Mathematical models of nonlinear oscillators are used to describe a wide variety of physical and biological phenomena that exhibit self-sustained oscillatory behavior. When these oscillators are strongly driven by forces that are periodic in time, they often exhibit a remarkable “mode-locking” that synchronizes the nonlinear oscillations to the driving force. The purpose of this paper is to demonstrate that a similar phenomenon occurs when nonlinear oscillators are strongly driven by a force that is varying randomly in time. In this case the synchronization is less obvious for a single oscillator, but when several oscillators with different initial conditions or phases are driven with the same aperiodic force, their fluctuating behavior may reliably converge to an identical response. Analytical estimates are derived for the conditions, rates, and structural stability for the synchronization of a broad class of aperiodically driven nonlinear oscillators.
REFERENCES
1.
C. Huygens, Horologium Oscillatorium (1673), translated by R. Blackwell (Iowa U.P., Ames, IA, 1986), p. 30.
2.
G. B.
Ermentrout
and J.
Rinzel
, “Beyond a pacemaker’s entrainment limit: Phase walk through
,” Am. J. Physiol.
246
, R102
–R106
(1984
).3.
S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison–Wesley, New York, 1994).
4.
C. Hayashi, Nonlinear Oscillations in Physical Systems (McGraw–Hill, New York, 1964).
5.
J. A. Murdock, Perturbations: Theory and Methods (Wiley, New York, 1991).
6.
D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations (Clarendon, Oxford, 1987).
7.
R.
He
and P. G.
Vaidya
, “Analysis and synthesis of synchronous periodic and chaotic systems
,” Phys. Rev. A
46
, 7387
–7392
(1992
).8.
A. S.
Pikovskii
, “Synchronization and stochastization of nonlinear oscillators by external noise” in Nonlinear and Turbulent Processes, edited by R. Z. Sagdeev (Harwood, New York, 1984), Vol. 3, pp. 1601–1604;A. S.
Pikovskii
, “Synchronization and stochastization of the ensemble of autogenerators by external noise
,” Radiophys. Quantum Electron.
27
, 576
–581
(1984
).9.
R. V.
Jensen
, “Synchronization of randomly driven nonlinear oscillators
,” Phys. Rev. E
58
, R6907
–R6910
(1998
).10.
Methods in Neuronal Modeling, edited by C. Koch and I. Segev (MIT, Cambridge, MA, 1989).
11.
R. V. Jensen, L. Jones, and D. Gartner, “Synchronization of randomly driven nonlinear oscillators and the reliable firing of cortical neurons” in Computational Neuroscience: Trends in Research, 1998, edited by J. M. Bower (Plenum, New York, 1998), pp. 403–407.
12.
Z. F.
Mainen
and T.
Sejnowski
, “Reliability of spike timing in neocortical neurons
,” Science
268
, 1503
–1506
(1995
);A. C.
Tang
, A. M.
Bartels
, and T.
Sejnowski
, “Effects of cholinergic modulation on neocortical neurons in responses to fluctuating inputs
,” Cereb. Cortex
7
, 502
–509
(1997
).13.
R.
de Ruyter van Steveninck
, D.
Lewen
, S. P.
Strong
, R.
Koberle
, and W.
Bialek
, “Reproducibility and variability in neural spike trains
,” Science
275
, 1805
–1808
(1997
);F. Rieke, D. Warland, R. de Ruyter van Steveninck, and W. Bialek, Spikes: Exploring the Neural Code (MIT, Cambridge, MA, 1997).
14.
See Focus Issue on Control and Synchronization of Chaos,
Chaos
7
, 509
–687
(1997
).15.
L. M.
Pecora
and T. L.
Carroll
, “Synchronization in chaotic systems
,” Phys. Rev. Lett.
64
, 821
–824
(1990
);L. M.
Pecora
and T. L.
Carroll
, “Driving systems with chaotic signals
,” Phys. Rev. A
44
, 2374
–2383
(1991
).16.
K. M.
Cuomo
and A. V.
Oppenheim
, “Circuit implementation of synchronized chaos with applications to communications
,” Phys. Rev. Lett.
71
, 65
–68
(1993
).17.
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, New York, 1984).
18.
See
S. H.
Strogatz
, “Norbert Wiener’s Brain Waves,” in Frontiers in Mathematical Biology, edited by S. Levin, Lecture Notes in Biomathematics Vol. 100 (Springer-Verlag, New York, 1994), p. 122;“From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators,”
S. H.
Strogatz
, Physica D
143
, 1
–20
(2000
) for references to the most recent work on the synchronization of arrays of coupled nonlinear oscillators.19.
See, for example, the classic work by R. L. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1967), Vol. 2.
20.
A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw–Hill, New York, 1965).
21.
S. O. Rice, in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954), pp. 133–284.
22.
P.
Hänggi
and P.
Riseborough
, “Dynamics of nonlinear dissipative oscillators
,” Am. J. Phys.
51
, 347
–352
(1983
).23.
R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford U.P., Oxford, 1994).
24.
R. V. Jensen (unpublished).
25.
M. G.
Rosenblum
, A. S.
Pikovsky
, and J.
Kurths
, “Phase synchronization of chaotic oscillators
,” Phys. Rev. Lett.
76
, 1804
–1807
(1996
).26.
J. D.
Hunter
, J. G.
Milton
, P. J.
Thomas
, and J. D.
Cowan
, “Resonance effect for neural spike time reliability
,” J. Neurophysiol.
80
, 1427
–1438
(1998
).27.
M. T. Roberts and R. V. Jensen (unpublished).
28.
P. C. Bressloff, J. D. Cowan, R. V. Jensen, and P. J. Thomas (unpublished).
29.
K.
Wiesenfeld
and F.
Moss
, “Stochastic resonance and the benefits of noise: From ice ages to crayfish and SQUIDs
,” Nature (London)
373
, 33
–36
(1995
).30.
A.
Longtin
and D. R.
Chialvo
, “Stochastic and deterministic resonances for excitable systems
,” Phys. Rev. Lett.
81
, 4012
–4015
(1998
).
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