Delays and weakly coupled neuronal oscillators

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 367, Issue 1891, 2009, Pages 1097-1115

  Ermentrout, Bard ^a   Ko, Tae Wook ^a  

^a UNIVERSITY OF PITTSBURGH   (United States)

Author keywords

Delay; Oscillations; Phase locking; Synchrony

Indexed keywords

DENDRIMERS; MEAN FIELD THEORY; MOTION ESTIMATION; NEURAL NETWORKS; STATISTICAL MECHANICS;

CONSTANT DELAYS; COUPLED OSCILLATORS; COUPLED SYSTEMS; DELAY; DENDRITIC BRANCHES; DISTRIBUTED DELAYS; IN NETWORKS; MEAN FIELDS; NEURONAL OSCILLATORS; OSCILLATIONS; PHASE LOCKING; PHASE MODELS; STATE VARIABLES; SYNCHRONY; TRAVELLING WAVES;

OSCILLATORS (ELECTRONIC);

BEHAVIOR; BIOLOGICAL MODEL; CIRCADIAN RHYTHM; DENDRITE; HUMAN; NERVE CELL; NERVOUS SYSTEM FUNCTION; NONLINEAR SYSTEM; OSCILLOMETRY; PERIODICITY; PHYSIOLOGY; REACTION TIME; REVIEW;

BEHAVIOR; CIRCADIAN RHYTHM; DENDRITES; HUMANS; MODELS, NEUROLOGICAL; NERVOUS SYSTEM PHYSIOLOGICAL PHENOMENA; NEURONS; NONLINEAR DYNAMICS; OSCILLOMETRY; PERIODICITY; REACTION TIME;

EID: 60749112869     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2008.0259     Document Type: Article

References (35)

