Slow waves in mutually inhibitory neuronal networks

Physica D: Nonlinear Phenomena

Volumn 192, Issue 1-2, 2004, Pages 95-122

  Jalics, Jozsi ^a  

^a USA   (United States)

Author keywords

Singular perturbation; Synaptic coupling; Thalamus; Traveling waves

Indexed keywords

DIFFERENTIAL EQUATIONS; INTEGRAL EQUATIONS; MATHEMATICAL MODELS; NEURAL NETWORKS; OSCILLATIONS; PERTURBATION TECHNIQUES; PROBLEM SOLVING;

SYNAPTIC COUPLING; TRAVELING WAVES;

WAVE PROPAGATION;

EID: 1842842203     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2004.01.001     Document Type: Article

References (32)

1. Angstadt J.D., Calabrese R.L., A hyperpolarization-activated inward current in heart interneurons of the medicinal leech. J. Neurosci. 9:1989;2846-2857.
2. Bazhenov M., Timofeev I., Steriade M., Sejnowski J., Self-sustained rhythmic activity in the thalamic reticular nucleus mediated by depolarizing GABA-A receptor potentials. Nature Neurosci. 2:1999;168-174.
3. Bose A., Kopell N., Terman D., Almost-synchronous solutions for mutually coupled excitatory neurons. Physica D. 140:2000;69-94.
4. Deng B., The existence of infinitely many traveling front and back waves in the FitzHugh-Nagumo equations. SIAM J. Math. Anal. 22:1991;1631-1650.
5. Destexhe A., Bal T., McCormick D.A., Sejnowski T.J., Ionic mechanisms underlying synchronized oscillations and propagating waves in a model of ferret thalamic slices. J. Neurophys. 76:1996;2049-2070.
6. Doelman A., Gardner R., Kaper T., Large stable pulse solutions in reaction-diffusion equations. Ind. Univ. Math. J. 50(1):2001;443-507.
7. F. Dumortier, R. Roussarie, Geometric singular perturbation theory beyond normal hyperbolicity, in: Multiple Time-scale Dynamical Systems, IMA Volumes in Mathematics and its Applications, vol. 122, 2001, pp. 29-63.
8. Fenichel N., Persistence and smoothness of invariant manifolds for flows. Ind. Univ. Math. J. 21:1971;193-226.
9. Fenichel N., Geometric singular perturbation theory. J. Diff. Equ. 31:1979;53-98.
10. Golomb D., Ermentrout B., Continuous and lurching traveling pulses in neuronal networks with spatially-decaying connectivity and delay. Proc. Natl. Acad. Sci. U.S.A. 96:1999;13480-13485.
11. Golomb D., Wang X.-J., Rinzel J., Propagation of spindle waves in a thalamic slice model. J. Neurophysiol. 75:1996;750-769.
12. J. Jalics, Existence of slow waves in mutually inhibitory thalamic neuronal networks, Ph.D. Thesis, The Ohio State University, 2002.
13. C.K.R.T. Jones, Geometric singular perturbation theory, in: Dynamical Systems, Springer Lecture Notes in Mathematics, vol. 1609, 1995, pp. 44-120.
14. C.K.R.T. Jones, N. Kopell, R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, in: H. Swinney, G. Aris, D.G. Aronson (Eds.), IMA Volumes in Mathematics and its Applications, vol. 37, Springer, New York, 1991.
15. Kim U., Bal T., McCormick D.A., Spindle waves are propagating synchronized oscillations in the ferret LGNd in vitro. J. Neurophysiol. 74:1995;1301-1323.
16. Krupa M., Sandstede B., Szmolyan P., Fast and slow waves in the FitzHugh-Nagumo equation. J. Diff. Equ. 133:1997;49-97.
17. Krupa M., Szmolyan P., Extending geometric singular perturbation theory to non-hyperbolic points-fold and canard points in two dimensions. SIAM J. Math. Anal. 33(2):2001;286-314.
18. R. Langer, Existence of homoclinic traveling wave solutions to the FitzHugh-Nagumo equations, Ph.D. Thesis, Northeastern University, 1980.
19. McCormick D.A., Pape H.C., Properties of a hyperpolarization-activated cation current and its role in rhythmic oscillation in thalamic relay neurone. J. Physiol. 431:1990;291-318.
20. Pape H.C., Queer current and pacemaker: the hyperpolarization-activated cation current in neurons. Ann. Rev. Physiol. 58:1996;299-327.
21. Pinto D.J., Ermentrout G.B., Spatially structured activity in synaptically coupled neuronal networks. I. Traveling fronts and pulses. SIAM J. Appl. Math. 62(1):2001;206-225.
22. Rinzel J., Terman D., Wang X.-J., Ermentrout B., Propagating activity patterns in large-scale inhibitory neuronal networks. Science. 279:1998;1351-1355.
23. Rubin J.E., Terman D., Geometric analysis of population rhythms in synaptically coupled neuronal networks. Neural Comput. 12(3):2000;597-645.
24. J. Rubin, D. Terman, Geometric singular perturbation analysis of neuronal dynamics, in: B. Fiedler, G. Iooss, N. Kopell (Eds.), Handbook of Dynamical Systems, vol. 3: Toward Applications, Elsevier, Amsterdam, 2002.
25. Skinner F.K., Kopell N., Marder E., Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks. J. Comput. Neurosci. 1:1994;69-87.
26. 3, preprint.
27. V. Sohal, J. Huguenard, Long-range connections synchronize rather than spread intrathalamic oscillations: computational modeling and in vitro electrophysiology, J. Neurophysiol. 80 (1998) 1736-1751.
28. Terman D.H., Ermentrout G.B., Yew A.C., Propagating activity patterns in thalamic neuronal networks. SIAM J. Appl. Math. 61:2001;1578-1604.
29. Terman D., Kopell N., Bose A., Dynamics of two mutually coupled inhibitory neurons. Physica D. 117:1998;241-275.
30. Wang X.-J., Rinzel J., Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Comput. 4:1992;84-97.
31. 3, Thesis, TU-Wien, 1998.
32. L. Zhang, Existence and stability of traveling wave solutions of neuronal network equations, Ph.D. Thesis, The Ohio State University, 1999.