Dominant ionic mechanisms explored in spiking and bursting using local low-dimensional reductions of a biophysically realistic model neuron

  • Robert Clewley
  • Cristina Soto-Treviño
  • Farzan Nadim
Article

Abstract

The large number of variables involved in many biophysical models can conceal potentially simple dynamical mechanisms governing the properties of its solutions and the transitions between them as parameters are varied. To address this issue, we extend a novel model reduction method, based on “scales of dominance,” to multi-compartment models. We use this method to systematically reduce the dimension of a two-compartment conductance-based model of a crustacean pyloric dilator (PD) neuron that exhibits distinct modes of oscillation—tonic spiking, intermediate bursting and strong bursting. We divide trajectories into intervals dominated by a smaller number of variables, resulting in a locally reduced hybrid model whose dimension varies between two and six in different temporal regimes. The reduced model exhibits the same modes of oscillation as the 16 dimensional model over a comparable parameter range, and requires fewer ad hoc simplifications than a more traditional reduction to a single, globally valid model. The hybrid model highlights low-dimensional organizing structure in the dynamics of the PD neuron, and the dependence of its oscillations on parameters such as the maximal conductances of calcium currents. Our technique could be used to build hybrid low-dimensional models from any large multi-compartment conductance-based model in order to analyze the interactions between different modes of activity.

Keywords

Model reduction Compartmental modeling Oscillations Stomatogastric Hybrid dynamical system 

