45 research outputs found
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 analogue simulation results by Gingl et al. in Refs. [18,
19].Comment: 22 pages, 5 eps figures, submitted to PR
Data driven optimal filtering for phase and frequency of noisy oscillations: application to vortex flowmetering
A new method for extracting the phase of oscillations from noisy time series
is proposed. To obtain the phase, the signal is filtered in such a way that the
filter output has minimal relative variation in the amplitude (MIRVA) over all
filters with complex-valued impulse response. The argument of the filter output
yields the phase. Implementation of the algorithm and interpretation of the
result are discussed. We argue that the phase obtained by the proposed method
has a low susceptibility to measurement noise and a low rate of artificial
phase slips. The method is applied for the detection and classification of mode
locking in vortex flowmeters. A novel measure for the strength of mode locking
is proposed.Comment: 12 pages, 10 figure
Theory and computation of covariant Lyapunov vectors
Lyapunov exponents are well-known characteristic numbers that describe growth
rates of perturbations applied to a trajectory of a dynamical system in
different state space directions. Covariant (or characteristic) Lyapunov
vectors indicate these directions. Though the concept of these vectors has been
known for a long time, they became practically computable only recently due to
algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and
by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in
covariant Lyapunov vectors and their wide range of potential applications, in
this article we summarize the available information related to Lyapunov vectors
and provide a detailed explanation of both the theoretical basics and numerical
algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The
angles between these vectors and the original covariant vectors are
norm-independent and can be considered as characteristic numbers. Moreover, we
present and study in detail an improved approach for computing covariant
Lyapunov vectors. Also we describe, how one can test for hyperbolicity of
chaotic dynamics without explicitly computing covariant vectors.Comment: 21 pages, 5 figure
Dynamical mean-field theory of spiking neuron ensembles: response to a single spike with independent noises
Dynamics of an ensemble of -unit FitzHugh-Nagumo (FN) neurons subject to
white noises has been studied by using a semi-analytical dynamical mean-field
(DMF) theory in which the original -dimensional {\it stochastic}
differential equations are replaced by 8-dimensional {\it deterministic}
differential equations expressed in terms of moments of local and global
variables. Our DMF theory, which assumes weak noises and the Gaussian
distribution of state variables, goes beyond weak couplings among constituent
neurons. By using the expression for the firing probability due to an applied
single spike, we have discussed effects of noises, synaptic couplings and the
size of the ensemble on the spike timing precision, which is shown to be
improved by increasing the size of the neuron ensemble, even when there are no
couplings among neurons. When the coupling is introduced, neurons in ensembles
respond to an input spike with a partial synchronization. DMF theory is
extended to a large cluster which can be divided into multiple sub-clusters
according to their functions. A model calculation has shown that when the noise
intensity is moderate, the spike propagation with a fairly precise timing is
possible among noisy sub-clusters with feed-forward couplings, as in the
synfire chain. Results calculated by our DMF theory are nicely compared to
those obtained by direct simulations. A comparison of DMF theory with the
conventional moment method is also discussed.Comment: 29 pages, 2 figures; augmented the text and added Appendice