Cognitive and neural basis of numerical abilities

Manuela Piazza

Course plan

Topic More detailed Reading list

1 class

- General introduction.- The hypothesis of cultural recycling of cortical maps.

- Dehaene and Cohen, Neuron 2007.

- The notion of number.

- Ethnological aspects. -

- The perception of number.

- Numerical estimation and subitizing.

- Revkin et al., Pyschological Science 2009.

- Piazza, Trend In Cognitive Sscience, 2010.

2 class

- Number neurons in humans and other animals. Parietal cortex and numerical

quantity coding.

- Brief introduction to neuroimagingand neurophysiological methods and discussion of studies.

- Piazza et al., Neuron 2004.

- Nieder., Neuron 2003.

- Piazza and Izard, The neuroscientist, 2009.

3 class

- Cerebral circuits of simple arithmetic. Cultural variability and universal mechanisms. Numbers and its relation to language, and spatial abilities.

- Discussion of studies using fMRI, EEG, behavioral, and neuropsychological methods.

- Dehaene et al., Cognitive Neuropsychology, 2003.

- Handbook of Mathematical Cognition (2004), J.Campbell. Ed.: Chapter 10. Learning Mathematics in China and the United States.

- Hubbard et al., Nature Reviews Neuroscience, 2005

4 class

- Normal and abnormal development of elementary numerical abilities

- Discussion of behavioral and neuroimaging studies.

- Ansari, Nature Reviews Neuroscience, 2008.

- Piazza et al., Cognition 2010.

- Handbook of Mathematical Cognition (2004), J.Campbell. Ed.: Chapter 10. -Developmental Dyscalculia.

IntroducionThe hypothesis of cultural

“recylcing” of pre-exising neural circuits

Or: cultural traditions are such becuase they fund adequante “neuronal nich” in our brains

- Other primates are able to acquire new abilities, the use of new tools (Iriki, 2005), and even of symbols like Arabic digits (Matsuzawa, 1985)

- They also show some rudiment of cultural skills that are locally trasnmitted (Whiten et al., 1999).

A thirteen-month-old chimpanzee traces curves on a graphic tablet (Tanaka et al., 2003)

Composition produced by an adult chimpanzee living semi-independently in the Mefou Forest Reserve inCameroon (© Canadian Ape Alliance)

The Primate human’s specificity for cultural acqusition

- Which are the brain architectures that support extraordinary cultural inventions (such as reading and arithmetic)?- Which are the new caracteristics in brain architectures that makes the human species the only real “cultural species”, the only one capable of real cultural inventions?

Which are the biological basis of human culture?

According to some positions, and in particular by scholars in the classical social sciences, brain architecture is completely irrelevant when it comes to understanding “high level” cultural acquisitions.

Many scientists implicitely or explicitely agree with the idea of a “generalized brain plasticity”, or “cultural relativism”, which reminds the idea of “tabula rasa” of Aristotelian memory (according to which humans are born with no innate abilities, and thus are equipotential learners).

Does the brain constrain cultural acquisitions?

Brain is irrelevant• Constructivism in neurobiologye.g. Quartz & Sejnowski (1997) (“The neural basis of cognitive development: a constructivist manifesto”): «cerebral cortex during development is equipotential and free to devote specific structures to different domains»• Functionalism in psychology:e.g. Fodor, Johnson-Laird: « the physical nature of the huan brain does not impose any constraint on thought»

Receptors

Synapsis

Neurons

Cortical columnsCortical regions

Behaviour

Education and cognition

Our long term objective is to characterize the

conversion laws between incapsulated levels

• Technique of non invasive neuro-immaging allow us to investigate the neural mechanisms involved in the manipulation of cultural tools.• For both reading and mathematics, even though there are racross cultures a large variability, there are in all individuals brain regions specialized for each of these abilities.• These results may seem paradoxical, since the natural evolution does not seem to have had the time sufficient to select brain architectures specifically to support recent cultural objects.

The “neural recycling” model (“exaptation”: modification of the function of a pre-selected given trait –

concept already introduced by Darwin):

• The architecture of our brain is limited.• It is determined by genetic/epigenetic laws, but they include a certain variability and plasticity (itself genetically determined).• The new cultural acquisitions are possible only inhasmuch they capitalize on this variability. Each cultural object is such because it finds its NEURONAL NICHE.• Our brain is NOT a tabula rasa, but it adapts to the cultural environment to which it is exposed to by a minimal riconversion or “recycling” of pre-existing systems, even thought they were selected for a different function.

Neurons and culture. Which links?

Perception of numerical quantity

Humans have two systems for perceiving numerical quantity:

1. The approximate number system

2. The object tracking system

The approximate number system

Allows estimation of the approximate number of items in

collections

Demonstration

Which set contains more dots?

Two sets of different number

Ratio (S/L) = 0.5

Less errors and faster reaction times

Ratio = ~0.9

More errors and slower reaction times

12 24

5 10

22 24

10 11

Weber law

Weight Loudness Brightness Numerosity

A psychophysical law describing the relationship between the physical and the perceived magnitude of a stimulus.

