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Bursting in Neurons and Small Networks

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References

  • Abbott LF, Marder E, Hooper SL (1991) Oscillating networks: control of burst duration by electrically coupled neurons. Neural Comput 3:487–497

    Article  Google Scholar 

  • Adams WB, Levitan IB (1985) Voltage and ion dependences of the slow currents which mediate bursting in Aplysia neurone R15. J Physiol 360:69–93

    CAS  PubMed Central  PubMed  Google Scholar 

  • Buchholtz F, Golowasch J, Epstein IR, Marder E (1992) Mathematical model of an identified stomatogastric ganglion neuron. J Neurophysiol 67:332–340

    CAS  PubMed  Google Scholar 

  • Butera RJ Jr, Clark JW Jr, Byrne JH (1996) Dissection and reduction of a modeled bursting neuron. J Comput Neurosci 3:199–223

    Article  PubMed  Google Scholar 

  • Calabrese RL (1995) Oscillation in motor pattern-generating networks. Current Opin Neurobiol 5:816–823

    Article  CAS  Google Scholar 

  • Calabrese RL, Nadim F, Olsen OH (1995) Heartbeat control in the medicinal leech: a model system for understanding the origin, coordination, and modulation of rhythmic motor patterns. J Neurobiol 27:390–402

    Article  CAS  PubMed  Google Scholar 

  • Canavier CC, Clark JW, Byrne JH (1991) Simulation of the bursting activity of neuron R15 in Aplysia: role of ionic currents, calcium balance, and modulatory transmitters. J Neurophysiol 66:2107–2124

    CAS  PubMed  Google Scholar 

  • Chow CC, Kopell N (2000) Dynamics of spiking neurons with electrical coupling. Neural Comput 12:1643–1678

    Article  CAS  PubMed  Google Scholar 

  • Cymbalyuk GS, Gaudry Q, Masino MA, Calabrese RL (2002) Bursting in leech heart interneurons: cell-autonomous and network-based mechanisms. J Neurosci 22:10580–10592

    CAS  PubMed  Google Scholar 

  • Desroches MK, Kaper TJ, Krupa M (2013) Mixed-mode bursting oscillations: dynamics created by a slow passage through spike-adding canard explosion in a square wave burster. Chaos 23:1–13

    Article  Google Scholar 

  • Ermentrout GB, Kopell N (1991) Multiple pulse interactions and averaging in systems of coupled neural oscillators. J Math Biol 29:195–217

    Article  Google Scholar 

  • Feng HF (2001) Is the integrate-and-fire model good enough? A review. Neural Netw 14:955–975

    Article  CAS  PubMed  Google Scholar 

  • Fitzhugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1:445–466

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Fitzhugh R (1969) Mathematical models for excitation and propagation in nerve. McGraw Hill, New York

    Google Scholar 

  • Friesen WOP, Pearce RA (1993) Mechanisms of intersegmental coordination in leech locomotion. Semin Neurosci 5:41–47

    Article  Google Scholar 

  • Gola M, Selverston A (1981) Ionic requirements for bursting activity in lobster stomatogastric neurons. J Comp Physiol 145:191–207

    Article  CAS  Google Scholar 

  • Golowasch J, Buchholtz F, Epstein IR, Marder E (1992) Contribution of individual ionic currents to activity of a model stomatogastric ganglion neuron. J Neurophysiol 67:341–349

    CAS  PubMed  Google Scholar 

  • Graham Brown TG (1911) The intrinsic factors in the act of progression in the mammal. Proc Royal Soc Lond B 84:308–319

    Article  Google Scholar 

  • Grillner S, Matsushima T, Wadden T, Tegner J, El Manira A, Wallen P (1993) The neurophysiological bases of undulatory locomotion in vertebrates. Semin Neurosci 5:17–27

    Article  Google Scholar 

  • Guckenheimer J, Gueron S, Harris-Warrick RM (1993) Mapping the dynamics of a bursting neuron. Philos Trans R Soc Lond B Biol Sci 341:345–359

    Article  CAS  PubMed  Google Scholar 

  • Guckenheimer J, Harris-Warrick R, Peck J, Willms A (1997) Bifurcation, bursting, and spike frequency adaptation. J Comput Neurosci 4:257–277

    Article  CAS  PubMed  Google Scholar 

  • Harris-Warrick RM, Flamm RE (1987) Multiple mechanisms of bursting in a conditional bursting neuron. J Neurosci 7:2113–2128

