On the Phase Reduction and Response Dynamics of Neural Oscillator Populations

Neural Computation

Volumn 16, Issue 4, 2004, Pages 673-715

  Brown, Eric ^a   Moehlis, Jeff ^a,c   Holmes, Philip ^b  

^a PRINCETON UNIVERSITY   (United States)

^b Princeton University   (United States)

^c University of California at Santa Barbara   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ACTION POTENTIAL; ANIMAL; ARTICLE; BIOLOGICAL RHYTHM; BRAIN; ELECTROSTIMULATION; HUMAN; NERVE CELL; NERVE CELL NETWORK; PHYSIOLOGY; STATISTICAL MODEL; SYNAPTIC TRANSMISSION;

ACTION POTENTIALS; ANIMALS; BIOLOGICAL CLOCKS; BRAIN; ELECTRIC STIMULATION; HUMANS; MODELS, STATISTICAL; NERVE NET; NEURONS; SYNAPTIC TRANSMISSION;

EID: 1542351229     PISSN: 08997667     EISSN: None     Source Type: Journal    
DOI: 10.1162/089976604322860668     Document Type: Article

References (64)

1. Aston-Jones, G., Chen, S., Zhu, Y., & Oshinsky, M. (2001). A neural circuit for circadian regulation of arousal. Nature Neurosci., 4, 732-738.
2. Aston-Jones, G., Rajkowski, J., & Cohen, J. (2000). Locus coeruleus and regulation of behavioral flexibility and attention. Prog. Brain Res., 126, 165-182.
3. Aston-Jones, G., Rajkowski, J., Kubiak, P., & Alexinsky, T. (1994). Locus coeruleus neurons in the monkey are selectively activated by attended stimuli in a vigilance task. J. Neurosci., 14, 4467-4480.
4. Bogacz, R., Yeung, N., & Holroyd, C. (2002). Detection of phase resetting in the electroencephalogram: An evaluation of methods. Abstract, Society for Neuroscience, Washington DC.
5. Bressloff, P., & Coombes, S. (2000). Dynamics of strongly coupled spiking neurons. Neural Comp., 12, 91-129.
6. Brown, E., Holmes, P., & Moehlis, J. (2003a). Globally coupled oscillator networks. In E. Kaplan, J. Marsden, & K. Sreenivasan (Eds.), Problems and perspectives in nonlinear science, A celebratory volume in honor of Lawrence Sirovich (pp. 183-215). New York: Springer-Verlag.
7. Brown, E., Moehlis, J., Holmes, P., Clayton, E., Rajkowski, J., & Aston-Jones, G. (2003b). The influence of spike rate and stimulus duration on noradrenergic neurons. Unpublished manuscript, Program in Applied and Computational Mathematics, Princeton University.
8. Brunel, N., Chance, F., Fourcaud, N., & Abbott, L. (2001). Effects of synaptic noise and filtering on the frequency response of spiking neurons. Phys. Rev. Lett., 86, 2186-2189.
9. Casti, A., Omurtag, A., A., S., Kaplan, E., Knight, B., Sirovich, L., & Victor, J. (2001). A population study of integrate-and-fire-or-burst neurons. Neural Comp., 14, 957-986.
10. Coddington, E., & Levinson, N. (1955). Theory of ordinary differential equations. New York: McGraw-Hill.
11. Eckhorn, R. (1999). Neural mechanisms of scene segmentation: Recordings from the visual cortex suggest basic circuits for linking field models. IEEE Trans. Neural Networks, 10, 464-479.
12. Ermentrout, G. (1981). n: m phase locking of weakly coupled oscillators. J. Math. Biol., 12, 327-342.
13. Ermentrout, G. (1996). Type I membranes, phase resetting curves, and synchrony. Neural Comp., 8, 979-1001.
14. Ermentrout, G. (2002). Simulating, analyzing, and animating dynamical systems: A guide to XPPAUT for researchers and students. Philadelphia: SIAM.
15. Ermentrout, G., & Kopell, N. (1984). Frequency plateaus in a chain of weakly coupled oscillators, I. SIAM J. Math. Anal., 15, 215-237.
16. Ermentrout, G., & Kopell, N. (1990). Oscillator death in systems of coupled neural oscillators. SIAM J. Appl. Math., 50, 125-146.
17. Ermentrout, G., & Kopell, N. (1991). Multiple pulse interactions and averaging in coupled neural oscillators. J. Math. Biol., 29, 195-217.
18. Evans, L. (1998). Partial differential equations. Providence, RI: American Mathematical Society.
19. Fenichel, N. (1971). Persistence and smoothness of invariant manifolds for flows. Ind. Univ. Math. J., 21, 193-225.
20. Fetz, E., & Gustaffson, B. (1983). Relation between shapes of post-synaptic potentials and changes in firing probability of cat motoneurones. J. Physiol., 341, 387-410.
21. Fourcaud, N., & Brunel, N. (2002). Dynamics of the firing probability of noisy integrate-and-fire neurons. Neural Comp., 14, 2057-2110.
22. Gerstner, W. (2000). Population dynamics of spiking neurons: Fast transients, synchronous states, and locking. Neural Comp., 12, 43-89.
23. Gerstner, W., & Kistler, W. (2002). Spiking neuron models. Cambridge: Cambridge University Press.
24. Gerstner, W., van Hemmen, L., & Cowan, J. (1996). What matters in neuronal locking? Neural Comp., 8, 1653-1676.
25. Glass, L., & Mackey, M. (1988). From Clocks to chaos. Princeton, NJ: Princeton Paperbacks.
26. Gray, C. (2000). The temporal correlation hypothesis of visual feature integration: Still alive and well. Neuron, 24, 31-47.
27. Guckenheimer, J. (1975). Isochrons and phaseless sets. J. Math. Biol., 1, 259-273.
