Magnitudes for Nervous Systems: Theoretical Issues and Experimental Evidence

Quantity, Discrete and Continuous

Physical quantity is a property of a system or object that can be measured. Quantity can be continuous or discrete. Discrete quantity is characterized by a plurality of elements or units that is divisible ultimately into indivisibles, whereas continuous magnitude is extensive and unitary and can be infinitely subdivided into smaller continua. The number of leaves on the crown of a tree is an example of discrete quantity, whereas the length of a bough is an example of a continuous quantity.

The assessment of continuous quantity by animals has been widely investigated, in studies ranging from investigations of estimates of distances during navigation to studies on the neural circuits filtering small versus large moving objects (Preuss et al., 2014). Also, research has shown that several animal species of very distant taxonomic groups—both vertebrates and invertebrates—can assess discrete quantities of food items, social partners, or even artificial objects (Nieder, 2019). Animals can manipulate these quantities, performing arithmetic operations on them, such as summation or computation of ratios, to accomplish behaviors crucial for survival and reproduction. However, few studies have investigated neural circuits involved in discriminating the number of elements in a group.

Humans and other animals spontaneously perceive quantities in a noisy and approximate way, an ability likely having a long evolutionary history (Lorenzi et al., 2021). Instruments and symbols (e.g., numerals) enable humans to measure and numerate quantities precisely, using units of measurement (Pitt & Casasanto, 2020). Nonetheless, even without arbitrary symbols, all animals—including humans—can rely on an approximate sensitivity to discrete quantity that obeys Weber’s law, which states that the magnitude of a just-noticeable difference in the intensity of a stimulus is a constant proportion of the original magnitude of the stimulus. For example, if a person requires an increase or a decrease of 1 g to detect a change in a weight of 10 g, Weber’s law predicts that the person would require a 10-g increment or decrement to detect a change in a weight of 100 g.

Subpallial and Sensory Involvement: Neurobiological Evidence

Although some researchers (e.g., Leibovich et al., 2017) have argued that it may be difficult to disentangle animals’ ability to assess discrete quantity from their ability to assess confounding continuous quantities, accurate controls of continuous physical variables in some investigations have made it possible to conclude that animals can estimate pure discrete quantity (numerosity; see, e.g., Potrich et al., 2022, which is noteworthy in that the role of spatial frequency was also controlled for). The idea of a neat separation of discrete from continuous magnitudes seems to go hand in hand with the idea that selectivity to discrete quantities (numerosities) would be apparent only at later stages of processing, in pallial (cortical) areas of the brain, whereas selectivity to the size of stimuli and other continuous quantities would be apparent already at earlier stages, in subpallial (subcortical) areas. In this article, we refer to “cortical” areas to indicate the more dorsal portions of the brain that have a laminated structure in mammals and to “pallial” areas to indicate homologous or analogous areas that have a nuclear structure in other vertebrates (see Lorenzi et al., 2021). For invertebrates, we refer more generically to higher levels of integration in the brain (see Bortot et al., 2021; Giurfa, 2019).

The existing literature on the neurobiology of quantity cognition has devoted most attention to cortical or pallial associative regions in humans, nonhuman primates, and corvids (a family of birds that includes crows and ravens), with a special emphasis on prefrontal regions in mammals and allegedly equivalent regions in other animals (Nieder, 2019). However, the ability to discriminate different discrete and continuous magnitudes shows up ubiquitously (e.g., in newborn humans—Izard et al., 2009; insects—Bortot et al., 2021; and even single neurons—e.g., Rapp et al., 2020), which suggests that cortical and pallial regions might not be essential for such skills (see Fig. 1).

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Fig. 1.

Evidence for the brain’s involvement in magnitude estimation. The top four panels represent the location and relative size of key regions in the brains of (a) fish, (b) amphibians, (c), birds, and (d) mammals. The pallium (cortex in mammals) is shown in beige, the thalamus in yellow, and the midbrain in orange. The bottom panel (e) is a schematic representation of the main sensory pathways, from the input (e.g., visual or auditory) at the top to the motoric output at the bottom. The diagram shows animals for which involvement of each brain region has been implicated in magnitude estimation, together with labels indicating the kind of magnitude estimation that the experimental tasks involved (discrete, continuous).

