# 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

## Access Full Article

top## Abstract

top## How to cite

topLee 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- L. F. Abbott C. van Vreeswijk. Asynchronous states in networks of pulse-coupled oscillators. Phys. Rev. E, 48 (1993), No. 2, 1483–1490.
- 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
- N. Alon, J. H. Spencer. The probabilistic method. Wiley & Sons Inc., Hoboken, NJ, 2008. Zbl1148.05001
- 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.
- 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
- Béla Bollobás. Random graphs. Cambridge Studies in Advanced Mathematics, Vol. 73, Cambridge University Press, Cambridge, 2001.
- 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.
- 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.
- J. Buck E. Buck. Mechanism of rhythmic synchronous flashing of fireflies, Science159 (1968), No. 3821, 1319–1327.
- 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
- 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.
- 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.
- 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
- 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.
- M. de Sousa Vieira. Chaos and synchronized chaos in an earthquake model. Phys. Rev. Lett., 82 (1999), No. 1, 201–204.
- 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
- B. Doiron, J. Rinzel A. Reyes. Stochastic synchronization in finite size spiking networks. Phys. Rev. E (3), 74 (2006), No. 3, 030903
- P. Erdős A. Rényi. On random graphs. I. Publ. Math. Debrecen, 6 (1959), 290–297. Zbl0092.15705
- P. Erdős A. Rényi. On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl., 5 (1960), 17–61. Zbl0103.16301
- G. B. Ermentrout J. Rinzel. Reflected waves in an inhomogeneous excitable medium. SIAM J. Appl. Math., 56 (1996), No. 4, 1107–1128. Zbl0858.92009
- 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.
- 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
- 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
- D. Hansel H. Sompolinsky. Synchronization and computation in a chaotic neural network. Phys. Rev. Lett., 68 (1992), No. 5, 718–721.
- 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
- C. Huygens. Horoloquium oscilatorium. Parisiis, Paris, 1673.
- R. Kapral, K. Showalter (eds.). Chemical waves and patterns. Springer, 1994.
- B. W. Knight. Dynamics of encoding in a population of neurons. J. Gen. Phys., 59 (1972), No. 6, 734–766.
- Y. Kuramoto. Chemical oscillations, waves, and turbulence. Springer Series in Synergetics, Vol. 19, Springer-Verlag, Berlin, 1984. Zbl0558.76051
- Y. Kuramoto. Collective synchronization of pulse-coupled oscillators and excitable units. Phys. D, 50 (1991), No. 1, 15–30. Zbl0736.92001
- T. G. Kurtz. Relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys., 57 (1972), No. 7, 2976–2978.
- T. G. Kurtz. Strong approximation theorems for density dependent Markov chains. Stoch. Proc. Appl., 6 (1977/78), No. 3, 223–240. Zbl0373.60085
- Z.-H. Liu P.M. Hui. Collective signaling behavior in a networked-oscillator model. Phys. A, 383 (2007), No. 2, 714
- R. E. Mirollo S. H. Strogatz. Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math., 50 (1990), No. 6, 1645–1662. Zbl0712.92006
- 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.
- K. Pakdaman D. Mestivier. Noise induced synchronization in a neuronal oscillator. Phys. D, 192 (2004), No. 1-2, 123–137. Zbl1055.92013
- C. S. Peskin. Mathematical aspects of heart physiology. Courant Institute, New York University, New York, 1975. Zbl0301.92001
- A. Pikovsky, M. Rosenblum, J. Kurths. Synchronization: A universal concept in nonlinear sciences. Cambridge University Press, 2003. Zbl1219.37002
- 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
- A. Shwartz, A. Weiss. Large deviations for performance analysis. Chapman & Hall, London, 1995. Zbl0871.60021
- L. Sirovich. Dynamics of neuronal populations: eigenfunction theory; some solvable cases. Network-computation in Neural Systems, 14 (2003), No. 2, 249–272.
- 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
- S. Strogatz. Sync: The emerging science of spontaneous order. Hyperion, 2003.
- C Sulem, P.-L. Sulem. The nonlinear Schrödinger equation. Applied Mathematical Sciences, Vol. 139, Springer-Verlag, New York, 1999. Zbl0928.35157
- R. Temam, A. Miranville. Mathematical modeling in continuum mechanics. Cambridge University Press, Cambridge, 2005. Zbl1077.76001
- 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
- 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.
- 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.
- 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
- C. van Vreeswijk, L. Abbott G. Ermentrout. When inhibition not excitation synchronizes neural firing. J. Comp. Neurosci., 1 (1994), No. 4, 313–322.
- C. van Vreeswijk H. Sompolinsky. Chaotic balance state in a model of cortical circuits. Neur. Comp., 10 (1998), No. 6, 1321–1372.
- A. T. Winfree. The geometry of biological time. Interdisciplinary Applied Mathematics, Vol. 12, Springer-Verlag, New York, 2001. Zbl1014.92001

## Citations in EuDML Documents

top## NotesEmbed ?

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