title: Spontaneous movements and linear response of a noisy oscillator.
abstract: A deterministic system that operates in the vicinity of a Hopf bifurcation can be described by a single equation of a complex variable, called the normal form. Proximity to the bifurcation ensures that on the stable side of the bifurcation (i.e. on the side where a stable fixed point exists), the linear-response function of the system is peaked at the frequency that is characteristic of the oscillatory instability. Fluctuations, which are present in many systems, conceal the Hopf bifurcation and lead to noisy oscillations. Spontaneous hair bundle oscillations by sensory hair cells from the vertebrate ear provide an instructive example of such noisy oscillations. By starting from a simplified description of hair bundle motility based on two degrees of freedom, we discuss the interplay of nonlinearity and noise in the supercritical Hopf normal form. Specifically, we show here that the linear-response function obeys the same functional form as for the noiseless system on the stable side of the bifurcation but with effective, renormalized parameters. Moreover, we demonstrate in specific cases how to relate analytically the parameters of the normal form with added noise to effective parameters. The latter parameters can be measured experimentally in the power spectrum of spontaneous activity and linear-response function to external stimuli. In other cases, numerical solutions were used to determine the effects of noise and nonlinearities on these effective parameters. Finally, we relate our results to experimentally observed spontaneous hair bundle oscillations and responses to periodic stimuli.
authors.affiliation: Max Planck Institut für Physik komplexer Systeme, Dresden, Germany.
authors.forename: F
authors.initials: F
authors.lastname: Jülicher
authors.forename: K
authors.initials: K
authors.lastname: Dierkes
authors.forename: B
authors.initials: B
authors.lastname: Lindner
authors.forename: J
authors.initials: J
authors.lastname: Prost
authors.forename: P
authors.initials: P
authors.lastname: Martin
mesh_terms: D016014:Linear Models; D008956:Models, Chemical; D009038:Motion; D017711:Nonlinear Dynamics
references.citation: Proc Natl Acad Sci U S A. 2001 Dec 4;98(25):14380-5
references.pmid: 11724945
references.citation: Proc Natl Acad Sci U S A. 1999 Dec 7;96(25):14306-11
references.pmid: 10588701
references.citation: Hear Res. 1992 Jun;60(1):89-102
references.pmid: 1500380
references.citation: Proc Natl Acad Sci U S A. 1987 May;84(9):3064-8
references.pmid: 3495007
references.citation: Proc Natl Acad Sci U S A. 1998 Dec 22;95(26):15321-6
references.pmid: 9860967
references.citation: Proc Natl Acad Sci U S A. 2004 Aug 17;101(33):12195-200
references.pmid: 15302928
references.citation: Phys Rev Lett. 2005 Sep 16;95(12):123903
references.pmid: 16197076
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Apr;77(4 Pt 1):041901
references.pmid: 18517650
references.citation: Phys Rev Lett. 2000 May 29;84(22):5232-5
references.pmid: 10990910
references.citation: Proc Natl Acad Sci U S A. 2000 Mar 28;97(7):3183-8
references.pmid: 10737791
references.citation: Proc Natl Acad Sci U S A. 2000 Oct 24;97(22):12026-31
references.pmid: 11027302
references.citation: J Neurosci. 2003 Jun 1;23(11):4533-48
references.pmid: 12805294
references.citation: J Physiol. 1985 Jul;364:359-79
references.pmid: 4032304
references.citation: Proc Natl Acad Sci U S A. 2001 Dec 4;98(25):14386-91
references.pmid: 11724944