A visual sense of number David Burr & John Ross The University of Florence Burr & Ross Current Biology 2008

Many agree, that besides a mechanism for precise calculation based on symbolic representation, there exists an evolutionarily ancient approximate number system shared by and non-human animals for rough estimation of numerosity.

Estimation of numerosity in rats (A) The probability of rats breaking off a sequence of lever presses as a function of the number of presses in the sequence and the number required to get the reward. The inset shows the mean number of lever presses (circles), standard deviation (squares). The coefficient of variation (CV), which is the ratio between the mean and the standard deviation, is constant, indicating Weber’s law (redrawn from Mechner, 1958 and from Gallistel & Gelman, 2000).

Monkeys: same/different task (B) Behavioral performance of two monkeys in a same-different task where they judged whether a test stimulus contained the same or a different number of items as the sample display. Each curve represents the percentage of “same” response as a function of test numerosity, for a given sample numerosity (modified from Nieder, 2003).

Number production by key-press 0.15 0,30 CV 10 20 30 Mean SD 2 4 Number of level presses Behavioral performance of human adults that were asked to produce a given number of key presses. The mean number of presses (circles), standard deviation (squares), and the coefficient of variation are striking similar to the rats’ performance drawn above.

Human estimation of prices Frequency (%) 2 4 6 8 10 12 14 16 18 20 1 3 Normal distribution. Log-normal distribution. Normalized price (linear scale) D Distribution of human adults’ estimates of prices of items, after normalization by the mean price. The distribution is consistently skewed and is better fitted by a log-normal than by a normal curve (from Dehaene & Marques, 2002).

Brain imaging A B C Right hemisphere Left hemisphere Top view L CS IPS Right hemisphere Left hemisphere left angular gyrus (AG) bilateral posterior superior parietal lobe (PSPL) bilateral horizontal segment of intraparietal sulcus (HIPS) Top view A L C B (A) Three-dimensional representation of the three parietal sites of major activation in number processing individuated by a recent meta-analysis of fMRI studies of number processing. CS, central sulcus; IPS, intraparietal sulcus. (from Dehaene et al., 2003). (B) Regions whose activation increases with number size during calculation (from Stanescu et al., 2001), including left HIPS, left premotor, and left inferior prefrontal areas. (C) Region of reduced grey matter in a population of subjects with developmental dyscalculia (from Isaacs et al., 2000). The location of impairment coincides with the left HIPS.

Neurons in monkey pre-frontal and parietal cortex B Time N u m b e r o f i t s ( l g c a ) 2 5 7 1 z d p n % 3 4 A C D Fixation 500 ms Sample 800 ms Delay 1000 ms Test 1200 ms Match Non-Match P=0.25 P=0.50 Spike rate (Hz) Selectivity follows a log scale

Number neurons cover a large range Nieder & Merten J Neuroscience 2007

Could numerosity be a visual attribute? If so it should be subject to adaptation.

If so it should be subject to adaptation. Is numerosity a primary visual attribute, or quale, like colour or motion? If so it should be subject to adaptation. John Mollon (1974). After-effects and the brain. New Scientist 61: ‘If you can adapt it, it’s there.’

Complementary after-images If you fixate a strongly-coloured image on a constant region of the retina, the sensitivity of the absorbing cones will be temporarily reduced and they will respond to the incident light much less. It’s very important that you fixate the image. The treshold of sensibility of your blue cones are increased in this part.

Complementary after-images If you fixate a strongly-coloured image on a constant region of the retina, the sensitivity of the absorbing cones will be temporarily reduced and they will respond to the incident light much less. It’s very important that you fixate the image. The treshold of sensibility of your blue cones are increased in this part.

Complementary after-images If you fixate a strongly-coloured image on a constant region of the retina, the sensitivity of the absorbing cones will be temporarily reduced and they will respond to the incident light much less. It’s very important that you fixate the image. The treshold of sensibility of your blue cones are increased in this part.

Adaptation demo

Adaptation demo

Where did the other dots go? (We’ll come back to that)

Adaptation: 45 sec + 8 sec top-up

Test stimulus (500 ms)

0.5 sec pause

Probe stimulus (500 ms)

Psychometric functions with adaptation

Adaptation vs dot number Adapt to 400 dots

Effect of number of adaptor dots

Adaptation: magnitude estimation No adapt Adapt 120

Numerosity or texture?

Size of rectangular elements: paired comparisons

Adaptation does not depend on element orientation

Effect of the test contrast

Effect of adaptor contrast

Numerosity or texture Neither PSE nor Weber fractions depend on: Size or shape of elements Orientation of elements Fourier sprectra of stimuli Contrast, or contrast sign Chromaticity

Colour-contingency after-effect

Colour-contingency after-effect

Colour-contingency after-effect

What are the neural mechanisms underlying numerosity adaptation?

“Number-neurons” in monkey pre-frontal and parietal cortex Time N u m b e r o f i t s ( l g c a ) 2 5 7 1 z d p n % 3 4 A C D Fixation 500 ms Sample 800 ms Delay 1000 ms Test 1200 ms Match Non-Match P=0.25 P=0.50 Spike rate (Hz)

LIP neurons respond in graded fashion to total number in RF Roitman, Brannon &Platt PLoS 2007

Implications for adaptation LIP VIP

Data

Attention and subitizing

Weber’s law for numerosity 25% Weber fraction explains the subitizing limit of 4 Ross, Perception, 2003

Numerosity and subitizing Rather than counting them, economist William Jevons estimated numbers of beans thrown into a dish, and made errors when there were more than 4 beans. Errors in estimate varied with bean number: Weber’s law. William Stanley Jevons

The attentional blink: slow motion Giovanni Anobile

The attentional blink: real time Giovanni Anobile

Attention affects subitizing but not estimation

Spatial attention: slow motion demo Marco Turi

Spatial attention: real time demo Marco Turi

Attention affects subitizing but not estimation

Attention affects subitizing but not estimation

Attention affects subitizing but not estimation

Mental abacus represents large exact numerosities using pre-existing visual resources  Frank, M.C.., & Barner, D.

Abacus

Mental abacus