Counting and timing models in psychophysics and the conjoint Weber's law

J. C. Falmagne, G. Iverson, and S. Marcovici ((1979). Psychological Review, 86, 25–43) proposed a generalization of Weber's law, which they called the conjoint Weber's law. Empirically, the law sometimes holds. When it fails, the data satisfy a relation that Falmagne et al. identify as the conjoint Weber's inequality. This paper investigates the ability of counting and timing models of psychophysics to predict Weber's law and the conjoint Weber's law. It is shown that although the timing model naturally predicts both laws to hold, all reasonable counting models predict them to fail. Instead, counting models naturally predict Weber's inequality and the conjoint Weber's inequality.     

Do perception and motor production share common timing mechanisms: A correlational analysis

The two experiments of this study exploited individual variation in timing ability to ask whether the production of time intervals by different motor effectors and the judgement of perceptually based time intervals all share common timing mechanisms. In one task subjects produced a series of taps, attempting to maintain constant intervals between them. Individual differences in variability of the produced intervals correlated across the effectors of finger and foot. That is, people that were ‘good timers’ with one effector tended to be ‘good timers’ with another. Besides timing motor production, the subjects also judged durations of brief perceptual events. The acuity of perceptual judgements correlate substantially with regularity of motor production. Further results involving maximum speed of motor production suggested that variability of motor timing comes from two sources, one source in common with perception, and hence called clock variability, and the other source in common with motor speed, and hence called motor implementation variability. The second experiment showed that people high in skill on the piano were better at both types of timing on the average than control subjects with no expertise.     

Timing processes in motor imagery

Previous research shows inconsistencies in the timing of imagined and actual actions. Little is known about the timing in imagery, or how it relates to other forms of timing. Two studies examined whether imagery timing followed Weber's law, where variations in judgements grow linearly as the interval duration increases, or Vierordt's law, where short durations are overestimated and longer durations underestimated. In Study 1 participants (n=22) mentally walked and estimated journey times for flat paths and stairways, with and without a load. The timing patterns that emerged did not conform to Weber's law. In Study 2 participants (n=20) completed imagery, reproduction, production, and estimation timing tasks. Timing errors for imagery along a straight path, reproduction, estimation, and production all showed “Vierordt-like” effects. However, when imagining walking in a square participants consistently overestimated. It was concluded that imagery and interval timing processes are similar, but imagery timing is task dependent.     

Correlations between intelligence and components of serial timing variability

Psychometric intelligence correlates with reaction time in elementary cognitive tasks, as well as with performance in time discrimination and judgment tasks. It has remained unclear, however, to what extent these correlations are due to top–down mechanisms, such as attention, and bottom–up mechanisms, i.e. basic neural properties that influence both temporal accuracy and cognitive processes. Here, we assessed correlations between intelligence (Raven SPM Plus) and performance in isochronous serial interval production, a simple, automatic timing task where participants first make movements in synchrony with an isochronous sequence of sounds and then continue with self-paced production to produce a sequence of intervals with the same inter-onset interval (IOI). The target IOI varied across trials. A number of different measures of timing variability were considered, all negatively correlated with intelligence. Across all stimulus IOIs, local interval-to-interval variability correlated more strongly with intelligence than drift, i.e. gradual changes in response IOI. The strongest correlations with intelligence were found for IOIs between 400 and 900 ms, rather than above 1 s, which is typically considered a lower limit for cognitive timing. Furthermore, poor trials, i.e. trials arguably most affected by lapses in attention, did not predict intelligence better than the most accurate trials. We discuss these results in relation to the human timing literature, and argue that they support a bottom–up model of the relation between temporal variability of neural activity and intelligence.     

