Threshold learning and Brownian motion (Corresp.)

An approach to the analysis of threshold learning suggested by the classical theory of Brownian motion is presented. In particular, it is shown how a nonlinear Langevin equation represents the motion of the threshold of a trainable signal detector, and how a Fokker-Planck diffusion equation yields an estimate of the shape of the probability density of the threshold. Our results are applicable to all trainable signal detectors in which the training procedure 1) raises the threshold in response to a false alarm, lowers the threshold in response to a false rest, and keeps the threshold unchanged in response to a correct decision, and 2) adjusts the size of the threshold increment by an amount that depends only on the trial number, and such that the threshold can eventually reach any real number.