«Proefschrift ter verkrijging van de graad van doctor aan de Radboud Universiteit Nijmegen op gezag van de rector magnificus prof. mr. S.C.J.J. ...»
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© 2014 Florian Krause
Numbers and magnitude in the brain:
A sensorimotor grounding of numerical cognition
ter verkrijging van de graad van doctor
aan de Radboud Universiteit Nijmegen
op gezag van de rector magnificus prof. mr. S.C.J.J. Kortmann,
volgens besluit van het college van decanen
in het openbaar te verdedigen op vrijdag 10 oktober 2014 om 10.30 uur precies door Florian Krause geboren op 25 mei 1983 te Giessen (Duitsland)
Prof. dr. H. Bekkering Prof. dr. I. Toni
Dr. O. Lindemann (Universiteit van Potsdam, Duitsland)
Prof. dr. W.P. Medendorp Prof. dr. M.H. Fischer (Universiteit van Potsdam, Duitsland) Dr. S.M. Göbel (Universiteit van York, Verenigd Koninkrijk)
Numbers and magnitude in the brain:
A sensorimotor grounding of numerical cognition Doctoral Thesis to obtain the degree of doctor from Radboud University Nijmegen on the authority of the Rector Magnificus prof. dr. S.C.J.J. Kortmann, according to the decision of the Council of Deans to be defended in public on Friday, October 10, 2014 at 10.30 hours by Florian Krause Born on May 25, 1983 in Giessen (Germany)
Prof. dr. H. Bekkering Prof. dr. I. Toni
Dr. O. Lindemann (University of Potsdam, Germany)
Doctoral Thesis Committee:
Prof. dr. W.P. Medendorp Prof. dr. M.H. Fischer (University of Potsdam, Germany) Dr. S.M. Göbel (University of York, United Kingdom) Contents Chapter 1 9 Introduction Chapter 2 29
A common magnitude metric in perception:
interference between numbers and size during visual search Chapter 3 51
A feeling for numbers:
shared metric for tactile and symbolic numerosities Chapter 4 71 A shared representation of perceptual and motor magnitudes in early childhood Chapter 5 89
Different brains process numbers differently:
structural bases of individual differences in spatial and nonspatial number representations
Numbers and analogue magnitude In our modern society we are surrounded by numerical information every day – all day long. We start our day by staring at the digits of our digital alarm clock and we end it by counting the bills for paying the 3-course menu in the fancy restaurant around the corner. In between, we process several thousands of numbers (Butterworth, 1999). We remember and dial phone numbers, we receive, spend and transfer money, etc. Processing all these numerical information is an essential cognitive ability, and having deficits in this domain makes individuals less employable, significantly reduces lifetime earnings and is even a risk factor for depression (Butterworth, 2010). Understanding how our brain represents and processes numerical information is important for diagnostic and educational purposes alike.
When we want to investigate the cognitive and neural mechanisms of processing numerical information it is important to have a clear idea on what kind of information we are dealing with. In fact, numerical information, as we use it, is best considered as a composition of several components with different informational content that might well be represented mentally by distinct mechanisms (e.g. Dehaene & Cohen, 1995; Dehaene, 1993; Campbell & Clark, 1992; McCloskey, 1992). For instance, McCloskey (1992) proposed that numerical information is processed via three distinct and independent modules: a comprehension module transforms an initial notation-specific representation into an amodal
(i.e. notation-independent) format on which a calculation module performs all mental calculations, while a production module provides the opposite functionality of transforming the internal abstract representation back into a notation-specific output format. Campbell & Clark (1992) on the other hand suggested that number comprehension is mediated by an interconnected network of format-specific representations, rejecting the hypothesis of a central abstract number representation.
While both of the above mentioned models aim to explain the syntactic mechanisms of manipulating number notations, they unfortunately do not speak to the actual semantics of numerical information (e.g. what is the informational content of the different representations). The perhaps most influential model of how the human brain represents numerical information, the triple code model, addresses this question in more detail and differentiates between three kinds of interconnected cardinal number representations (Dehaene, 1993; Dehaene &
1. Symbolic representation For as long as numbers have been part of human culture, numerical information is referred to by using some form of formal notation. While the earliest notations were simple, with a number of similar tokens referring to the set with this numerosity (e.g. I, II, III; see also Ifrah, 1981), they soon evolved into more complex systems with new symbols and special fundamental numbers, such as Roman digits (I, V, X, L, C, M) or Arabic digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). A symbolic representation of this visual number form is thought to be neurally realized in the occipito-temporal region of the brain, as part of the ventral visual pathway which is involved in visual object recognition (Ungerleider & Mishkin, 1982).
2. Verbal representation
Just as we have symbols for numbers, we also have words for them. A verbal representation of spoken number words and their sequence allows us to count discrete items and to retrieve arithmetic facts from memory (e.g. 7 * 7 = 49).
The system is assumed to be part of a left-lateralized network of language areas, which include the inferior frontal, superior and middle temporal gyri, as well as the angular gyrus.
