Critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling

Neural Computation

Volumn 19, Issue 8, 2007, Pages 2032-2092

  Ly, Cheng ^a   Tranchina, Daniel ^a  

^a NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34548013119     PISSN: 08997667     EISSN: 1530888X     Source Type: Journal    
DOI: 10.1162/neco.2007.19.8.2032     Document Type: Article

References (66)

1. Abbott, L. F., & van Vreeswijk, C. (1993). Asynchronous states in networks of pulse-coupled oscillators. Physical Review E, 48, 1483-1490.
2. Amit, D., & Brunel, N. (1997). Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex. Cerebral Cortex, 7, 237-252.
3. Apfaltrer, F., Ly, C., & Tranchina, D. (2006). Population density methods for stochastic neurons with a 2-D state space: Application to neural networks with realistic synaptic kinetics. Network: Computation in Neural Systems, 17, 373-418.
4. Bardos, C., Galose, F., & Levermore, D. (1991). Fluid dynamic limit of kinetic equations 1: Formal derivations. Journal of Statistical Physics, 63, 323-344.
5. Bardos, C., Galose, F., & Levermore, D. (1993). Fluid dynamic limit of kinetic equations 2: Convergence proofs for the Boltzmann equation. Communication on Pure and Applied Math, 46, 667-753.
6. Biella, G., Uva, L., & Curtis, M. D. (2001). Network activity evoked by neocortical stimulation in area 36 of the guinea pig perirhinal cortex. Journal of Neurophysiology, 86, 164-172.
7. Brown, E., Moehlis, J., & Holmes, P. (2004). On the phase reduction and response dynamics of neural oscillator populations. Neural Computation, 16, 673-715.
8. Brunel, N. (2000). Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience, 8, 183-208.
9. Brunel, N., Chance, F., Fourcaud, N., & Abbott, L. (2001). Effects of synaptic noise and filtering on the frequency response of spiking neurons. Physical Review Letter, 86, 2186-2189.
10. Brunel, N., & Hakim, V. (1999). Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Computation, 11, 1621-1671.
11. Cai, D., Tao, L., Rangan, A., & McLaughlin, D. (2006). Maximum-entropy closures for kinetic theories of neuronal network dynamics. Communications in Mathematical Sciences, 4, 97-127.
12. Cai, D., Tao, L., Shelley, M., & McLaughlin, D. (2004). An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex. Proceedings of the National Academy of Science, 101, 7757-7762.
13. Casti, A. R., Omurtag, A., Sornborger, A., Kaplan, E., Knight, B., Sirovich, L., & Victor, J. (2002). A population study of integrate-and-fire-or-burst neurons. Neural Computation, 14, 957-986.
14. Chapman, S. I., & Cowling, T. (1970). The mathematical theory of non-uniform gases. Cambridge: Cambridge University Press.
15. Doiron, B., Rinzel, J., & Reyes, A. (2006). Stochastic synchronization in finite size spiking networks. Physical Review E, 74, 030903.
16. Dreyer, W., Herrmann, M., & Kunik, M. (2004). Kinetic solutions of the Boltzmann-Peierls equation and its moment systems. Continuum Mechanics and Thermodynamics, 16, 453-469.
17. Dreyer, W., Junk, M., & Kunik, M. (2001). On the approximation of the Fokker-Planck equation by moment systems. Institute of Physics Publishing, 14, 881-906.
18. Dyan, P., & Abbott, L. (2001). Theoretical neuroscience: Computational and mathematical modeling of neural systems. Cambridge, MA: MIT Press.
19. Edmonds, B., Bigg, A. J., & Colquhoun, D. (1995). Mechanisms of activation of glutamate receptors and the time course of excitatory synaptic Currents. Annual Review of Physiology, 57, 495-519.
20. Fourcaud, N., & Brunel, N. (2002). Dynamics of the firing probability of noisy integrate-and-fire neurons. Neural Computation, 14, 2057-2110.
21. Fourcard-Trocme, N., Hansel, D., van Vreeswijk, C., & Brunel, N. (2003). How spike generation mechanisms determine the neuronal response to fluctuating inputs. Journal of Neuroscience, 23, 11628-11640.
22. Fricker, D., & Miles, R. (2000). EPSP amplification and the precision of spike timing in hippocampal neurons. Neuron, 28, 559-569.