Interval timing predicts impulsivity in intertemporal choice: combined behavioral and drift-diffusion model evidence

ABSTRACT

This study investigated the relationship between interval timing and impulsivity in intertemporal choice in a healthy population. A duration production task was used to assess interval timing. Choice impulsivity was assessed using a hypothetical money choice task. Results from 134 participants indicated that faster internal clock significantly predicted lower choice impulsivity. A subsequent drift-diffusion model analysis of the behavioural data revealed that in the sub-group of relatively farsighted participants, faster internal clock predicted consideration of more information before making a choice, which in turn was associated with lower choice impulsivity. In the sub-group of relatively impatient participants too, faster internal clock predicted consideration of more information, but which in turn was associated with higher choice impulsivity. It is concluded that among relatively farsighted individuals in a normal population, faster internal clock favours a more deliberate processing of the options at hand, thus eliciting less impulsive choices.     

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.     

Variability of timing in expressive piano performance increases with interval duration

In simple motor tasks such as finger tapping at different constant rates, within-trial variability of response interonset intervals (IOIs) increases with IOI duration (which varies between trials). In expressive piano performance, the rate of key depressions is not constant, in part due to compositional structure and in part due to expressive timing, so that IOIs of many different durations occur within a single “trial.” Nevertheless, across repeated performances of the same music (Schumann’s “Träumerei” and Debussy’s “La fille aux cheveux de lin”) at the same intended tempo, the standard deviations of individual IOIs tend to increase linearly with their average duration. This is also true when the variation is due to expressive timing alone and when unintended differences in basic tempo between performances are taken into account. In the music studied here, at least, there was no evidence of compensatory timing. The results suggest that the pianists employed a continuously variable tempo governed by a flexible internal timekeeper whose variability follows a generalized Weber’s law (for IOIs longer than about 300 msec).     

“One-thousand<Emphasis Type="Italic">one</Emphasis> … one-thousand<Emphasis Type="Italic">two</Emphasis> …”: Chronometric counting violates the scalar property in interval timing

Medical College of Wisconsin, Milwaukee, Wisconsin Weber’s law applied to interval timing is called thescalar property. A hallmark of timing in the secondsto-minutes range, the scalar property is characterized by proportionality between the standard deviation of a response distribution and the duration being timed. In this temporal reproduction study, we assessed whether the scalar property was upheld when participants chronometrically counted three visually presented durations (8, 16, and 24 sec) as compared with explicitly timing durations without counting. Accuracy for timing and accuracy for counting were similar. However, whereas timing variability showed the scalar property, counting variability did not. Counting variability across intervals was accurately modeled by summing a random variable representing an individual count. A second experiment replicated the first and demonstrated that task differences were not due to presentation order or practice effects. The distinct psychophysical properties of counting and timing behaviors argue for greater attention to participant strategies in timing studies.     

Long-range correlation properties in motor timing are individual and task specific

1/f (β) noise represents a specific form of (long-range) correlations in a time series that is pervasive across many sensorimotor variables. Recent studies have shown that the precise properties of the correlations demonstrated by a group of test participants may vary as a function of experimental conditions or factors characterizing the group. Our purpose in the present study was to clarify whether long-range correlations affect sensorimotor performance generally or in a task-specific manner and whether each individual produces characteristic long-range correlations that are reliable across several runs of the same task. We analyzed the series of time intervals produced by 43 participants in two timing tasks: unimanual rhythmic tapping and circle drawing. We found that a participant's 1/f (β) properties in tapping were not related to the 1/f (β) properties in circle drawing. However, within each task, individual differences were reliable, and a Cronbach's alpha of .59 showed a high degree of within-subjects reproducibility of the long-range correlations. Thus, long-range correlations represent a consistent and distinctive characteristic of individuals performing a particular task, rather than a ubiquitous generic property of sensorimotor time series. The implications of these results are discussed from both a theoretical and a methodological perspective.     

