Significance: The ability to map numbers onto space is fundamental to measurement and mathematics. The mental “numberline” is an important predictor of math ability, thought to reflect an internal, native logarithmic representation of number, later becoming linearized by education. Here we demonstrate that the nonlinearity results not from a static logarithmic transformation but from dynamic processes that incorporate past history into numerosity judgments. We show strong and significant correlations between the response to the current trial and the magnitude of the previous stimuli and that subjects respond with a weighted average of current and recent stimuli, explaining completely the logarithmic-like nonlinearity. We suggest that this behavior reflects a general strategy akin to predictive coding to cope adaptively with environmental statistics.

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 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.

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.

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/deg²), 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.