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Stochastic resonance. From suprathreshold stochastic resonance to stochastic signal quantization. (English) Zbl 1157.94003

Cambridge: Cambridge University Press (ISBN 978-0-521-88262-0/hbk; 978-0-511-42658-2/ebook). xix, 425 p. (2008).
The presented book is firmly situated within the fields of signal processing and mathematical physics. It is interdisciplinary in nature. There are two main topics that are considered in its content: (a) the use of a phenomenon known as “stochastic resonance” (SR), and (b) the path towards its engineering applications.
In the beginning of the book, some explanations are provided about the real meaning of the stochastic resonance as a counter-intuitive phenomenon where the presence of noise in a nonlinear system is essential for optimal system performance. It is underlined that this 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.
The book is particularly focused on an exciting new development in the field of SR, known as “suprathreshold stochastic resonance” (SSR). Suprathreshold stochastic resonance occurs in a parallel array of simple threshold devices. Each individual threshold device receives the same signal, but is subject to independent additive random noise. The output of each device is a binary signal, which is unity when the input is greater than the threshold value, and zero otherwise. The overall output of the SSR model is the sum of the individual binary signals. Originally, threshold devices were considered to have the same threshold value. The aim of the presented book is to comprehensively outline known theoretical and numerical results on SSR and to extend this theory. It is anticipated that the presented landscape will form a launching pad for future research into the specific role that SSR may play in real neural coding. The main goal of the authors is not to prove that living systems actively exploit SR or SSR – these are ongoing research areas in the domain of neurophysiology and biophysics. Their attention is focused on the theoretical mathematical underpinning of SR-type effects in the very simple McCulloch-Pitts model. Following this direction, the authors are showing that the analysis of SR in arrays of such simplified neural models gives rise to rich complex phenomena and also to a number of surprises. Analysis of the simple model lays the foundation for adding further complexities in the future. The mathematical foundation provided by the book assist future neurophysiologists in asking the right questions and in performing the right experiments when establishing if real neurons actively exploit SSR. In the meantime, the book also contributes to the application of SSR in artificial neural and electronic systems. To this end, the book culminates in a chapter on the application of SSR to electronic cochlear implants.
Another motivation for the authors is the importance of the problem of overcoming the effects of noise in sensors and signal and data processing applications. Because of small signal-to-noise ratios (SNRs), it becomes impossible to operate with the traditional noise reduction methods. It becomes actual to use the effects of SR, especially for the need of the optimal circuit design. Two experimental examples are provided in the book to illustrate this application.
A second area of the presented research is that of distributed sensor networks. Of particular interest to the book is the problem where it is not necessarily a network of complete sensors that are distributed, but it is actually the data acquisition, or compression, that is distributed. A key aspect of such a scenario is that data are acquired from a number of independently noisy sources that do not cooperate, and are then fused by some central processing unit. In the information theory and signal processing literature, this is referred to as distributed source coding or distributed compression. Given that the SSR effect overcomes a serious limitation of all previously studied forms of SR, a complete theoretical investigation of its behavior may lead to new design approaches to new SNR systems or data acquisition and compression in distributed sensor networks.
Working on the book, the authors were aware that the SSR effect is equivalent to a noisy, or stochastic, quantization of a signal. Consequently, as well as describing its behavior from the perspective of information transmission, it is equally valid to describe it from the perspective of information compression or, more specifically, lossy compression. Note that quantization of a signal is a form of lossy compression. The distinguishing feature of SSR, that sets it apart from standard forms of quantization, is that conventionally the rules that specify a quantizer’s operation are considered to be fixed and deterministic. In contrast, when the SSR effect is viewed as quantization, the governing rules lead to a set of parameters that are independent random variables. Hence, the authors often refer to the SSR model’s output as a stochastic quantization. Given this perspective, there are three immediate questions that can be asked: is it possible to describe the SSR effect in terms of conventional quantization and compression theory? Given that a central SR question is that of finding the optimal noise conditions, what noise intensity optimizes the performance of the SSR model when it is described as a quantizer? How good is the SSR effect at quantization when compared with conventional quantizers? The underlying theme of this book is to address these three questions. It consists of ten chapters, as follows:
Chapter 1 provides the background and motivation for the work described in this book.
Chapter 2 contains an overview of the historical landscape against which this book is set. It defines stochastic resonance as it is most widely understood, and gives a broad literature review of SR, with particular emphasis on aspects relevant to quantization. This chapter is deliberately sparse in equations and devoid of quantitative results, but does provide qualitative illustrations of how SR works. It also provides some discussion that is somewhat peripheral to the main scope of the book, but they prove to be useful for readers unfamiliar with, or confused about, SR.
Chapter 3 contains the information-theoretic definitions required for the remainder of the book, and discusses the differences between dithering and stochastic resonance.
Chapters 4 and 5 begin the main focus of the book, by defining the SSR model, giving a detailed literature review of all previous research on SSR, and replicating all the most significant theoretical results to date. These two chapters consider only the original concept of the SSR model as a communications channel. In particular, the authors are examining how the mutual information between the input and output of the SSR model varies with noise intensity. A subset of these results pertains specifically to a large number of individual threshold devices in the SSR model. Chapter 5 is devoted to elaborating on results in this area, as well as developing new results, while Chapter 4 focuses on more general behavior.
Chapters 6 and 7 contain work on the description of the SSR model as a quantizer. Two main aspects of such a description are revealed. First, quantizers are specified by two operations: an encoding operation and a decoding operation. The encoding operation assigns ranges of values of the quantizer’s input signal to one of a finite number of output states. The decoding operation approximately reconstructs the original signal by assigning “reproduction values” to each encoded state. In contrast to conventional quantizers, the SSR model’s encoding is stochastic, as the output state for given input signal values is nondeterministic. However, it is possible to decode the SSR output in a similar manner to conventional quantizers, and the authors examine various ways to achieve this. The second aspect consider in the book is the performance of the decoded SSR model. Since the decoding is designed to approximate the original signal, performance is measured by the average properties of the error between this original signal and the quantizer’s output approximation. Conventionally, mean square error distortion is used to measure this average error, and the authors are examining in detail how this measure varies with noise intensity, the decoding scheme used, and the number of threshold devices in the SSR model. As with Chapters 4 and 5, Chapter 6 focuses on general behavior, while Chapter 7 is devoted to discussion of the SSR model in the event of a large number of individual threshold devices.
Chapter 8 expounds work that extends the SSR model beyond its original specification. The authors are relaxing the constraint that all individual threshold devices must have identical threshold values, and allow each device to have an arbitrary threshold value. We then consider how to optimally choose the set of threshold values as the noise intensity changes. The most important result of this study is the numerical demonstration that the SSR model, where all threshold devices have the same threshold value - is optimal, for sufficiently large noise intensity.
Chapter 9 is motivated by recent achievements of the neural-coding research. The SSR model is further extended, by including a constraint on the energy available to the system. The performance of a quantizer is characterized by two opposing factors: rate and distortion. Also, the SSR model and its extension are explored from the point of view of rate-distortion theory, to arbitrary thresholds.
Chapter 10 concludes the content of the book, by pointing to a concrete application of SSR theory, in the area of cochlear implants. It illustrates the key principles, with a view to allowing the reader to understand the ideas presented in the text.
Finally, Chapter 11 contains some speculations on the future of stochastic resonance and suprathreshold stochastic resonance.
The book will be of interest for all specialists and researchers, working on the use of the scientific problems related to the “stochastic resonance” and its engineering applications. It also is suitable for self-study by practitioners who wish to brush up on fundamental concepts.

MSC:

94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
60K40 Other physical applications of random processes
93E10 Estimation and detection in stochastic control theory
93E11 Filtering in stochastic control theory
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
94A13 Detection theory in information and communication theory
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