Elsevier

Microelectronics Journal

Volume 31, Issue 7, 30 July 2000, Pages 553-559
Microelectronics Journal

Simulation of circuits demonstrating stochastic resonance

https://doi.org/10.1016/S0026-2692(00)00029-XGet rights and content

Abstract

In certain dynamical systems, the addition of noise can assist the detection of a signal and not degrade it as normally expected. This is possible via a phenomenon termed stochastic resonance (SR), where the response of a nonlinear system to a subthreshold periodic input signal is optimal for some non-zero value of noise intensity. We investigate the SR phenomenon in several circuits and systems. Although SR occurs in many disciplines, the sinusoidal signal by itself is not information bearing. To greatly enhance the practicality of SR, an (aperiodic) broadband signal is preferable. Hence, we employ aperiodic stochastic resonance (ASR) where noise can enhance the response of a nonlinear system to a weak aperiodic signal. We can characterize ASR by the use of cross-correlation-based measures. Using this measure, the ASR in a simple threshold system and in a FitzHugh–Nagumo neuronal model are compared using numerical simulations. Using both weak periodic and aperiodic signals, we show that the response of a nonlinear system is enhanced, regardless of the signal.

Introduction

Noise is usually considered a nuisance in communication and signal processing systems, but via a phenomenon known as stochastic resonance (SR) noise, can assist the detection of a signal. Using the signal-to-noise ratio (SNR) as a measure of the coherence of the output signal, the signature of SR is a sharp increase in the SNR followed by a gradual decrease as the noise is increased. The three main ingredients usually required to observe SR are noise (with correlation time much lower than that of the system), subthreshold periodic signal and a nonlinear system. The nonlinear system is essential since the output would be defined by linear response theory for a linear system, thus the SNR would be proportional at the input and output of such a system. The simplest way to provide a nonlinear system is by introducing a threshold element.

Since its emergence as an explanation for the periodic recurrences on the Earth's climate [1], [2], [3], [4] where the term SR was first coined, SR has traversed many disciplines. These range from electronic systems [5], [6], sensing neurons [7], [8], visual perception [9], [10], [11], [12], bidirectional ring lasers [13] and super conducting quantum loops (SQUIDS) [14] to name a few. For further background, Gammaitoni et al. have written an extensive review [15]. More recently, SR is believed to assist with hearing systems in the auditory nerve [16], [17], [18] while adaptive systems can learn to add the optimal amount of noise to some nonlinear feedback systems [19].

As an example of SR, we will consider the work by Simonotto et al. [9], which deals with the human visual system. This is closely related to the dithering effect [10]. Consider a system that is capable of transmitting single bits of information, each of which marks a threshold crossing. A visual realization of this is shown in Fig. 1 that was generated following the procedure in Ref. [9]. The original gray scale image shown in Fig. 1a is depressed beneath a threshold, white noise added to the gray value in each pixel, and the result compared to the threshold. Thus, the noise is incoherent with that in all other pixels. Pixels with a value above the threshold are made black and the others below are made white. Every pixel contains one bit of information, whether or not the threshold has been crossed. Fig. 1b–d shows the result of adding noise of three intensities, increasing from left to right. There is an optimal noise intensity in (b) that maximizes the information content. Additional improvement in perceived picture quality can be gained by varying the noise temporally [9]. The images in Fig. 1 have been averaged over five ensembles.

A limitation of SR is that it only considers periodic signals; this shortcoming has lead to the development of a method for characterizing SR with aperiodic stimuli [20], where the term aperiodic stochastic resonance (ASR) was coined. Most of the literature regarding ASR to date has considered neuronal models [20], [21], [22], [23], [24], [25], [26], [27].

In this paper, we first describe the types of nonlinear systems and noise that are used. This is followed by algorithms used for numerical simulations. The next two sections replicate SR and ASR, which includes applying ASR to the simplest nonlinear system.

Section snippets

Nonlinear systems and noise

We used noise given by the Ornstein–Uhlenbeck (OU) stochastic process of the formζ̇(t)=λζ(t)+λξ(t),where ξ(t) is the white Gaussian noise with mean 〈ξ(t)〉=0 and autocorrelation 〈ξ(t)ξ(s)〉=2Dδ(t−s). The angled brackets 〈·〉 denote an ensemble average. The correlation time of the OU process is τc=λ−1 and the autocorrelation is given by〈ζ(t)ζ(s)〉=Dτcexp−|t−s|τc,with a variance of D/τc. The OU process provides control over both noise intensity D, and correlation time τc.

In biological systems, SR has

Numerical simulations

This section explicitly details the algorithms used for numerical analysis. By showing the algorithms, it should enable the reader to easily replicate the simulations.

We can approximate Gaussian white noise by choosing τc equal to the integration step size. The Box–Muller algorithm [32] (, ) were used to generate normalized Gaussian random variables from uniform random variables, and the algorithm described by Eq. (7) was used to integrate the OU process to produce colored noise [33]. This has

Replicating SR

The most common way to quantify SR is through the SNR [36], which is the method used in this paper. The SNR is defined as the ratio of the signal power spectral density to the broadband background noise taken at the signal frequency and is given in decibels asSNR=10log10S(f0)B(f0),where S and B are the signal and background noise at the fundamental frequency f0, respectively. The process used to calculate the SNR given the output signal from a system is shown in Fig. 4. If SR exists we expect

Replicating ASR

A large proportion of work in SR has been limited to systems with periodic stimulus. Although it has served useful in many areas (Section 1), the applicability of SR to in practice is limited. This is due to many real world stimuli being aperiodic.

This limitation leads to the concept of ASR, first coined by Collins et al. [20]. ASR introduces another hindrance, namely, how to measure it. Both the methods used for SR in Section 4 assess the coherence of the response from the system with the

Conclusion

We have explicitly provided the algorithms for the numerical simulations. This should enable the reader to easily replicate a system that exhibits ASR. Using these, it was shown that ASR is present in even the simplest nonlinear system—the LCC. This exposes a wide variety of systems where ASR can be used, not just neuron models. For the first time, we have explicitly demonstrated ASR in an LCC. This is of importance for motion detection models, for example, and for making a clearer performance

Acknowledgements

This work was funded by the Australian Research Council and the Sir Ross and Sir Keith Smith Fund.

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