23. Gerstner, W. (1995). Time structure of the activity in neural network models. Phys. Rev. E, 51, 738-758.
24. Gerstner, W. (1999). Population dynamics of spiking neurons: Fast transients, asynchronous states, and locking. Neural Computation, 12, 43-90.
25. Hansel, D., Mato, G., Meunier, C., & Neltner, L. (1998). On numerical simulations of integrate-and-fire neural networks. Neural Computation, 10, 467-483.
26. Haskell, E., Nykamp, D. Q., & Tranchina, D. (2001). Population density methods for large-scale modeling of neuronal networks with realistic synaptic kinetics: Cutting the dimension down to size. Network Compututation in Neural Systems, 12, 141-174.
27. Huertas, M. A., & Smith, G. D. (2006). A multivariate population density model of the dLGN/PGN relay. Journal of Computational Neuroscience, 21(2), 171-189.
28. Izhikevich, E. (2004). Which model to use for cortical spiking neurons? IEEE Transactions on Neural Networks, 15, 1063-1070.
29. Jolivet, R., Lewis, T., & Gerstner, W. (2004). Generalized integrate-and-fire models of neuronal activity approximate spike trains of a detailed model to a high degree of accuracy. Journal of Neurophysblogy, 92, 952-976.
30. Junk, M., & Unterreiter, A. (2002). Maximum entropy moment systems and Galilean invariance. Continuum Mechanics and Thermodynamics, 14, 536-576.
31. Kitano, K., Okamoto, H., & Fukai, T. (2003). Time representing cortical activities: Two models inspired by prefrontal persistent activity. Biological Cybernetics, 88, 387-394.
32. Knight, B. W. (1972). Dynamics of encoding in a population of neurons. Journal of General Physiology, 59, 734-766.
33. Knight, B. W. (2000). Dynamics of encoding in neuron populations: Some general mathematical features. Neural Computation, 12, 473-518.
34. Knight, B. W., Manin, D., & Sirovich, L. (1996). Dynamical models of interacting neuron populations. In E. C. Gerf (Ed.), Symposium on Robotics and Cybernetics: Computational Engineering in Systems Applications. Lille, France: Cite Scientifique.
35. Knight, B. W., Omurtag, A., & Sirovich, L. (2000). The approach of a neuron population firing rate to a new equilibrium: An exact theoretical result. Neural Computation, 12, 1045-1055.
36. Krukowski, A. E., & Miller, K. D. (2001). Thalamocortical NMDA conductances and intracortical inhibition can explain cortical temporal tuning. Nature Neuroscience, 4, 424-430.
37. Kuramoto, Y. (1991). Collective synchronization of pulse-coupled oscillators and excitable units. Physica D, 50, 15-30.
38. Mattia, M., & Giudice, P. D. (2002). Population dynamics of interacting spiking neurons. Physical Review E, 66, 1-19.
39. McLaughlin, D., Shapley, R., & Shelley, M. (2003). Large-scale modeling of the primary visual cortex: Influence of cortical architecture upon neuronal response. Journal of Physiology-Paris, 97, 237-252.
40. McLaughlin, D., Shapley, R., Shelley, M., & Wielaard, D. J. (2000). A neuronal network model of macaque primary visual cortex (V1): Orientation tuning and dynamics in the input layer 4Cα. Proceedings of National Academy of Science, 97, 8087-8092.
41. McNaughton, B. L., Barnes, C. A., & Anderson, P. (1981). Synaptic efficacy and EPSP summation in granule cells of rat fascia dentata studied in vitro. Journal of Neurophysiology, 46, 952-966.
42. Miller, P., & Wang, X. (2006). Power-law neuronal fluctuations in a recurrent network model of parametric working memory. Journal of Neurophysiology, 95, 1099-1114.
43. Moreno-Bote, R., & Parga, N. (2004). Role of synaptic filtering on the firing response of simple model neurons. Physical Review Letters, 92, 028102.
44. Moreno-Bote, R., & Parga, N. (2005). Membrane potential and response properties of populations of cortical neurons in the high conductance state. Physical Review Letters, 94, 088103.