    CAS  PubMed  Google Scholar 

  • Hill AA, Masino MA, Calabrese RL (2002) Model of intersegmental coordination in the leech heartbeat neuronal network. J Neurophysiol 87:1586–1602

    PubMed  Google Scholar 

  • Hindmarsh J, Cornelius P (2005) The development of the Hindmarsh-Rose model for bursting. In: Bursting: the genesis of rhythm in the nervous system. World Scientific, Hackensack

    Google Scholar 

  • Hindmarsh JL, Rose RM (1984) A model of neuronal bursting using three coupled first order differential equations. Proc R Soc Lond B 221:87–102

    Article  CAS  PubMed  Google Scholar 

  • Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544

    CAS  PubMed Central  PubMed  Google Scholar 

  • Hooper SL, Marder E (1987) Modulation of the lobster pyloric rhythm by the peptide proctolin. J Neurosci 7:2097–2112

    CAS  PubMed  Google Scholar 

  • Hoppensteadt FC, Izhikevich EM (1997) Weakly connected neural networks. Springer, New York

    Book  Google Scholar 

  • Izhikevich EM (2000a) Subcritical elliptic bursting of Bautin type. SIAM J Appl Math 60:503–535

    Article  Google Scholar 

  • Izhikevich EM (2000b) Neural excitability, spiking, and bursting. Int J Bifurcat Chaos 10:1171–1266

    Article  Google Scholar 

  • Izhikevich EM (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. MIT Press, Cambridge

    Google Scholar 

  • Jones SR, Mulloney B, Kaper TJ, Kopell N (2003) Coordination of cellular pattern-generating circuits that control limb movements: the sources of stable differences in intersegmental phases. J Neurosci 23:3457–3468

    CAS  PubMed  Google Scholar 

  • Kepler TB, Marder E, Abbott LF (1990) The effect of electrical coupling on the frequency of model neuronal oscillators. Science 248:83–85

    Article  CAS  PubMed  Google Scholar 

  • Kepler TB, Abbott LF, Marder E (1992) Reduction of conductance-based neuron models. Biol Cybern 66:381–387

    Article  CAS  PubMed  Google Scholar 

  • Kispersky TW, White JA, Rotstein HG (2010) The mechanism of abrupt transition between theta and hyper-excitable spiking activity in medial entorhinal cortex layer II stellate cells. PloS One 5:1–21

    Article  Google Scholar 

  • Kopell N, Abbott LF, Soto-Trevino C (1998) On the behavior of a neural oscillator electrically coupled to a bistable element. Phys D 121:367–395

    Article  Google Scholar 

  • Kramer RH, Zucker RS (1985) Calcium-induced inactivation of calcium current causes the inter-burst hyperpolarization of Aplysia bursting neurones. J Physiol 362:131–160

    CAS  PubMed Central  PubMed  Google Scholar 

  • Lapicque L (1907) Recherches quantitatives sur l’excitation e’lectrique des nerfs traite’e comme une polarization. J Physiol Pathol Gen 9:620–635

    Google Scholar 

  • Malashchenko T, Shilnikov A, Cymbalyuk G (2011) Six types of multistability in a neuronal model based on slow calcium current. PloS One 6:e21782

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Manor Y, Rinzel J, Segev I, Yarom Y (1997) Low-amplitude oscillations in the inferior olive: a model based on electrical coupling of neurons with heterogeneous channel densities. J Neurophysiol 77:2736–2752

    CAS  PubMed  Google Scholar 

  • Marder E, Calabrese RL (1996) Principles of rhythmic motor pattern generation. Physiol Rev 76:687–717

    CAS  PubMed  Google Scholar 

  • Mulloney B, Smarandache C (2010) Fifty years of CPGs: two neuroethological papers that shaped the course of neuroscience. Front Behav Neurosci 4(45):1–8

    Google Scholar 

  • Nadim F, Olsen Ø, Schutter E, Calabrese R (1995a) The interplay of intrinsic and synaptic currents in a half-center oscillator. In: Bower J (ed) The neurobiology of computation. Springer, US, pp 269–274

    Google Scholar 

  • Nadim F, Olsen OH, De Schutter E, Calabrese RL (1995b) Modeling the leech heartbeat elemental oscillator. I. Interactions of intrinsic and synaptic currents. J Comput Neurosci 2:215–235

