zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

82C31Stochastic methods in time-dependent statistical mechanics
35B35Stability of solutions of PDE
37N35Dynamical systems in control
Full Text: DOI
[1] Kuramoto, Y.: Chemical oscillations, waves and turbulance, (1984) · Zbl 0558.76051
[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) · Zbl 0957.92026
[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) · Zbl 0464.92001
[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) · Zbl 0521.93002
[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) · Zbl 0729.60112
[32] Risken, H.: The Fokker--Planck equation, (1989) · Zbl 0665.60084
[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