A tuned-trace theory of interval-timing dynamics

Animals on interval schedules of reinforcement can rapidly adjust a temporal dependent variable, such as wait time, to changes in the prevailing interreinforcement interval. We describe data on the effects of impulse, step, sine-cyclic, and variable-interval schedules and show that they can be explained by a tuned-trace timing model with a one-back threshold-setting rule. The model can also explain steady-state timing properties such as proportional and Weber law timing and the effects of reinforcement magnitude. The model assumes that food reinforcers and other time markers have a decaying effect (trace) with properties that can be derived from the rate-sensitive property of habituation (the multiple-time-scale model). In timing experiments, response threshold is determined by the trace value at the time of the most recent reinforcement. The model provides a partial account for the learning of multiple intervals, but does not account for scalloping and other postpause features of responding on interval schedules and has some problems with square-wave schedules.     

Time, trace, memory

Objections to a trace hypothesis for interval timing do not apply to the multiple-time-scale (MTS) theory, which incorporates a dynamic trace tuned by the system history and can easily accommodate interval timing over a 1,000:1 range. The MTS model can also account for Weber's law as well as systematic deviations from it. Contrary to our critics, we contend that patterns of variance in interval timing experiments are not fully described by scalar expectancy theory, and that attempting to understand timing by assigning variance to different elements of a flexible model that lacks inductive support is a flawed strategy, because the attempt may be successful even if the model is wrong. We further argue that biological plausibility is an unreliable guide to the development of behavioral theory, that prediction is not the same as test, that induction should precede deduction, and that a rat is not a clock.     

Response differentiation: A psychophysical method for response produced stimuli

Twenty-four rats were reinforced for releasing a lever between t and t + .20 sec after pressing it. Response duration differentiation performance decreased monotonically as t was increased in length from .10 sec to 1.60 sec. A response analogue of Weber’s ratio was constant throughout the middle range of duration bandwidth requirements and increased at the ex tremely short bandwidth—a response-differentiation finding analogous to Weber’s law of stimulus discrimination. This finding was interpreted as support for the conceptualization that response differentiation represents a special instance of response discrimination or discrimination based on response-produced stimuli.     

The choose-short effect and trace models of timing

The tuned-trace multiple-time-scale (MTS) theory of timing can account both for the puzzling choose-short effect in time-discrimination experiments and for the complementary choose-long effect. But it cannot easily explain why the choose-short effect seems to disappear when the intertrial and recall intervals are signaled by different stimuli. Do differential stimuli actually abolish the effect, or merely improve memory? If the latter, there are ways in which an expanded MTS theory might explain differential-context effects in terms of reduced interference. If the former, there are observational and experimental ways to determine whether differential context favors prospective encoding or some other nontemporal discrimination.     

Why 'Sounds Are Judged Longer Than Lights': Application of a Model of the Internal Clock in Humans

Three experiments, using temporal generalization and verbal estimation methods, studied judgements of duration of auditory (500-Hz tone) and visual (14-cm blue square) stimuli. With both methods, auditory stimuli were judged longer, and less variable, than visual ones. The verbal estimation experiments used stimuli from 77 to 1183 msec in length, and the slope of the function relating mean estimate to real length differed between modalities (but the intercept did not), consistent with the idea that a pacemaker generating duration representations ran faster for auditory than for visual stimuli. The different variability of auditory and visual stimuli was attributed to differential variability in the operation of a switch of a pacemaker-accumulator clock, and experimental datasuggested that such switch effects were separable from changes in pacemaker speed. Overall, the work showed how a clock model consistent with scalar timing theory, the leading account of animal timing, can address an issue derived from the classical literature on human time perception.     