It states that the threshold of discrimination (also referred to as ‘smallest noticeable difference’) between two stimuli increases linearly with stimulus intensity.

Weber’s law can be accounted for by postulating a logarithmic relation between the physical stimulus and its internal representation.

8 16 32 640

20

40

60

80

100

0.5 1 20

20

40

60

80

100

8 16 32 640

20

40

60

80

100

Deviation ratio (log scale)

Deviant numerosity (log scale)

Nhabit 16Nhabit 32

Nhabit 16Nhabit 32

Deviant numerosity (linear scale)3 exemplars of a given number (16 or 32)

Followed by a single deviant number(8-32 and 16-64)

Weber law in numerosity judgements

On a log scale the two curves have the same width !!! This indicates that numerosity is mentally represented on a compressed scale

The ANS is universal: acrossspecies

The ability to discriminate between two quantities depends upon their ratio, according to Weber’s law.

This law holds for humans but also for animals’ numerosity judgments (here, macaque monkeys vs. humans).

RatsThe number of presses produced as a function of the number of presses requested [Mechner, 1958]

HumansErrors in a dots comparison task as a function of the reference number [Van Oeffelen and Vos, 1982]

The ANS is universal: across species

The Munduruku have number wordsonly up to 5.

-They have a perfectly normal non-verbal magnitude system, even for very large quantities

-They have a spontaneous capacity for estimation, comparison, addition

-They fail in tasks of exact calculation

[Pica, Lemer, Izard, & Dehaene, Science, 2004]

The ANS isuniversal: across

cultures

monolingual, no instruction

Adults Children

bilingual, no instruction

monolingual, with instruction

bilingual, with instruction

n=9 (55.5 y)

n=10 (59.3 y)

n=7 (38.7 y)

n=9 (4.7 y)

n=7 (8.6 y)

n=13 (9.6 y)

Brasil

Transamazonianhighway

Rio Tapajos

Rio Juruena

Mundurúkú Territory

Rio Cururu

Rio Teles Pires

Missão Velha

Jacarecanga

Rio das tropasKatõSai Cinza

40 km

+ 10 French adults [Pica, Lemer, Izard, & Dehaene, Science, 2004]

Numerosity naming

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

stimulus numerosity

resp

onse

freq

uenc

ypug ma = onexep xep = twoebapug = threeebadipdip = fourpug põgbi = one handxep xep põgbi = two handsadesu/ade gu = some, not manyade/ade ma = many, really many~

[Pica, Lemer, Izard, & Dehaene, Science, 2004]

Approximate addition and comparison

[Pica, Lemer, Izard, & Dehaene, Science, 2004]

+

n1 n2

n3

Approximation addition and comparison

adul

tsch

ildre

n

B,I

M,NI

M,NI

B,I

B,NI

M,I

Ratio of n1+n2 and n3 (L/S)

All Munduruku

French controls

[Pica, Lemer, Izard, & Dehaene, Science, 2004]

Internal representation of numerosity: a model

0

1 2 3 5 84 6 7 9…

w

w = sd of the gaussian distribution of the internal representation of numerical quantity (on a log scale!). The larger w the worst the discriminability between two close numbers.

w is a measure of the precision of the internal representation of numerosity

Numerosity

Activation

Log scale

48 Newborn babiesMean age= 49 h [7-100 h]

[Izard et al., PNAS 2009]

12

4

Human newborns spontaneously match approximate number(violation of expectation paradigm)

The ANS is active in infants

SENSITIVE TO A MINIMAL 1:3 RATIO

Habituation

Test

8 16

816

ou

F. Xu, E. S. Spelke, Cognition 74, B1-B11 (2000).

The ANS is active in infants

8 vs. 16 but not 8 vs. 12, nor 4 vs. 5…

Also older babies (6 months) spontaneously match number

SENSITIVE TO A MINIMAL 1:2 RATIO

5 years old children

The ANS is active in children

SENSITIVE TO A MINIMAL 4:5 RATIO

Barth (2005)

The ANS acuity improves during development

10 20 30 40 500

0.2

0.4

0.6

0.8

1

2

Round numbers accurately discriminated

Age in years

Age in years

ANS acuity (Weber fraction)

0 1 2 3 4 5 6 7 10

5:6

4:5

3:4

2:3

1:2

The ANS acts on numerical quantity independent from the nature of the stimulus

(Example: Jordan et al., 2008)

(S/L)

(1/9) (8/9)Match SEQUENCIAL to SIMULTANEOUS

Match AUDITORY to VISUAL

Vedi video:http://www.duke.edu/web/mind/level2/faculty/liz/xmodal.htm

Adds AUDITORY to VISUAL

Conclusion:• A system for extracting the approximate number (ANS)

– present universally in the animal world– active early during development in humans– represents number independently from the stimulus mode

(simultaneous or sequential)– represents number independently from the stimulus modality

(visual, auditory, motor, ...)– is used to perform simple operations (comparison, additions,

subtractions, ...)

