Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory

R. E. Lee DeVille; C. S. Peskin; J. H. Spencer

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 2, page 26-66
  • ISSN: 0973-5348

Abstract

top
We analyze a stochastic neuronal network model which corresponds to an all-to-all network of discretized integrate-and-fire neurons where the synapses are failure-prone. This network exhibits different phases of behavior corresponding to synchrony and asynchrony, and we show that this is due to the limiting mean-field system possessing multiple attractors. We also show that this mean-field limit exhibits a first-order phase transition as a function of the connection strength — as the synapses are made more reliable, there is a sudden onset of synchronous behavior. A detailed understanding of the dynamics involves both a characterization of the size of the giant component in a certain random graph process, and control of the pathwise dynamics of the system by obtaining exponential bounds for the probabilities of events far from the mean.

How to cite

top

Lee DeVille, R. E., Peskin, C. S., and Spencer, J. H.. "Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory." Mathematical Modelling of Natural Phenomena 5.2 (2010): 26-66. <http://eudml.org/doc/197622>.

@article{LeeDeVille2010,
abstract = {We analyze a stochastic neuronal network model which corresponds to an all-to-all network of discretized integrate-and-fire neurons where the synapses are failure-prone. This network exhibits different phases of behavior corresponding to synchrony and asynchrony, and we show that this is due to the limiting mean-field system possessing multiple attractors. We also show that this mean-field limit exhibits a first-order phase transition as a function of the connection strength — as the synapses are made more reliable, there is a sudden onset of synchronous behavior. A detailed understanding of the dynamics involves both a characterization of the size of the giant component in a certain random graph process, and control of the pathwise dynamics of the system by obtaining exponential bounds for the probabilities of events far from the mean.},
author = {Lee DeVille, R. E., Peskin, C. S., Spencer, J. H.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {neural network; neuronal network; synchrony; mean-field analysis; integrate-and-fire; random graphs; limit theorem},
language = {eng},
month = {3},
number = {2},
pages = {26-66},
publisher = {EDP Sciences},
title = {Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory},
url = {http://eudml.org/doc/197622},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Lee DeVille, R. E.
AU - Peskin, C. S.
AU - Spencer, J. H.
TI - Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/3//
PB - EDP Sciences
VL - 5
IS - 2
SP - 26
EP - 66
AB - We analyze a stochastic neuronal network model which corresponds to an all-to-all network of discretized integrate-and-fire neurons where the synapses are failure-prone. This network exhibits different phases of behavior corresponding to synchrony and asynchrony, and we show that this is due to the limiting mean-field system possessing multiple attractors. We also show that this mean-field limit exhibits a first-order phase transition as a function of the connection strength — as the synapses are made more reliable, there is a sudden onset of synchronous behavior. A detailed understanding of the dynamics involves both a characterization of the size of the giant component in a certain random graph process, and control of the pathwise dynamics of the system by obtaining exponential bounds for the probabilities of events far from the mean.
LA - eng
KW - neural network; neuronal network; synchrony; mean-field analysis; integrate-and-fire; random graphs; limit theorem
UR - http://eudml.org/doc/197622
ER -