Les effets de la modalité sensorielle sur la perception du temps

The subjective duration of an event can be influenced by the modality of the signal demarcating that event. For example, auditory signals are often judged as of longer duration than equivalent duration visual signals. Within the framework of pacemaker-accumulator models of timing, such modality effects raise the question of whether the internal clock is modality specific or amodal. The answer to this question is quite important for the development of cognitively and neurologically realistic models of timing and time perception. Here, we argue that the internal clock has both modality specific and amodal components. Specifically, we claim there is a common amodal pacemaker, but the switch-accumulator systems are modality specific. Moreover, we also posit that long-term memory representations of duration are stored amodally.     

Pure timing in temporal differentiation

Temporal control appears to depend on whether the critical durations are those of stimuli or those of responses. Stimulus timing (temporal discrimination) supports Weber's law, whereas response timing (temporal differentiation) indicates decreasing relative sensitivity with longer time intervals. The two types of procedure also yield different conclusions in scaling experiments designed to study the functional midpoint of two or more durations (temporal bisection procedures). In addition, the fractional-exponent power relation between emitted and required duration usually found with animals in differentiation experiments conflicts with deductions from formal analyses. The experiment reported here derived from considering differentiation arrangements as schedules of reinforcement. When analyzed from this perspective, the procedures are tandem schedules involving a required pause followed by a response, and it is the pause alone that involves temporal control. A choice procedure separated timing from responding, and enabled observations of pause timing in isolation. Pure temporal control in differentiation consisted of linear overestimation of the standard duration, and Weber's law described sensitivity. These results indicate that the two problems, the fractional-exponent power relations and the apparently different nature of sensitivity in differentiation and discrimination, disappear when temporal control is observed alone in differentiation.     

When are semantic targets detected faster than visual or acoustic ones?

Duration-discrimination data from an experiment using empty auditory intervals in a two-alternative forced-choice paradigm are presented. The observed functional relationship between standard deviation of the psychometric density function and stimulus duration is shown to be fit significantly better by a Weber’s law model of duration discrimination than by Creelman’s counter model. Both models fail to predict the rapid rise in the Weber fraction observed for durations longer than about 2 sec. However, the Weber’s law model, based on a generalization of Weber’s law, accurately predicts the initial drop in the Weber fraction for very short durations and the observed constancy of the Weber fraction for durations up to 2 sec.     

Reinforcement value and fixed‐interval performance

The concept of reinforcement value summarizes the effect of different variables, such as reinforcement delay, reinforcement magnitude, and deprivation level, on behavior. In the present set of experiments, we evaluated the effect of reinforcement devaluation on performance under FI schedules. The literature on timing and reinforcement value suggests that devaluation generates longer expected times to reinforcement than the same intervals trained under control conditions. We devalued reinforcement with delay in Experiments 1A, 1B, and 2, and diminished deprivation in Experiments 3A and 3B. Devaluation reduced response rates, increased the number of one‐response intervals, and lengthened postreinforcement pauses, but had inconsistent effects on other timing measures such as quarter life and breakpoint. The results of delayed reinforcement and diminished deprivation manipulations are well summarized as reinforcement devaluation effects. These results suggest that devaluation may reduce stimulus control. In addition, we argue that the process by which delayed reinforcement affects behavior might also explain some effects observed in other devaluation procedures through the concept of reinforcement value.     

Discrimination based on response-produced stimuli

This study outlines a simplified operant conditioning procedure for investigating discrimination based on response-produced stimuli. The method, using concurrent schedules of reinforcement, greatly reduced the complexity of the procedures and the number of training sessions necessitated by a recently offered alternative technique. Calculations of Weber’s Constant for discrimination between response-produced stimuli are presented and the relevance for learning theory of empirical studies concerning response-produced stimuli is indicated.     

A probabilistic model for the discrimination of visual number

This paper proposes a probabilistic model of how humans identify the number of dots within a briefly presented visual display. The model is an application of Thurstone’s law of comparative judgment, and it is assumed that the internal representation of numerosity consists of log-spaced random variables. The discrimination between any two different numerosities is consequently described as a function of max/rain, where max and rain are the larger and smaller numbers, respectively. The model was tested in two experiments in which the Weber fraction for numerosity, corresponding with the critical ratio of max and min, was found to have the value of 162. It was concluded that the classical span of subitizing numerosity is but a special case of the span of discrimination.     