28. Guckenheimer, J., & Holmes, P. (1983). Nonlinear osculations, dynamical systems and bifurcations of vector fields. New York: Springer-Verlag.
29. Hansel, D., Mato, G., & Meunier, C. (1993). Phase dynamics for weakly coupled Hodgkin-Huxley neurons. Europhys. Lett., 25(5), 367-372.
30. Hansel, D., Mato, G., & Meunier, C. (1995). Synchrony in excitatory neural networks. Neural Comp., 7, 307-337.
31. Herrmann, A., & Gerstner, W. (2001). Noise and the PSTH response to current transients: I. General theory and application to the integrate-and-fire neuron. J. Comp. Neurosti., 11, 135-151.
32. Hirsch, M., Pugh, C., & Shub, M. (1977). Invariant manifolds. New York: SpringerVerlag.
33. Hodgkin, A., & Huxley, A. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 117, 500-544.
34. Hoppensteadt, P., & Izhikevich, E. (1997). Weakly connected neural networks. New York: Springer-Verlag.
35. Izhikevich, E. (2000a). Neural excitability, spiking and bursting. Int. J. Bif. Chaos, 10, 1171-1266.
36. Izhikevich, E. (2000b). Phase equations for relaxation oscillators. SIAM J. Appl. Math., 60, 1789-1804.
37. Keener, J., & Sneyd, J. (1998). Mathematical physiology. New York: Springer-Verlag.
38. Kim, S., & Lee, S. (2000). Phase dynamics in the biological neural networks. Physica D, 288, 380-396.
39. Kuramoto, Y. (1984). Chemical oscillations, waves, and turbulence. Berlin: Springer-Verlag.
40. Kuramoto, Y. (1997). Phase- and center-manifold reductions for large populations of coupled oscillators with application to non-locally coupled systems. M. J. Bif. Chaos, 7, 789-805.
41. Kuznetsov, Y. (1998). Elements of applied bifurcation theory (2nd ed.). New York: Springer-Verlag.
42. Lewis, T., & Rinzel, J. (2003). Dynamics of spiking neurons connected by both inhibitory and electrical coupling. J. Comp. Neurosci., 14, 283-309.
43. Mainen, Z., & Sejnowski, T. (1995). Reliability of spike timing in neocortical neurons. Science, 268, 1503-1506.
44. Makeig, S., Westerfield, M., Jung, T.-P., Enghoff, S., Townsend, J., Courchesne, E., & Sejnowski, T. (2002). Dynamic brain sources of visual evoked responses. Science, 295, 690-694.
45. Malkin, I. (1949). Methods ofPoincaré and Liapunov in the theory of nonlinear oscillations. Moscow: Gostexizdat.
46. Matell, M., & Meck, W. (2000). Neurophysiological mechanisms of interval timing behavior. BioEssays, 22, 94-103.
47. Moore, R., & Bloom, R (1979). Central catecholamine neuron systems: Anatomy and physiology of the norepinephrine and epinephrine systems. Ann. Rev. Neurosci., 2, 113-168.
48. Murray, J. (2002). Mathematical biology I: An introduction. (3rd ed.). New York: Springer-Verlag.
49. Nykamp, D., & Tranchina, D. (2000). A population density approach that facilitates large-scale modeling of neural networks: Analysis and application to orientation tuning. J. Comp. Neurosci., 8, 19-50.
50. Omurtag, A., Knight, B., & Sirovich, L. (2000). On the simulation of large populations of neurons. J. Comp. Neurosci., 8, 51-63.
51. Rinzel, J., & Ermentrout, G. (1998). Analysis of neural excitability and oscillations. In C. Koch & I. Segev (Eds.), Methods in neuronal modeling (pp. 251-291). Cambridge, MA: MIT Press.
52. Rinzel, J., & Miller, R. (1980). Numerical calculations of stable and unstable periodic solutions to the Hodgkin-Huxley equations. Math. Biosci., 49, 27-59.
53. Ritt, J. (2003). A probabilistic analysis of forced oscillators, with application to neuronal response reliability. Unpublished doctoral dissertation, Boston University.
54. Rose, R., & Hindmarsh, J. (1989). The assembly of ionic currents in a thalamic neuron I. The three-dimensional model. Proc. R. Soc. Lond. B, 237, 267-288.
55. Servan-Schreiber, D., Printz, H., & Cohen, J. (1990). A network model of catecholamine effects: Gain, signal-to-noise ratio, and behavior. Science, 249, 892-895.
56. Stein, R. (1965). A theoretical analysis of neuronal variability. Biophys. J., 5, 173-194.
57. Strogatz, S. (2000). From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143, 1-20.
58. Tass, P. (1999). Phase resetting in medicine and biology. New York: Springer-Verlag.
59. Usher, M., Cohen, J., Servan-Schreiber, D., Rajkowsky, J., & Aston-Jones, G. (1999). The role of locus coeruleus in the regulation of cognitive performance. Science, 283, 549-554.
60. van Vreeswijk, C., Abbot, L., & Ermentrout, G. (1994). When inhibition not excitation synchronizes neural firing. J. Comp. Neurosci., 1, 313-321.
61. Whitham, G. (1974). Linear and nonlinear waves. New York: Wiley.
62. Wilson, H., & Cowan, J. (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J., 22, 1-24.
63. Winfree, A. (1974). Patterns of phase compromise in biological cycles, J. Math. Biol., 1, 73-95.
64. Winfree, A. (2001). The geometry of biological time. Second Edition. New York: Springer-Verlag.