Systematic fluctuation expansion for neural network activity equations

Neural Computation

Volumn 22, Issue 2, 2010, Pages 377-426

  Buice, Michael A ^a   Cowan, Jack D ^b   Chow, Carson C ^a  

^a NATIONAL INSTITUTE OF DIABETES AND DIGESTIVE AND KIDNEY DISEASES   (United States)

^b UNIVERSITY OF CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ACTION POTENTIAL; ALGORITHM; ANIMAL; ARTIFICIAL INTELLIGENCE; ARTIFICIAL NEURAL NETWORK; BRAIN; HUMAN; LETTER; MATHEMATICAL COMPUTING; MATHEMATICAL PHENOMENA; NERVE CELL; NERVE CELL NETWORK; PHYSIOLOGY;

ACTION POTENTIALS; ALGORITHMS; ANIMALS; ARTIFICIAL INTELLIGENCE; BRAIN; HUMANS; MATHEMATICAL COMPUTING; MATHEMATICAL CONCEPTS; NERVE NET; NEURAL NETWORKS (COMPUTER); NEURONS;

EID: 77649304853     PISSN: 08997667     EISSN: 1530888X     Source Type: Journal    
DOI: 10.1162/neco.2009.02-09-960     Document Type: Letter

References (60)

1. Abbott, L., & Van Vreeswijk, C. (1993). Asynchronous states in networks of pulsecoupled oscillators. Phys. Rev. E, 48(2), 1483-1490.
2. Amari, S. (1975). Homogeneous nets of neuron-like elements. Biological Cybernetics, 17, 211-220.
3. Amari, S. (1977). Dynamics of pattern formation in lateral-inhibition type neural fields. Biological Cybernetics, 27, 77-87.
4. Amit, D., & Brunel, N. (1997a). Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex. Cereb. Cortex, 7(3), 237-252.
5. Amit, D. J., & Brunel, N. (1997b). Dynamics of a recurrent network of spiking neurons before and following learning. Network: Computation in Neural Systems, 8(4), 373-404.
6. Amit, D. J., Gutfreund, H., & Sompolinsky, H. (1985). Spin-glass models of neural networks. Physical Review A, 32(2), 1007-1018.
7. Beggs, J. M. (2008). The criticality hypothesis: How local cortical networks might optimize information processing. Philosophical Transactions of the Royal Society A, 366(1864), 329-343.
8. Beggs, J. M. & Plenz, D. (2003). Neuronal avalanches in neocortical circuits. Journal of Neuroscience, 23(35), 11167-11177.
9. Beshel, J., Kopell, N., and Kay, L. (2007). Olfactory bulb gamma oscillations are enhanced with task demands. Journal of Neuroscience, 27(31), 8358-8365.
10. Bressloff, P., Cowan, J. D., Golubitsky, M., Thomas, P. J., & Wiener, M. C. (2001). Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex. Philosophical Transactions of the Royal Society of London B, 356, 299-330.
11. Brunel, N. (6 May 2000). Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience, 8, 183-208.
12. Brunel, N., & Hakim, V. (1999). Fast global oscillations in networks of integrate-andfire neurons with low firing rates. Neural Computation, 11(7), 1621-1671.
13. Brunel, N., & Hansel, D. (2006). How noise affects the synchronization properties of recurrent networks of inhibitory neurons. Neural Comput., 18(5), 1066- 1110.
14. Buice, M. A., & Chow, C. C. (2007). Correlations, fluctuations, and stability of a finite-size network of coupled oscillators. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 76(3), 031118.
15. Buice, M. A., & Cowan, J. D. (2007). Field theoretic approach to fluctuation effects for neural networks. Physical Review E, 75, 051919.
16. Cai, D., Tao, L., Shelley, M., & McLaughlin, D. W. (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 Sciences of the United States of America, 101(20), 7757-7762.
17. Cohen, M., & Grossberg, S. (1983). Absolute stability and global pattern formation and parallel storage by competitive neural networks. IEEE Transactions on Systems, Man, and Cybernetics, 13, 815-826.
18. Coombes, S. (2005). Waves, bumps, & patterns in neural field theories. Biological Cybernetics, 93, 91-108.
19. Cornwall, J. M., Jackiw, R., & Tomboulis, E. (1974). Effective action for composite operators. Physical Review D, 10(8), 2428-2445.
20. Cowan, J. D. (1991). Stochastic neurodynamics. In D. S. Touretzky, R. P. Lippman, & J. E. Moody (Eds.), Advances in neural information processing systems, 3. San Francisco: Morgan Kaufmann.
21. Doi, M. (1976a). Second quantization representation for classical many-particle system. Journal of Physics A: Mathematical and General, 9(9), 1465-1477.
22. Doi, M. (1976b). Stochastic theory of diffusion controlled reaction. Journal of Physics A: Mathematical and General, 9(9), 1479-1495.
23. El Boustani, S., & Destexhe, A. (2009). A master equation formalism for macroscopic modeling of asynchronous irregular activity states. Neural Computation, 21, 46- 100.
24. Ermentrout, B. (1998). Neural networks as spatio-temporal pattern-forming systems. Reports on Progress in Physics, 61(4), 353-430.
25. Ermentrout, G. B., & Cowan, J. D. (1979). A mathematical theory of visual hallucination patterns. Biological Cybernetics, 34, 137-150.