References

  1. Bose, A., & Booth, V. (2005). Bursting in two-compartment neurons: a case study of the Pinsky–Rinzel model. In Bursting: The genesis of rhythm in the nervous system (pp. 123–144). Singapore: World Scientific.Google Scholar
  2. Bucher, D., Thirumalai, V., & Marder, E. (2003). Axonal dopamine receptors activate peripheral spike initiation in a stomatogastric motor neuron. Journal of Neuroscience, 23, 6866–6875.PubMedGoogle Scholar
  3. Butera, R. J., Clark, J. W., Byrne, J. H., & Rinzel, J. (1996). Dissection and reduction of a modeled bursting neuron. Journal of Computational Neuroscience, 3, 199–223.PubMedCrossRefGoogle Scholar
  4. Chow, C. C., & Kopell, N. (2000). Dynamics of spiking neurons with electrical coupling. Neural Computation, 12, 1643–1678.PubMedCrossRefGoogle Scholar
  5. Clewley, R., Rotstein, H. G., & Kopell, N. (2005). A computational tool for the reduction of nonlinear ODE systems possessing multiple scales. Multiscale Modeling and Simulation, 4, 732–759.CrossRefGoogle Scholar
  6. de Vries, G., & Sherman, A. (2001). From spikers to bursters via coupling: help from heterogeneity. Bulletin of Mathematical Biology, 63, 371–391.PubMedCrossRefGoogle Scholar
  7. Fitzhugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophysics Journal, 1, 445–466.CrossRefGoogle Scholar
  8. Golomb, D., Guckenheimer, J., & Gueron, S. (1993). Reduction of a channel-based model for a stomatogastric ganglion LP neuron. Biological Cybernetics, 69, 129–137.CrossRefGoogle Scholar
  9. Guckenheimer, J., & Holmes, P. (1983). Nonlinear oscillations, dynamical systems and bifurcations of vector fields. New York: Springer-Verlag.Google Scholar
  10. Guckenheimer, J., Tien, J. H., & Willms, A. R. (2005). Bifurcations in the fast dynamics of neurons: implications for bursting. In S. Coombes, & P. Bressloff (Eds.), Bursting: The Genesis of Rhythm in the Nervous System. Singapore: World Scientific.Google Scholar
  11. Hines, M. L., Morse, T., Migliore, M., Carnevale, N. T., & Shepherd, G. M. (2004). ModelDB: a database to support computational neuroscience. Journal of Computational Neuroscience, 17, 7–11.PubMedCrossRefGoogle Scholar
  12. Johnson, B. R., Kloppenburg, P., & Harris-Warrick, R. M. (2003). Dopamine modulation of calcium currents in pyloric neurons of the lobster stomatogastric ganglion. Journal of Neurophysiology, 90, 631–643.PubMedCrossRefGoogle Scholar
  13. Kepler, T. B., Abbott, L. F., & Marder, E. (1992). Reduction of conductance-based neuron models. Biological Cybernetics, 66.Google Scholar
  14. Kopell, N., & LeMasson, G. (1994). Rhythmogenesis, amplitude modulation, and multiplexing in a cortical architecture. Proceedings of the National Academy of Sciences of the United States of America, 91, 10586–10590.PubMedCrossRefGoogle Scholar
  15. Marder, E. (1984). Mechanisms underlying neurotransmitter modulation of neuronal circuit. Trends in Neurosciences, 7, 48–53.CrossRefGoogle Scholar
  16. Marder, E., & Bucher, D. (2007). Understanding circuit dynamics using the stomatogastric nervous system of lobsters and crabs. Annual Review of Physiology, 69, 1–26.CrossRefGoogle Scholar
  17. Medvedev, G. S., & Kopell, N. (2001). Synchronization and transient dynamics in chains of electrically coupled Fitzhugh-Nagumo oscillators. SIAM Journal on Applied Mathematics, 61, 1762–1801.CrossRefGoogle Scholar
  18. Meunier, C. (1992). Two and three dimensional reductions of the Hodgkin–Huxley system: separation of time scales and bifurcation schemes. Biological Cybernetics, 67, 461–468.PubMedCrossRefGoogle Scholar
  19. Miller, J. P., & Selverston, A. I. (1982). Mechanisms underlying pattern generation in lobster stomatogastric ganglion as determined by selective inactivation of identified neurons. II. Oscillatory properties of pyloric neurons. Journal of Neurophysiology, 48, 1378–1391.PubMedGoogle Scholar
  20. Murray, J. D. (1989). Mathematical biology. New York: Springer-Verlag.Google Scholar
  21. Nusbaum, M., & Beenhakker, M. (2002). A small-systems approach to motor pattern generation. Nature, 417, 343–350.PubMedCrossRefGoogle Scholar
  22. Prinz, A. A., Bucher, D., & Marder, E. (2004a). Similar network activity from disparate circuit parameters. Nature Neuroscience, 7, 1345–1353.PubMedCrossRefGoogle Scholar
  23. Prinz, A. A., Abbott, L. F., & Marder, E. (2004b). The dynamic clamp comes of age. Trends in Neurosciences, 27, 218–224.PubMedCrossRefGoogle Scholar
  24. Shelley, M., McLaughlin, D., Shapley, R., & Wielaard, J. (2002). States of high conductance in a large-scale model of the visual cortex. Journal of Computational Neuroscience, 13, 93–109.PubMedCrossRefGoogle Scholar
  25. Sherman, A., & Rinzel, J. (1992). Rhythmogenic effects of weak electrotonic coupling in neuronal models. Proceedings of the National Academy of Sciences of the United States of America, 89, 2471–2474.PubMedCrossRefGoogle Scholar
  26. Soto-Treviño, C., Rabbah, P., Marder, E., & Nadim, F. (2005). A computational model of electrically coupled, intrinsically distinct pacemaker neurons. Journal of Neurophysiology, 94, 590–604.PubMedCrossRefGoogle Scholar
  27. Strogatz, S. H. (2001). Nonlinear Dynamics and Chaos: Perseus Books.Google Scholar
  28. Suckley, R., & Biktashev, V. (2003). The asymptotic structure of the Hodgkin–Huxley equations. International Journal of Bifurcation and Chaos, 13, 3805–3826.CrossRefGoogle Scholar
  29. Terman, D. (1991). Chaotic spikes arising from a model of bursting in excitable membranes. SIAM Journal on Applied Mathematics, 51, 1418–1450.CrossRefGoogle Scholar
  30. Van Der Schaft, A., & Schumacher, J. M. (2000). An Introduction to hybrid systems. London: Springer-Verlag.Google Scholar
  31. Wilson, H. R., & Cowan, J. D. (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophysics Journal, 12, 1–24.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Robert Clewley
    • 1
  • Cristina Soto-Treviño
    • 2
  • Farzan Nadim
    • 2
    • 3
  1. 1.Department of Mathematics and StatisticsGeorgia State UniversityAtlantaUSA
  2. 2.Department of Mathematical SciencesNew Jersey Institute of TechnologyNewarkUSA
  3. 3.Department of Biological SciencesRutgers UniversityNewarkUSA

Personalised recommendations