The behavioral theory of timing: Reinforcer rate determines pacemaker rate

In the behavioral theory of timing, pulses from a hypothetical Poisson pacemaker produce transitions between states that are correlated with adjunctive behavior. The adjunctive behavior serves as a discriminative stimulus for temporal discriminations. The present experiments tested the assumption that the average interpulse time of the pacemaker is proportional to interreinforcer interval. Responses on a left key were reinforced at variable intervals for the first 25 s since the beginning of a 50-s trial, and right-key responses were reinforced at variable intervals during the second 25 s. Psychometric functions relating proportion of right-key responses to time since trial onset, in 5-s intervals across the 50-s trial, were sigmoidal in form. Average interpulse times derived by fitting quantitative predictions from the behavioral theory of timing to obtained psychometric functions decreased when the interreinforcer interval was decreased and increased when the interreinforcer interval was increased, as predicted by the theory. In a second experiment, average interpulse times estimated from trials without reinforcement followed global changes in interreinforcer interval, as predicted by the theory. Changes in temporal discrimination as a function of interreinforcer interval were therefore not influenced by the discrimination of reinforcer occurrence. The present data support the assumption of the behavioral theory of timing that interpulse time is determined by interreinforcer interval.     

Duration discrimination in a series of rhythmic events

Duration discrimination of the last of a series of four clicks was investigated. Examination of psychophysical functions from eight subjects revealed evidence for a Weber’s law model relating discrimination to base interclick interval. Also, the point of subjective equality was seen to change reliably as a function of base rate.     

Intensity discrimination for pulsed sinusoids of various frequencies

Studies of pure-tone intensity discrimination have shown that Weber’s law fails for tones in the region of 1 kHz. In this experiment, intensity discrimination of pulsed sinusoids ranging in frequency from 0.15 to 12 kHz is investigated. For each tone in this region, Weber’s law is found to fail. Some theoretical implications of these results are discussed.     

Numerical discrimination is mediated by neural coding variation

One foundation of numerical cognition is that discrimination accuracy depends on the proportional difference between compared values, closely following the Weber–Fechner discrimination law. Performance in non-symbolic numerical discrimination is used to calculate individual Weber fraction, a measure of relative acuity of the approximate number system (ANS). Individual Weber fraction is linked to symbolic arithmetic skills and long-term educational and economic outcomes. The present findings suggest that numerical discrimination performance depends on both the proportional difference and absolute value, deviating from the Weber–Fechner law. The effect of absolute value is predicted via computational model based on the neural correlates of numerical perception. Specifically, that the neural coding “noise” varies across corresponding numerosities. A computational model using firing rate variation based on neural data demonstrates a significant interaction between ratio difference and absolute value in predicting numerical discriminability. We find that both behavioral and computational data show an interaction between ratio difference and absolute value on numerical discrimination accuracy. These results further suggest a reexamination of the mechanisms involved in non-symbolic numerical discrimination, how researchers may measure individual performance, and what outcomes performance may predict.     

Interval production as an analogue of the peak procedure: Evidence for similarity of human and animal timing processes

Humans subjects repeatedly produced durations ranging from 0.5 to 1.3 sec, with each trial being followed by accurate feedback as to the time produced. An analogy was drawn between this production task and the peak procedure, a technique used to examine temporal control in rats which typically produces data conforming to scalar timing theory. Human production data were analysed in a way similar to that employed in peak procedure experiments. Using non-linear regression, it was found that the peak of relative frequency distributions of times produced varied accurately with the target time, and that the coefficient of variation of fitted curves was approximately constant as peak location and target time varied. In both these respects human production data were very similar to those collected from rats in peak procedure experiments, and thus compatible with scalar timing and Weber's law. Weber fractions obtained in this experiment were about half the value of those produced by rats in peak procedure studies, but only slightly lower than those obtained from animals in some other experimental situations.     

