Journal of Computational Neuroscience

, Volume 21, Issue 3, pp 307–328 | Cite as

Differential control of active and silent phases in relaxation models of neuronal rhythms

  • Joël TabakEmail author
  • Michael J. O'Donovan
  • John Rinzel


Rhythmic bursting activity, found in many biological systems, serves a variety of important functions. Such activity is composed of episodes, or bursts (the active phase, AP) that are separated by quiescent periods (the silent phase, SP). Here, we use mean field, firing rate models of excitatory neural network activity to study how AP and SP durations depend on two critical network parameters that control network connectivity and cellular excitability. In these models, the AP and SP correspond to the network's underlying bistability on a fast time scale due to rapid recurrent excitatory connectivity. Activity switches between the AP and SP because of two types of slow negative feedback: synaptic depression—which has a divisive effect on the network input/output function, or cellular adaptation—a subtractive effect on the input/output function. We show that if a model incorporates the divisive process (regardless of the presence of the subtractive process), then increasing cellular excitability will speed up the activity, mostly by decreasing the silent phase. Reciprocally, if the subtractive process is present, increasing the excitatory connectivity will slow down the activity, mostly by lengthening the active phase. We also show that the model incorporating both slow processes is less sensitive to parameter variations than the models with only one process. Finally, we note that these network models are formally analogous to a type of cellular pacemaker and thus similar results apply to these cellular pacemakers.


Excitatory network Synaptic depression Cellular adaptation Episodic activity Mean field model 


