Killeen & Taylor NSF Proposal Supplement

1. How the propagation of error through stochastic counters affects time discrimination and other psychophysical judgements, Psychological Review 107 (2000), with Peter Killeen

The performance of fallible counters is investigated in the context of pacemaker-counter models of interval timing.Failure to reliably transmit signals from one stage of a counter to the next generates periodicity in mean and variance of counts registered,with means approximately power functions of input,and standard deviations proportional to the means (Weber 1s law).The transition diagrams and matrices of the counter are self-similar:Their eigenvalues have a fractal form,and closely approximate Julia sets.The distributions of counts registered and of hitting times approximate Weibull densities,which provide the foundation for a signal-detection model of discrimination.Different schemes for weighting the values of each stage may be established by conditioning.As higher-order stages of a cascade come on-line the veridicality of lower-order stages degrades,leading to scale-invariance in error.The capacity of a counter is more likely to be limited by fallible transmission between stages than by a paucity of stages.Probabilities of successful transmission between stages of a binary counter around 0.98 yield predictions consistent with performance in temporal discrimination and production,and with channel capacities for identification of unidimensional stimuli.

2. A stochastic adding machine and complex dynamics, Nonlinearity 13, 1889-1904 (2000), also with Peter Killeen

This paper considers properties of a markov chain on the natural numbers which models a binary adding machine in which there a nonzero probability of failure each time a register attempts to increment the succeeding register and resets. This chain has a family of natural quotient markov chains, and extends naturally to a chain on the 2-adic integers. The transition operators of these chains have a self similar structure, and have a spectrum which is, variously, the Julia set or filled Julia set of a quadratic map of the complex plane.