Bulletin of Mathematical Biology

, Volume 70, Issue 6, pp 1608–1633 | Cite as

Synchrony and Asynchrony in a Fully Stochastic Neural Network

  • R. E. Lee DeVille
  • Charles S. Peskin
Original Article


We describe and analyze a model for a stochastic pulse-coupled neural network, in which the randomness in the model corresponds to synaptic failure and random external input. We show that the network can exhibit both synchronous and asynchronous behavior, and surprisingly, that there exists a range of parameters for which the network switches spontaneously between synchrony and asynchrony. We analyze the associated mean-field model and show that the switching parameter regime corresponds to a bistability in the mean field, and that the switches themselves correspond to rare events in the stochastic system.


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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


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

Copyright information

© Society for Mathematical Biology 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations