## Introduction

Number cognition, broadly speaking, includes numerical estimation, simple arithmetic operations, magnitude judgments, and counting amongst other skills. There is a long history of research on number cognition, including the cognitive and neural processes involving numerical magnitude. Research includes behavioral studies of number development (e.g. Gelman & Gallistel, 1978; Piaget, 1954, amongst others) and more recently a large number of neural studies relevant to number cognition (e.g. Ansari & Dhital, 2006; Ansari, Garcia, Lucas, Hamon & Dhital, 2005; Cantlon, Brannon, Carter & Pelphrey, 2006; Cantlon, Libertus, Pinel, Dehaene, Brannon & Pelphrey, 2008; Cohen Kadosh & Walsh, 2009; Dehaene, Piazza, Pinel & Cohen, 2003; Göbel, Calabria, Farnè & Rossetti, 2006; Pesenti, Thioux, Samson, Bruyer & Seron, 2000; Walsh, 2003; Whalen, McCloskey, Lesser & Gordon, 1997). This increasingly large literature involving humans has been supplemented by research with non-human primates (e.g. Brannon & Terrace, 1998; Nieder & Miller, 2003; Roitman, Brannon & Platt, 2007) and by computational methods that incorporate neural principles (e.g. Ahmad, Casey & Bale, 2002; Dehaene, 2007; Dehaene & Changeux, 1993; Verguts & Fias, 2004; Zorzi, Stoianov & Umiltà, 2004).

Significant contributions to cognitive development have been made through computational modeling that connects neural and behavioral data – in areas of language learning (Elman, 1993), motor development (e.g. Spencer, Simmering, Schutte & Schöner, 2007), and visual development (e.g. Mareschal & Johnson, 2002). For example, Spencer and colleagues use neurocomputational modeling to provide evidence for a novel interpretation of the classic A-not-B error developmental phenomenon (Piaget, 1954). By modeling of visual-motor neural processes Spencer and colleagues conclude that the A-not-B phenomenon is an example of a broader class of errors that occur in development. The current study presents a series of simulations based on recent advances in the study of the neural coding of numerical magnitude that offer new insights into behavioral phenomena described in the developmental literature.

### Neural coding of number

A variety of investigations with both humans and non-human primates have characterized the neural activity related to the perception of number. First, research has focused on the localization of neural activity specific to number. There has been convergence on the intraparietal sulcus and areas of prefrontal cortex (e.g. middle frontal gyrus) from both humans (e.g. Ansari & Dhital, 2006; Ansari *et al*., 2005; Cantlon *et al*., 2006; Cantlon *et al*., 2008; Dehaene *et al*., 2003) and non-human primates (Nieder, Freedman & Miller, 2002; Nieder & Miller, 2003; Nieder & Merten, 2007; Sawamura, Shima & Tanji, 2002). Numerical coding activity has been recorded in both intraparietal sulcus and prefrontal cortex; two areas that have been found to be functionally connected (Cavada & Goldman-Rakic, 1989; Chafee & Goldman-Rakic, 2000; Quintana, Fuster & Yajeya, 1989). Neural activity in these areas has been recorded in tasks such as number magnitude comparison, arithmetic operations and even the perception of a digit. The basic result has been replicated across a variety of presentation formats, such as dot displays and written digits (Eger, Sterzer, Russ, Giraud & Kleinschmidt, 2003) and cultures (Tang, Zhang, Chen, Feng, Ji, Shen, Reiman & Liu, 2006).

Second, studies have described in detail neural responses to number with the use of direct neural recording. Two types of neural coding have been described: number selective coding and summation coding. Summation, or monotonic coding, of number includes graded coding that increases as the perceived number magnitude increases (Roitman *et al*., 2007). This type of coding is consistent with the accumulator model of number representation; that number is represented by accumulating a fixed number of pulses produced serially by some pacemaker (Meck & Church, 1983). There is also evidence of number specific activity in that the spiking rate of a given set of neurons is correlated maximally to a particular value *N*, and less so for *N* + 1, *N*– 1 and so on (Nieder *et al*., 2002; Nieder & Miller, 2003; Nieder & Merten, 2007; Sawamura *et al*., 2002). This holds across presentation format (e.g. dot displays, written digits) of the numerical values. This type of coding creates Gaussian-like neural tuning function (see Figure 1). Each number magnitude is not coded exactly, but in a manner that is consistent with Weber-Fechner’s law (Fechner, 1966 [1860]); that noticeable differences between perceptual stimuli are a function of the proportional difference. As the magnitude of the number increases the neural tuning function width increases proportionally. For example, the width of the tuning function for the magnitude 5 is half that of the magnitude 10, which is half of 20. Thus differences in the perceived value are a function of the proportional stimulus differences, as with Weber-Fechner’s law.

