Gain in stochastic resonance: precise numerics versus linear response theory beyond the two-mode approximation.

In the context of the phenomenon of stochastic resonance (SR), we study the correlation function, the signal-to-noise ratio (SNR), and the ratio of output over input SNR, i.e., the gain, which is associated to the nonlinear response of a bistable system driven by time-periodic forces and white Gaussian noise. These quantifiers for SR are evaluated using the techniques of linear response theory (LRT) beyond the usually employed two-mode approximation scheme. We analytically demonstrate within such an extended LRT description that the gain can indeed not exceed unity. We implement an efficient algorithm, based on work by Greenside and Helfand (detailed in the Appendix), to integrate the driven Langevin equation over a wide range of parameter values. The predictions of LRT are carefully tested against the results obtained from numerical solutions of the corresponding Langevin equation over a wide range of parameter values. We further present an accurate procedure to evaluate the distinct contributions of the coherent and incoherent parts of the correlation function to the SNR and the gain. As a main result we show for subthreshold driving that both the correlation function and the SNR can deviate substantially from the predictions of LRT and yet the gain can be either larger or smaller than unity. In particular, we find that the gain can exceed unity in the strongly nonlinear regime which is characterized by weak noise and very slow multifrequency subthreshold input signals with a small duty cycle. This latter result is in agreement with recent analog simulation results by Gingl et al. [ICNF 2001, edited by G. Bosman (World Scientific, Singapore, 2002), pp. 545-548; Fluct. Noise Lett. 1, L181 (2001)].

[1]  M. Feit,et al.  Solution of the Schrödinger equation by a spectral method , 1982 .

[2]  Zoltan Gingl,et al.  A stochastic resonator is able to greatly improve signal-to- noise ratio , 1996 .

[3]  P. Hänggi,et al.  Hopping and phase shifts in noisy periodically driven bistable systems , 1993 .

[4]  Peter Hänggi,et al.  Stochastic Nonlinear Dynamics Modulated by External Periodic Forces , 1989 .

[5]  William Bialek,et al.  Information flow in sensory neurons , 1995 .

[6]  Zoltan Gingl,et al.  Signal-to-noise ratio gain by stochastic resonance in a bistable system , 2000 .

[7]  Péter Makra,et al.  HIGH SIGNAL-TO-NOISE RATIO GAIN BY STOCHASTIC RESONANCE IN A DOUBLE WELL , 2001 .

[8]  F Liu,et al.  Signal-to-noise ratio gain in neuronal systems. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  E. Helfand,et al.  Numerical integration of stochastic differential equations — ii , 1979, The Bell System Technical Journal.

[10]  Frank Moss,et al.  Stochastic resonance: noise-enhanced order , 1999 .

[11]  Jung,et al.  Amplification of small signals via stochastic resonance. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[12]  Bulsara,et al.  Nonlinear stochastic resonance: the saga of anomalous output-input gain , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  Kurt Wiesenfeld,et al.  Minireview of stochastic resonance. , 1998, Chaos.

[14]  Peter Hänggi,et al.  Stochastic resonance in biology. How noise can enhance detection of weak signals and help improve biological information processing. , 2002, Chemphyschem : a European journal of chemical physics and physical chemistry.

[15]  William H. Press,et al.  Numerical recipes , 1990 .

[16]  P. Hānggi,et al.  Rocking bistable systems: Use and abuse of linear response theory , 2002, cond-mat/0202258.

[17]  P. Hänggi,et al.  CHECKING LINEAR RESPONSE THEORY IN DRIVEN BISTABLE SYSTEMS , 2002 .

[18]  Hermann Haken,et al.  A study of stochastic resonance without adiabatic approximation , 1992 .

[19]  Adi R. Bulsara,et al.  Tuning in to Noise , 1996 .

[20]  Wiesenfeld,et al.  Theory of stochastic resonance. , 1989, Physical review. A, General physics.

[21]  M. Gell-Mann,et al.  Physics Today. , 1966, Applied optics.

[22]  François Chapeau-Blondeau,et al.  Theory of stochastic resonance in signal transmission by static nonlinear systems , 1997 .

[23]  Peter Hänggi,et al.  Stochastic processes: Time evolution, symmetries and linear response , 1982 .

[24]  Morillo,et al.  Amplification and distortion of a periodic rectangular driving signal by a noisy bistable system. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  武者 利光,et al.  Noise in physical systems and 1/f fluctuations , 1992 .