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

Journal of Computational Neuroscience

Volumn 23, Issue 2, 2007, Pages 169-187

  Matveev, Victor ^a   Bose, Amitabha ^a   Nadim, Farzan ^a,b  

^a NEW JERSEY INSTITUTE OF TECHNOLOGY   (United States)

^b RUTGERS UNIVERSITY   (United States)

Author keywords

Dimensional reduction; Half center bursting; Multistability; Poincar return map; T type calcium current

Indexed keywords

CALCIUM; CYTOLOGY;

ANTIPHASE; CALCIUM CURRENT; DIMENSIONAL REDUCTION; HALF-CENTER BURSTING; MULTISTABILITY; ONE DIMENSIONAL MAP; POINCARÉ RETURN MAP; PROPERTY; RETURN MAP; T-TYPE CALCIUM CURRENT;

CELLS;

CALCIUM ION; POTASSIUM ION;

ACCURACY; ARTICLE; CALCIUM CURRENT; CELL KINETICS; CONTROLLED STUDY; DEPOLARIZATION; HYPERPOLARIZATION; MATHEMATICAL COMPUTING; NERVE CELL EXCITABILITY; NERVE CELL MEMBRANE STEADY POTENTIAL; NERVE CELL NETWORK; POTASSIUM CURRENT; PRIORITY JOURNAL; REGULATORY MECHANISM; SENSITIVITY ANALYSIS; SPIKE WAVE; SYNAPTIC INHIBITION;

EID: 34548231662     PISSN: 09295313     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10827-007-0026-x     Document Type: Article

References (38)

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.
2. Bertram, R., & Sherman, A. (2000). Dynamical complexity and temporal plasticity in pancreatic beta-cells. Journal of Biosciences, 25, 197-209.
3. Bose, A., Manor, Y., & Nadim, F. (2001). Bistable oscillations arising from synaptic depression. SIAM Journal on Applied Mathematics, 62, 706-727.
4. Butera, R. J. (1998). Multirhythmic bursting. Chaos, 8, 274-284.
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.
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.
7. Chay, T. R., & Rinzel, J. (1985). Bursting, beating, and chaos in an excitable membrane model. Biophysical Journal, 47, 357-366.
8. Coombes, C., & Bressloff, P. (Eds.) (2005). Bursting: The genesis of rhythm in the nervous system. London: World Scientific.
9. Destexhe, A., & Sejnowski, T. J. (2003). Interactions between membrane conductances underlying thalamocortical slow-wave oscillations. Physiological Reviews, 83, 1401-1453.
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.
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.
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.
13. Huguenard, J. R. (1996). Low-threshold calcium currents in central nervous system neurons. Annual Review of Physiology, 58, 329-48.
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.
15. Izhikevich, E. M., & Hoppensteadt, F. C. (2004). Classification of bursting mappings. International Journal of Bifurcation and Chaos, 14, 3847-3854.
16. Keener, J., & Sneyd, J. (1998). Mathematical physiology (pp. 154-155). New York: Springer-Verlag.
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.
18. Lee, E., & Terman, D. (1999). Uniqueness and stability of periodic bursting solutions. Journal of Difference Equations, 158, 48-78.
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.
20. Llinas, R. R., & Steriade, M. (2006). Bursting of thalamic neurons and states of vigilance. Journal of Neurophysiology, 95, 3297-3308.
21. Manor, Y., & Nadim, F. (2001). Synaptic depression mediates bistability in neuronal networks with recurrent inhibitory connectivity. Journal of Neuroscience, 21, 9460-9470.
22. Marder, E., & Calabrese, R. (1996). Principles of rhythmic motor pattern generation. Physiological Reviews, 76, 687-717.
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.
24. Medvedev, G. (2005). Reduction of a model of an excitable cell to a one-dimensional map. Physica D, 202, 37-59.
25. Morris, C., & Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal, 35, 193-213.
26. Perkel, D. H., & Mulloney, B. (1974). Motor pattern production in reciprocally inhibitory neurons exhibiting postinhibitory rebound. Science, 185, 181-183.
27. Rubin, J., & Terman, D. (2000). Geometric analysis of population rhythms in synaptically coupled neuronal networks. Neural Computation, 12, 597-645.
28. Satterlie, R. (1985). Reciprocal inhibition and postinhibitory rebound produce reverberation in a locomotor pattern generator. Science, 229, 402-404.
29. Selverston, A., & Moulins, M. (1986). The Crustacean stomatogastric system: A model for the study of central nervous systems. Berlin Heidelberg New York: Springer.
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.
31. Sohal, V., & Huguenard, J. (2001). It takes T to tango. Neuron, 31, 35-45.
32. Terman, D. (1994). Chaotic spikes arising from a model of bursting in excitable membranes. SIAM Journal on Applied Mathematics, 51, 1418-1450.
33. Terman, D., Kopell, N., & Bose, A. (1998). Dynamics of two mutually coupled slow inhibitory neurons. Physica D, 117, 241-275.
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.
35. Van Vreeswijk, C., Abbott, L. F., & Ermentrout, B. (1994). When inhibition not excitation synchronizes neural firing. Journal of Computational Neuroscience, 1, 313-321.
36. Wang, X. J., & Buzsaki, G. (1996). Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. Journal of Neuroscience, 16, 6402-6413
37. Wang, X. J., & Rinzel, J. (1992). Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Computation, 4, 84-97.
38. Wang, X. J., & Rinzel, J. (1994). Spindle rhythmicity in the reticularis thalami nucleus: Synchronization among mutually inhibitory neurons. Neuroscience, 53, 899-904.