Synchrony and asynchrony in a fully stochastic neural network

Bulletin of Mathematical Biology

Volumn 70, Issue 6, 2008, Pages 1608-1633

  DeVille, R E Lee ^a   Peskin, Charles S ^b  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b NEW YORK UNIVERSITY   (United States)

Author keywords

Bistability; Mean field analysis; Neural network; Neuronal network; Rare events; Stochastic integrate and fire; Synchronization

Indexed keywords

ALGORITHM; ANIMAL; ARTICLE; BIOLOGICAL MODEL; COMPUTER SIMULATION; NERVE CELL NETWORK; PHYSIOLOGY; STATISTICS; SYNAPTIC TRANSMISSION;

ALGORITHMS; ANIMALS; COMPUTER SIMULATION; MODELS, NEUROLOGICAL; NERVE NET; STOCHASTIC PROCESSES; SYNAPTIC TRANSMISSION;

EID: 48449104886     PISSN: 00928240     EISSN: 15229602     Source Type: Journal    
DOI: 10.1007/s11538-008-9311-8     Document Type: Article

References (38)

1. L.F. Abbott C. van Vreeswijk 1993 Asynchronous states in networks of pulse-coupled oscillators Phys. Rev. E 48 2 1483 1490
2. F. Apfaltrer C. Ly D. Tranchina 2006 Population density methods for stochastic neurons with realistic synaptic kinetics: Firing rate dynamics and fast computational methods Netw. Comput. Neural Syst. 17 4 373 418
3. J.M. Beggs D. Plenz 2003 Neuronal avalanches in neocortical circuits J. Neurosci. 23 35 11167 11177
4. J.M. Beggs D. Plenz 2004 Neuronal avalanches are diverse and precise activity patterns that are stable for many hours in cortical slice cultures J. Neurosci. 24 22 5216 5229
5. P.C. Bressloff S. Coombes 1998 Desynchronization, mode locking, and bursting in strongly coupled integrate-and-fire oscillators Phys. Rev. Lett. 81 10 2168 2171
6. N. Brunel V. Hakim 1999 Fast global oscillations in networks of integrate-and-fire neurons with low firing rates Neural Comput. 11 7 1621 1671
7. D. Cai L. Tao M. Shelley D.W. McLaughlin 2004 An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex Proc. Natl. Acad. Sci. USA 101 20 7757 7762
8. D. Cai L. Tao A.V. Rangan D.W. McLaughlin 2006 Kinetic theory for neuronal network dynamics Commun. Math. Sci. 4 1 97 127
9. S.R. Campbell D.L.L. Wang C. Jayaprakash 1999 Synchrony and desynchrony in integrate-and-fire oscillators Neural Comput. 11 7 1595 1619
10. B. Doiron J. Rinzel A. Reyes 2006 Stochastic synchronization in finite size spiking networks Phys. Rev. E (3) 74 3 030903 030904
11. G.B. Ermentrout N. Kopell 1984 Frequency plateaus in a chain of weakly coupled oscillators, I SIAM J. Math. Anal. 15 2 215 237
12. Franklin, J.N., 1968. Matrix Theory. Prentice-Hall, Englewood Cliffs.
13. Freidlin, M.I., Wentzell, A.D., 1998. Random Perturbations of Dynamical Systems, 2nd edn. Springer, New York.
14. W. Gerstner J.L. van Hemmen 1993 Coherence and incoherence in a globally-coupled ensemble of pulse-emitting units Phys. Rev. Lett. 71 3 312 315
15. I. Ginzburg H. Sompolinsky 1994 Theory of correlations in stochastic neural networks Phys. Rev. E 50 4 3171 3191
16. P. Goel B. Ermentrout 2002 Synchrony, stability, and firing patterns in pulse-coupled oscillators Physica D Nonlinear Phenom. 163 3-4 191 216
17. D. Hansel G. Mato C. Meunier 1993 Clustering and slow switching in globally coupled phase oscillators Phys. Rev. E 48 5 3470 3477
18. E. Haskell D.Q. Nykamp D. Tranchina 2001 Population density methods for large-scale modelling of neuronal networks with realistic synaptic kinetics: cutting the dimension down to size Netw. Comput. Neural Syst. 12 2 141 174
19. Huygens, C., 1673. Horoloquium Oscilatorium. Parisiis, Paris.
20. B.W. Knight 1972 Dynamics of encoding in a population of neurons J. Gen. Physiol. 59 6 734 766
21. Kuramoto, Y., 1984. Chemical Oscillations, Waves, and Turbulence. Springer Series in Synergetics, vol. 19. Springer, Berlin.
22. Y. Kuramoto 1991 Collective synchronization of pulse-coupled oscillators and excitable units Physica D 50 1 15 30
23. B. Lindner J. García-Ojalvo A. Neiman L. Schimansky-Geier 2004 Effects of noise in excitable systems Phys. Rep. 392 6 321 424
24. M. Mattia P. Del Giudice 2002 Population dynamics of interacting spiking neurons Phys. Rev. E 66 5 051917
25. R.E. Mirollo S.H. Strogatz 1990 Synchronization of pulse-coupled biological oscillators SIAM J. Appl. Math. 50 6 1645 1662
26. Peskin, C.S., 1975. Mathematical aspects of heart physiology. Courant Institute of Mathematical Sciences New York University, New York. Notes based on a course given at New York University during the year 1973/74, available at http://math.nyu.edu/faculty/peskin/heartnotes/index.html.
27. Pikovsky, A., Rosenblum, M., Kurths, J., 2003. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge.
28. W. Senn R. Urbanczik 2000/2001 Similar nonleaky integrate-and-fire neurons with instantaneous couplings always synchronize SIAM J. Appl. Math. 61 4 1143 1155 (electronic)
29. Shwartz, A., Weiss, A., 1995. Large Deviations for Performance Analysis. Chapman & Hall, London.
30. L. Sirovich 2003 Dynamics of neuronal populations: eigenfunction theory; some solvable cases Netw. Comput. Neural Syst. 14 2 249 272
31. L. Sirovich A. Omrtag B.W. Knight 2000 Dynamics of neuronal populations: The equilibrium solution SIAM J. Appl. Math. 60 6 2009 2028
32. H. Soula C.C. Chow 2007 Stochastic dynamics of a finite-size spiking neural network Neural Comput. 19 12 3262 3292
33. Strogatz, S., Sync: The Emerging Science of Spontaneous Order. Hyperion, 2003.
34. D. Terman N. Kopell A. Bose 1998 Dynamics of two mutually coupled slow inhibitory neurons Physica D 117 1-4 241 275
35. M. Tsodyks I. Mitkov H. Sompolinsky 1993 Pattern of synchrony in inhomogeneous networks of oscillators with pulse interactions Phys. Rev. Lett. 71 8 1280 1283
36. C. van Vreeswijk H. Sompolinsky 1998 Chaotic balance state in a model of cortical circuits Neural Comput. 10 6 1321 1372
37. van Vreeswijk, C., Abbott, L., Ermentrout, G., 1994. When inhibition not excitation synchronizes neural firing. J. Comput. Neurosci. 313-322.
38. Winfree, A.T., 2001. The Geometry of Biological Time, 2nd edn. Interdisciplinary Applied Mathematics, vol. 12. Springer, New York.