3.4 Subitizing effect
4.1.2 Numerosity code a.k.a thermometer representation, a.k.a sum-
summation coding, a.k.a. monotonic coding
The variety of names epitomizes the popularity of this representation schema: a linear, discrete implementation of cardinal semantics. Originally proposed in Zorzi and But- terworth [187, 186], a thermometer representation encodes numerosity magnitude in a standard set theoretical way s.t. numbers are represented by sets and bigger numbers contains smaller numbers. In particular, a given number is defined as the set of active nodes, and forN>M sets of active nodes, there is a K⊂N s.t.M,K are in one-to-one correspondence. The size and distance effects are explained by referring to the similarity of the corresponding representation vectors. In particular, in the case of binary vectors, we have the following relations between the cosine similarity, the Tanimoto coefficient and the Jaccard coefficient.
LetN andMbe two numerosity vectors, then
cos(θ)= N·M ||N|| · ||M||∝ N·M ||N||2· ||M||2−M·N = N∩M N∪M=sim(N,M)
This gives us an extremely concise way to see the similarity between two numbersnand
m, s.t.n>masm/n. The distance effect is readily explained. Let’s defined the Jaccard distance as the complement of the Jaccard coefficientd ist(N,M)=1−sim(N,M). For an arbitrary numbermand a set of successors ofm,N={m+1=n1<n2<...<nk}we have d ist(m,n1)>d ist(m,n2)>...>d ist(m,nk), for anyk∈N. If we define the subject’s Weber’s coefficient as w=W R−1, with W R= qp,q>p being the smallest ratio that results in a probability of discrimination greater than 75% (Just Noticeable Difference), then for every pairm,nthe error rate is proportional tod ist(m,n)∗W R−w. Importantly the Weber’s parameter wis contributed by the decision process. Similarly in Zorzi and Butterworth [186] the size effect is a byproduct of the decreasing weight connections learned via Hebbian Rule, s.t. for two pairs of numbers with the same ratio, the larger are connected to the output node (in the response system) via a weaker connection. Consequently the time the net needs to cycle to win the competition increases for larger numbers.
4.1.3
Number line models (a.k.a. place code, a.k.a. analog
magnitude)
Analog magnitude representations predate the literature, and it’s often considered the standard model for numerosity representation. A tutorial overview of the model is
CHAPTER 4. FUNCTIONAL AND COMPUTATIONAL MODELS
available In the online support material, NumberLineModel.ipynb.
In [32] Dehaene narrates that the SNARC effect, the association of number and space, promoted the metaphor of an oriented “number line”. The reports of subjects’ capacity of automatically visualize numbers as disposed in a line, and the positioning of number in a given line, are interpreted as decoders of an inner “number line”, a conscious and enriched version of it. Less metaphorically, within the field of signal detection theory (SDT, see Macmillan and Creelman [98]), it is assumed that numerosity judgment follows Thurstone’s law of comparative judgment (Thurstone [165]). Briefly, the law states that a stimulus triggers a process in which the external stimulus value n is transformed into some value on an internalpsychological continuum.
Appreciating the model requires a little digression in SDT. The basic components of STD are a sensory process (which transforms physical stimulation into internal sensations/representations) and a decision process (which decides on responses based on this internal representation). In the simplest scenario, the response is a simple yes or no (“yes, the stimulus was larger” or “no, the stimulus was not larger”). From the hitting rate (HR) and the false alarm rate (FAR) detection sensitivity and response criterion are computed. The specific way in which they are computed from the HR and FAR depends on the specific model one adopts for the sensory process, and for the decision process. The leading model is the Gaussian model, according to which the sensory process is assumed to have a continuous output based on random Gaussian noise and that when a signal is present it combines with that noise (i.e. it is assumed that noise and stimuli are i.i.d.). By assumption, the noise distribution has a mean,µn, of 0 and a standard deviation,
σn, of 1. The mean of the signal-plus-noise distribution,µs , and its standard deviation,
σs, depend on the sensitivity of the sensory process and the strength of the signal. The observer is credited with computing the log posterior ratio of two alternatives, and the decision arises by comparing its value to a criterion. The noise in the observation used to compute the log posterior ratio determines the observer’s errors.
Coding numerosity representation as Gaussians was originally proposed in Van Oeffelen and Vos [176], and it has been extended during a 20 years period and put together in Dehaene [30].
The ‘internal representation’ of numerosity is represented as a random variable X with distribution p(X∈[x,x+dx])= 1 σn p 2πex p− (x−qn)2 2σ2
4.1. FUNCTIONAL MODELS
Weber’s law. The simplest scenario, assumed in Van Oeffelen and Vos [176] and taken up in Dehaene [33, 31], assumes the internal variability to be the same for anyn, (σn=σ) that take the value of the Weber’s ratio, therefore qn=l o g(n). Briefly this assumption is referred to as logarithmic scale and fixed variance or as compressed number line model (Figure 4.1[b.]). Alternatively, as proposed in Gallistel and Gelman [50], assuming the internal variable scales linearly (qn=n) requires that also the internal variability scales linearly (σn=σ∗n), (Figure 4.1[a.]).
When taken as phenomenological models, the two choices make practically identical prediction w.r.t. discrimination and comparison behavior. The Gaussian hypothesis is moreover transferred to neuronal representations. In Dehaene and Changeux [35] for example the neuron tuning curves (the encoding part of the representation) mirror exactly the internal representation. For a neuron that responds preferentially to a numerosity p, the tuning curve is given by
f(n,p)= 1
wp2πex p−
(log(n)−log(p))2 2w2
The only difference w.r.t. the behavioral model lies in the exact value of the Weber ratiow(the coarseness of the representation).
Subitizing
There is some debate over whether subitizing and estimation might be explained by a common mechanism. In Vetter, Butterworth, and Bahrami [180] is proposed that the ANS mechanism might explain the subitizing phenomena, given that neighboring numerosities are more discriminable in the small number range. This implies that the subitizing range is dependent on the Gaussian standard deviation (Weber coefficient), therefore subjects with more acute number sense should have a larger subitizing range, and infants a smaller one. In a study by Revkin et al. [135]), aimed at testing this hypothesis. However, the authors found particularly low variability in the 1-4 range, suggesting a separate mechanism is indeed in place. As implied by Sengupta, Surampudi, and Melcher [153] study, the inhibition parameter alone in the decision system, starting from a thermometer representation, can explain the subitizing phenomena.
CHAPTER 4. FUNCTIONAL AND COMPUTATIONAL MODELS
[a.]
[b.]
Figure 4.1: Number Line Model linear scale and normalized logarithmic scale. Figure b) makes apparent that the width of the Gaussians is constant when plotted on a log scale
4.1. FUNCTIONAL MODELS