How the propagation of error through stochastic counters affects time discrimination and other psychophysical judgments.

Abstract

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 power functions of input and standard deviations approximately proportional to the means (Weber's 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.

Writing and overwriting short-term memory

“…In other summation paradigms-for instance, duration discrimination-they may be asked whether the sum of one type of stimulus exceeds a criterion (e.g., Loftus & McLean, 1999;Meck & Church, 1983). Counting is summation with multiple criteria corresponding to successive numbers (Davis & Pérusse, 1988;Killeen & Taylor, 2000). Effects analogous to those reported here have been discussed under the rubric response summation (e.g., Aydin & Pearce, 1997).…”

Duration perception in crossmodally-defined intervals

“…Namely, summative distortion predicts a constant bias across all interval durations leading to an identical slope when the order of onset and offset of an interval is reversed. Multiplicative distortion, in contrast, predicts that the bias increases linearly with the interval duration because according to the pacemaker-accumulator model the longer the switch is activated the larger is the difference in the accumulated pulses between pacemakers with high pulse emission frequency and pacemakers with low pulse emission frequency (see Killeen & Taylor, 2000). It should be noted that the effect of multiplicative distortions is additional to the one of latencies in the activation/deactivation of the switch.…”

Timing and time perception: A review of recent behavioral and neuroscience findings and theoretical directions

“…As compared with the pacemaker case, a more limited number of models have emphasized the fallibility of the counter (Killeen & Fetterman, 1988), although Killeen and Taylor (2000) proposed a model involving a cascade of counter systems. If counting is hierarchical, as it is when decimal or binary systems are used, the magnitude of the timing error increases disproportionately each time the next stage in the counter must be set.…”