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  1. Bertam R, Previte J, Sherman A, Kinard TA, Satin LS (2000) The phantom burster model for pancreatic β-cells. Biophys J 79: 2880–2892.Google Scholar
  2. Borodinsky LN, Root CM, Cronin JA, Sann SB, Gu X, Spitzer NC (2004) Activity-dependent homeostatic specification of transmitter expression in embryonic neurons. Nature 429: 523–530.Google Scholar
  3. Bracci E, Ballerini L, Nistri A (1996) Spontaneous bursts induced by pharmacological block of inhibition in lumbar motoneurons of the neonatal rat spinal cord. J Neurophysiol 75: 640–647.Google Scholar
  4. Butera RJ, Rinzel J, Smith JC (1999) Models of respiratory rhythm generation in the pre-Bötzinger complex: I. Bursting pacemaker neurons. J Neurophysiol 82: 382–397.Google Scholar
  5. Butts DA, Feller MB, Shatz CJ, Rokhsar DS (1999) Retinal waves are governed by collective network properties. J Neurosci 19: 3580–3593.Google Scholar
  6. Compte A, Sanchez-Vives MV, McCormick DA, Wang X-J (2003) Cellular and network mechanisms of slow oscillatory activity (<1 Hz) and wave propagations in a cortical network model. J Neurophysiol 89: 2707–2725.Google Scholar
  7. Coombes S, Bressloff PC (eds.) (2005) Bursting: The genesis of rhythm in the nervous system. World Scientific.Google Scholar
  8. Ermentrout B (1998) Linearization of f-i curves by adaptation. Neural Comput 10: 1721–1729.Google Scholar
  9. Ermentrout B (2002) Simulating, analyzing, and animating dynamical systems. SIAM.Google Scholar
  10. Ermentrout G, Chow CC (2002) Modeling neural oscillations. Physiol and Behav 77: 629–633.Google Scholar
  11. Friesen WO, Block GD (1984) What is a biological oscillator? Am J Physiol 246: R847–853.Google Scholar
  12. Giugliano M, Darbon P, Arsiero M, Lüscher H-R, Streit J (2004) Single-neuron discharge properties and network activity in dissociated cultures of neocortex. J Neurophysiol 92: 977–996.Google Scholar
  13. Gu X, Spitzer NC (1995) Distinct aspects of neuronal differentiation encoded by frequency of spontaneous Ca2+ transients. Nature 375: 784–787.Google Scholar
  14. Hanson MG, Landmesser LT (2004) Normal patterns of spontaneous activity are required for correct motor axon guidance and the expression of specific guidance molecules. Neuron 43: 687–701.Google Scholar
  15. Heeger DJ (1993) Modeling simple-cell direction selectivity with normalized, half-squared, linear operators. J Neurophysiol 70: 1885–1898.Google Scholar
  16. Holt GR, Koch C (1997) Shunting inhibition does not have a divisive effect on firing rates. Neural Comput 9: 1001–1013.Google Scholar
  17. Latham PE, Richmond BJ, Nelson PG, Nirenberg S (2000) Intrinsic dynamics in neuronal networks. I. Theory. J Neurophysiol 83: 808–827.Google Scholar
  18. Marchetti C, Tabak J, Chub N, O'Donovan MJ, Rinzel J (2005) Modeling spontaneous activity in the developing spinal cord using activity-dependent variations of intracellular chloride. J Neurosci 25: 3601–3612.Google Scholar
  19. Marder E, Calabrese R (1996) Principles of rhythmic motor pattern generation. Physiol Rev 76: 687–717.Google Scholar
  20. Netoff TI, Clewley R, Arno S, Keck T, White JA (2004) Epilepsy in small-world networks. J Neurosci 24: 8075–8083.Google Scholar
  21. O'Donovan MJ (1999) The origin of spontaneous activity in developing networks of the vertebrate nervous system. Curr Opin Neurobiol 9: 94–104.Google Scholar
  22. Pena F, Parkis MA, Tryba AK, Ramirez J-M (2004) Differential contribution of pacemaker properties to the generation of respiratory rhythms during normoxia and hypoxia. Neuron 43: 105–117.Google Scholar
  23. Pinto DJ, Brumberg JC, Simons DJ, Ermentrout GB (1996) A quantitative population model of whisker barrels: Re-examining the wilson-cowan equations. J Comput Neurosci 3: 247–264.Google Scholar
  24. Rinzel J, Ermentrout GB (1998) Analysis of neural excitability and oscillations. In Methods in Neuronal Modeling. MIT Press.Google Scholar
  25. Rinzel J (1985) Excitation dynamics: Insights from simplified membrane models. Fed Proc 44: 2944–2946.Google Scholar
  26. Rozzo A, Ballerini L, Abbate G, Nistri A (2002) Experimental and modeling studies of novel bursts induced by blocking Na+ pump and synaptic inhibition in the rat spinal cord. J Neurophysiol 88: 676–691.Google Scholar
  27. Sanchez-Vives MV, Nowak LG, McCormick DA (2000) Cellular mechanisms of long lasting adaptation in visual cortical neurons in vitro. J Neurosci 20: 4286–4299.Google Scholar
  28. Skinner FK, Turrigiano GG, Marder E (1993) Frequency and burst duration in oscillating neurons and two-cell networks. Biol Cybern 69: 375–383.Google Scholar
  29. Smolen P, Sherman A (1994) Phase independent resetting in relaxation and bursting oscillators. J theor Biol 169: 339–348.Google Scholar
  30. Staley KJ, Longacher M, Bains JS, Yee A (1998) Presynaptic modulation of CA3 network activity. Nature Neurosci 1: 201–209.Google Scholar
  31. Stellwagen D, Shatz CJ (2002) An instructive role for retinal waves in the development of retinogeniculate connectivity. Neuron 33: 357–367.Google Scholar
  32. Tabak J, Latham PE (2003) Analysis of spontaneous bursting activity in random neural networks. Neuroreport 14: 1445–1449.Google Scholar
  33. Tabak J, Rinzel J, O'Donovan MJ (2001) The role of activity-dependent network depression in the expression and self-regulation of spontaneous activity in the developing spinal cord. J Neurosci 21: 8966–8978.Google Scholar
  34. Tabak J, Rinzel J (2005) Bursting in excitatory neural networks. In Coombes S, Bressloff, P (eds.) Bursting: The Genesis of Rhythm in the Nervous System, World Scientific, pp. 273–301.Google Scholar
  35. Tabak J, Senn W, O'Donovan MJ, Rinzel J (2000) Modeling of spontaneous activity in the developing spinal cord using activity-dependent depression in an excitatory network. J Neurosci 20: 3041–3056.Google Scholar
  36. Tsodyks M, Uziel A, Markram H (2000) Synchrony generation in recurrent networks with frequency-dependent synapses. J Neurosci 20: RC50.Google Scholar
  37. Viemari J-C, Ramirez J-M (2004) Role of norepinephrine in respiratory rhythm generation. Soc Neurosci Abs 755.1.Google Scholar
  38. Wiedemann UA, Lüthi A (2003) Timing of network synchronization by refractory mechanisms. J Neurophysiol 90: 3902–3911.Google Scholar
  39. Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12: 1–24.Google Scholar
  40. Wilson HR (2003) Computational evidence for a rivalry hierarchy in vision. Proc Natl Acad Sci 100: 14499–14503.Google Scholar
  41. Yee AS, Longacher M, Staley KJ (2003) Convulsant and anticonvulsant effects on spontaneous CA3 population bursts. J Neurophysiol 89: 427–441.Google Scholar

Copyright information

© Springer Science Business Media, LLC 2006

Authors and Affiliations

  • Joël Tabak
    • 1
    • 3
    Email author
  • Michael J. O'Donovan
    • 1
  • John Rinzel
    • 2
  1. 1.Laboratory of Neural ControlNINDS/NIHBethesda
  2. 2.Center for Neural Science and Courant Institute of Mathematical SciencesNew York University
  3. 3.Department of Biological ScienceFlorida State UniversityTallahassee

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