In many of the systems and models in which stochastic resonance has been observed, the essential nonlinearity is effectively a single threshold. Usually SR occurs when an entirely subthreshold signal is subjected to additive noise, which allows threshold crossings to occur that otherwise would not have. In such systems, it is generally thought that when the input signal is suprathreshold, then the addition of noise will not have any beneficial effect on the system output.

However, the 1999 discovery of a novel form of SR in simple threshold-based systems showed that this is not the case. This phenomenon is known as suprathreshold stochastic resonance, and occurs in arrays of identical threshold devices subject to independent additive noise. In such arrays, SR can occur regardless of whether the signal is entirely subthreshold or not, hence the name suprathreshold SR. The SSR effect is quite general, and is not restricted to any particular type of signal or noise distribution.

This chapter reviews the early theoretical work on SSR. Recent theoretical extensions are also presented, as well as numerical analysis of previously unstudied input and noise signals, a new technique for calculating the mutual information by integration, and an investigation of a number of channel capacity questions for SSR. Finally, this chapter shows how SSR can be interpreted as a stochastic quantization scheme.

Introduction

Suprathreshold stochastic resonance (SSR) is a form of stochastic resonance (SR) that occurs in arrays of identical threshold devices. A schematic model of the system is shown in Fig. 4.1, and is described in detail in Section 4.3.

The aim of this chapter is to find asymptotic large N approximations to the mean square error distortion for the suprathreshold stochastic resonance model. In particular, we are interested in how the distortion varies with noise intensity and how it scales with the number of threshold devices. That is, does the distortion become asymptotically small for large N?

Introduction

Chapter 6 developed the idea of treating the SSR model as a lossy source coding or quantization model. We saw that such a treatment requires specification of reproduction points corresponding to each of the N + 1 discrete output states. Once specified, an approximate reconstruction of the input signal can be made from a decoding, and the average error between this approximation and the original signal subsequently measured by the mean square error (MSE) distortion. We saw also in Chapter 5 that asymptotic approximations to the output probability mass function, Py(n), output entropy, average conditional output entropy, and mutual information can be found if the number of threshold devices, N, in the SSR model is allowed to become very large. The aim of this chapter is to again allow N to become very large, and develop asymptotic approximations to the MSE distortion for the cases of optimal linear and optimal nonlinear decodings of the SSR model.

Chapter structure

This chapter has three main sections. We begin in Section 7.2 by letting N become large in the formulas derived in Chapter 6 and analyzing the result. Next, Section 7.3 takes the same approach from the estimation theory perspective.

Background and motivation

We begin by briefly outlining the background and motivation for this book, before giving an overview of each chapter, and pointing out the most significant questions addressed.

Although the methodology used is firmly within the fields of signal processing and mathematical physics, the motivation is interdisciplinary in nature.

The initial open questions that inspired this direction were:

1. (i) How might neurons make use of a phenomenon known as stochastic resonance?

2. (ii) How might a path towards engineering applications inspired by these studies be initiated?

Stochastic resonance and sensory neural coding

Stochastic resonance (SR) is a counter-intuitive phenomenon where the presence of noise in a nonlinear system is essential for optimal system performance. It is not a technique. Instead, it is an effect that might be observed and potentially exploited or induced. It has been observed to occur in many systems, including in both neurons and electronic circuits.

A motivating idea is that since we know the brain is far better at many tasks compared to electronic and computing devices, then maybe we can learn something from the brain. If we can ultimately better understand the possible exploitation of SR in the brain and nervous system, we may also be able to improve aspects of electronic systems.

Although it is important to have an overall vision, in practical terms it is necessary to consider a concrete starting point. This book is particularly focused on an exciting new development in the field of SR, known as suprathreshold stochastic resonance (SSR) (Stocks 2000c). Suprathreshold stochastic resonance occurs in a parallel array of simple threshold devices.

Engineered systems usually require finding the right tradeoff between cost and performance. Communications systems are no exception, and much theoretical work has been undertaken to find the limits of achievable performance for the transmission of information. For example, Shannon's celebrated channel capacity formula and coding theorems say that there is an upper limit on the average amount of information that can be transmitted in a channel for error-free communication. This limit can be increased if the power of the signal is increased, or the bandwidth in the channel is increased. However, nothing comes for free, and increasing either power or bandwidth can be expensive; hence there is a tradeoff between cost and performance in such a communications system – performance (measured by bit rates) can be increased by increasing the cost (power or bandwidth). This chapter discusses several problems related to the tradeoff between cost and performance in the SSR model. We are interested in the SSR model as a channel model, from an energy efficient neural coding point of view, as well as the lossy source coding model, where there is a tradeoff between rate and distortion.

Introduction

Chapter 8 introduces an extension to the suprathreshold stochastic resonance (SSR) model by allowing all thresholds to vary independently, instead of all having the same value. This chapter further extends the SSR model by introducing an energy constraint into the optimal stochastic quantization problem. We also examine the tradeoff between rate and distortion, when the SSR model is considered as a stochastic quantizer.