26. Fourcaud, N., & Brunel, N. (2002). Dynamics of the firing probability of noisy integrate-and-fire neurons. Neural Comput., 14(9), 2057-2110.
27. Fusi, S., & Mattia, M. (1999). Collective behavior of networks with linear (VLSI) integrate-and-fire neurons. Neural Computation, 11(3), 633-652.
28. Gerstner, W. (1995). Time structure of the activity in neural network models. Phys. Rev. E, 51(1), 738-758.
29. Gerstner,W. (2000). Population dynamics of spiking neurons: Fast transients, asynchronous states, and locking. Neural Computation, 12, 43-89.
30. Gerstner,W., & Kistler,W. M. (2002). Spiking neuron models. Cambridge: Cambridge University Press.
31. Ginzburg, I., & Sompolinsky, H. (1994). Theory of correlations in stochastic neural networks. Physical Review E, 50(4), 3171-3191.
32. Golomb, D., & Hansel, D. (2000). The number of synaptic inputs and the synchrony of large, sparse neuronal networks. Neural Comput., 12(5), 1095-1139.
33. Gray,C.M.,&Singer,W. (1989). Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proceedings of the National Academy of Sciences of the United States of America, 86(5), 1698-1702.
34. Hildebrand, E. J., Buice, M. A., & Chow, C. C. (2006). Kinetic theory of coupled oscillators. Physical Review Letters, 98, 1-4.
35. Hopfield, J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences, 79, 2445-2558.
36. Hopfield, J. (1984). Neurons with graded response have collective computational properties like those of two state neurons. Proceedings of the National Academy of Sciences, 81, 3088-3092.
37. Ichimaru, S. (1973). Basic principles of plasma physics: A Statistical Approach. Reading, MA: W. A. Benjamin Advanced Book Program.
38. Janssen, H.-K., & Tauber, U. C. (2005). The field theory approach to percolation processes. Annals of Physics, 315(1), 147-192.
39. Laing, C., & Chow, C. (2001). Stationary bumps in networks of spiking neurons. Neural Computation, 13(7), 1473-1494.
40. Lefévre, A., & Biroli, G. (2007). Dynamics of interacting particle systems: Stochastic process and field theory. J. Stat. Mech., 2007(7), P07024-P07024.
41. Ly, C., & Tranchina, D. (2007). Critical analysis of dimension reduction by amoment closure method in a population density approach to neural network modeling. Neural Computation, 19(8), 2032-2092.
42. Martin, P. C., Siggia, E. D., & Rose, H. A. (1973). Statistical dynamics of classical systems. Physical Review A, 8(1), 423-437.
43. Mattia, M. & Del Giudice, P. (2002). Population dynamics of interacting spiking neurons. Phys. Rev. E, 66(5), 051917.
44. Nicholson, D. R. (1992). Introduction to plasma theory. Malabar, FL: Krieger.
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. J. Comp. Neurosci., 8, 19-50.
46. Ohira, T.,&Cowan, J. (1993).Master equation approach to stochastic neurodynamics. Phys. Rev. E, 48, 2259-2266.
47. Peliti, L. (1985). Path integral approach to birth-death processes on a lattice. Journal de Physique, 46, 1469-1483.
48. Peretto, P. (1984). Collective properties of neural networks: A statistical physics approach. Biological Cybernetics, 50, 51-62.
49. Plesser, H. E., & Gerstner, W. E. (2000). Noise in integrate-and-fire neurons: From stochastic input to escape rates. Neural Comput., 12(2), 367-384.
50. Salinas, E., & Sejnowski, T. J. (2002). Integrate-and-fire neurons driven by correlated stochastic input. Neural Comput., 14(9), 2111-2155.
51. Softky, W. R., & Koch, C. (1993). The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPS. Journal of Neuroscience, 13(1), 334-350.
52. Soula, H., Beslon, G., & Mazet, O. (2006). Spontaneous dynamics of asymmetric random recurrent spiking neural networks. Neural Comput., 18(1), 60-79.
53. Soula, H., & Chow, C. C. (2007). Stochastic dynamics of a finite-size spiking neural network. Neural Comput., 19(12), 3262-3292.
54. Tauber, U. C., Howard, M., & Vollmayr-Lee, B. P. (2005). Applications of fieldtheoretic renormalization group methods to reaction-diffusion problems. Journal of Physics A: Mathematical and General, 38(17), R79-R131.
55. Treves, A. (1993). Mean-field analysis of neuronal spike dynamics. Network: Computation in Neural Systems, 4, 259-284.
56. Van Vreeswijk, C., & Sompolinsky, H. (1996). Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science, 274, 1724-1726.
57. Van Vreeswijk, C., & Sompolinsky, H. (1998). Chaotic balanced state in a model of cortical circuits. Neural Computation, 10, 1321-1371.
58. Wilson, H. R., & Cowan, J. D. (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical Journal, 12, 1-24.
59. Wilson, H. R., & Cowan, J. D. (1973). A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik, 13, 55-80.
60. Zinn-Justin, J. (2002). Quantum field theory and critical phenomena (4th ed.). New York: Oxford Science Publications.