A midbrain (subpallial) role in magnitude representation is suggested by the involvement of the optic tectum (superior colliculus in mammals) in size discrimination (see Fig. 1). The optic tectum, situated toward the back of the midbrain, plays a central role in interfacing sensory stimuli and behavioral motor patterns. In animals ranging from amphibians to pigeons, size selectivity emerges from neurons in the optic tectum that modulate their responses in accordance with the size of the stimulus (small or large; Preuss et al., 2014).

In addition, zebrafish habituated to a certain stimulus size show a change in the expression of c-fos (a gene used as a marker of neuronal activity) in the optic tectum when the stimulus size is changed (Messina et al., 2020). In humans, the pathway that connects the superior colliculus with more anterior brain regions develops earlier than the pathway that connects the thalamus with the cortex, and 2-day-old human newborns, who have mainly the former pathway in use, are capable of discriminating discrete visual quantities (Izard et al., 2009). Human adults are facilitated in discriminating two quantities when they are presented sequentially to only one eye (monocular condition; Collins et al., 2017), and this facilitation must arise from a substantial subcortical contribution because monocular neurons can be found only in the first four layers of visual cortex, where early processing of subcortical input takes place.

The midbrain seems to be responsive to magnitude even in other sensory modalities. Rose (2018) found that in the torus semicircularis, a midbrain structure involved in sound perception, some neurons are selective to specific calls crucial for female toads’ reproductive choice: Interval-counting neurons spike (“fire”) when the number of calls reaches a certain threshold.

Evidence for a thalamic (subpallial) involvement in magnitude estimation comes from work carried out in our laboratory. Expression of the c-fos gene in a portion of zebrafish thalamus appears to be modulated by the magnitude of change in both continuous (size) and discrete (numerosity) visual quantity (Messina et al., 2020; see Fig. 1). Moreover, differential gene expression in response to changes in visual discrete quantity (number of dots) has also been observed in a central division of zebrafish pallium (Messina et al., 2022), which could be the result of thalamic input. Supporting such an early pallial involvement in the processing of discrete magnitude are findings indicating that the primary sensory cortices in humans may be involved in perception of discrete quantities, in both the visual and the auditory domains (Fornaciai et al., 2017).

Recently, Bengochea et al. (2022) demonstrated that lobula columnar neurons that connect early stages of visual neural processing with higher brain structures are causally involved in fruit flies’ discrimination of the number of visual stimuli (stripes). Silencing these neurons reduced or removed a preference for the more numerous stimuli. In contrast, the authors did not find any involvement of more integrative higher-order brain regions.

Overall, these studies, together with evidence from sensory adaptation (Burr & Ross, 2008), suggest that both discrete and continuous aspects of magnitude are encoded mainly as primary sensory features.

Interacting Systems for Magnitude

The continuous and discrete aspects of quantitative features not only are somewhat intertwined in theory but also can interact with one another in cognition. For example, when shown larger objects that are black and have stripes and smaller objects that are white and have dots, 9-month-old infants expect the same color-plus-pattern mapping to hold for discrete quantity (i.e., larger discrete quantity: black with stripes; smaller discrete quantity: white with dots) and duration (i.e., longer-lasting stimuli: black with stripes; shorter-lasting stimuli: white with dots; Longo & Lourenco, 2010).

Comparative evidence for interaction among magnitude systems is widespread in vertebrates (for a review, see Vallortigara, 2018). Recently, we showed that honeybees can transfer rules involving relations such as “larger than” and “smaller than” from discrete (numerosity) to continuous (size) magnitude. We trained bees to discriminate between different sets of elements having either a 0.5 ratio (2 vs. 4; 4 vs. 8) or a 0.67 ratio (2 vs. 3; 4 vs. 6), and then we tested the bees’ spontaneous choice between sets that had the same number of elements but elements of different sizes. We found that, irrespective of the ratio of the training stimuli, bees trained to select the smaller number of elements chose the set with elements of a smaller size, and bees trained to select the larger set of elements chose the set with elements of a larger size (Bortot et al., 2020). This evidence for a cross-dimensional transfer between discrete (numerical) and continuous (spatial) dimensions supports the hypothesis of a common cognitive currency for general magnitude (Gallistel, 2011).

