Applications of Forbidden Interval Theorems in Stochastic Resonance

  • Bart Kosko
  • Ian Lee
  • Sanya Mitaim
  • Ashok Patel
  • Mark M. Wilde
Part of the Understanding Complex Systems book series (UCS)


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.


Neuron Model Stochastic Resonance Threshold System Levy Process Normal Inverse Gaussian 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Bart Kosko
    • 1
  • Ian Lee
  • Sanya Mitaim
  • Ashok Patel
  • Mark M. Wilde
  1. 1.Department of Electrical EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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