The Neural Basis of the Weber-Fechner Law: A Logarithmic Mental Number Line PDF

Update TRENDS in Cognitive Sciences Vol.7 No.4 April 2003 145 | Research Focus The neural basis of the Weber – Fechner law: a logarithmic mental number line Stanislas Dehaene INSERM Unit 56...

Update TRENDS in Cognitive Sciences Vol.7 No.4 April 2003 145 | Research Focus The neural basis of the Weber – Fechner law: a logarithmic mental number line Stanislas Dehaene INSERM Unit 562, Cognitive Neuroimaging, Service Hospitalier Frédéric Joliot, Orsay, France The recent discovery of number neurons allows for a how Weber’s law could be accounted for by postulating dissection of the neuronal implementation of number that the external stimulus is scaled into a logarithmic representation. In a recent article, Nieder and Miller internal representation of sensation. More recently, demonstrate a neural correlate of Weber’s law, and Stevens discussed the possibility that the internal scale thus resolve a classical debate in psychophysics: the is a power function rather than a logarithm, and mental number line seems to be logarithmic rather than Shepard introduced the multidimensional scaling linear. method as a means of estimating, without a priori assumptions, the geometrical organization of an Some twenty years ago, it was fashionable for many internal continuum. Although Weber and Fechner scientists to separate psychology from the study of the concentrated on perceptual continua such as loudness, brain. Functionalist philosophers such as Jerry Fodor Stevens and Shepard showed that more abstract convinced a generation of psychologists that the compre- parameters, including our sense of number , also hension of the mind called for the development of purely followed Weber’s law. computational theories, without concern for their biologi- In spite of these brilliant analyses, often based on cal implementation. The computer metaphor promoted a solid mathematical foundations, the Fechner– Weber – logical separation of the software from the hardware, and Stevens debate was never fully resolved. One of the lead inevitably to the conclusion that the details of the reasons is that there are basic mathematical ambi- neural machinery were irrelevant to the psychological guities in the modeling of behavioral data. In particu- enterprise. lar, given suitable assumptions, both logarithmic and Today, however, we know that this view was unnecess- linear models of the internal scale are tenable. arily narrow. The new cognitive neuroscience routinely Fechner’s logarithmic scale easily accounts for Weber’s mixes psychological and neural observations in the same finding: if the scale has a fixed internal variability, experiments. Psychological concepts are not ruthless then doubling the value of the compared quantities eliminated, as was initially foreseen by the most opinio- leads to a corresponding halving of discrimination nated anti-functionalist philosophers. Rather, they are power. However, the same discrimination function can enriched, constrained and transformed by the accruing also be accounted for by postulating a linear internal neural data. In a recent article, Andreas Nieder and Earl scale with a corresponding linear increase in the Miller provide a beautiful illustration of this by standard deviation of the internal noise. Here too, showing how the study of the neural coding of number doubling the comparanda leads to a doubling of the can resolve a classical problem in psychophysics: what is variability and therefore to a halving of the the mental scale for number? discriminability. In the case of the mental representation of number, Mental scaling: linear, logarithmic, or power function? Gallistel has argued that the linear model should be The ‘scaling problem’ was integral to the birth of preferred because it allows for a simpler calculation of psychology as a scientific discipline. Founding fathers sums and differences. Contrary to that, Changeux and I of experimental psychology, inlcuding Weber and have proposed a simple neural network of numerosity Fechner considered as one of their central goals the detection that assumes a logarithmic encoding of number, mathematical description of how a continuum of thus avoiding an explosion in the number of neurons sensation, such as loudness or duration, is represented needed as the range of internally represented numbers in the mind. By careful psychophysical experiments, increases. I have also argued, however, that the often requiring thousands of discrimination trials on psychological predictions of the linear and logarithmic pairs of stimuli, they identified basic regularities of our models are essentially equivalent. With the possible psychological apparatus. Ernst Weber discovered what exception of a novel psychophysical paradigm , it is hard we now know as Weber’s Law: over a large dynamic to see how behavioral observations alone could ever