Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-17T05:49:13.711Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  10 March 2010

R. E. Lee DeVille ,
C. S. Peskin  and
J. H. Spencer
R. E. Lee DeVille*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 60801
C. S. Peskin
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012
J. H. Spencer
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012
*
* Corresponding author. E-mail: rdeville@illinois.edu
Get access

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.

Keywords

neural networkneuronal networksynchronymean-field analysisintegrate-and-firerandom graphslimit theorem
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 2: Mathematics and neuroscience , 2010 , pp. 26 - 66
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbott, L. F. van Vreeswijk, C.. Asynchronous states in networks of pulse-coupled oscillators . Phys. Rev. E, 48 (1993), No. 2, 1483–1490.CrossRefGoogle ScholarPubMed
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.
N. Alon, J. H. Spencer. The probabilistic method. Wiley & Sons Inc., Hoboken, NJ, 2008.
Apfaltrer, F., Ly, C. Tranchina, D.. 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.CrossRefGoogle ScholarPubMed
Bennett, M., Schatz, M. F., Rockwood, H. Wiesenfeld, K.. Huygens’s clocks . R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458 (2002), No. 2019, 563–579.CrossRefGoogle Scholar
Béla Bollobás. Random graphs. Cambridge Studies in Advanced Mathematics, Vol. 73, Cambridge University Press, Cambridge, 2001.
Bressloff, P. C. Coombes, S.. Desynchronization, mode locking, and bursting in strongly coupled integrate-and-fire oscillators . Phys. Rev. Lett., 81 (1998), No. 10, 2168–2171.CrossRefGoogle Scholar
Brunel, N. Hakim, V.. Fast global oscillations in networks of integrate-and-fire neurons with low firing rates . Neural Comp., 11 (1999), No. 7, 1621–1671.CrossRefGoogle ScholarPubMed
Buck, J. Buck, E.. Mechanism of rhythmic synchronous flashing of fireflies , Science 159 (1968), No. 3821, 1319–1327.CrossRefGoogle ScholarPubMed
Cai, D., Tao, L., Rangan, A. V. McLaughlin, D. W.. Kinetic theory for neuronal network dynamics . Comm. Math. Sci., 4 (2006), No. 1, 97–127.CrossRefGoogle Scholar
Cai, D., Tao, L., Shelley, M. McLaughlin, D. W.. 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.CrossRefGoogle ScholarPubMed
Campbell, S. R., Wang, D. L. L. Jayaprakash, C.. Synchrony and desynchrony in integrate-and-fire oscillators . Neur. Comp., 11 (1999), No. 7, 1595–1619.CrossRefGoogle ScholarPubMed
Cartwright, J. H. E., Eguíluz, V. M., Hernández-García, E. Piro, O.. Dynamics of elastic excitable media . Int. J. Bif. Chaos, 9 (1999), No. 11, 2197–2202.CrossRefGoogle Scholar
Czeisler, C. A., Weitzman, E., Moore-Ede, M. C., Zimmerman, J. C. Knauer, R. S.. Human sleep: its duration and organization depend on its circadian phase . Science 210 (1980), No. 4475, 1264–1267.CrossRefGoogle ScholarPubMed
de Sousa Vieira, M.. Chaos and synchronized chaos in an earthquake model . Phys. Rev. Lett., 82 (1999), No. 1, 201–204.CrossRefGoogle Scholar
DeVille, R. E. L., Peskin, C. S.. Synchrony and asynchrony in a fully stochastic neural network . Bull. Math. Bio., 70 (2008), No. 6, 1608–1633. CrossRefGoogle Scholar
Doiron, B., Rinzel, J. Reyes, A.. Stochastic synchronization in finite size spiking networks . Phys. Rev. E (3), 74 (2006), No. 3, 030903 CrossRefGoogle ScholarPubMed
Erdős, P. Rényi, A.. On random graphs. I . Publ. Math. Debrecen, 6 (1959), 290–297.Google Scholar
Erdős, P. Rényi, A.. On the evolution of random graphs . Magyar Tud. Akad. Mat. Kutató Int. Közl., 5 (1960), 17–61.Google Scholar
Ermentrout, G. B. Rinzel, J.. Reflected waves in an inhomogeneous excitable medium . SIAM J. Appl. Math., 56 (1996), No. 4, 1107–1128.CrossRefGoogle Scholar
Gerstner, W. van Hemmen, J. L.. Coherence and incoherence in a globally-coupled ensemble of pulse-emitting units . Phys. Rev. Lett., 71 (1993), No. 3, 312–315.CrossRefGoogle Scholar
Glass, L., Goldberger, A. L., Courtemanche, M. Shrier, A.. Nonlinear dynamics, chaos and complex cardiac arrhythmias . Proc. Roy. Soc. London Ser. A, 413 (1987), No. 1844, 9–26.CrossRefGoogle Scholar
