Applications of Forbidden Interval Theorems in Stochastic Resonance

Abstract

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.

Algebraic Information Theory and Kosko's Forbidden Interval Theorem

“…, which continuously increases from 0 to 1/3, as 4 3 , we see that N σ (1.7) ≡ 1. So, now we see that the continuously parametrized channel matrix is of the form :…”

“…This variance is considered to be the noise power. The standard deviation σ may be interpreted [11] as the noise rms amplitude 4 . Furthermore, we require a continuous "family" of noise distributions with a fixed mean µ.…”

“…Kosko's Forbidden Interval Theorem (FIT) [1,2,3,4] gives guidance on when additive noise will improve signal strength in noisy subthreshold systems (the motivation for such threshold systems being neuronal spikes [5]). This counterintuitive effect is due to the phenomenon of Stochastic Resonance (SR).…”

“…We give a more general definition of SR than Kosko [1,2,3,4], and re-examine his FIT in that light. With our definition of SR we find that the FIT must be relaxed.…”

Aperiodic stochastic resonance in neural information processing with Gaussian colored noise

“…As a starting point, we will generalize the existing perturbation results (Freidlin and Wentzell 2012) of nonlinear dynamical systems from Gaussian white noise to Gaussian colored noise. And then, we follow the "forbidden interval" theorem (Kosko et al 2009) and direct simulation to explore the aperiodic stochastic resonance in bistable and excitable neural systems.…”