The influence of dominant versus non-dominant hand on event and emergent motor timing

It has been hypothesized that timing in tapping utilizes event timing; a clock-like process, whereas timing in circle drawing is emergent. Three experiments examined timing in tapping and circle drawing by the dominant and non-dominant hand. Participants were right-hand dominant college aged males and females. The relationship between variance and the square of the timed interval (the Weber fraction), thought to capture clock-like timekeeping processes, was compared. Furthermore, timing variance was decomposed into a clock and a motor component. The slopes for timing were different for dominant hand tapping and circle drawing, but equal for non-dominant and dominant hand tapping. Negative lag one covariance, consistent with motor implementation variability, was found for non-dominant but not for dominant hand circle drawing (Experiment 1). Practice did not influence this relation (Experiment 2). A significant correlation for clock variability was found between non-dominant hand circle drawing and tapping (Experiment 3). Collectively, these findings indicate that event timing is shareable across hands while emergent timing is specific to an effector. Emergent timing does not appear to be obligatory for the non-dominant hand in circle drawing. We suggest that the use of emergent timing might depend upon the extensive practice experienced by a person's dominant hand.     

Comparison of timing and classical conditioning

Four experiments with rats investigated if the timing of a stimulus (sound) correlated with the strength of a conditioned response (CR) to the stimulus. The timing (effective duration) of the stimulus was measured using the peak procedure, similar to a discrete-trials fixed-interval procedure. The rats were trained so that their response rate reached a maximum about 40 s or 60 s after the onset of a light; the time of the maximum measured from the start of the light (peak time) was the measure of timing. On some trials, the light was preceded by a short (5 s) or long (20 s or 30 s) interval of sound. We assumed that the difference in peak time after long and short sounds reflected the timing of the sound--if the sound was timed, the longer sound would produce a lower peak time; if the sound was not timed, the two durations of sound would produce the same peak time. The CR was lever-pressing during the sound. The sound was treated in various ways: presented alone (Experiments 1, 3, and 4), followed by food (Experiments 1, 3, and 4), preceded by food (Experiment 3), and followed by food after 20 s (Experiment 4). Treatments that produced no timing of sound produced no CR, and treatments that increased (or diseased) timing also increased (or decreased) the CR. The results suggest that there is overlap between the mechanisms that produce time discrimination and the mechanisms that produce classical conditioning.     

Sequential and Biomechanical Factors Constrain Timing and Motion in Tapping

The authors examined how timing accuracy in tapping sequences is influenced by sequential effects of preceding finger movements and biomechanical interdependencies among fingers. Skilled pianists tapped sequences at 3 rates; in each sequence, a finger whose motion was more or less independent of other fingers' motion was preceded by a finger to which it was more or less coupled. Less independent fingers and those preceded by a more coupled finger showed large timing errors and change in motion because of the preceding finger's motion. Motion change correlated with shorter intertap intervals and increased with rate. Thus, timing of sequence elements is not independent of the motion trajectories that individuals use to produce them. Neither motion nor its relation to timing is invariant across rates.     

Evidence of Common Timing Processes in the Control of Manual,Orofacial, and Speech Movements

Recent investigations of timing in motor control have been interpreted as support for the concept of brain modularity. According to this concept, the brain is organized into functional modules that contain mechanisms responsible for general processes. Keele and colleagues (Keele & Hawkins, 1982; Keele & Ivry, 1987; Keele, Ivry, & Pokorny, 1987; Keele, Pokorny, Corcos, & Ivry, 1985) demonstrated that the within-subject variability in cycle duration of repetitive movements is correlated across finger, forearm, and foot movements, providing evidence in support of a general timing module. The present study examines the notion of timing modularity of speech and nonspeech movements of the oral motor system as well as the manual motor system. Subjects produced repetitive movements with the finger, forearm, and jaw. In addition, a fourth task involved the repetition of a syllable. All tasks were to be produced with a 400-ms cycle duration; target duration was established with a pacing tone, which then was removed. For each task, the within-subject variability of the cycle duration was computed for the unpaced movements over 20 trials. Significant correlations were found between each pair of effectors and tasks. The present results provide evidence that common timing processes are involved not only in movements of the limbs, but also in speech and nonspeech movements of oral structures.     