Guevara, M. R. Glass, L.. 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.CrossRefGoogle Scholar
Hansel, D. Sompolinsky, H.. Synchronization and computation in a chaotic neural network . Phys. Rev. Lett., 68 (1992), No. 5, 718–721.CrossRefGoogle Scholar
Haskell, E., Nykamp, D. Q. Tranchina, D.. 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.CrossRefGoogle Scholar
C. Huygens. Horoloquium oscilatorium. Parisiis, Paris, 1673.
R. Kapral, K. Showalter (eds.). Chemical waves and patterns. Springer, 1994.
Knight, B. W.. Dynamics of encoding in a population of neurons . J. Gen. Phys., 59 (1972), No. 6, 734–766.CrossRefGoogle Scholar
Y. Kuramoto. Chemical oscillations, waves, and turbulence. Springer Series in Synergetics, Vol. 19, Springer-Verlag, Berlin, 1984.
Kuramoto, Y.. Collective synchronization of pulse-coupled oscillators and excitable units . Phys. D, 50 (1991), No. 1, 15–30.CrossRefGoogle Scholar
Kurtz, T. G.. Relationship between stochastic and deterministic models for chemical reactions . J. Chem. Phys., 57 (1972), No. 7, 2976–2978.CrossRefGoogle Scholar
Kurtz, T. G.. Strong approximation theorems for density dependent Markov chains . Stoch. Proc. Appl., 6 (1977/78), No. 3, 223–240. CrossRefGoogle Scholar
Liu, Z.-H. Hui, P.M.. Collective signaling behavior in a networked-oscillator model . Phys. A, 383 (2007), No. 2, 714 CrossRefGoogle Scholar
Mirollo, R. E. Strogatz, S. H.. Synchronization of pulse-coupled biological oscillators . SIAM J. Appl. Math., 50 (1990), No. 6, 1645–1662.CrossRefGoogle Scholar
Olami, Z., Feder, H. J. S. Christensen, K.. Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes . Phys. Rev. Lett., 68 (1992), No. 8, 1244–1247.CrossRefGoogle Scholar
Pakdaman, K. Mestivier, D.. Noise induced synchronization in a neuronal oscillator . Phys. D, 192 (2004), No. 1-2, 123–137.CrossRefGoogle Scholar
C. S. Peskin. Mathematical aspects of heart physiology. Courant Institute, New York University, New York, 1975.
A. Pikovsky, M. Rosenblum, J. Kurths. Synchronization: A universal concept in nonlinear sciences. Cambridge University Press, 2003.
Senn, W., Urbanczik, R. . Similar nonleaky integrate-and-fire neurons with instantaneous couplings always synchronize . SIAM J. Appl. Math., 61 (2000/01), No. 4, 1143–1155 (electronic). Google Scholar
A. Shwartz, A. Weiss. Large deviations for performance analysis. Chapman & Hall, London, 1995.
Sirovich, L.. Dynamics of neuronal populations: eigenfunction theory; some solvable cases. Network-computation in Neural Systems, 14 (2003), No. 2, 249–272. CrossRefGoogle ScholarPubMed
Sirovich, L., Omrtag, A. Knight, B. W.. Dynamics of neuronal populations: The equilibrium solution . SIAM J. Appl. Math., 60 (2000), No. 6, 2009–2028.Google Scholar
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.
R. Temam, A. Miranville. Mathematical modeling in continuum mechanics. Cambridge University Press, Cambridge, 2005.
Terman, D., Kopell, N. Bose, A.. Dynamics of two mutually coupled slow inhibitory neurons . Phys. D, 117 (1998), No. 1-4, 241–275.CrossRefGoogle Scholar
Tsodyks, M., Mitkov, I. Sompolinsky, H.. Pattern of synchrony in inhomogeneous networks of oscillators with pulse interactions . Phys. Rev. Lett., 71 (1993), No. 8, 1280–1283.CrossRefGoogle ScholarPubMed
Tyson, J. J., Hong, C. I., Thron, C. D. Novak, B.. A Simple Model of Circadian Rhythms Based on Dimerization and Proteolysis of PER and TIM . Biophys. J., 77 (1999), No. 5, 2411–2417.CrossRefGoogle ScholarPubMed
Tyson, J. J. Keener, J. P.. Singular perturbation theory of traveling waves in excitable media (a review) . Phys. D, 32 (1988), No. 3, 327–361.CrossRefGoogle Scholar
van Vreeswijk, C., Abbott, L. Ermentrout, G.. When inhibition not excitation synchronizes neural firing . J. Comp. Neurosci., 1 (1994), No. 4, 313–322.CrossRefGoogle Scholar
van Vreeswijk, C. Sompolinsky, H.. Chaotic balance state in a model of cortical circuits . Neur. Comp., 10 (1998), No. 6, 1321–1372.CrossRefGoogle Scholar
A. T. Winfree. The geometry of biological time. Interdisciplinary Applied Mathematics, Vol. 12, Springer-Verlag, New York, 2001.