. S. Saha, R. W. Pace, H. Zhang, G. D. Prestwich, C. A. Del Negro, Phosphatidylinositol 4,5-bisphosphate regulates inspiratory burst activity in the neonatal mouse preBötzinger complex, J. Physiol., 582 (2007), 1047-1058. https://doi.org/10.1113/jphysiol.2007.134577

[26] J. E. Rubin, J. A. Hayes, J. L. Mendenhall, C. A. Del Negro, Calcium-activated nonspecific cation current and synaptic depression promote network-dependent burst oscillations, Proc. Natl. Acad. Sci. U.S.A., 106 (2009), 2939-2944. https://doi.org/10.1073/pnas.0808776106

[27] J. R. Dunmyre, C. A. Del Negro, J. E. Rubin, Interactions of persistent sodium and calcium- activated nonspecific cationic currents yield dynamically distinct bursting regimes in a model of respiratory neurons, J. Comput. Neurosci., 31 (2011), 305-328. https://doi.org/10.1007/s10827-010-0311-y

[28] A. Ben-Tal, J. C. Smith, A model for control of breathing in mammals: coupling neural dynamics to peripheral gas exchange and transport, J. Theor. Biol., 251 (2008), 480-497. https://doi.org/10.1016/j.jtbi.2007.12.018

[29] C. O. Diekman, P. J. Thomas, C. G. Wilson, Eupnea, tachypnea, and autoresuscitation in a closed-loop respiratory control model, J. Neurophysiol., 118 (2017), 2194-2215. https://doi.org/10.1152/jn.00170.2017

[30] L. Duan, X. Chen, L. Xia, Z. Wang, Dynamics and control of mixed bursting in nonlinear pre-Bötzinger complex systems, Nonlinear Dyn., 112 (2024), 8539-8556. https://doi.org/10.1007/s11071-024-09473-3

[31] N. Toporikova, R. J. Butera, Two types of independent bursting mechanisms in in- spiratory neurons: an integrative model, J. Comput. Neurosci., 30 (2011), 515-528. https://doi.org/10.1007/s10827-010-0274-z

[32] Y. Wang, J. E. Rubin, Multiple timescale mixed bursting dynamics in a respiratory neuron model, J. Comput. Neurosci., 41 (2016), 245-268. https://doi.org/10.1007/s10827-016-0616-6

[33] D. D. Guo, Z. S. Lü, Effect of magnetic flow and external forcing current on mixed bursting in the pre-Bötzinger complex, Chin. Phys. B, 28 (2019), 110501. https://doi.org/10.1088/1674-1056/ab43b9

[34] Y. Yang, Y. Li, H. Gu, Synchronization transition from bursting to spiking and bifur- cation mechanism of the pre-Bötzinger complex, Acta Phys. Sin., 69 (2020), 040501. https://doi.org/10.7498/aps.69.20191509

[35] L. Duan, T. Liang, Y. Zhao, H. Xi, Multi-time scale dynamics of mixed depolarization block bursting, Nonlinear Dyn., 103 (2021), 1043-1053. https://doi.org/10.1007/s11071-020-05744-x

[36] Y. Shao, F. Wu, Q. Wang, Bursting dynamics and synchronization of neuromorphic systems with VO₂ memristors and Josephson junctions, Nonlinear Dyn., 113 (2025), 33907-33926. https://doi.org/10.1007/s11071-025-11757-1

[37] M. Liu, L. Duan, In-phase and anti-phase spikes synchronization within mixed bursters of the pre-Bötzinger complex, Electron. Res. Arch., 30 (2022), 961-977. https://doi.org/10.3934/era.2022050

[38] H. Hua, H. Gu, Y. Jia, Y. Li, Opposite stimulations can induce a same mixed-mode neuronal bursting containing four phases: underlying two-parameter bifurcations and threshold, Nonlinear Dyn., 113 (2025), 25153-25173. https://doi.org/10.1007/s11071-025-11387-7

[39] M. Liu, L. Duan, Dynamics and bifurcation mechanisms of respiratory patterns under optogenetic intervention, Nonlinear Dyn., 113 (2025), 20149-20168. https://doi.org/10.1007/s11071-025-11179-z

[40] J. A. Collins, A. Rudenski, J. Gibson, L. Howard, R. O’Driscoll, Relating oxygen partial pressure, saturation and content: the haemoglobin-oxygen dissociation curve, Breathe, 11 (2015), 194-201. https://doi.org/10.1183/20734735.001415

[41] A. Rizvi, P. Macedo, L. Babawale, H. C. Tighe, J. M. B. Hughes, J. E. Jackson, et al., Hemoglobin is a vital determinant of arterial oxygen content in hypoxemic patients with pulmonary arteriovenous malformations, Ann. Am. Thorac. Soc., 14 (2017), 903-911. https://doi.org/10.1513/AnnalsATS.201611-872OC

[42] D. R. Ouellette, The impact of anemia in patients with respiratory failure, Chest, 128 (2005), 576S-582S. https://doi.org/10.1378/chest.128.5suppl 2.576S

[43] J. S. Windsor, G. W. Rodway, Heights and haematology: the story of haemoglobin at altitude, Postgrad. Med. J., 83 (2007), 148-151. https://doi.org/10.1136/pgmj.2006.049734

[44] S. Tang, W. Zhou, L. Chen, H. Yan, L. Chen, F. Luo, High altitude polycythemia and its maladaptive mechanisms: an updated review, Front. Med., 11 (2024), 1448654. https://doi.org/10.3389/fmed.2024.1448654

[45] C. O. Diekman, P. J. Thomas, C. G. Wilson, COVID-19 and silent hypoxemia in a minimal closed-loop model of the respiratory rhythm generator, Biol. Cybern., 118 (2024), 145-163. https://doi.org/10.1007/s00422-024-00989-w

[46] B. Ermentrout, Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students, Appl. Mech. Rev., 56 (2003), B53. https://doi.org/10.1115/1.1579454

[47] E. M. Izhikevich, Neural excitability, spiking and bursting, Int. J. Bifurcation Chaos, 10 (2000), 1171-1266. https://doi.org/10.1142/S0218127400000840

[48] M. Chevalier, N. Toporikova, J. Simmers, M. Thoby-Brisson, Development of pacemaker properties and rhythmogenic mechanisms in the mouse embryonic respiratory network, eLife, 5 (2016), e16125. https://doi.org/10.7554/eLife.16125

[49] Y. I. Molkov, N. A. Shevtsova, C. Park, A. Ben-Tal, J. C. Smith, J. E. Rubin, A closed-loop model of the respiratory system: focus on hypercapnia and active expiration, PLoS One , 9 (2014), e109894. https://doi.org/10.1371/journal.pone.0109894

[50] Y. I. Molkov, J. E. Rubin, I. A. Rybak, J. C. Smith, Computational models of the neural control of breathing, WIREs Syst. Biol. Med., 9 (2017), e1371. https://doi.org/10.1002/wsbm.1371

[51] A. Buscarino, L. Fortuna, M. Frasca, Essentials of Nonlinear Circuit Dynamics with MATLAB R⃝ and Laboratory Experiments , CRC Press, 2017. https://doi.org/10.1201/b22063