We measured temporal reproduction in human subjects with various levels of musical expertise: expert drummers, string musicians, and non-musicians. While duration reproduction of the non-percussionists showed a characteristic central tendency or regression to the mean, drummers responded veridically. Furthermore, when the stimuli were auditory tones rather than flashes, all subjects responded veridically. The behavior of all three groups in both modalities is well explained by a Bayesian model that seeks to minimize reproduction errors by incorporating a central tendency prior, a probability density function centered at the mean duration of the sample. We measured separately temporal precision thresholds with a bisection task; thresholds were twice as low in drummers as in the other two groups. These estimates of temporal precision, together with an adaptable Bayesian prior, predict well the reproduction results and the central tendency strategy under all conditions and for all subject groups. These results highlight the efficiency and flexibility of sensorimotor mechanisms estimating temporal duration.

The mapping of number onto space is fundamental to measurement and mathematics. However, the mapping of young children, unschooled adults, and adults under attentional load shows strong compressive nonlinearities, thought to reflect intrinsic logarithmic encoding mechanisms, which are later "linearized" by education. Here we advance and test an alternative explanation: that the nonlinearity results from adaptive mechanisms incorporating the statistics of recent stimuli. This theory predicts that the response to the current trial should depend on the magnitude of the previous trial, whereas a static logarithmic nonlinearity predicts trialwise independence. We found a strong and highly significant relationship between numberline mapping of the current trial and the magnitude of the previous trial, in both adults and school children, with the current response influenced by up to 15% of the previous trial value. The dependency is sufficient to account for the shape of the numberline, without requiring logarithmic transform. We show that this dynamic strategy results in a reduction of reproduction error, and hence improvement in accuracy.numerical cognition | predictive coding | approximate number system | Weber-Fechner law | serial dependency H umans have a strong intuition of the spatial nature of numbers, usually (but not always) a horizontal "mental numberline," with numbers increasing from left to right (1-4). However, the nature of number mapping is not identical for all, but changes during development, starting from a nonlinear representation, well characterized as logarithmic (placing, for example, the number 10 near the midpoint of a 1-100 scale), then becoming more linear over the first years of schooling (3,5,6). Similarly, logarithmic-like numberlines have been demonstrated in indigenous Amazonian populations without formal mathematical schooling (4).Several recent studies have shown that under certain circumstances even the math-educated tend to reproduce numbers logarithmically. For example, we showed that depriving attentional resources leads to logarithmic-like numberline responses (7), consistent with the possibility that the native logarithmic encoding emerges when attention is deprived. Other studies have shown that the use of unfamiliar numerical format (such as exponential) can induce a switch from a linear to a logarithmic-like response, even in math-educated adults (7,8). Most recently, Dotan and Dehaene (9) have devised a clever technique to record the whole trajectory of the pointing response (across the face of a touchscreen), rather than just the endpoint: The response begins quite logarithmically, then corrects toward linear mapping by the time contact is made. All these studies have led many to interpret the logarithmic map as the direct reflection of the internal native number representation (4, 10-12) that becomes corrected over time by education but can emerge under special circumstances.Whereas the nonlinear numberline is consistent with intrinsic logarithmic processes, other explanat...

The world tends to be stable from moment to moment, leading to strong serial correlations in natural scenes. As similar stimuli usually require similar behavioural responses, it is highly likely that the brain has developed strategies to leverage these regularities. A good deal of recent psychophysical evidence is beginning to show that the brain is sensitive to serial correlations, causing strong drifts in observer responses towards previously seen stimuli. However, it is still not clear that this tendency leads to a functional advantage. Here, we test a formal model of optimal serial dependence and show that as predicted, serial dependence in an orientation reproduction task is dependent on current stimulus reliability, with less precise stimuli, such as low spatial frequency oblique Gabors, exhibiting the strongest effects. We also show that serial dependence depends on the similarity between two successive stimuli, again consistent with the behaviour of an ideal observer aiming at minimizing reproduction errors. Lastly, we show that serial dependence leads to faster response times, indicating that the benefits of serial integration go beyond reproduction error. Overall our data show that serial dependence has a beneficial role at various levels of perception, consistent with the idea that the brain exploits the temporal redundancy of the visual scene as an optimization strategy.

