KINETIC THEORY FOR NEURONAL NETWORK DYNAMICS*

Communications in Mathematical Sciences

Volumn 4, Issue 1, 2006, Pages 97-127

  Cai, David ^a   Tao, Louis ^a   Rangan, Aaditya V ^a   Mclaughlin, David W ^a,b  

^a NEW YORK UNIVERSITY   (United States)

^b NEW JERSEY INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

coarse grain; correlation; diffusion; fluctuation; Visual cortex

Indexed keywords

EID: 33845421047     PISSN: 15396746     EISSN: 19450796     Source Type: Journal    
DOI: 10.4310/CMS.2006.v4.n1.a4     Document Type: Article

References (39)

1. [1] D. Somers, S. Nelson and M. Sur, An emergent model of orientation selectivity in cat visual cortical simple cel ls, J. Neurosci., 15, 5448-5465, 1995.
2. [2] T. Troyer, A. Krukowski, N. Priebe and K. Miller, Contrast invariant orientation tuning in cat visual cortex with feedforward tuning and correlation based intracortical connectivity, J. Neurosci., 18, 5908-5927, 1998.
3. [3] D. McLaughlin, R. Shapley, M. Shelley and J. Wielaard, A neuronal network model of macaque primary visual cortex (V1): orientation selectivity and dynamics in the input layer 4Ca, Proc. Nat'l. Acad. Sci. USA, 97, 8087-8092, 2000.
4. [4] D. Cai, A. V. Rangan and D. W. McLaughlin, Architectural and synaptic mechanisms under lying coherent spontaneous activity in V1, Proc. Nat'l Acad. Sci. USA, 102, 5868-5873, 2005.
5. [5] A. V. Rangan, D. Cai and D. W. McLaughlin, Modeling the spatiotemporal cortical activity associated with the line-motion illusion in primary visual cortex, Proc. Nat'l Acad. Sci. USA, 102, 18793-18800, 2005.
6. [6] T. Bonhoeffer and A. Grinvald, Iso-orientation domains in cat visual cortex are arranged in pinwheel like patterns, Nature, 353, 429-431, 1991.
7. [7] G. Blasdel, Differential imaging of ocular dominance and orientation selectivity in monkey striate cortex, J. Neurosci., 12, 3115-3138, 1992.
8. [8] G. Blasdel, Orientation selectivity, preference, and continuity in the monkey striate cortex, J. of Neurosci., 12, 3139-3161, 1992.
9. [9] R. Everson, A. Prashanth, M. Gabbay, B. Knight, L. Sirovich and E. Kaplan, Representation of spatial frequency and orientation in the visual cortex, Proc. Nat'l. Acad. Sci. USA, 95, 8334-8338, 1998.
10. [10] P. Maldonado, I. Godecke, C. Gray and T. Bonhoeffer, Orientation selectivity in pinwheel centers in cat striate cortex, Science, 276, 1551-1555, 1997.
11. [11] J. Anderson, I. Lampl, D. Gillespie and D. Ferster, The contribution of noise to contrast invariance of orientation tuning in cat visual cortex, Science, 290, 1968-1972, 2000.
12. [12] E. A. Stern, A. E. Kincaid and C. J. Wilson, Spontaneous subthreshold membrane potential fluctuations and action potential variability of rat corticostriatal and striatal neurons in vivo, J. Neurophysiol., 77, 1697-1715, 1997.
13. [13] M. Shelley and D. McLaughlin, Coarse-grained reduction and analysis of a network model of cortical response. I. drifting grating stimuli, J. Comp. Neurosci., 12, 97-122, 2002.
14. [14] N. Fourcaud and N. Brunel, Dynamics of the firing probability of noisy integrate-and-fire neu rons, Neural Comp., 14, 2057-2110, 2002.
15. [15] C. van Vreeswijk and H. Sompolinsky, Chaos in neuronal networks with balanced excitatory and inhibitory activity, Science, 274, 1724-1726, 1996.
16. [16] C. van Vreeswijk and H. Sompolinsky, Chaotic balanced state in a model of cortical circuits, Neural Comput., 15, 1321-1371, 1998.
17. [17] D. Cai, L. Tao, M. Shelley and D. W. Mclaughlin, An effective representation of fluctuation- driven neuronal networks with application to simple and complex cells in visual cortex, Proc. Nat'l. Acad. Sci. USA, 101, 7757-7762, 2004.