range, and for many parameters, the threshold of disentangle the linear and logarithmic hypotheses. discrimination between two stimuli increases linearly with stimulus intensity. Later, Gustav Fechner showed The neuronal code for number The ability to record from neurons that are assumed to Corresponding author: Stanislas Dehaene ([email protected]). constitute the neural basis of the psychological number http://tics.trends.com 146 Update TRENDS in Cognitive Sciences Vol.7 No.4 April 2003 (a) Anatomy (c) Neural tuning curves Arcuate sulcus 100 Principal 75 sulcus 50 25 0 100 75 50 Normalized response (%) 25 0 00 75 1 2 (b) Number discrimination performance 50 3 25 4 5 0 100 Monkey T 100 Monkey P 75 (% same as sample) 80 Average 50 Performance 25 60 0 100 40 75 20 50 25 0 0 1 5 10 1 2 3 4 5 Number of items (log scale) Number of items (log scale) TRENDS in Cognitive Sciences Fig. 1. Evidence for logarithmic coding of number in the monkey brain. (a) The anatomical location in monkey prefrontal cortex where Nieder and Miller recorded number neurons. In their experiments, monkeys were presented with a first set of dots, which they were then asked to discriminate from a second set of dots. (b) The percentage of trials on which they responded ‘same’ is plotted as a function of the second number (abscissa) for different values of the first number, which ranged from 2– 6 during beha- vioral testing (color of plot). Performance decreased smoothly with the distance between the two numbers (i.e. the peak occurs when the two numbers are the same). This distance effect assumed a Gaussian shape when plotted on a logarithmic scale. (c) So did the tuning curves of individual number neurons (shown for 1– 5). scale now brings direct physiological evidence to bear on stimuli such as Arabic digits [3,12]. Thus, it is likely that this issue. In the early days of neurophysiology, a few Weber’s law for numbers is determined solely by the neurons that encoded number were reported in the internal organization of cortical representations. association cortex of the cat , although this initial In their paper, Nieder and Miller analyzed in minute discovery was quickly forgotten. In 2002, however, two detail the behavioral and neural response curves of two papers, one recording in parietal cortex and the other in monkeys, which had been engaged in a task of discrimin- prefrontal cortex, reported the observation of neurons ating the numerosity of two visually presented sets whose firing rate was tuned to a specific numerosity (Fig. 1). They found clear evidence for Weber’s law. Both [10,11]. A given neuron, for instance, might respond animals showed a linear increase in their discrimination optimally to three visual objects, a little less to displays thresholds as the numerosity increased. Furthermore, the or two or four objects, and not at all to displays of one or five data were sufficiently regular to allow for a detailed objects. This offered a unique opportunity to examine the analysis of the exact shape of the response distributions. neural code for an abstract psychological continuum. When plotted on a linear scale, both behavioral and neural As noted by Nieder and Miller, it was particularly tuning curves were asymmetrical, and assumed a different interesting to investigate the neural basis of Weber’s law width for each number. Both sets of curves, however, with an abstract dimension such as number. For para- became simpler when plotted on a logarithmic scale: they meters that are more closely dependent on sensory were fitted by a Gaussian with a fixed variance across the physiology, such as loudness, weight or brightness, there entire range of numbers tested (Fig. 1b,c). Thus, the neural is often evidence that the stimulus compression occurs at a code for number can be described in a more parsimonious peripheral sensory level. In the case of number, however, way by a logarithmic than by a linear scale. there are no obvious limitations in our ability to perceive It should be stressed that this form of internal multiple objects or sounds. Furthermore, in human representation was not imposed by the training scheme subjects, Weber’s law is even observed with symbolic the monkeys had. Training was based solely on the http://tics.trends.com  Update TRENDS in Cognitive Sciences Vol.7 No.4 April 2003 147 numbers 1 to 5, which were presented with roughly equal First, it was compact enough to allow the processing of frequency. The optimal coding scheme would therefore arbitrarily large numbers with a pocket-sized device. have been a linear code with an exact encoding of each Second, it ensured an accuracy proportional to the size of number 1, 2, 3, 4 and 5. The fact that the monkeys could the numbers involved, something that was pertinent for not help but encode the numerosities on an approximate real-life engineering applications. Perhaps the very same compressed scale confirms that