Applications of Forbidden Interval Theorems in Stochastic Resonance

Forbidden interval theorems state whether a stochastic-resonance noise benefit occurs based on whether the average noise value falls outside or inside an interval of parameter values. Such theorems act as a type of screening device for mutual-information noise benefits in the detection of subthreshold signals. Their proof structure reduces the search for a noise benefit to the often simple task of showing that a zero limit exists. This chapter presents the basic forbidden interval theorem for threshold neurons and four applications of increasing complexity. The first application shows that small amounts of electrical noise can help a carbon nanotube detect faint electrical signals. The second application extends the basic forbidden interval theorem to quantum communication through the judicious use of squeezed light. The third application extends the theorems to noise benefits in standard models of spiking retinas. The fourth application extends the noise benefits in retinal and other neuron models to Levy noise that generalizes Brownian motion and allows for jump and impulsive noise processes. 1 Forbidden Interval Theorems for Stochastic Resonance Stochastic resonance (SR) occurs in a nonlinear system when noise benefits the system [3, 17, 33]. The noise benefit can take the form of an increase in mutual information or a signal-to-noise ratio or correlation or a decrease in an error measure. But when will such a noise benefit occur? Forbidden interval theorems answer that SR question for several nonlinear systems. The theorems act as a type of SR screening device because they can give sufficient or necessary conditions for an SR noise benefit. We here restrict noise benefits B. Kosko (B) Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA, e-mail: [email protected] V. In et al. (eds.), Applications of Nonlinear Dynamics, Understanding Complex Systems, 71 DOI 10.1007/978-3-540-85632-0 7, c © Springer-Verlag Berlin Heidelberg 2009