18. [18] L. Tao, D. Cai, D. W. Mclaughlin, M. Shelley and R. Shapley, Orientation selectivity in primary visual cortex by fluctuation-controlled criticality, Proc. Nat'l. Acad. Sci. USA, submitted, 2006.
19. [19] D. Cai, L. Tao and D. W. McLaughlin, An embedded network approach for scale-up of fluctuation-driven systems with preservation of spike information, Proc. Nat'l. Acad. Sci. USA, 101, 14288-14293, 2004.
20. [20] B. Knight, Dynamics of encoding in a populaton neurons, J. Gen. Physiol., 59, 734-766, 1972.
21. [21] W. J. Wilbur and J. Rinzel, A theoretical basis for large coefficient of variation and bimodality in neuronal interspike interval distributions, J. Theor. Biol, 105, 345-368, 1983.
22. [22] L. F. Abbott and C. van Vreeswijk, Asynchronous states in networks of pulse-coupled oscil la-tors, Phys. Rev. E, 48, 1483-1490, 1993.
23. [23] A. Treves, Mean field analysis of neuronal spike dynamics, Network, 4, 259-284, 1993.
24. [24] T. Chawanya, A. Aoyagi, T. Nishikawa, K. Okuda and Y. Kuramoto, A model for feature linking via col lective oscil lations in the primary visual cortex, Biol. Cybern., 68, 483-90, 1993.
25. [25] G. Barna, T. Grobler and P. Erdi, Statistical model of the hippocampal CA3 region, II. the population framework: model of rhythmic activity in CA3 slice, Biol. Cybern., 79, 309321, 1998.
26. [26] J. Pham, K. Pakdaman, J. Champagnat and J. Vibert, Activity in sparsely connected excitatory neural networks: effect of connectivity neural networks, Neural Networks, 11, 415-434, 1998.
27. [27] N. Brunel and V. Hakim, Fast global oscil lations in networks of integrate-and-fire neurons with low firing rates, Neural Comp., 11, 1621-1671, 1999.
28. [28] W. Gerstner, Population dynamics of spiking neurons: fast transients, asynchronous states, and locking, Neural Comp., 12, 43-80, 2000.
29. [29] A. Omurtag, B. W. Knight and L. Sirovich, On the simulation of large populations of neurons, J. Comp. Neurosci., 8, 51-63, 2000.
30. [30] A. Omurtag, E. Kaplan, B. W. Knight and L. Sirovich, A population approach to cortical dynamics with an application to orientation tuning, Network: Comput. Neural Syst., 11, 247-260, 2000.
31. [31] D. Nykamp and D. Tranchina, A population density method that facilitates large-scale modeling of neural networks: analysis and application to orientation tuning, J. Comp. Neurosci., 8, 19-50, 2000.
32. [32] D. Nykamp and D. Tranchina, A population density method that facilitates large-scale modeling of neural networks: extension to slow inhibitory synapses, Neural Comput., 13, 511-546, 2001.
33. [33] E. Haskell, D. Q. Nykamp and D. Tranchina, Population density methods for large-scale modeling of neuronal networks with realistic synaptic kinetics: cutting the dimension down to size, Network: Compt. Neural. Syst, 12, 141-174, 2001.
34. [34] H. Wilson and J. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 13, 55-80, 1973.
35. [35] B. Knight, D. Manin and L. Sirovich, Dynamical models of interacting neuron populations, In E. Gerf, editor, Symposium on Robotics and Cybernetics: Computational Engineering in Systems Applications. Cite Scientifique, Lille, France, 1996.
36. [36] A. Casti, A. Omurtag, A. Sornborger, E. Kaplan, B. Knight, J. Victor and L. Sirovich, A population study of integrate-and-fire-or-burst neurons, Neural Computation, 14, 957-986, 2002.
37. [37] A. V. Rangan, D. Cai and L.Tao, Numerical methods for solving kinetic equations of neuronal network dynamics, preprint, 2005.
38. [38] C. Koch, Biophysics of Computation, Oxford University Press, Oxford, 1999.
39. [39] E. Cinlar, Superposition of point processes, In P. A. W. Lewis, editor, Stochastic Point Processes: Statistical Analysis, Theory, and Applications, Wiley, New York, NY, 549-606, 1972.