this approximation mode is reasons can explain why nature selected an ‘internal slide the natural way that number is encoded in a brain without rule’ as its most efficient way of doing mental arithmetic. language. References Future prospects 1 Churchland, P.S. (1986) Neurophilosophy: Toward a Unified Science of The monkey data of Nieder and Miller are just a first stab the Mind/Brain, MIT Press at the problem from the neurophysiological standpoint, 2 Nieder, A. and Miller, E.K. (2003) Coding of cognitive magnitude: Compressed scaling of numerical information in the primate pre- and do not fully resolve the Fechner– Weber– Stevens frontal cortex. Neuron 37, 149 – 157 debate yet. When Nieder and Miller fitted their data with a 3 Shepard, R.N. et al. (1975) The internal representation of numbers. power function, they obtained only a slightly worse fit than Cogn. Psychol. 7, 82 – 138 that with the logarithmic scale. To discriminate the power 4 Krueger, L.E. (1989) Reconciling Fechner and Stevens: Toward a and the logarithmic functions in future experiments, it will unified psychophysical law. Behav. Brain Sci. 12, 251 – 267 5 Gallistel, C.R. and Gelman, R. (1992) Preverbal and verbal counting be important to increase the range of numbers tested. We and computation. Cognition 44, 43 – 74 know from behavioral paradigms that, once trained with 6 Dehaene, S. and Changeux, J.P. (1993) Development of elementary small numerosities, monkeys generalize to larger numbers numerical abilities: A neuronal model. J. Cogn. Neurosci. 5, 390 – 407 up to 10 or more. This is another proof that the 7 Dehaene, S. (2001) Subtracting pigeons: logarithmic or linear? numerical ability of animals is not merely inculcated in Psychol. Sci. 12, 244– 246 them by laboratory training, but is inherent in their 8 Gorea, A. and Sagi, D. (2001) Disentangling signal from noise in visual contrast discrimination. Nat. Neurosci. 4, 1146 – 1150 mental toolkit. It is already remarkable that one can 9 Thompson, R.F. et al. (1970) Number coding in association cortex of the discriminate linear and logarithmic coding schemes with a cat. Science 168, 271 – 273 range of numbers as small as 1 to 5. By testing the neurons 10 Nieder, A. et al. (2002) Representation of the quantity of visual items in with a greater range of numbers, it should be easier to see the primate prefrontal cortex. Science 297, 1708 – 1711 if the small advantage of the logarithmic fit over the power 11 Sawamura, H. et al. (2002) Numerical representation for action in the parietal cortex of the monkey. Nature 415, 918 – 922 function fit found over the range 1 to 5 will continue to hold 12 Dehaene, S. (1997) The Number Sense, Oxford University Press with larger numerosities. 13 Dehaene, S. et al. (1999) Sources of mathematical thinking: behavioral Overall, Nieder and Miller’s recordings confirm and brain-imaging evidence. Science 284, 970 – 974 Fechner’s intuitions of 130 years ago. The neural repre- 14 Brannon, E.M. and Terrace, H.S. (2000) Representation of the sentation of number is comparable to the slide rule that numerosities 1 – 9 by rhesus macaques (Macaca mulatta). J. Exp. Psychol. Anim. Behav. Process. 26, 31– 49 some of us learned to use before the advent of electronic calculators, which was also graduated with a logarithmic 1364-6613/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. scale. The advantages of this instrument were two-fold. doi:10.1016/S1364-6613(03)00055-X Structure and pragmatics in informal argument: circularity and question-begging Sarah K. Brem Division of Psychology in Education, Mail Code 0611, Arizona State University, Tempe, AZ 85287-0611, USA Most everyday arguments are informal, as contrasted given that the premises are true, any conclusion made with the formal arguments of logic and mathematics. without violating any rules of logic is also necessarily true. Whereas formal argument is well understood, the nature Formal argument plays an essential role in areas such as of informal argument is more elusive. A recent study by mathematics and logic [1,2]. Informal arguments, however, Rips (2002) provides further evidence regarding the roles are based on induction and marked by uncertainty. All of structure and pragmatics in informal argument. propositions about the world are inherently imperfect; we believe that the sun will rise tomorrow, but we can as easily An exemplar of formal argument is the syllogism: Socrates conceive of the alternative. Countless events could turn us is a man, all men are mortal, therefore we may conclude on our head, literally and metaphorically. No theory about the that Socrates is mortal. Formal argument is deductive; world can ever be fully proven or refuted; all are built upon countless unstated assumptions. When we argue about Corresponding author: Sarah K. Brem ([email protected]). whom to vote for or the best recycling policy, the imperfections http://tics.trends.com