Maximising information transfer through nonlinear noisy devices Mark D. McDonnell a , Nigel G. Stocks b , Charles E. M. Pearce c and Derek Abbott a a Centre for Biomedical Engineering (CBME) and Department of Electrical & Electronic Engineering, The University of Adelaide, SA 5005, Australia b School of Engineering, The University of Warwick, Coventry CV4 7AL, UK c Department of Applied Mathematics The University of Adelaide, SA 5005, Australia ABSTRACT Consider an array of parallel comparators (threshold devices) receiving the same input signal, but subject to independent noise, where the output from each device is summed to give an overall output. Such an array is a good model of a number of nonlinear systems including flash analogue to digital converters, sonar arrays and parallel neurons. Recently, this system was analysed by Stocks in terms of information theory, who showed that under certain conditions the transmitted information through the array is maximised for non-zero noise. This phenomenon was termed Suprathreshold Stochastic Resonance (SSR). In this paper we give further results related to the maximisation of the transmitted information in this system. Keywords: optimal quantisation, stochastic resonance, ADC 1.INTRODUCTION The problem we examine in this paper is to maximise the information flow through the array of N comparators (threshold devices) shown in Figure 1. All comparators receive the same input signal, x and the i–th device is subject to independent continuously valued additive noise, η i (i =1,..,N ). The output from each comparator is unity if the input signal plus the noise is greater than the threshold, θ i , of that device and zero otherwise. The outputs from each comparator are summed to give the overall output signal, y. Hence, y is a discrete signal taking on integer values from 0 to N and can be considered as the number of devices that are currently “on”. Such arrays can model various devices such as flash analog to digital converters (ADCs) 1 (when the thresholds are uniformly distributed across the signal space), DIMUS (Digital Multibeam Steering) sonar arrays, in the “on target” position 2, 3 or a summing network of N FitzHugh-Nagumo neurons. 4 In recent work, Stocks analysed this system using Shannon information theory. For the case of all thresh- olds set equal to the mean, it was shown that the maximum transmitted information (also known as mutual information, i.e. the information, in bits per sample, about the input contained in the output) has a maximum for nonzero noise. This phenomenon was termed Suprathreshold Stochastic Resonance (SSR). 5–7 Conventional Stochastic Resonance (SR) occurs when a nonlinear system is optimised by a nonzero value of noise. 8 For a single threshold SR only occurs for subthreshold signals. By contrast, SSR occurs for any magnitude of signal, due to the presence of more than one threshold. More recently, the principle of SSR has been applied to cochlear implants. 9, 10 Further author information: (Send correspondence to Mark D McDonnell.) Mark D. McDonnell.: E-mail: mmcdonne@eleceng.adelaide.edu.au, Telephone: +61 8 83036296 Fax: +61 8 8303 4360 Nigel G. Stocks.: E-mail: es2003@eng.warwick.ac.uk Charles E. M. Pearce.: E-mail: cpearce@maths.adelaide.edu.au Derek Abbott.: E-mail: dabbott@eleceng.adelaide.edu.au, Telephone: +61 8 83035748 Biomedical Applications of Micro- and Nanoengineering, Dan V. Nicolau, Abraham P. Lee, Editors, Proceedings of SPIE Vol. 4937 (2002) © 2002 SPIE · 0277-786X/02/$15.00 254