    Article  CAS  PubMed  Google Scholar 

  • Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line stimulating nerve axon. Proc IRE 50:2061–2070

    Article  Google Scholar 

  • Nusbaum MP, Beenhakker MP (2002) A small-systems approach to motor pattern generation. Nature 417:343–350

    Article  CAS  PubMed  Google Scholar 

  • Rinzel J (1986) A formal classification of bursting mechanisms in excitable systems. In: Proceedings of the international congress of mathematics, AMS, Providence, pp 1578–1593

    Google Scholar 

  • Rinzel JE, Ermentrout B (1998) Analysis of neural excitability and oscillation. In: Koch CS, Segev I (eds) Methods in neuronal modeling: from ions to networks, 2nd edn. MIT Press, Cambridge, pp 251–291

    Google Scholar 

  • Rinzel J, Lee YS (1987) Dissection of a model for neuronal parabolic bursting. J Math Biol 25:653–675

    Article  CAS  PubMed  Google Scholar 

  • Schwemmer M, Lewis T (2012) The theory of weakly coupled oscillators. In: Schultheiss NW, Prinz AA, Butera RJ (eds) Phase response curves in neuroscience. Springer, New York, pp 3–31

    Chapter  Google Scholar 

  • Selverston AI (2005) A neural infrastructure for rhythmic motor patterns. Cell Mol Neurobiol 25:223–244

    Article  PubMed  Google Scholar 

  • Selverston AI, Szucs A, Huerta R, Pinto R, Reyes M (2009) Neural mechanisms underlying the generation of the lobster gastric mill motor pattern. Front Neural Circuit 3:12

    Google Scholar 

  • Sherman A, Rinzel J (1992) Rhythmogenic effects of weak electrotonic coupling in neuronal models. Proc Natl Acad Sci U S A 89:2471–2474

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Skinner FK, Mulloney B (1998) Intersegmental coordination of limb movements during locomotion: mathematical models predict circuits that drive swimmeret beating. J Neurosci 18:3831–3842

    CAS  PubMed  Google Scholar 

  • Skinner FK, Kopell N, Marder E (1994) Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks. J Comput Neurosci 1:69–87

    Article  CAS  PubMed  Google Scholar 

  • Skinner FK, Kopell N, Mulloney B (1997) How does the crayfish swimmeret system work? Insights from nearest-neighbor coupled oscillator models. J Comput Neurosci 4:151–160

    Article  CAS  PubMed  Google Scholar 

  • Smarandache C, Hall WM, Mulloney B (2009) Coordination of rhythmic motor activity by gradients of synaptic strength in a neural circuit that couples modular neural oscillators. J Neurosci 29:9351–9360

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Smith GD, Cox CL, Sherman SM, Rinzel J (2000) Fourier analysis of sinusoidally driven thalamocortical relay neurons and a minimal integrate-and-fire-or-burst model. J Neurophysiol 83:588–610

    CAS  PubMed  Google Scholar 

  • Soto-Trevino C, Rabbah P, Marder E, Nadim F (2005) Computational model of electrically coupled, intrinsically distinct pacemaker neurons. J Neurophysiol 94:590–604

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Taylor AL, Goaillard JM, Marder E (2009) How multiple conductances determine electrophysiological properties in a multicompartment model. J Neurosci 29:5573–5586

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Tazaki K, Cooke IM (1990) Characterization of Ca current underlying burst formation in lobster cardiac ganglion motorneurons. J Neurophysiol 63:370–384

    CAS  PubMed  Google Scholar 

  • Wang X-J, Rinzel J (1992) Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Comput 4:84–97

    Article  Google Scholar 

  • Zhang B, Wootton JF, Harris-Warrick RM (1995) Calcium-dependent plateau potentials in a crab stomatogastric ganglion motor neuron. II. Calcium-activated slow inward current. J Neurophysiol 74:1938–1946

    CAS  PubMed  Google Scholar 

Further Reading

  • Coombes S, Bressloff PC (2005) Bursting: the genesis of rhythm in the nervous system. World Scientific, Hackensack

    Book  Google Scholar 

  • Ermentrout GB (1992) Stable periodic-solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM J Appl Math 52:1665–1687

    Article  Google Scholar 

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Acknowledgment

Supported by grants NIH MH060605 (FN), NSF DMS1313861(HGR).

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Fox, D.M., Rotstein, H.G., Nadim, F. (2014). Bursting in Neurons and Small Networks. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7320-6_454-1

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