Abstract
In this chapter, we consider a very different approach to studying networks of neurons from that presented in Chap. 8. In Chap. 8, we assumed each cell is an intrinsic oscillator, the coupling is weak, and details of the spikes are not important. By assuming weak coupling, we were able to exploit powerful analytic techniques such as the phase response curve and the method of averaging. In this chapter, we do not assume, in general, weak coupling or the cells are intrinsic oscillators. The main mathematical tool used in this chapter is geometric singular perturbation theory. Here, we assume the model has multiple timescales so we can dissect the full system of equations into fast and slow subsystems. This will allow us to reduce the complexity of the full model to a lower-dimensional system of equations. We have, in fact, introduced this approach in earlier chapters when we discussed bursting oscillations and certain aspects of the Morris–Lecar model.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Bose, N. Kopell, and D. Terman. Almost-synchronous solutions for mutually coupled excitatory neurons. Phys. D, 140:69–94, 2000.
T. G. Brown. The intrinsic factors in the act of progression in the mammal. Proc. R. Soc. Lond. B, 84:308–319, 1911.
W. Gerstner, J. L. van Hemmen, and J. Cowan. What matters in neuronal locking? Neural Comput., 8:1653–1676, 1996.
N. Kopell and B. Ermentrout. Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators. In B. Fiedler, G. Iooss, and N. Kopell, editors, Handbook of Dynamical Systems II: Towards Applications. Elsevier, Amsterdam, 2002.
N. Kopell and D. Somers. Anti-phase solutions in relaxation oscillators coupled through excitatory interactions. J. Math. Biol., 33:261–280, 1995.
D. H. Perkel and B. Mulloney. Motor pattern production in reciprocally inhibitory neurons exhibiting postsynaptic rebound. Science, 145:61–63, 1974.
P. Pinsky and J. Rinzel. Intrinsic and network rhythmogenesis in a reduced traub model of ca3 neurons. J. Comput. Neurosci., 1:39–60, 1994.
J. Rubin and D. Terman. Analysis of clustered firing patterns in synaptically coupled networks of oscillators. J. Math. Biol., 41:513–545, 2000.
C. B. Saper, T. C. Chou, and T. E. Scammell. The sleep switch: hypothalamic control of sleep and wakefulness. Trends Neurosci., 24:726–731, 2001.
F. Skinner, N. Kopell, and E. Marder. Mechanisms for oscillation and frequency control in networks of mutually inhibitory relaxation oscillators. J. Comput. Neurosci., 1:69–87, 1994.
M. Steriade. Neuronal Substrates of Sleep and Epilepsy. Cambridge University Press, Cambridge, 2003.
D. Terman and D. L. Wang. Global competition and local cooperation in a network of neural oscillators. Phys. D, 81:148–176, 1995.
D. Terman, G. Ermentrout, and A. Yew. Propagating activity patterns in thalamic neuronal networks. SIAM J. Appl. Math., 61:1578–1604, 2001.
D. Terman, S. Ahn, X. Wang, and W. Just. Reducing neuronal networks to discrete dynamics. Phys. D, 237:324–338, 2008.
C. Van Vreeswijk, L. F. Abbott, and G. B. Ermentrout. When inhibition not excitation synchronizes neural firing. J. Comput. Neurosci., 1:313–321, 1994.
X.-J. Wang and J. Rinzel. Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Comput., 4:84–97, 1992.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Ermentrout, G.B., Terman, D.H. (2010). Neuronal Networks: Fast/Slow Analysis. In: Mathematical Foundations of Neuroscience. Interdisciplinary Applied Mathematics, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-0-387-87708-2_9
Download citation
DOI: https://doi.org/10.1007/978-0-387-87708-2_9
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-87707-5
Online ISBN: 978-0-387-87708-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)