Theories of how number sensitive neural activity develops have been supported by computational models (e.g. Ahmad *et al*., 2002; Dehaene, 2007; Dehaene & Changeux, 1993; Miller & Kenyon, 2007; Pearson, Roitman, Brannon, Platt & Raghavachari, 2010; Verguts & Fias, 2004). These studies demonstrate the development of number selective activity from other inputs, such as perceptual object tracking, or accumulator-like summation coding (Miller & Kenyon, 2007; Verguts & Fias, 2004). Computational results show number selective activity coded with tuning functions that are proportional to the number magnitude, skewed on the linear scale and symmetric on the log scale, similar to the neural data (Dehaene, 2007).

### The current simulations

The current simulations are in part based on prior neural and computational work. General aspects of the model such as Gaussian tuning curves for number values have been illustrated in prior neural (e.g. Nieder & Miller, 2003) and computational work (Dehaene, 2007; Verguts & Fias, 2004). The current model posits these basic aspects and focuses on developmental change in both the neural activity and behavior. Prior computational work has not provided a clear mechanism of how the neural coding of number may influence developmental behavioral phenomena, such as the apparent log to linear shift in number line estimations; ‘what triggers the conceptual shift from logarithmic to linear in children remains unknown’ (Dehaene, 2007, p. 557). The current focus on how changes in neural activity may influence behavioral changes provides possible answers to this and other questions of numerical development.

The current model focuses on two aspects of the neural tuning curves. First, the width of the function depends on the magnitude of the value being coded. Thus the tuning function for the value 10 is narrower than the function for the value 30, on a linear scale. The functions are proportionally similar, and thus similar on a log scale (Nieder & Miller, 2003; see Figure 1). Second, the tuning functions, though resembling Gaussian distributions, are positively skewed *on a linear scale*. The positive skew also results from the transformation from a logarithmic scale to a linear scale; if the tuning function is symmetric on a log scale it will be positively skewed on a linear scale. In their studies of non-human primates, Nieder and Miller (2003) reported that neural responses are positively skewed on a linear scale. In addition, Nieder and Merten (2007) found that in the coding of values 1–30, smaller values are clearly positively skewed, and larger values are not skewed as much. Computational accounts (Dehaene, 2007; Verguts & Fias, 2004) have shown positive skew in number coding that arises through unsupervised learning with number magnitudes. Thus these two properties – the logarithmic scale and the positive skew – may be fundamental aspects of the human number system. Although both positive skew and proportional tuning functions have been reported in the literature, their role in number cognition has not been well studied.

The current study includes a series of computational simulations that explore how the properties of the neural coding of number may contribute to the development of number cognition. More specifically, the simulations provide a likely neural mechanism for several phenomena previously only described behaviorally. The tasks used in the simulations reflect the tasks used in behavioral investigations of number line estimation and operational momentum. Within the simulations, for a given set of numerical values there is a corresponding set of neural tuning functions that resemble Gaussian distributions with peak activity corresponds to the number being coded (see Figure 2). The simulations specifically examine the relation in coding between the positive skew and the varying width of the tuning function. Building on the neural evidence (Nieder & Miller, 2003), it is assumed that the more narrow distributions that characterize small number values are more skewed than the wider distributions that represent larger numbers. Thus, the tuning functions resemble a Poisson distribution in that both displays attenuate positive skew. Poisson distributions have a history of use in neurocomputational work in describing neural spike trains (Ashby & Valentin, 2007; Boccaletti, Latora, Moreno, Chavez & Hwang, 2006; Song, Miller & Abbott, 2000). The tuning curves presented in prior work (Nieder & Miller, 2002) are arranged to show one particular neural population’s relative activation to varied numerical stimuli. The tuning curves used in the current work represent the relative activation of a range of neural populations in response to one specific numerical stimulus. The shape and characteristics of the neural tuning curves, if viewed this way, retain the identical shape of a positively skewed Gaussian curve.

Prior research has also reported that when behavioral errors occur, the neural activity for the preferred quantity is significantly reduced compared to correct trials (Nieder *et al*., 2002; Nieder, Diester & Tudusciuc, 2006; Nieder & Miller, 2004; Nieder & Merten, 2007). Errors in neural coding of number were linked to errors in the behavioral task. This is key to the current framework. Errors or lack of precision in neural coding may occur and give rise to these same properties in numerical judgments.