Shifts in the psychometric function and their implications for models of timing

This study examined how two models of timing, scalar expectancy theory (SET) and learning to time (LeT), conceptualize the learning process in temporal tasks, and then reports two experiments to test these conceptualizations. Pigeons responded on a two-alternative free-operant psychophysical procedure in which responses on the left key were reinforceable during the first two, but not the last two, quarters of a 60-s trial, and responses on the right key were reinforceable during the last two, but not the first two, quarters of the trial. In Experiment 1 three groups of birds experienced a difference in reinforcement rates between the two keys only at the end segments of the trial (i.e., between the first and fourth quarters), only around the middle segments of the trial (i.e., between the second and third quarters), or in both end and middle segments. In Condition 1 the difference in reinforcement rate favored the left key; in Condition 2 it favored the right key. When the reinforcement rates differed in the end segments of the trial, the psychometric function--the proportion of right responses across the trial--did not shift across conditions; when it occurred around the middle of the trial or in both end and middle segments, the psychometric function shifted across conditions. Experiment 2 showed that the psychometric function shifts even when the overall reinforcement rate for the two keys is equal, provided the rates differ around the middle of the trial. This pattern of shifts of the psychometric function is inconsistent with SET. In contrast, LeT provided a good quantitative fit to the data.     

Duration discrimination by musicians and nonmusicians

This study investigated the effects of stimulus modality, standard duration, sex, and laterality in duration discrimination by musicians and nonmusicians. Seventeen musicians (M age = 24.1 yr.) and 22 nonmusicians (M age = 26.8 yr.) participated. Auditory (1,000 Hz) and tactile (250 Hz) sinusoidal suprathreshold stimuli with varying durations were used. The standard durations tested were 0.5 and 3.0 sec. Participants discriminated comparison stimuli which had durations slightly longer and shorter than the standard durations. Difference limens were found by the method of limits and converted to Weber fractions based on the standard durations. Musicians had lower, i.e., better, Weber fractions than nonmusicians in the auditory modality, but there was no significant difference between musicians and nonmusicians in the tactile modality. Auditory discrimination was better than tactile discrimination. Discrimination improved when the standard duration was increased both for musicians and nonmusicians. These results support previous findings of superior auditory processing by musicians. Significant differences between discrimination in the millisecond and second ranges may be due to a deviation from Weber's law and the discontinuity of timing in different duration ranges reported in the literature.     

Contingency-shaped and rule-governed behavior: instructional control of human loss avoidance

Instructions can override the influence of programmed schedules of reinforcement. Although this finding has been interpreted as a limitation of reinforcement schedule control in humans, an alternative approach considers instructional control, itself, as a phenomenon determined by subjects' reinforcement histories. This approach was supported in a series of experiments that studied instructional and schedule control when instructions either did or did not accord with the schedule of reinforcement. Experiment I demonstrated that accurate instructions control discriminative performances on multiple avoidance schedules, and that such control persists in a novel discrimination. Experiments II and III showed that elimination of instruction-following occurs when inaccurate instructions cause subjects to contact a monetary loss contingency. Experiment IV demonstrated the reinforcing properties of accurate instructions. Skinner's view of rule-governed behavior is consistent with these findings, and can be extended to account for many aspects of instructional control of human operant behavior.     

Time scales in motor learning and development

A theoretical framework based on the concepts and tools of nonlinear dynamical systems is advanced to account for both the persistent and transitory changes traditionally shown for the learning and development of motor skills. The multiple time scales of change in task outcome over time are interpreted as originating from the system's trajectory on an evolving attractor landscape. Different bifurcations between attractor organizations and transient phenomena can lead to exponential, power law, or S-shaped learning curves. This unified dynamical account of the functions and time scales in motor learning and development offers several new hypotheses for future research on the nature of change in learning theory.