WHAT (IF ANY) IS IT’S ROLE IN NUMERACY ACQUISITION?

Mechanisms of numerosity extraction: Models

• 1. “Numerosity detector” (Dehaene, 1993 version)

1. (Visual) Input2. Object Location and Normalization

3. Numerosity Accumulation

1 2 3 4 5 6

4. NumerosityDetection(« band-passfilters »)

Operations that occur in parallel

• 1. “Numerosity detector” (Verguts, 2006 version)

Mechanisms of numerosity extraction: Models

• 2. “Accumulator”(Gallistel, 2002)

Operations occurring serially

Mechanisms of numerosity extraction: Models

The object tracking system (OTS)

Allows exact apprehension of the number of items in collections (but

limited in capacity to 3-4 items)

Demonstration :

How many dots?

The OTS• OTS is capacity limited “subitizing”

0

0,1

0,2

0,3

0,4

0,5

1 2 3 4 5 6 7 8

Set size

Err

or

rate

500

600

700

800

900

1000

Res

po

nse

Tim

e (m

s)

AccuracySpeed

Mandler & Shebo, 1982, Trick & Pylyshyn, 1994,…

A discuntinuity in both accuracy and response time

SUBITIZING

This phenomenon:

1) Does not depend on the spatial layout of the items (no “patter recognition”)II III IIIIIIIIII

2) Does not depend on the nature of the itemsOX OXO OXOXOX

3) It is not altered by tasks tapping on verbal memory

BUT

1) It disappears if the elements do not “pop-out” from the background

2) It is altered by a concurrent taslk which occupies visuo-spatial memory

Count the number of horizontal bars

Count the number of horizontal red bars

What is the origin of SUBITIZING?

1. DOMAIN-SPECIFIC HYPOTHESIS (Butterworth, ...):

Subitizing reflects the sensitivity of the ANS (1:2, 2:3, 3:4 are easily discriminable ratios)

PREDICTIONS:(A) We should find a similar discuntinuity in RTs and errors for similar ratios in the large number quantification 1 2 3 4 …8, 10 20 30 40 ... 80(B) Subitizing SPAN (number of objects that can be readily quantified with no errors) should correlate with other measures of the ANS precision.

2. DOMAIN-GENERAL HYPOTHESIS (Phylyshyn, ...):

Subitizing is reflects our ability to “index” (individuate, or pay attention to individuals with their basic features, like color shape position) multiple objects at once.

PREDICTIONS:(A) Subitizing SPAN should correlate with other measures of multiple object tracking (e.g., visuo-spatial working memory span)(B) Subitizing SPAN should be altered by a concurrent visuo-spatial task tapping on similar visuo-attentive resources

• Hp1. The OTS is “just” the ANS that is of higher precision for small numbers?

Characterizing the OTS

[Revkin, et al., Psychological Science 2008]

i The Subitizing range across subjects does NOT correlate with the ANS acuity (measured with an independent large numerosity comparison task)

RANGE [1-8] RANGE [10-80] RANGE [1-8] RANGE [10-80]

• Hp 2. The OTS is not number-specific, but it reflects the capacity of the visual object individuation system, used for processing multiple items individually

0

0,1

0,2

0,3

0,4

0,5

1 2 3 4 5 6 7 8

Set size

Erro

r rat

e

500

600

700

800

900

1000

Res

pons

e Ti

me

(ms)

AccuracySpeed

200 ms

How many?

250 ms

Time

TASK 1 -Enumeration task

Characterizing the OTS

10 adult subjects; 3 tasks

[Piazza, Fumarola, and Melcher, under revision]

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8

Set size

Num

ber o

f obj

ects

enc

oded

(K)

700 ms

Same or different?

1000 ms

Time

200 ms

2000 ms

TASK 2-VWM task

Cowan’s K=S (hits - f.alarms) 16 320

20

40

60

80

100

TASK 3 -Large numerosity comparison task

W = 0.19

*

“choose the larger”

Results

(1) High correlation across capacity limited systems, and NO correlation between both these systems and the acuity of the ANS. 1,5

2,5

3,5

4,5

3,5 4 4,5 5

VWM

cap

acity

Subitizing capacity

If they tap on same mechanisms then they should interfere selectively with each other.

TASK 4 – Dual task : Enumeration + VWM task. Results: Subitizing range is reduced by increasing working memory load

TASK 5 – Dual task : Large number comparison + VWM task. Results: Weber fraction is unchanged by increasing working memory load

CONCLUSIONS:

Subitizing is based on object tracking mechanisms (OTS) and reflects the general architecture of our visuo-attentive system, which cannot individuate more then 3-4 items at once. It is a separate function from the ANS

WHAT (IF ANY) IS IT’S ROLE IN NUMERACY ACQUISITION?

OTS developmental trajectory