Evidence of Common Timing Processes in the Control of Manual,Orofacial, and Speech Movements

Recent investigations of timing in motor control have been interpreted as support for the concept of brain modularity. According to this concept, the brain is organized into functional modules that contain mechanisms responsible for general processes. Keele and colleagues (Keele & Hawkins, 1982; Keele & Ivry, 1987; Keele, Ivry, & Pokorny, 1987; Keele, Pokorny, Corcos, & Ivry, 1985) demonstrated that the within-subject variability in. cycle duration of repetitive movements is correlated across finger, forearm, and foot movements, providing evidence in support of a general timing module. The present study examines the notion of timing modularity of speech and nonspeech movements of the oral motor system as well as the manual motor system. Subjects produced repetitive movements with the finger, forearm, and jaw. In addition, a fourth task involved the repetition of a syllable. All tasks were to be produced with a 400-ms cycle duration; target duration was established with a pacing tone, which then was removed. For each task, the within-subject variability of the cycle duration was computed for the unpaced movements over 20 trials. Significant correlations were found between each pair of effectors and tasks. The present results provide evidence that common timing processes are involved not only in movements of the limbs, but also in speech and nonspeech movements of oral structures.     

Weber (slope) analyses of timing variability in tapping and drawing tasks

Timing variability in continuous drawing tasks has not been found to be correlated with timing variability in repetitive finger tapping in recent studies (S. D. Robertson et al., 1999; H. N. Zelaznik, R. M. C. Spencer, & R. B. Ivry, 2002). Furthermore, the central component of timing variability, as measured by the slope of the timing variance versus the square of the timed interval, differed for tapping and drawing tasks. On the basis of those results, the authors posited that timing in tapping is explicit and as such uses a central representation of the interval to be timed, whereas timing in drawing tasks is implicit, that is, the temporal component is an emergent property of the trajectory produced. The authors examined that hypothesis in the present study by determining the linear relationship between timing variance and squared duration for tapping, circle-drawing, and line-drawing tasks. Participants (N = 50) performed 1 of 5 tasks: finger tapping, line drawing in the x dimension, line drawing in the y dimension, continuous circle drawing timed in the x dimension, or continuous circle drawing timed in the y dimension. The slopes differed significantly between finger tapping, line drawing, and circle drawing, suggesting separable sources of timing variability. The slopes of the 2 circle-drawing tasks did not differ from one another, nor did the slopes of the 2 line-drawing tasks differ significantly, suggesting a shared timing process within those tasks. Those results are evidence of a high degree of specificity in timing processes.     

The beneficial effect of synchronized action on motor and perceptual timing in children

We examined the role of action in motor and perceptual timing across development. Adults and children aged 5 or 8 years old learned the duration of a rhythmic interval with or without concurrent action. We compared the effects of sensorimotor versus visual learning on subsequent timing behaviour in three different tasks: rhythm reproduction (Experiment 1), rhythm discrimination (Experiment 2) and interval discrimination (Experiment 3). Sensorimotor learning consisted of sensorimotor synchronization (tapping) to an isochronous visual rhythmic stimulus (ISI = 800 ms), whereas visual learning consisted of simply observing this rhythmic stimulus. Results confirmed our hypothesis that synchronized action during learning systematically benefitted subsequent timing performance, particularly for younger children. Action‐related improvements in accuracy were observed for both motor and perceptual timing in 5 years olds and for perceptual timing in the two older age groups. Benefits on perceptual timing tasks indicate that action shapes the cognitive representation of interval duration. Moreover, correlations with neuropsychological scores indicated that while timing performance in the visual learning condition depended on motor and memory capacity, sensorimotor learning facilitated an accurate representation of time independently of individual differences in motor and memory skill. Overall, our findings support the idea that action helps children to construct an independent and flexible representation of time, which leads to coupled sensorimotor coding for action and time.     

