Journal of Computational Neuroscience

, Volume 23, Issue 2, pp 169–187 | Cite as

Capturing the bursting dynamics of a two-cell inhibitory network using a one-dimensional map

  • Victor MatveevEmail author
  • Amitabha Bose
  • Farzan Nadim


Out-of-phase bursting is a functionally important behavior displayed by central pattern generators and other neural circuits. Understanding this complex activity requires the knowledge of the interplay between the intrinsic cell properties and the properties of synaptic coupling between the cells. Here we describe a simple method that allows us to investigate the existence and stability of anti-phase bursting solutions in a network of two spiking neurons, each possessing a T-type calcium current and coupled by reciprocal inhibition. We derive a one-dimensional map which fully characterizes the genesis and regulation of anti-phase bursting arising from the interaction of the T-current properties with the properties of synaptic inhibition. This map is the burst length return map formed as the composition of two distinct one-dimensional maps that are each regulated by a different set of model parameters. Although each map is constructed using the properties of a single isolated model neuron, the composition of the two maps accurately captures the behavior of the full network. We analyze the parameter sensitivity of these maps to determine the influence of both the intrinsic cell properties and the synaptic properties on the burst length, and to find the conditions under which multistability of several bursting solutions is achieved. Although the derivation of the map relies on a number of simplifying assumptions, we discuss how the principle features of this dimensional reduction method could be extended to more realistic model networks.


