The dynamic range of bursting in a model respiratory pacemaker network

SIAM Journal on Applied Dynamical Systems

Volumn 4, Issue 4, 2005, Pages 1107-1139

  Best, Janet ^a   Borisyukt, Alla ^a   Rubin, Jonathan ^b   Terman, David ^a   Wechselberger, Martin ^a  

^a OHIO STATE UNIVERSITY   (United States)

^b UNIVERSITY OF PITTSBURGH   (United States)

Author keywords

Averaged equations; Bifurcation analysis; Fast slow decomposition; Respiratory pacemaker; Square wave bursting; Synaptic coupling

Indexed keywords

AVERAGED EQUATIONS; BIFURCATION ANALYSIS; FAST/SLOW DECOMPOSITION; SQUARE-WAVE BURSTING;

BIFURCATION (MATHEMATICS); BRAIN; RESPIRATORY MECHANICS;

PACEMAKERS;

EID: 33644924105     PISSN: 15360040     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/050625540     Document Type: Article

References (33)

1. R.J. BUTERA, J. RINZEL, AND J.C. SMITH, Models of respiratory rhythm generation in the pre-Bötzinger complex. I. Bursting pacemaker neurons, J. Neurophysiol., 82 (1999), pp. 382-397.
2. R.J. BUTERA, J. RINZEL, AND J.C. SMITH, Models of respiratory rhythm generation in the pre-Bötzinger complex. II. Populations of coupled pacemaker neurons, J. Neurophysiol., 82 (1999), pp. 398-415.
3. T.R. CHAY AND J. RINZEL, Bursting, beating, and chaos in an excitable membrane model, Biophys. J., 47 (1985), pp. 357-366.
4. G. CYMBALYUK AND A. SHILNIKOV, Coexistence of tonic, spiking oscillations in a leech neuron model, J. Comp. Neurosci., 18 (2005), pp. 255-263.
5. G. DE VRIES AND A. SHEHMAN, Beyond synchronization: Modulatory and emergent effects of coupling in square-wave bursting, in BURSTING: The Genesis of Rhythm in the Nervous System, S. Coombes and P. C. Bressloff, eds., World Scientific, River Edge, NJ, 2005, pp. 243-272.
6. C.A. DEL NEGRO, C. MORGADO-VALLE, AND J.L. FELDMAN, Respiratory rhythm: An emergent network property?, Neuron, 34 (2002), pp. 821-830.
7. C.A. DEL NEGRO, C. MORGADO-VALLE, J.A. HAYES, D.D. MACKAY, R.W. PACE, E.A. CROWDER, AND J.L. FELDMAN, Sodium and calcium current-mediated pacemaker neurons and respiratory rhythm generation, J. Neurosci., 25 (2005), pp. 446-453.
8. C.A. DEL NEGRO, C.G. WILSON, R.J. BUTERA, H. RIGATTO, AND J.C. SMITH, Periodicity, mixed-mode oscillations, and quasiperiodicity in a rhythm-generating neural, network, Biophys. J., 82 (2002), pp. 206-214.
9. B. ERMENTROUT, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, Software Environ. Tools 14, SIAM, Philadelphia, 2002.
10. G.B. ERMENTROUT AND N. KOPELL, Multiple pulse interactions and averaging in systems of coupled neural oscillators, J. Math. Biol., 29 (1991), pp. 195-217.
11. E. IZHIKEVICH, Neural, excitability, spiking, and bursting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), pp. 1171-1266.
12. S.M. JOHNSON, J.C. SMITH, G.D. FUNK, AND J.L. FELDMAN, Pacemaker behavior of respiratory neurons in medullary slices from neonatal rat, J. Neurophysiol., 72 (1994), pp. 2598-2608.
13. N. KOPELL, Toward a theory of modelling central pattern generators, in The Neural Control of Rhythmic Movements in Vertebrates, A. Cohen, H. Rossignol, and S. Grillner, eds., Wiley, New York, 1988, pp. 369-413.
14. N. KOSHIYA AND J.C. SMITH, Neuronal pacemaker for breathing visualized in vitro, Nature, 400 (1999), pp. 360-363.