45. Nykamp, D. Q., & Tranchina, D. (2000). A population density approach that facilitates large-scale modeling of neural networks: Analysis and an application to orientation tuning. Journal of Computational Neuroscience, 8, 19-50.
46. Nykamp, D. Q., & Tranchina, D. (2001). A population density approach that facilitates large-scale modeling of neural networks: Extension to slow inhibitory synapses. Neural Computation, 13, 511-546.
47. Omurtag, A., Knight, B. W., & Sirovich, L. (2000). On the simulation of large populations of neurons. Journal of Computational Neuroscience, 8, 51-63.
48. Rangan, A., & Cai, D. (2006). Maximum-entropy closures for kinetic theories of neuronal network dynamics. Physical Review Letters, 96, 178101.
49. Rangan, A., & Cai, D. (2007). Fast numerical methods for simulating large-scale integrate-and-fire neuronal networks. Journal of Computational Neuroscience, 22, 81-100.
50. Rangan, A., Cai, D., & McLaughlin, D. (2005). Modeling the spatiotemporal cortical activity associated with the line-motion illusion in primary visual cortex. Proceedings of the National Academy of Science, 102, 18793-18800.
51. Shadlen, M. N., & Newsome, W. T. (1998). The variable discharge of cortical neurons: Implications for connectivity, computation, and information coding. Journal of Neuroscience, 18, 3870-3896.
52. Silberberg, G., Wu, C., & Markram, H. (2004). Synaptic dynamics control the timing of neuronal excitation in the activated neocortical microcircuit. Journal of Physiology, 556, 19-27.
53. Sirovich, L. (2003). Dynamics of neuronal populations: Eigenfunction theory: Some solvable cases. Network: Computational Neural Systems, 14, 249-272.
54. Sirovich, L., Omurtag, A. & Knight, B., (2000). Dynamics of neuronal populations: The equilibrium solution. SIAM Journal of Applied Mathematics, 60, 2009-2028.
55. Somers, D. C., Nelson, S. B., & Sur, M. (1995). An emergent model of orientation selectivity in cat visual cortical simple cells. Journal of Neuroscience, 15, 5448-5465.
56. Teramae, J., & Fukai, T. (2005). A cellular mechanism for graded persistent activity in a model neuron and its implication on working memory. Journal of Computational Neuroscience, 18, 105-121.
57. Thompson, A. M., Girdlestone, D., & West, D. C. (1988). Voltage-dependent currents prolong single-axon postsynaptic potentials in layer 3 pyramidal neurons in rate neocortical slices. Journal of Neurophysiology, 60, 1896-1907.
58. Treves, A. (1993). Mean-field analysis of neuronal spike dynamics. Network: Compputation in Neural Systems, 4, 259-284.
59. Tuckwell, H. C. (1988). Introduction to theoretical neurobiology, Vol 1: Linear cable themory and dendritic structure and stochastic theories. Cambridge: Cambridge University Press.
60. Ursino, M., & La Cara, G. (2005). Dependence of visual cell properties on intracortical synapses among hypercolumns: Analysis by a computer model. Journal of Computational Neuroscience, 19, 291-310.
61. Wang, X. J. (1999). Synaptic basis of cortical persistent activity: The importance of NMDA receptors to working memory. Journal of Neuroscience, 19, 9587-9603.
62. Whittington, M. A., Traub, R., Kopell, N., Ermentrout B., & Buhl, E. H. (2000). Inhibition-based rhythms: Experimental and mathematical observations on network dynamics. International Journal of Psychophysiology, 38, 315-336.
63. Wilbur, W., & Rinzel, J. (1983). A theoretical basis for large coefficient of variation and bimodality in neuronal interspike interval distributions. Journal of Theoretical Biology, 105, 345-368.
64. Wilson, H. R., & Cowan, J. D. (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical journal, 12, 1-24.
65. Wilson, H. R., & Cowan, J. D. (1973). A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Biological Cybernetics, 13, 55-80.
66. Yoshimura, Y., Sato, H., Imamura, K., & Watanabe, Y. (2000). Properties of horizontal and vertical inputs to pyramidal cells in the superficial layers of the cat visual cortex. Journal of Neuroscience, 20, 1931-1940.