First return maps for the dynamics of synaptically coupled conditional bursters

Springer Science and Business Media LLC - Tập 103 Số 2 - Trang 87-104 - 2010
Evandro Manica1, Georgi S. Medvedev2, Jonathan E. Rubin3
1Departamento de Matematica, Universidade Federal do Rio Grande do Sul, Porte Alegre, Brazil
2Department of Mathematics, Drexel University, Philadelphia, USA
3Department of Mathematics and Center for the Neural Basis of Cognition, University of Pittsburgh, Pittsburgh, USA

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Tài liệu tham khảo

Altendorfer R, Koditschek D, Holmes P (2003) Stability analyisis of legged locomotion by symmetry-factored return maps. Int J Robot Res 23: 979–999

Best J, Borisyuk A, Rubin J, Terman D, Welschselberger M (2005) The dynamic range of bursting in a model respiratory pacemaker network. SIAM J Appl Dyn Syst 4: 1107–1139

Butera R, Rinzel J, Smith J (1999a) Models of respiratory rhythm generation in the pre-Bötzinger complex. I. Bursting pacemaker neurons. J Neurophysiol 82: 382–397

Butera R, Rinzel J, Smith J (1999b) Models of respiratory rhythm generation in the pre-Bötzinger complex. II. Population of coupled pacemaker. J Neurophysiol 82: 398–415

Channell P Jr., Cymbalyuk G, Shilnikov A (2007) Applications of the poincaré mapping technique to analysis of neuronal dynamics. Neurocomputing 70: 2107–2111

Chay TR, Keizer J (1983) Minimal model for membrane oscillations in the pancreatic β-cell. Biophys J 42: 181–190

Ditto WL, Rauseo SN, Spano ML (1990) Experimental control of chaos. Phys Rev Lett 65: 3211–3214

Dunmyre JR, Rubin JE (2009) Optimal intrinsic dynamics for bursting in a three-cell network. SIAM J Dyn Syst 9: 154–187

Ermentrout B (2002) Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students. Software Environ. Tools 14, SIAM, Philadelphia

Ermentrout GB, Kopell N (1998) Fine structure of neural spiking and synchronization in the presence of conduction delays. Proc Natl Acad Sci USA 95(3): 1259–1264

Gomes AA, Manica E, Varriale MC (2008) Applications of chaos control techniques to a three-species food chain. Chaos Solitons Fractals 35: 432–441

Hitczenko P, Medvedev GS (2009) Bursting oscillations induced by small noise. SIAM J Appl Math 69: 1359–1392

Innocenti G, Genesio R (2009) On the dynamics of chaotic spiking- bursting transition in the Hindmarsh-Rose neuron. Chaos 19: 023124

Izhikevich E (2000) Neural excitability, spiking, and bursting. Int J Bifurc Chaos 10: 1171–1266

Kuznetsov YA (1995) Elements of applied bifurcation theory, vol 112 of Applied Mathematical Sciences. Springer-Verlag, Berlin

LoFaro T, Kopell N (1999) Timing regulation in a network reduced from voltage-gated equations to a one-dimensional map. J Math Biol 38: 479–533

Medvedev GS (2005) Reduction of a model of an excitable cell to a one-dimensional map. Physica D 202: 37–59

Medvedev GS (2006) Transition to bursting via deterministic chaos. Phys Rev Lett 97: 048102

Medvedev GS, Yoo Y (2008) Chaos at the border of criticality. Chaos 18(3): 033105,7

Milik A, Szmolyan P, Löffelmann H, Gröller E (1998) Geometry of mixed-mode oscillations in the 3-d autocatalator. Int J Bifurc Chaos 8: 505–519

Pedersen MG Sørensen MP (2006/07) The effect of noise in β-cell burst period. SIAM J Appl Math 67(2):530–542 (electronic), 2006/07

Pontryagin LS, Rodygin LV (1960) Periodic solution of a system of ordinary differential equations with a small parameter in the terms containing the derivatives. Sov Math Dokl 1: 611–614

Rinzel J (1985) Bursting oscillations in an excitable membrane model. In: Sleeman BD, Jarvis RJ (eds) Ordinary and partial differential equations. Springer-Verlag, Berlin, pp 304–316

Rinzel J, Troy WC (1982a) Bursting phenomena in a simplified oregonator flow system d activity an model. J Chem Phys 76: 1775–1789

Rinzel J, Troy WC (1982b) A one-variable map analysis of bursting in the belousov-zhabotinskii reaction. Lect Notes Biomath 51: 1–23

Rubin JE (2006) Bursting induced by excitatory synaptic coupling in non-identical conditional relaxation oscillators or square-wave bursters. Phys Rev E 74: 021917

Rulkov NF (2002) Modeling of spiking-bursting neural behavior using two-dimensional map. Phys Rev E 65: 041922–041930

Rulkov NF (2004) Oscillations in large-scale cortical networks: map-based model. J Comput Neurosci 17: 203–223

Smith JC, Ellenberger HH, Ballayi K, Richter DW, Feldman JL (1991) Pre-Bötzinger complex: a brainstem region that may generate respiratory rhythm in mammals. Science 254: 726–729

Su J, Rubin J, Terman D (2004) Effects of noise on elliptic bursters. Nonlinearity 17: 133–157

Terman D (1991) Chaotic spikes arising from a model for bursting in excitable membranes. SIAM J Appl Math 51: 1418–1450

Terman D (1992) The transition from bursting to continuous spiking in an excitable membrane model. J Nonlinear Sci 2: 133–182

The MathWorks, Inc (2008) MATLAB The Language of Technical Computing, version 7.6.0.324(r2008a) edition

Wang X (1991) Period-doublings to chaos in a simple neural network. IEEE, pp 333–339