15. Y.A. KUZNETSOV, Elements of Applied Bifurcation Theory, 2nd ed., Springer-Verlag, New York, 1998.
16. E. LEE AND D. TERMAN, Uniqueness and stability of periodic bursting solutions, J. Differential Equations, 158 (1999), pp. 48-78.
17. F. PEÑA, M.A. PARKIS, A.K. TRYBA, AND J.-M. RAMIREZ, Differential contribution of pacemaker properties to the generation of respiratory rhythms during normoxia and hypoxia, Neuron, 43 (2004), pp. 105-117.
18. J.-M. RAMIREZ, U.J.A. QUELLMALZ, AND D.W. RICHTER, Postnatal changes in the mammalian respiratory network as revealed by the transverse brain stem slice of mouse, J. Physiol. (Lond.), 491 (1996), pp. 799-812.
19. J. RINZEL, Bursting oscillations in an excitable membrane model, in Ordinary and Partial Differential Equations, B.D. Sleeman and R.J. Jarvis, eds., Springer-Verlag, Berlin, 1985, pp. 304-316.
20. J. RINZEL, A formal classification of bursting mechanisms in excitable systems, in Proceedings of the International Congress of Mathematics, A.M. Gleason, ed., AMS, Providence, RI, 1987, pp. 1578-1593.
21. J. RUBIN AND D. TERMAN, Geometric analysis of population rhythms in synaptically coupled neuronal networks, Neural Comput., 12 (2000), pp. 597-645.
22. J. RUBIN AND D. TERMAN, Geometric singular perturbation analysis of neuronal dynamics, in Handbook of Dynamical Systems, Vol. 2: Towards Applications, B. Fiedler, ed., North-Holland, Amsterdam, 2002, pp. 93-146.
23. J. RUBIN AND D. TERMAN, Synchronized activity and loss of synchrony among heterogeneous conditional oscillators, SIAM J. Appl. Dyn. Syst., l (2002), pp. 146-174.
24. I.A. RYBAK, N.A. SHEVTSOVA, W.M. ST-JOHN, J.F.R. PATON, AND O. PIERREFICHE, Endogeneous rhythm generation in the pre-Bötzinger complex and ionic currents: Modelling and in vitro studies, European J. Neurosci., 18 (2003), pp. 239-257.
25. A. SHERMAN, Antiphase, asymmetric, and aperiodic oscillations in excitable cells. 1. Coupled bursters, Bull. Math. Biol., 56 (1994), pp. 811-835.
26. A. SHILNIKOV, R. CALABRESE, AND G. CYMBALYUK, Mechanism of bistability: Tonic spiking and bursting in a neuron model, Phys. Rev. E, 71 (2005), 056214.
27. A. SHILNIKOV AND G. CYMBALYUK, Homoclinic bifurcations of periodic orbits en a route from tonicspiking to bursting in neuron models, Reg. Chaotic Dyn., 9 (2004), pp. 281-297.
28. J.C. SMITH, H.H. ELLENBERGER, K. BALLANYI, D.W. RICHTER, AND J.L. FELDMAN, Pre-Bötzinger complex: A brainstem region that may generate respiratory rhythm in mammals, Science, 254 (1991), pp. 726-729.
29. D. SOMERS AND N. KOPELL, Rapid synchronization through fast threshold modulation, Biol. Cybernet., 68 (1993), pp. 393-407.
30. D. TERMAN, Chaotic spikes arising from a model of bursting in excitable membranes, SIAM J. Appl. Math., 51 (1991), pp. 1418-1450.
31. D. TERMAN, The transition from bursting to continuous spiking in an excitable membrane model, J. Nonlinear Sci., 2 (1992), pp. 133-182.
32. D. TERMAN, N. KOPELL, AND A. BOSE, Dynamics of two mutually coupled inhibitory neurons, Phys. D, 117 (1998), pp. 241-275.
33. D. TERMAN AND D. L. WANG, Global competition and local cooperation in a network of neural oscillators, Phys. D, 81 (1995), pp. 148-176.