References

top
  1. L. F. Abbott C. van Vreeswijk. Asynchronous states in networks of pulse-coupled oscillators. Phys. Rev. E, 48 (1993), No. 2, 1483–1490. 
  2. M. Abramowitz, I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, Washington, D.C., 1964.  Zbl0171.38503
  3. N. Alon, J. H. Spencer. The probabilistic method. Wiley & Sons Inc., Hoboken, NJ, 2008.  Zbl1148.05001
  4. F. Apfaltrer, C. Ly D. Tranchina. Population density methods for stochastic neurons with realistic synaptic kinetics: Firing rate dynamics and fast computational methods. Network-computation in Neural Systems, 17 (2006), No. 4, 373–418. 
  5. M. Bennett, M. F. Schatz, H. Rockwood K. Wiesenfeld. Huygens’s clocks. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458 (2002), No. 2019, 563–579. Zbl1026.01007
  6. Béla Bollobás. Random graphs. Cambridge Studies in Advanced Mathematics, Vol. 73, Cambridge University Press, Cambridge, 2001.  
  7. P. C. Bressloff S. Coombes. Desynchronization, mode locking, and bursting in strongly coupled integrate-and-fire oscillators. Phys. Rev. Lett., 81 (1998), No. 10, 2168–2171. 
  8. N. Brunel V. Hakim. Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comp., 11 (1999), No. 7, 1621–1671. 
  9. J. Buck E. Buck. Mechanism of rhythmic synchronous flashing of fireflies, Science159 (1968), No. 3821, 1319–1327. 
  10. D. Cai, L. Tao, A. V. Rangan D. W. McLaughlin. Kinetic theory for neuronal network dynamics. Comm. Math. Sci., 4 (2006), No. 1, 97–127. Zbl1107.82037
  11. D. Cai, L. Tao, M. Shelley D. W. McLaughlin. An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex. Proc. Nat. Acad. Sci. USA, 101 (2004), No. 20, 7757–7762. 
  12. S. R. Campbell, D. L. L. Wang C. Jayaprakash. Synchrony and desynchrony in integrate-and-fire oscillators. Neur. Comp., 11 (1999), No. 7, 1595–1619. 
  13. J. H. E. Cartwright, V. M. Eguíluz, E. Hernández-García O. Piro. Dynamics of elastic excitable media. Int. J. Bif. Chaos, 9 (1999), No. 11, 2197–2202. Zbl1192.74204
  14. C. A. Czeisler, E. Weitzman, M. C. Moore-Ede, J. C. Zimmerman R. S. Knauer. Human sleep: its duration and organization depend on its circadian phase. Science210 (1980), No. 4475, 1264–1267. 
  15. M. de Sousa Vieira. Chaos and synchronized chaos in an earthquake model. Phys. Rev. Lett., 82 (1999), No. 1, 201–204. 
  16. R. E. L. DeVille, C. S. Peskin. Synchrony and asynchrony in a fully stochastic neural network. Bull. Math. Bio., 70 (2008), No. 6, 1608–1633.  Zbl1166.92010
  17. B. Doiron, J. Rinzel A. Reyes. Stochastic synchronization in finite size spiking networks. Phys. Rev. E (3), 74 (2006), No. 3, 030903 
  18. P. Erdős A. Rényi. On random graphs. I. Publ. Math. Debrecen, 6 (1959), 290–297. 
  19. P. Erdős A. Rényi. On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl., 5 (1960), 17–61. Zbl0103.16301
  20. G. B. Ermentrout J. Rinzel. Reflected waves in an inhomogeneous excitable medium. SIAM J. Appl. Math., 56 (1996), No. 4, 1107–1128. Zbl0858.92009
  21. W. Gerstner J. L. van Hemmen. Coherence and incoherence in a globally-coupled ensemble of pulse-emitting units. Phys. Rev. Lett., 71 (1993), No. 3, 312–315. 
  22. L. Glass, A. L. Goldberger, M. Courtemanche A. Shrier. Nonlinear dynamics, chaos and complex cardiac arrhythmias. Proc. Roy. Soc. London Ser. A, 413 (1987), No. 1844, 9–26. Zbl0626.92001
  23. M. R. Guevara L. Glass. Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: A theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias. J. Math. Bio.14 (1982), No. 1, 1–23. Zbl0489.92007
  24. D. Hansel H. Sompolinsky. Synchronization and computation in a chaotic neural network. Phys. Rev. Lett., 68 (1992), No. 5, 718–721. 
  25. E. Haskell, D. Q. Nykamp D. Tranchina. Population density methods for large-scale modelling of neuronal networks with realistic synaptic kinetics: cutting the dimension down to size. Network-Computation in Neural Systems, 12 (2001), No. 2, 141–174. Zbl1005.92007