A dominant view in cognitive sciences is that, although there is a natural disposition to associate domains of discrete and continuous quantities, specific directionalities (biases) for such associations will be observed only in humans, as a result of social and cultural habits. A well-known example is provided by the association between number and space, the so-called mental number line (Galton, 1880). There is evidence, however, that this disposition to associate small numbers with the left and large numbers with the right is observed also in nonhuman species and human neonates (Vallortigara, 2021). This evidence does not, however, rule out an important role of directional habits associated with writing and reading. There is plenty of evidence for such a modulatory and plastic role of experience: In adult organisms (strikingly, both nonhumans and humans), mental-number-line phenomena appear to be somewhat elusive and variable, though cases of clear directional biases have been reported (e.g., in adult monkeys; Drucker & Brannon, 2014). We have argued (Vallortigara, 2018), therefore, that there could be biological mechanisms underlying the association between number and space (and maybe other quantities) and that these mechanisms are most apparent in young and little-experienced organisms (e.g., in newly hatched chicks, as reported in Rugani et al., 2015, and in human newborns, as reported in Di Giorgio et al., 2019).

According to the valence theory (Davidson, 2004), activation of the left and right hemispheres is associated with positive and negative valence, respectively. Assuming that changes in discrete quantity toward larger magnitudes are associated with positive valence, and changes toward smaller magnitudes are associated with negative valence, the former changes would be expected to lead to greater activation of the left hemisphere, and the latter changes would be expected to lead to greater activation of the right hemisphere. In turn, such changes would be expected to result in attention to the opposite site of the body, given that this is how the hemispheres primarily allocate attention (Vallortigara, 2018). This would fit nicely with observed choice preferences for the left side when there is a transition from large to smaller magnitude and for the right side when there is a transition from small to larger magnitude (as revealed by approach in chicks and by fixation times in newborns; Rugani et al., 2015; Di Giorgio et al., 2019). Thus, valence theory would provide a simple and mechanistic explanation for the association between number and space shown by newborn babies and chicks without any need for mentalistic constructs such as that of a number line.

Conclusions and Future Directions

It is worth highlighting that several different terms are employed to refer to discrete quantities in cognitive neuroscience, “number” and “numerosity” being among the most used. The high polysemy of the word “number” has generated some confusion in this area of research, thus motivating recent debates (Clarke & Beck, 2021).

Although “number,” “numerosity,” and “numerousness” are often used interchangeably (see Dos Santos, 2022, for a review), these terms were originally introduced and differentiated by Stevens (1939/2006) in the domain of psychophysics in an attempt at rigorous terminology. In most of the philosophy of mathematics, numbers are considered abstract concepts that denote physical ones and that can be used to measure any quantity, discrete or continuous (Frege, 1980). The empirical studies on number cognition that we have discussed in this review, however, dealt with perception of the discrete quantity of objects and not numbers as such.

These theoretical and terminological issues are related to the evidence we have discussed here, because the search for the neural processes underlying the ability to extract and process discrete quantity has been guided misleadingly by the idea that discrete quantity, being more abstract than continuous quantity, needs to be processed by higher (e.g., cortical) brain areas. However, the distinction between numbers as abstract concepts and discrete quantities as percepts points to the necessity of studying the encoding of the latter in early sensory areas and circuits, the same ones that are known to be involved in encoding continuous quantity. Doing so might lead to a better account of the differences between the processing of continuous versus discrete quantity.

Recommended Reading

Dos Santos, C. F. (2022). (See References). A thoughtful discussion of the issues associated with the use of different terminologies in studies of quantity cognition.

Gallistel, C. R. (2011). (See References). A theoretical account in favor of a common analog representation of quantity, with discussion of the constraints on the implementation of computable magnitudes in nervous systems.

Messina, A., Potrich, D., Schiona, I., Sovrano, V. A., Fraser, S. E., Brennan, C. H., & Vallortigara, G. (2022). (See References). A presentation of the first evidence of a selective responsivity to the discrete quantity of visual items in the pallium of zebrafish.

Stevens, S. S. (2006). (See References). The original discussion on the distinctions among numerosity, numerousness, and numerals in psychophysical studies.