1. Barahona, M. & Pecora, L. M. 2002 Synchronization in small-world systems. Phys. Rev. Lett. 89, 054 101. (doi:10.1103/PhysRevLett. 89.054101)
2. Borisyuk, A., Ermcntrout, B., Friedman, A. & Terman, D. 2005 Tutorials in mathematical biosciences I. New York, NY: Springer.
3. Bressloff, P. C. & Coombes, S. 1997a Physics of the extended neuron. Int. J. Mod. Phys. B 11, 2343. (doi:10.1142/ S0217979297001209)
4. Bressloff, P. C. & Coombes, S. 1997b Synchrony in an array of integrate-and-fire neurons with dendritic structure. Phys. Rev. Lett. 78, 4665-4668. (doi:10.1103/PhysRcvLctt.78.4665)
5. Buszaki, G. 2006 Rhythms of the brain. Oxford, UK: Oxford University Press.
6. Choi, M. Y., Kim, H. J., Kim, D. & Hong, H. 2000 Synchronization in a system of globally coupled oscillators with time delay. Phys. Rev. E 61, 371. (doi:10.1103/PhysRevE.61.371)
7. Cohen, A. H., Ermentrout, G. B., Kiemel, T., Kopell, N., Sigvardt, K. A. & Williams, T. L. 1992 Modelling of intersegmental coordination in the lamprey central pattern generator for locomotion. Trends Neurosci. 15, 434-438. (doi:10.1016/0166-2236(92)90006-T)
8. Crook, S. M., Ermcntrout, G. B., Vender, M. C. & Bower, J. M. 1997 The role of axonal delay in the synchronization of networks of coupled cortical oscillators. J. Comput. Neurosci. 4, 161-172. (doi:10.1023/A: 1008843412952)
9. Crook, S. M., Ermentrout, G. B. & Bower, J. M. 1998 Dendritic and synaptic effects in systems of coupled cortical oscillators. J. Comput. Neurosci. 5, 315-329. (doi:10.1023/A:1008839112707)
10. Dayan, P. & Abbott, L. 2001 Theoretical neuroscience. Cambridge, MA: MIT Press.
11. Doiron, B., Chacron, M. J., Maler, L., Longtin, A. & Bastian, J. 2003 Inhibitory feedback required for network oscillatory responses to communication but not prey stimuli. Nature 421, 539-543. (doi:10.1038/ nature01360)
12. Earl, M. G. & Strogatz, S. H. 2003 Synchronization in oscillator networks with delayed coupling: a stability criterion. Phys. Rev. E 67. 036204. (doi:10.1103/PhysRevE.67.036204)
13. Ermentrout, B. 1995 A heuristic description of spiral wave instability in discrete media. Physica D 82, 154-164. (doi:10.1016/0167-2789(94) 00221-B)
14. Ermentrout, G. B. & Kleinfeld, D. 2001 Traveling electrical waves in cortex: insights from phase dynamics and speculation on a computational role. Neuron 29, 33-44. (doi:10.1016/S0896-6273(01)00178-7)
15. Ermentrout, G. B. & Kopell, N. 1984 Frequency plateaus in a chain of weakly coupled oscillators. I. SIAM J. Math. Anal. 15, 215-237. (doi:10.1137/0515019)
16. Ermentrout, G. B. & Kopell, N. 1994 Inhibition-produced patterning in chains of coupled nonlinear oscillators. SIAM J. Appl. Math. 54, 478-507. (doi:10.1137/S0036139992231964)
17. Glass, L. & Mackey, M. 1988 From clocks to chaos. Princeton, NJ: Princeton University Press.
18. Huang, X., Troy, W. C., Yang, Q., Ma, H., Laing, C. R., Schiff, S. J. & Wu, J. Y. 2004 Spiral waves in disinhibited mammalian neocortex. J. Neurosci. 24, 9897-9902. (doi:10.1523/JNEUROSCI. 2705-04.2004)
19. Izhikevich, E. M. 1998 Phase models with explicit time delays. Phys. Rev. E 58, 905-908. (doi:10. 1103/PhysRevE.58.905)
20. Izhikevich, E. M. 2007 Dynamical systems in neuroscience. Cambridge, MA: MIT Press.
21. Jeong, S.-O., Ko, T. W. & Moon, H. T. 2002 Time-delayed spatial patterns in a two-dimensional array of coupled oscillators. Phys. Rev. Lett. 89, 154104. (doi:10.1103/PhysRevLett.89.154104)
22. Kim, S., Park, S. H. & Ryu, C. S. 1997 Multistability in coupled oscillator systems with time delay. Phys. Rev. Lett. 79, 2911. (doi:10.1103/PhysRevLett.79.2911)
23. Ko, T.-W. & Ermentrout, G. B. 2007 Effects of axonal time delay on synchronization and wave formation in sparsely coupled neuronal oscillators. Phys. Rev. E 76, 056206. (doi:10.1103/PhysRcvE.76.056206)
24. Ko, T. W., Jeong, S. O. & Moon, H. T. 2004 Wave formation by time delays in randomly coupled oscillators. Phys. Rev. E 69, 056106. (doi:10.1103/PhysRevE.69.056106)
25. Lewis, T. J. & Rinzel, J. 2003 Dynamics of spiking neurons connected by both inhibitory and electrical coupling. J. Comput. Neurosci. 14, 283-309. (doi:10.1023/A:1023265027714)
26. Nakamura, Y., Tominaga, F. & Munakata, T. 1994 Clustering behavior of time-delayed nearestneighbor coupled oscillators. Phys. Rev. E 49, 4849. (doi:10.1103/PhysRevE.49.4849)
27. Niebur, E., Schuster, H. G. & Kammen, D. M. 1991 Collective frequencies and metastability in networks of limit-cycle oscillators with time delay. Phys. Rev. Lett. 67, 2753. (doi:10.1103/PhysRevLett.67. 2753)
28. Paullet, J. E. & Ermentrout, G. B. 1994 Stable rotating waves in two-dimensional discrete active media. SIAM J. Appl. Math. 54, 1720-1744. (doi:10.1137/S0036139993250683)
29. Schuster, H. G. & Wagner, P. 1989 Mutual entrainment of two limit cycle oscillators with time delayed coupling. Prog. Theor. Phys. 81, 939. (doi:10.1143/PTP.81.939)
30. Traub, R. D. et al. 2005 Single-column thalamocortical network model exhibiting gamma oscillations, sleep spindles, and epileptogenic bursts. J. Neurophysiol. 93, 2194-2232. (doi:10. 1152/jn.00983.2004)
31. Watts, D. J. & Strogatz, S. H. 1998 Collective dynamics of 'small-world' networks. Nature 393, 440-442. (doi:10.1038/30918)
32. Whittington, M. A., Traub, R. D., Kopell, N., Ermentrout, B. & Buhl, E. H. 2000 Inhibitionbased rhythms: experimental and mathematical observations on network dynamics. Int. J. Psychophysiol. 38, 315-336. (doi:10.1016/S0167-8760(00)00173-2)
33. Yeung, M. K. S. & Strogatz, S. H. 1999 Time delay in the Kuramoto model of coupled oscillators. Phys. Rev. Lett. 82, 648. (doi:10.1103/PhysRcvLett.82.648)
34. Yuste, R., MacLean, J. N., Smith, J. & Lansner, A. 2005 The cortex as a central pattern generator. Nat. Rev. Neurosci. 6, 477-483. (doi:10.1038/nrn1686)
35. Zanette, D. H. 2000 Propagating structures in globally coupled systems with time delays. Phys. Rev. E 62, 3167. (doi:10.1103/PhysRevE.62. 3167)