zbMATH — the first resource for mathematics

Clustering in globally coupled inhibitory neurons. (English) Zbl 0809.92003
Summary: A model of a large population of identical excitable neurons with a global slowly decaying inhibitory coupling is studied and its patterns of synchrony are examined. In addition to converging to a homogeneous fixed point and a homogeneous limit cycle, the system exhibits cluster states, in which it breaks spontaneously into a few macroscopically big clusters, each of which is fully synchronized. A method for calculating the stability of cluster states is described and used for investigating the dynamical behavior of the network versus the parameters that desribe the neurons and synapses. Effects of stochastic noise on the network dynamics are discussed. At large enough noise the system goes to a globally stationary state. Low levels of noise preserve the cluster-like neuron trajectories. In the regime where a noiseless system converges to a fully synchronized periodic state, relatively low noise levels cause neurons to burst only every two or more consecutive time periods.

92C20 Neural biology
Full Text: DOI
[1] Steriade, M.; Jones, E.G.; Llinás, R.R., Thalamic oscillations and signaling, (1990), Wiley New York
[2] Mulle, C.; Madariaga, A.; Deschenes, M., J. neurosci., 6, 2134, (1986)
[3] Steriade, M.; Domich, L.; Oakson, G.; Deschenes, M., J. neurophysiol., 57, 260, (1987)
[4] Wang, X.J.; Rinzel, J.; Rogawski, M.A., J. neurophysiol., 66, 839, (1991)
[5] Wang, X.J.; Rinzel, J., Neural comput., 4, 84, (1992)
[6] Wang, X.J.; Rinzel, J., Neuroscience, 53, 899, (1993)
[7] Perkel, D.H.; Mulloney, D., Science, 185, 181, (1974)
[8] Kaneko, K., Phys. rev. lett., Physica D, 41, 137, (1990)
[9] Golomb, D.; Hansel, D.; Shraiman, B.; Sompolinsky, H., Phys. rev. A, 45, 3516, (1992)
[10] Okuda, K., Physica D, 63, 424, (1993)
[11] Nakagawa, N.; Kuramoto, Y., Prog. theor. phys., 89, 313, (1993)
[12] Hansel, D.; Mato, G.; Meunier, C., Phys. rev. E, 48, 3470, (1993)
[13] Fisher, T.M.; Nicolellis, M.A.L.; Chapin, J.K., Neurosci. abstr., 18, 1392, (1992)
[14] Kuramoto, Y., Chemical oscillations, waves and turbulence, (1984), Springer New York · Zbl 0558.76051
[15] Kuramoto, Y.; Nishikawa, I., J. stat. phys., 49, 569, (1987)
[16] Hansel, D.; Mato, G.; Meunier, C., Europhys. lett., 23, 367, (1993)
[17] Amit, D.J., Modeling brain function, (), 65-66
[18] Van Kampen, N.G., Stochastic processes in physics and chemistry, (1981), North-Holland Amsterdam · Zbl 0511.60038
[19] Amit, D.J., Modeling brain function, (), 97-105
[20] (), 1
[21] Moss, F., ()
[22] Keener, J.P.; Hoppensteadt, F.C.; Rinzel, J., SIAM J. appl. math., 41, 503, (1981)
[23] Golomb, D.; Rinzel, J., Phys. rev. E, 48, 4810, (1993)
[24] D. Golomb, X.-J. Wang and J. Rinzel, in preparation.
[25] Golomb, D.; Wang, X.-J.; Rinzel, J., Neurosci. abstr., 19, 1584, (1993)
[26] Huguenard, J.R.; Prince, D.A., J. neurosci., 12, 3804, (1992)
[27] Huguenard, J.R.; McCormick, D.A., J. neurophys., 68, 1373, (1992)
[28] McCormick, D.A.; Huguenard, J.R., J. neurophys., 68, 1384, (1992)
[29] Rush, M.; Rinzel, J., Biol. cyber., (1994), in press
[30] Wang, X.J., Neuroscience, (1994), in press
[31] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer Berlin · Zbl 0515.34001
[32] Doedel, E., Cong. num., 30, 265, (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.