Robust stochastic resonance for simple threshold neurons

Abstract

Simulation and theoretical results show that memoryless threshold neurons benefit from small amounts of almost all types of additive noise and so produce the stochastic-resonance or SR effect. Input-output mutual information measures the performance of such threshold systems that use subthreshold signals. The SR result holds for all possible noise probability density functions with finite variance. The only constraint is that the noise mean must fall outside a "forbidden" threshold-related interval that the user can control--a new theorem shows that this condition is also necessary. A corollary and simulations show that the SR effect occurs for right-sided beta and Weibull noise as well. These SR results further hold for the entire uncountably infinite class of alpha-stable probability density functions. Alpha-stable noise densities have infinite variance and infinite higher-order moments and often model impulsive noise environments. The stable noise densities include the special case of symmetric bell-curve densities with thick tails such as the Cauchy probability density. The SR result for alpha-stable noise densities shows that the SR effect in threshold and thresholdlike systems is robust against occasional or even frequent violent fluctuations in noise. Regression analysis reveals both an exponential relationship for the optimal noise dispersion as a function of the alpha bell-curve tail thickness and an approximate linear relationship for the SR-maximal mutual information as a function of the alpha bell-curve tail thickness.

Noise Enhanced Hypothesis-Testing in the Restricted Bayesian Framework

“…11 Since , the first condition in Theorem 5 is automatically satisfied. For , and ; hence, the third bullet of the second condition implies that (68) is required for improvability.…”

“…For , and ; hence, the third bullet of the second condition implies that (68) is required for improvability. For and ; hence, the second bullet of the second condition becomes active, 11 Note that S = f1; 2;3g for z = 0, in which case the first condition in Theorem 5 cannot satisfied since F = f0; 0;0g. Therefore, z = 0 is not considered in obtaining improvability conditions.…”

“…This counterintuitive effect is mainly due to system nonlinearities and/or some parameters being suboptimal [14]. Improvements that can be obtained via SR can be in various forms, such as an increase in output signal-to-noise ratio (SNR) [1], [4], [5] or mutual information [6]- [11], [15], [16]. The first study of SR was performed in [1] to investigate the periodic recurrence of ice gases.…”

Design and characterization of nonlinear functions for the transmission of a small signal with non-Gaussian noise

“…The parameter that should be optimized is the threshold value θ . The threshold function is known to cause SR, and the optimum threshold value has been discussed for specific noises in terms of this phenomenon [10,12]. Note that, in the case of Gaussian noise, the SR in the static threshold function cannot exceed the response in the linear function from Eq.…”

“…The response is obviously correlated with the nonlinearity and noise characteristics, and understanding this relationship has been a key issue in both science and engineering. There have been several studies on the noise-characteristic dependence of the response [9][10][11][12], but in most cases, the behavioral approach has been used for analysis due to the difficulties of a full analytical treatment of a nonlinear function and noise. Recently, Ichiki and Tadokoro reported an analytical approach to finding a static nonlinear function that is optimized to enhance a small signal buried in a non-Gaussian noise [13].…”