Half-center bursting T-type calcium current Poincaré return map Multistability Dimensional reduction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bartos, M., Manor, Y., Nadim, F., Marder, E., & Nusbaum, M. P. (1999). Coordination of fast and slow rhythmic neuronal circuits. Journal of Neuroscience, 19, 6650–6660.PubMedGoogle Scholar
  2. Bertram, R., & Sherman, A. (2000). Dynamical complexity and temporal plasticity in pancreatic beta-cells. Journal of Biosciences, 25, 197–209.PubMedGoogle Scholar
  3. Bose, A., Manor, Y., & Nadim, F. (2001). Bistable oscillations arising from synaptic depression. SIAM Journal on Applied Mathematics, 62, 706–727.CrossRefGoogle Scholar
  4. Butera, R. J. (1998). Multirhythmic bursting. Chaos, 8, 274–284.PubMedCrossRefGoogle Scholar
  5. Canavier, C. C., Baxter, D. A., Clark, J. W., & Byrne, J. H. (1994). Multiple modes of activity in a model neuron suggest a novel mechanism for the effects of neuromodulators. Journal of Neurophysiology, 72, 872–882.PubMedGoogle Scholar
  6. Canavier, C. C., Clark, J. W., & Byrne, J. H. (1991). Simulation of the bursting activity of neuron R15 in Aplysia: Role of ionic currents, calcium balance, and modulatory transmitters. Journal of Neurophysiology, 66, 2107–2124.PubMedGoogle Scholar
  7. Chay, T. R., & Rinzel, J. (1985). Bursting, beating, and chaos in an excitable membrane model. Biophysical Journal, 47, 357–366.PubMedGoogle Scholar
  8. Coombes, C., & Bressloff, P. (Eds.) (2005). Bursting: The genesis of rhythm in the nervous system. London: World Scientific.Google Scholar
  9. Destexhe, A., & Sejnowski, T. J. (2003). Interactions between membrane conductances underlying thalamocortical slow-wave oscillations. Physiological Reviews, 83, 1401–1453.PubMedGoogle Scholar
  10. Ermentrout, G. B., & Kopell, N. (1998). Fine structure of neural spiking and synchronization in the presence of conduction delays. In Proceedings of the National Academy of Sciences of the United States of America, 95, 1259–1264.PubMedCrossRefGoogle Scholar
  11. Grillner, S., Markram, H., De Schutter, E., Silberberg, G., & LeBeau, F. E. (2005). Microcircuits in action—from CPGs to neocortex. Trends in Neurosciences, 28, 525–33.PubMedCrossRefGoogle Scholar
  12. Hines, M., Morse, T., Carnevale, N., & Shepard, G. (2004). Model DB: A database to support computational neuroscience. Journal of Computational Neuroscience, 17, 7–11.PubMedCrossRefGoogle Scholar
  13. Huguenard, J. R. (1996). Low-threshold calcium currents in central nervous system neurons. Annual Review of Physiology, 58, 329–48.PubMedCrossRefGoogle Scholar
  14. Huguenard, J. R., & McCormick, D. A. (1992). Simulation of the currents involved in rhythmic oscillations in thalamic relay neurons. Journal of Neurophysiology, 68, 1373–1383.PubMedGoogle Scholar
  15. Izhikevich, E. M., & Hoppensteadt, F. C. (2004). Classification of bursting mappings. International Journal of Bifurcation and Chaos, 14, 3847–3854.CrossRefGoogle Scholar
  16. Keener, J., & Sneyd, J. (1998). Mathematical physiology (pp. 154–155). New York: Springer-Verlag.Google Scholar
  17. Lechner, H. A., Baxter, D. A., Clark, J. W., & Byrne, J. H. (1996). Bistability and its regulation by serotonin in the endogenously bursting neuron R15 in Aplysia. Journal of Neurophysiology, 75, 957–962.PubMedGoogle Scholar
  18. Lee, E., & Terman, D. (1999). Uniqueness and stability of periodic bursting solutions. Journal of Difference Equations, 158, 48–78.CrossRefGoogle Scholar
  19. Lofaro, T., & Kopell, N. (1999). Timing regulation in a network reduced from voltage-gated equations to a one-dimensional map. Journal of Mathematical Biology, 38, 479–533.PubMedCrossRefGoogle Scholar
  20. Llinas, R. R., & Steriade, M. (2006). Bursting of thalamic neurons and states of vigilance. Journal of Neurophysiology, 95, 3297–3308.PubMedCrossRefGoogle Scholar
  21. Manor, Y., & Nadim, F. (2001). Synaptic depression mediates bistability in neuronal networks with recurrent inhibitory connectivity. Journal of Neuroscience, 21, 9460–9470.PubMedGoogle Scholar
  22. Marder, E., & Calabrese, R. (1996). Principles of rhythmic motor pattern generation. Physiological Reviews, 76, 687–717.PubMedGoogle Scholar
  23. Masino, M. A., & Calabrese, R. L. (2002). Period differences between segmental oscillators produce intersegmental phase differences in the leech heartbeat timing network. Journal of Neurophysiology, 87, 1603–1615.PubMedGoogle Scholar
  24. Medvedev, G. (2005). Reduction of a model of an excitable cell to a one-dimensional map. Physica D, 202, 37–59.CrossRefGoogle Scholar
  25. Morris, C., & Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal, 35, 193–213.PubMedCrossRefGoogle Scholar
  26. Perkel, D. H., & Mulloney, B. (1974). Motor pattern production in reciprocally inhibitory neurons exhibiting postinhibitory rebound. Science, 185, 181–183.PubMedCrossRefGoogle Scholar
  27. Rubin, J., & Terman, D. (2000). Geometric analysis of population rhythms in synaptically coupled neuronal networks. Neural Computation, 12, 597–645.PubMedCrossRefGoogle Scholar
  28. Satterlie, R. (1985). Reciprocal inhibition and postinhibitory rebound produce reverberation in a locomotor pattern generator. Science, 229, 402–404.CrossRefPubMedGoogle Scholar
  29. Selverston, A., & Moulins, M. (1986). The Crustacean stomatogastric system : A model for the study of central nervous systems. Berlin Heidelberg New York: Springer.Google Scholar
  30. Skinner, F. K., Kopell, N., & Marder, E. (1994). Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks. Journal of Computational Neuroscience, 1, 69–87.PubMedCrossRefGoogle Scholar
  31. Sohal, V., & Huguenard, J. (2001). It takes T to tango. Neuron, 31, 35–45.CrossRefGoogle Scholar
  32. Terman, D. (1994). Chaotic spikes arising from a model of bursting in excitable membranes. SIAM Journal on Applied Mathematics, 51, 1418–1450.CrossRefGoogle Scholar
  33. Terman, D., Kopell, N., & Bose, A. (1998). Dynamics of two mutually coupled slow inhibitory neurons. Physica D, 117, 241–275.CrossRefGoogle Scholar
  34. Traub, R. D., Whittington, M. A., Colling, S. B., Buzsaki, G., & Jefferys, J. G. (1996). Analysis of gamma rhythms in the rat hippocampus in vitro and in vivo. Journal of Physiology, 493, 471–484.PubMedGoogle Scholar
  35. Van Vreeswijk, C., Abbott, L. F., & Ermentrout, B. (1994). When inhibition not excitation synchronizes neural firing. Journal of Computational Neuroscience, 1, 313–321.PubMedCrossRefGoogle Scholar
  36. Wang, X. J., & Buzsaki, G. (1996). Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. Journal of Neuroscience, 16, 6402–6413PubMedGoogle Scholar
  37. Wang, X. J., & Rinzel, J. (1992). Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Computation, 4, 84–97.Google Scholar
  38. Wang, X. J., & Rinzel, J. (1994). Spindle rhythmicity in the reticularis thalami nucleus: Synchronization among mutually inhibitory neurons. Neuroscience, 53, 899–904.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNew Jersey Institute of TechnologyNewarkUSA
  2. 2.Department of Biological SciencesRutgers UniversityNewarkUSA

Personalised recommendations