Guido Marco Cicchini

Optimal Encoding of Interval Timing in Expert Percussionists

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.

Separate Mechanisms for Perception of Numerosity and Density

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.

Compressive mapping of number to space reflects dynamic encoding mechanisms, not static logarithmic transform.

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. 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.

Spatiotemporal Distortions of Visual Perception at the Time of Saccades

Both space and time are grossly distorted during saccades. Here we show that the two distortions are strongly linked, and that both could be a consequence of the transient remapping mechanisms that affect visual neurons perisaccadically. We measured perisaccadic spatial and temporal distortions simultaneously by asking subjects to report both the perceived spatial location of a perisaccadic vertical bar (relative to a remembered ruler), and its perceived timing (relative to two sounds straddling the bar). During fixation and well before or after saccades, bars were localized veridically in space and in time. In different epochs of the perisaccadic interval, temporal perception was subject to different biases. At about the time of the saccadic onset, bars were temporally mislocalized 50–100 ms later than their actual presentation and spatially mislocalized toward the saccadic target. Importantly, the magnitude of the temporal distortions co-varied with the spatial localization bias and the two phenomena had similar dynamics. Within a brief period about 50 ms before saccadic onset, stimuli were perceived with shorter latencies than at other delays relative to saccadic onset, suggesting that the perceived passage of time transiently inverted its direction. Based on this result we could predict the inversion of perceived temporal order for two briefly flashed visual stimuli. We developed a model that simulates the perisaccadic transient change of neuronal receptive fields predicting well the reported temporal distortions. The key aspects of the model are the dynamics of the “remapped” activity and the use of decoder operators that are optimal during fixation, but are not updated perisaccadically.

Number As a Primary Perceptual Attribute: A Review.

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.

Linear Mapping of Numbers onto Space Requires Attention.

Mapping of number onto space is fundamental to mathematics and measurement. Previous research suggests that while typical adults with mathematical schooling map numbers veridically onto a linear scale, pre-school children and adults without formal mathematics training, as well as individuals with dyscalculia, show strong compressive, logarithmic-like non-linearities when mapping both symbolic and non-symbolic numbers onto the numberline. Here we show that the use of the linear scale is dependent on attentional resources. We asked typical adults to position clouds of dots on a numberline of various lengths. In agreement with previous research, they did so veridically under normal conditions, but when asked to perform a concurrent attentionally-demanding conjunction task, the mapping followed a compressive, non-linear function. We model the non-linearity both by the commonly assumed logarithmic transform, and also with a Bayesian model of central tendency. These results suggest that veridical representation numerosity requires attentional mechanisms.

Temporal mechanisms of multimodal binding

The simultaneity of signals from different senses—such as vision and audition—is a useful cue for determining whether those signals arose from one environmental source or from more than one. To understand better the sensory mechanisms for assessing simultaneity, we measured the discrimination thresholds for time intervals marked by auditory, visual or auditory–visual stimuli, as a function of the base interval. For all conditions, both unimodal and cross-modal, the thresholds followed a characteristic ‘dipper function’ in which the lowest thresholds occurred when discriminating against a non-zero interval. The base interval yielding the lowest threshold was roughly equal to the threshold for discriminating asynchronous from synchronous presentations. Those lowest thresholds occurred at approximately 5, 15 and 75 ms for auditory, visual and auditory–visual stimuli, respectively. Thus, the mechanisms mediating performance with cross-modal stimuli are considerably slower than the mechanisms mediating performance within a particular sense. We developed a simple model with temporal filters of different time constants and showed that the model produces discrimination functions similar to the ones we observed in humans. Both for processing within a single sense, and for processing across senses, temporal perception is affected by the properties of temporal filters, the outputs of which are used to estimate time offsets, correlations between signals, and more.

Spontaneous perception of numerosity in humans

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.

Mechanisms for perception of numerosity or texture-density are governed by crowding-like effects

We have recently provided evidence that the perception of number and texture density is mediated by two independent mechanisms: numerosity mechanisms at relatively low numbers, obeying Webers law, and texture-density mechanisms at higher numerosities, following a square root law. In this study we investigated whether the switch between the two mechanisms depends on the capacity to segregate individual dots, and therefore follows similar laws to those governing visual crowding. We measured numerosity discrimination for a wide range of numerosities at three eccentricities. We found that the point where the numerosity regime (Webers law) gave way to the density regime (square root law) depended on eccentricity. In central vision, the regime changed at 2.3 dots/°2, while at 15° eccentricity, it changed at 0.5 dots/°2, three times less dense. As a consequence, thresholds for low numerosities increased with eccentricity, while at higher numerosities thresholds remained constant. We further showed that like crowding, the regime change was independent of dot size, depending on distance between dot centers, not distance between dot edges or ink coverage. Performance was not affected by stimulus contrast or blur, indicating that the transition does not depend on low-level stimulus properties. Our results reinforce the notion that numerosity and texture are mediated by two distinct processes, depending on whether the individual elements are perceptually segregable. Which mechanism is engaged follows laws that determine crowding.

The effects of cross-sensory attentional demand on subitizing and on mapping number onto space.

Various aspects of numerosity judgments, especially subitizing and the mapping of number onto space, depend strongly on attentional resources. Here we use a dual-task paradigm to investigate the effects of cross-sensory attentional demands on visual subitizing and spatial mapping. The results show that subitizing is strongly dependent on attentional resources, far more so than is estimation of higher numerosities. But unlike many other sensory tasks, visual subitizing is equally affected by concurrent attentionally demanding auditory and tactile tasks as it is by visual tasks, suggesting that subitizing may be amodal. Mapping number onto space was also strongly affected by attention, but only when the dual-task was in the visual modality. The non-linearities in numberline mapping under attentional load are well explained by a Bayesian model of central tendency.