3. Analogue magnitude representation
While we do formalize and verbalize them, at their very core, however, numbers are used to refer to quantities or magnitudes. A numerical magnitude representation allows us to deduce the actual size of a number and to compare it to the size of other numbers, to see if it is smaller or larger. The evolutionary significance of this capability is highlighted by the fact that basic sensitivity to quantity is not only present in early infancy (e.g. Feigenson, Dehaene, & Spelke, 2004), but also shared by non-human primates (e.g. Brannon, 2006) as well as other animals, such as rats (e.g. Mech and Church, 1983), parrots (e.g. Pepperberg & Gordon, 2005) and fish (e.g. Agrillo, Piffer, Bisazza, & Butterworth, 2012). Importantly, the mental representation of numerical size is hypothesized to be encoded in an analogue manner. That is, numerical size seems to be represented along a continuous domain, rather than in discrete steps, with the representation of one number co-activating the representations of the numbers that are semantically close with decreasing extent. Evidence for this analogue nature comes from the so-called ‘distance effect’, which CHAPTER 1 refers to the phenomenon that people are faster to compare two numbers, the further these numbers are away from each other (Moyer & Landauer, 1967). Neurally, this seems to be realized by number-sensitive neurons with overlapping Gaussian-shaped tuning curves (Nieder & Miller, 2004a). For instance, activating the representation of number 5, will also co-activate the representations of numbers 4 and 6, as well as 3 and 7, etc. The further away the surrounding numbers are, the lesser they are activated (e.g. 3 and 7 4 and 6 5 in the former example). Furthermore, this representational spread (i.e. how many surrounding numbers are co-activated) increases logarithmically with number size, resulting in a more precise representation of small numbers, compared to large numbers. Evidence for this property comes from the ‘size effect’, which shows that comparisons are in general (i.e. irrespective of semantic distance) faster for small numbers, compared to large numbers (Antell & Keating, 1983; Strauss & Curtis, 1981). The point at which the overlap between two representations is little enough to allow for a reliable discrimination between them is called the ‘Weber fraction’. The analogue magnitude representation is hypothesized to be implemented by brain areas in the posterior parietal cortex, such as the intraparietal sulcus.
This thesis focuses on the underlying mechanisms of an analogue magnitude representation, both on a behavioural as well as a neural level. In the following, I first review the current literature on this topic, followed by an outline of the thesis.
The implementation of analogue magnitude Over the last decades, different ideas on an analogue magnitude representation have been proposed to explain how we mentally represent and understand numerical size.
The accumulator model One of the historically first analogue models of numerical size representation is the accumulator model, which allows to map between the numerosity of a set and the mental representation of magnitude representing this numerosity (Meck & Church, 1983; Gallistel & Gelman, 1992). The model, which originally attempted to explain the (non-verbal) numerical abilities of animals, consists of
three components: the ‘pacemaker’ acts as source of a stream of impulses, the ‘accumulator’ is the destination of the stream, and a ‘gate’ will control the flow of the stream of impulses into the accumulator. The idea is as follows: for each item in a set whose numerosity is to be represented, the gate briefly closes for a fixed amount of time, letting a fixed amount of pulses flow through which will incrementally fill the accumulator. The final state of the accumulator will then represent a continuous measure of the total amount of items in the set.
Given that no system is perfectly accurate, neither the flow of impulses, nor the amount of time the gate is opened will be entirely constant. Hence, the amount of impulses entering the accumulator might vary slightly for each item, resulting in an approximate representation of numerical magnitude. The resulting increase in noise in the system can also explain the effect of a better discrimination of two small numbers, compared to two large numbers (i.e. the size effect; Antell & Keating, 1983; Strauss & Curtis, 1981). The accumulator model has furthermore been used successfully to explain empirical data on rats’ ability to discriminate between numerical as well as temporal quantities (Meck & Church, 1983).
Importantly, the accumulator model is of mathematical nature and as such does not speak to the actual neural realization of the suggested mechanism. The sequential accumulation of input items, however, seems to be in contrast to the finding of parallel numerosity detection in monkeys (e.g. Nieder & Miller, 2003).
Dehaene and Changeux (1993), were able to model the process of accumulating perceived objects (visual and auditory) in parallel, using networks of simulated nerve cells. In their neural model, all inputs are first normalized to a size- and shape-independent format and then summed by a layer of ‘numerosity detectors’.
Each detector responds preferably to one specific numerosity and less frequently to the surrounding numerosities, resulting in a Gaussian-shaped tuning curve.
The precision of a detector’s tuning decreases logarithmically with increasing numerosity (i.e. a detector with a large preferred numerosity has a wider Gaussian tuning curve than a detector with a small preferred numerosity). This inherent imprecision introduced by the Gaussian tuning curves and their non-linear compression – which leads to an approximate representation of numerosity – is in line with observations of numerosity selective neurons in the monkey brain (e.g. Nieder & Miller, 2004a).
The mental number line The perhaps most influential idea on how numerical size is cognitively represented in humans is the so-called mental number line hypothesis. In essence, the theory states that we represent numerical size in spatial terms – as positions on a horizontally oriented mental line – with small numbers on one side and large numbers on the other (e.g. Moyer, 1973; Moyer & Landauer, 1967). The most important empirical evidence for the existence of a mental number line (MNL) comes from the finding of a Spatial-Numerical Association of Response Codes (SNARC; Dehaene, Bossini, & Giraux, 1993). Dehaene and colleagues asked participants to judge the parity of single Arabic digits (i.e. whether a digit is even or odd) by pressing one of two buttons. Analysis of reaction times revealed an interference of numerical size with the execution of the spatial responses. On average, participants were faster when they responded to small numbers with a left button press compared to a right button press, and when they responded to large numbers with a right button press compared to a left button press.
Importantly, these spatial preferences followed a linear trend, with reaction times of the left hand becoming increasingly faster the smaller the presented number and reaction times of the right hand becoming increasingly faster the larger the presented number, revealing the analogue nature of this representation.
The SNARC effect is known as a very robust effect and has successfully been replicated various times over the last decades (for reviews see e.g. Wood, Nuerk, Willmes, & Fischer, 2008; Hubbard, Piazza, Pinel, & Dehaene, 2005).