Dissecting the clock: understanding the mechanisms of timing across tasks and temporal intervals

Currently, it is unclear what model of timing best describes temporal processing across millisecond and second timescales in tasks with different response requirements. In the present set of experiments, we assessed whether the popular dedicated scalar model of timing accounts for performance across a restricted timescale surrounding the 1-second duration for different tasks. The first two experiments evaluate whether temporal variability scales proportionally with the timed duration within temporal reproduction. The third experiment compares timing across millisecond and second timescales using temporal reproduction and discrimination tasks designed with parallel structures. The data exhibit violations of the assumptions of a single scalar timekeeper across millisecond and second timescales within temporal reproduction; these violations are less apparent for temporal discrimination. The finding of differences across tasks suggests that task demands influence the mechanisms that are engaged for keeping time.     

The effect of Parkinson's disease on time estimation as a function of stimulus duration range and modality

The present research sought to investigate the role of the basal ganglia in timing of sub- and supra-second intervals via an examination of the ability of people with Parkinson's disease (PD) to make temporal judgments in two ranges, 100-500 ms, and 1-5 s. Eighteen non-demented medicated patients with PD were compared with 14 matched controls on a duration-bisection task in which participants were required to discriminate auditory and visual signal durations within each time range. Results showed that patients with PD exhibited more variable duration judgments across both signal modality and duration range than controls, although closer analyses confirmed a timing deficit in the longer duration range only. The findings presented here suggest the bisection procedure may be a useful tool in identifying timing impairments in PD and, more generally, reaffirm the hypothesised role of the basal ganglia in temporal perception at the level of the attentionally mediated internal clock as well as memory retrieval and/or decision-making processes.     

Weber (Slope) Analyses of Timing Variability in Tapping and Drawing Tasks

Timing variability in continuous drawing tasks has not been found to be correlated with timing variability in repetitive finger tapping in recent studies (S. D. Robertson et al., 1999; H. N. Zelaznik, R. M. C. Spencer, & R. B. Ivry, 2002). Furthermore, the central component of timing variability, as measured by the slope of the timing variance versus the square of the timed interval, differed for tapping and drawing tasks. On the basis of those results, the authors posited that timing in tapping is explicit and as such uses a central representation of the interval to be timed, whereas timing in drawing tasks is implicit, that is, the temporal component is an emergent property of the trajectory produced. The authors examined that hypothesis in the present study by determining the linear relationship between timing variance and squared duration for tapping, circle-drawing, and line-drawing tasks. Participants (N = 501 performed 1 of 5 tasks: finger tapping, line drawing in the x dimension, line drawing in the y dimension, continuous circle drawing timed in the x dimension, or continuous circle drawing timed in the y dimension. The slopes differed significantly between finger tapping, line drawing, and circle drawing, suggesting separable sources of timing variability. The slopes of the 2 circle-drawing tasks did not differ from one another, nor did the slopes of the 2 line-drawing tasks differ significantly, suggesting a shared timing process within those tasks. Those results are evidence of a high degree of specificity in timing processes.     

The effect of alcohol administration on human timing: a comparison of prospective timing, retrospective timing and passage of time judgements

Previous research suggests that human timing may be affected by alcohol administration. The current study aimed to expand on previous research by examining the effect of alcohol on prospective timing, retrospective timing and passage of time judgements. A blind between-subjects design was employed in which participants were either administered 0 g of alcohol per kilogramme of body weight (placebo), 0.4 g/kg (low dose) or 0.6 g/kg (high dose). Participants completed four types of temporal task; verbal estimation and temporal generalisation, a retrospective timing task and a passage of time judgement task. A high dose of alcohol resulted in overestimations of duration relative to the low dose and placebo group in the verbal estimation task. A high dose of alcohol was also associated with time passing more quickly than normal. Alcohol had no effect on retrospective judgements. The results suggest that a high dose of alcohol increases internal clock speed leading to over-estimations of duration on prospective timing tasks, and the sensation of time passing more quickly than normal. The absence of an effect of alcohol on retrospective timing supports the suggestion that retrospective judgements are not based on the output of an internal clock.