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Synchronization of a large number of continuous one-dimensional stochastic elements with time-delayed mean-field coupling. (English) Zbl 1168.82020
The authors provide an approach to find the boundary of the synchronization domain for a system of stochastic one-dimensional elements with non-homogeneous mean-field coupling with delay. The synchronization threshold is obtained solving a boundary value problem for the Fokker-Planck equation. Both numerical and approximate analytical results are obtained. As a case study the authors consider bistable systems with a polynomial and a piece-wise linear potential.
MSC:
82C31Stochastic methods in time-dependent statistical mechanics
35B35Stability of solutions of PDE
37N35Dynamical systems in control
Software:
DDE-BIFTOOL
References:
[1]Kuramoto, Y.: Chemical oscillations, waves and turbulance, (1984)
[2]Tass, P.; Rosenblum, M. G.; Weule, J.; Kurths, J.; Pikovsky, A.; Volkmann, J.; Schnitzler, A.; Freund, H. -J.: Detection of n:m phase locking from noisy data: application to magnetoencephalography, Phys. rev. Lett. 81, 3291 (1998)
[3]Engel, A. K.; Fries, P.; Singer, W.: Dynamic predictions: oscillations and synchrony in top-down processing, Nature rev. Neurosci. (London) 2, 704 (2001)
[4]Winfree, A. T.: Biological rhythms and the behavior of populations of coupled oscillators, J. theoret. Biol. 16, 15 (1967)
[5]Winfree, A. T.: Integrated view of resetting a circadian clock, J. theoret. Biol. 28, 327 (1970)
[6]Winfree, A.: The geometry of biological time, (1980)
[7]Peskin, C. S.: Mathematical aspects of heart physiology, (1975) · Zbl 0301.92001
[8]Buck, J.: Synchronous rhythmic flashing of fireflies. Ii, Q. rev. Biol. 63, 265 (1988)
[9]Kiss, I. Z.; Zhai, Y.; Nudson, J. L.: Emerging coherence in a population of chemical oscillators, Science 296, 1676 (2002)
[10]Haken, H.: Advanced synergetics, (1983)
[11]Chatterjee, M.; Oba, S. I.: Noise improves modulation detection by cochlear implant listeners at moderate carrier levels, J. acoust. Soc. am. 118, 993 (2005)
[12]Stocks, N. G.: Information transmission in parallel arrays of threshold elements: suprathreshold stochastic resonance, Phys. rev. E 63, 041114 (2001)
[13]Dawson, D.: Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. stat. Phys. 29, 31 (1983)
[14]Jung, P.; Behn, U.; Pantazelou, E.; Moss, F.: Collective response in globally coupled bistable systems, Phys. rev. A 46, R1709 (1992)
[15]Koulakov, A.; Raghavachari, S.; Kepecs, A.; Lisman, J.: Model for a robust neural integrator, Nature neurosci. 5, 775 (2002)
[16]Camperi, M.; Wang, X.: A model of visuospatial working memory in prefrontal cortex: recurrent network and cellular bistability, J. comput. Neurosci. 5, 383 (1998) · Zbl 0918.92007 · doi:10.1023/A:1008837311948
[17]Sompolinsky, H.: Neural networks with nonlinear synapses and a static noise, Phys. rev. A 34, 2571 (1986)
[18]Zanette, D. H.: Dynamics of globally coupled bistable elements, Phys. rev. E 55, 5315 (1997)
[19]Shiino, M.: Dynamical behaviour of stochastic systems of infinitely many coupled nonlinear oscillators exhibiting phase transition of mean-field type: H theorem on asymptotic approach to equilibrium and critical slowing down of order-parameter fluctuations, Phys. rev. A 36, 2393 (1987)
[20]Huber, D.; Tsimring, L. S.: Dynamics of an ensemble of noisy bistable elements with global time delayed coupling, Phys. rev. Lett. 91, 260601 (2003)
[21]Huber, D.; Tsimring, L. S.: Cooperative dynamics in a network of stochastic elements with delayed feedback, Phys. rev. E 71, 036150 (2005)
[22]Zaks, M. A.; Neiman, A. B.; Feistel, S.; Schimansky-Geier, L.: Noise-controlled oscillations and their bifurcations in coupled phase oscillators, Phys. rev. E 68, 066206 (2003)
[23]Zaks, M. A.; Sailer, X.; Schimansky-Geier, L.; Neiman, A. B.: Noise induced complexity: from subthreshold oscillations to spiking in coupled excitable systems, Phys. rev. E 15, 026117 (2005) · Zbl 1080.82013 · doi:10.1063/1.1886386
[24]Frank, T. D.; Beek, P. J.: Fokker–Planck equations for globally coupled many-body systems with time delays, J. stat. Mech. 10010, 1742-5468 (2005)
[25]Reimann, P.; Den Broeck, C. Van; Kawai, R.: Nonequilibrium noise in coupled phase oscillators, Phys. rev. E 60, 6402 (1999)
[26]Park, S. Hee; Kim, S.: Noise-induced phase transitions in globally coupled active rotators, Phys. rev. E 53, 3425 (1996)
[27]Brunel, N.; Hakim, V.; Richardson, M. J. E.: Firing-rate resonance in a generalized integrate-and-fire neuron with subthreshold resonance, Phys. rev. E 67, 051916 (2003)
[28]Brunel, N.; Hansel, D.: How noise affects the synchronization properties of recurrent networks of inhibitory neurons, Neural comput. 18, 1066-1110 (2006) · Zbl 1088.92004 · doi:10.1162/089976606776241048
[29]Acebron, J. A.; Bulsara, A. R.; Rappel, W. -J.: Noisy Fitzhugh–Nagumo model: from single elements to globally coupled networks, Phys. rev. E 69, 026202 (2004)
[30]E. Doedel, R. Paffenroth, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B. Sandstede, X. Wang, Auto 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), Technical Report, Caltech
[31]Hu, G.; Nicolis, G.; Nicolis, C.: Periodically forced Fokker–Planck equation and stochastic resonance, Phys. rev. A 42, 2030 (1990)
[32]Risken, H.: The Fokker–Planck equation, (1989)
[33]Desai, R. C.; Zwansig, R.: Statistical mechanics of a nonlinear stochastic model, J. stat. Phys. 19, 1-24 (1978)
[34]Hu, G.; Haken, H.; Xie, F.: Stochastic resonance with sensitive frequency dependence in globally coupled continuous systems, Phys. rev. Lett. 77, 1925 (1996)
[35]K. Engelborghs, T. Luzyanina, G. Samaey, Dde-biftool v. 2.00: A matlab package for bifurcation analysis of delay differential equations, Technical Report TW-330, Department of Computer Science, K.U. Leuven, Leuven, Belgium
[36]R. Szalai, Pdde-cont: A continuation and bifurcation software for delay-differential equations. http://www.mm.bme.hu/szalai/pdde/, 2005
[37]Cohen-Tannoudji, C.; Diu, B.; Laloë, F.: Quantum mechanics II, (1977)
[38]Pikovsky, A.: Dynamics of globally coupled noisy oscillators, Lecture notes in physics, 210-219 (2007) · Zbl 0896.60050