Biological Cybernetics

, Volume 63, Issue 1, pp 25–34 | Cite as

Multiple modes of a conditional neural oscillator

  • I. R. Epstein
  • E. Marder


We present a model for a conditional bursting neuron consisting of five conductances: Hodgkin-Huxley type time- and voltage-dependent Na+ and K+ conductances, a calcium activated voltage-dependent K+ conductance, a calcium-inhibited time- and voltage-dependent Ca++ conductance, and a leakage Cl( conductance. With an initial set of parameters (versionS), the model shows a hyperpolarized steady-state membrane potential at which the neuron is silent. IncreasinggNa and decreasinggCl, whereg i , is the maximal conductance for speciesi, produces bursts of action potentials (BursterN). Alternatively, an increase ingCa produces a different bursting state (BursterC). The two bursting states differ in the periods and amplitudes of their bursting pacemaker potentials. They show different steady-stateI–V curves under simulated voltage-clamp conditions; in simulations that mimic a steady-stateI–V curve taken under experimental conditions only BursterN shows a negative slope resistance region. ModelC continues to burst in the presence of TTX, while bursting in ModelN is suppressed in TTX. Hybrid models show a smooth transition between the two states.


Calcium Membrane Potential Hybrid Model Versus Curve Negative Slope 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson WW (1980) Synaptic mechanisms generating nonspiking network oscillations in the stomatogastric ganglion of the lobster,Panulirus interrupts. Thesis, University of Oregon, EugeneGoogle Scholar
  2. Benson JA, Adams WB (1987) The control of rhythmic neuronal firing. In: Kaczmarek LK, Levitan IB (eds) Neuromodulation — the biochemical control of neuronal excitability. Oxford University Press, Oxford, pp 100–118Google Scholar
  3. Benson JA, Adams WB (1989) Ionic mechanisms of endogeneous activity in molluscan burster neurons. In: Jacklet JW (ed) Cellular and neuronal oscillators. Dekker, New York, pp 87–120Google Scholar
  4. Chay TR, Keizer J (1983) Minimal model for membrane oscillations in the pancreaticβ-cell. Biophys J 42:181–189Google Scholar
  5. Connor JA (1985) Neural pacemakers and rhythmicity. Annu Rev Physiol 47:17–28Google Scholar
  6. DiFrancesco D, Noble D (1989) The currenti F and its contribution to cardiac pacemaking. In: Jacklet JW (ed) Cellular and neuronal oscillators. Dekker, New York, pp 31–58Google Scholar
  7. Flamm RE, Harris-Warrick RM (1986) Aminergic modulation in lobster stomatogastric ganglion. II. Target neurons of dopamine, octopamine, and serotonin within the pyloric circuit. J Neurophysiol 55:866–881Google Scholar
  8. Gear CW (1971) Numerical initial value problems in ordinary differential equations. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  9. Getting PA (1988) Comparative analysis of invertebrate central pattern generators. In: Cohen AH, Rossignol S, Grillner S (eds) Neural control of rhythmic movements. Wiley, New York, pp 101–127Google Scholar
  10. Graubard K, Ross WN (1985) Regional distribution of calcium influx into bursting neurons detected with arsenazo III. Proc Natl Acad Sci USA 82:5565–5569Google Scholar
  11. Harris-Warrick RM, Flamm RE (1987) Multiple mechanisms of bursting in a conditional bursting neuron. J Neurosci 7:2113–2128Google Scholar
  12. Hindmarsh AC (1974) GEAR: Ordinary differential equation solver. Lawrence Livermore Laboratory Rept. No UCID-3001, Rev 3Google Scholar
  13. Hindmarsh JL, Rose RM (1984) A model of neural bursting using three coupled first order differential equations. Proc R Soc Lond B 221:87–102Google Scholar
  14. Hodgkin AL, Huxley AF (1952) A quantitative discription of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544Google Scholar
  15. Hooper SL, Marder E (1987) Modulation of a central pattern generator by the peptide, proctolin. J Neurosci 7:2097–2112Google Scholar
  16. Llinás RR (1988) Intrinsic electrophysiological properties of mammalian neurons: insights into central nervous system function. Science 242:1654–1664Google Scholar
  17. Marder E, Eisen JS (1984) Electrically coupled pacemaker neurons respond differently to same physiological inputs and neurotransmitters. J Neurophysiol 51:1362–1374Google Scholar
  18. Meyrand P, Marder E (1989) Endogeneous oscillatory properties of a striated muscle are activated by FMRFamide-like peptides. (submitted for publication)Google Scholar
  19. Meyrand P, Moulins M (1986) Myogenic oscillatory activity in the pyloric rhythmic motor system of crustacea. J Comp Physiol A 158:489–503Google Scholar
  20. Noble D, DiFrancesco D, Denyer J (1989) Ionic mechanisms in normal and abnormal cardiac pacemaker activity. In: Jacklet JW (ed) Cellular and neuronal oscillators. Dekker, New York, pp 59–86Google Scholar
  21. Petersen OH, Findlay I (1987) Electrophysiology of the pancreas. Physiol Rev 67:1054–1116Google Scholar
  22. Plant RE (1981) Bifurcation and resonance in a model for bursting nerve cells. J Math Biol 11:15–32Google Scholar
  23. Plant RE, Kim M (1976) Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations. Biophys J 16:227–244Google Scholar
  24. Rinzel J (1987) A formal classification of bursting mechanisms in excitable systems. In: Teramoto T, Yamaguti M (eds) Mathematical topics in population biology, morphogenesis and neuroscience. Springer, Berlin Heidelberg New York, pp 267–281Google Scholar
  25. Rinzel J, Baer SM (1988) Firing threshold of the Hodgkin-Huxley model for a slow current ramp: a memory effect and its dependence on fluctuations. Biophys J 53:551–556Google Scholar
  26. Rinzel J, Lee YS (1987) Dissection of a model for neuronal parabolic bursting. J Math Biol 25:653–675Google Scholar
  27. Robertson RM, Moulins M (1981) Firing between two spike thresholds: implications for oscillating lobster interneurons. Science 214:941–943Google Scholar
  28. Rose RM, Hindmarsh JL (1989a) The assembly of currents in a thalamic neuron. I. The three dimensional model. Proc R Soc Lond B 237:267–288Google Scholar
  29. Rose RM, Hindmarsh JL (1989b) The assembly of currents in a thalamic neuron. II. The stability and state diagrams. Proc R Soc Lond B 237:289–312Google Scholar
  30. Rose RM, Hindmarsh JL (1989c) The assembly of currents in a thalamic neuron. III. The seven dimensional model. Proc R Soc B 237:313–334Google Scholar
  31. Ross WN, Graubard K (1989) Spatially and temporally resolved calcium concentration changes in oscillating neurons of crab stomatogastric ganglion. Proc Natl Acad Sci USA 86:1679–1683Google Scholar
  32. Selverston AI, Moulins M (1985) Oscillatory neural networks. Annu Rev Physiol 47:29–48Google Scholar
  33. Van Renterghem C, Romey G, Lazdunski M (1988) Vasopressin modulates the spontaneous electrical activity in aortic cells (line A7r5) by acting on three different types of ionic channels. Proc Natl Acad Sci USA 85:9365–9369Google Scholar
  34. Wilson WA, Wachtel H (1974) Negative resistance characteristic essential for the maintenance of slow oscillations in bursting neurons. Science 186:932–934Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • I. R. Epstein
    • 1
  • E. Marder
    • 2
  1. 1.Department of ChemistryBrandeis UniversityWalthamUSA
  2. 2.Department of BiologyBrandeis UniversityWalthamUSA

Personalised recommendations