There is good evidence that biological perceptual systems exploit the temporal continuity in the world: When asked to reproduce or rate sequentially presented stimuli (varying in almost any dimension), subjects typically err toward the previous stimulus, exhibiting so-called "serial dependence." At this stage it is unclear whether the serial dependence results from averaging within the perceptual system, or at later stages. Here we demonstrate that strong serial dependencies occur within both perceptual and decision processes, with very little contribution from the response. Using a technique to isolate pure perceptual effects (Fritsche, Mostert, & de Lange, 2017), we show strong serial dependence in orientation judgements, over the range of orientations where theoretical considerations predict the effects to be maximal. In a second experiment we dissociate responses from stimuli to show that serial dependence occurs only between stimuli, not responses. The results show that serial dependence is important for perception, exploiting temporal redundancies to enhance perceptual efficiency.

Despite the existence of much evidence for a number sense in humans, several researchers have questioned whether number is sensed directly or derived indirectly from texture density. Here, we provide clear evidence that numerosity and density judgments are subserved by distinct mechanisms with different psychophysical characteristics. We measured sensitivity for numerosity discrimination over a wide range of numerosities: For low densities (less than 0.25 dots/deg(2)), thresholds increased directly with numerosity, following Weber's law; for higher densities, thresholds increased with the square root of texture density, a steady decrease in the Weber fraction. The existence of two different psychophysical systems is inconsistent with a model in which number is derived indirectly from noisy estimates of density and area; rather, it points to the existence of separate mechanisms for estimating density and number. These results provide strong confirmation for the existence of neural mechanisms that sense number directly, rather than indirectly from texture density.

Although humans are the only species to possess language-driven abstract mathematical capacities, we share with many other animals a nonverbal capacity for estimating quantities or numerosity. For some time, researchers have clearly differentiated between small numbers of items -less than about four-referred to as the subitizing range, and larger numbers, where counting or estimation is required. In this review, we examine more recent evidence suggesting a further division, between sets of items greater than the subitizing range, but sparse enough to be individuated as single items; and densely packed stimuli, where they crowd each other into what is better considered as a texture. These two different regimes are psychophysically discriminable in that they follow distinct psychophysical laws and show different dependencies on eccentricity and on luminance levels. But provided the elements are not too crowded (less than about two items per square degree in central vision, less in the periphery), there is little evidence that estimation of numerosity depends on mechanisms responsive to texture. The distinction is important, as the ability to discriminate numerosity, but not texture, correlates with formal maths skills.

Humans, including infants, and many other species have a capacity for rapid, nonverbal estimation of numerosity. However, the mechanisms for number perception are still not clear; some maintain that the system calculates numerosity via density estimates—similar to those involved in texture—while others maintain that more direct, dedicated mechanisms are involved. Here we show that provided that items are not packed too densely, human subjects are far more sensitive to numerosity than to either density or area. In a two-dimensional space spanning density, area and numerosity, subjects spontaneously react with far greater sensitivity to changes in numerosity, than either area or density. Even in tasks where they were explicitly instructed to make density or area judgments, they responded spontaneously to number. We conclude, that humans extract number information, directly and spontaneously, via dedicated mechanisms.

Central tendency, the tendency of judgements of quantities (lengths, durations etc.) to gravitate towards their mean, is one of the most robust perceptual effects. A Bayesian account has recently suggested that central tendency reflects the integration of noisy sensory estimates with prior knowledge representations of a mean stimulus, serving to improve performance. The process is flexible, so prior knowledge is weighted more heavily when sensory estimates are imprecise, requiring more integration to reduce noise. In this study we measure central tendency in autism to evaluate a recent theoretical hypothesis suggesting that autistic perception relies less on prior knowledge representations than typical perception. If true, autistic children should show reduced central tendency than theoretically predicted from their temporal resolution. We tested autistic and age- and ability-matched typical children in two child-friendly tasks: (1) a time interval reproduction task, measuring central tendency in the temporal domain; and (2) a time discrimination task, assessing temporal resolution. Central tendency reduced with age in typical development, while temporal resolution improved. Autistic children performed far worse in temporal discrimination than the matched controls. Computational simulations suggested that central tendency was much less in autistic children than predicted by theoretical modelling, given their poor temporal resolution.