  26. C. Huygens. Horoloquium oscilatorium. Parisiis, Paris, 1673.  
  27. R. Kapral, K. Showalter (eds.). Chemical waves and patterns. Springer, 1994.  
  28. B. W. Knight. Dynamics of encoding in a population of neurons. J. Gen. Phys., 59 (1972), No. 6, 734–766. 
  29. Y. Kuramoto. Chemical oscillations, waves, and turbulence. Springer Series in Synergetics, Vol. 19, Springer-Verlag, Berlin, 1984.  Zbl0558.76051
  30. Y. Kuramoto. Collective synchronization of pulse-coupled oscillators and excitable units. Phys. D, 50 (1991), No. 1, 15–30. Zbl0736.92001
  31. T. G. Kurtz. Relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys., 57 (1972), No. 7, 2976–2978. 
  32. T. G. Kurtz. Strong approximation theorems for density dependent Markov chains. Stoch. Proc. Appl., 6 (1977/78), No. 3, 223–240.  Zbl0373.60085
  33. Z.-H. Liu P.M. Hui. Collective signaling behavior in a networked-oscillator model. Phys. A, 383 (2007), No. 2, 714 
  34. R. E. Mirollo S. H. Strogatz. Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math., 50 (1990), No. 6, 1645–1662. Zbl0712.92006
  35. Z. Olami, H. J. S. Feder K. Christensen. Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes. Phys. Rev. Lett., 68 (1992), No. 8, 1244–1247. 
  36. K. Pakdaman D. Mestivier. Noise induced synchronization in a neuronal oscillator. Phys. D, 192 (2004), No. 1-2, 123–137. Zbl1055.92013
  37. C. S. Peskin. Mathematical aspects of heart physiology. Courant Institute, New York University, New York, 1975.  Zbl0301.92001
  38. A. Pikovsky, M. Rosenblum, J. Kurths. Synchronization: A universal concept in nonlinear sciences. Cambridge University Press, 2003.  Zbl1219.37002
  39. W. Senn, R. Urbanczik. Similar nonleaky integrate-and-fire neurons with instantaneous couplings always synchronize. SIAM J. Appl. Math., 61 (2000/01), No. 4, 1143–1155(electronic).  Zbl1013.92009
  40. A. Shwartz, A. Weiss. Large deviations for performance analysis. Chapman & Hall, London, 1995.  Zbl0871.60021
  41. L. Sirovich. Dynamics of neuronal populations: eigenfunction theory; some solvable cases. Network-computation in Neural Systems, 14 (2003), No. 2, 249–272.  
  42. L. Sirovich, A. Omrtag B. W. Knight. Dynamics of neuronal populations: The equilibrium solution. SIAM J. Appl. Math., 60 (2000), No. 6, 2009–2028. Zbl0991.92005
  43. S. Strogatz. Sync: The emerging science of spontaneous order. Hyperion, 2003.  
  44. C Sulem, P.-L. Sulem. The nonlinear Schrödinger equation. Applied Mathematical Sciences, Vol. 139, Springer-Verlag, New York, 1999.  
  45. R. Temam, A. Miranville. Mathematical modeling in continuum mechanics. Cambridge University Press, Cambridge, 2005.  Zbl1077.76001
  46. D. Terman, N. Kopell A. Bose. Dynamics of two mutually coupled slow inhibitory neurons. Phys. D, 117 (1998), No. 1-4, 241–275. Zbl0941.34027
  47. M. Tsodyks, I. Mitkov H. Sompolinsky. Pattern of synchrony in inhomogeneous networks of oscillators with pulse interactions. Phys. Rev. Lett., 71 (1993), No. 8, 1280–1283. 
  48. J. J. Tyson, C. I. Hong, C. D. Thron B. Novak. A Simple Model of Circadian Rhythms Based on Dimerization and Proteolysis of PER and TIM. Biophys. J., 77 (1999), No. 5, 2411–2417. 
  49. J. J. Tyson J. P. Keener. Singular perturbation theory of traveling waves in excitable media (a review). Phys. D, 32 (1988), No. 3, 327–361. Zbl0656.76018
  50. C. van Vreeswijk, L. Abbott G. Ermentrout. When inhibition not excitation synchronizes neural firing. J. Comp. Neurosci., 1 (1994), No. 4, 313–322. 
  51. C. van Vreeswijk H. Sompolinsky. Chaotic balance state in a model of cortical circuits. Neur. Comp., 10 (1998), No. 6, 1321–1372. 
  52. A. T. Winfree. The geometry of biological time. Interdisciplinary Applied Mathematics, Vol. 12, Springer-Verlag, New York, 2001.  Zbl1014.92001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.