Numbers in the brain

If you look around, number quantities are inescapable: our culture is based on precisely measuring time, distance, and quantities. However, even creatures that aren’t exposed to sophisticated human number systems are inevitably exposed to quantities of things. Animals in the wild come into contact with varying quantities of other animals and of their food, and differentiating between one predator and ten, for example, matters for their survival. Thus we might ask what is going on in our brains when we use this crucial number sense – also referred to as numerosity.

Humans have undoubtedly complex mathematical capabilities, many of which are possible because they’re grounded in symbolic systems. But humans also exhibit some lower-level numerical abilities that we share with other animals. It is well established that humans, animals, and pre-linguistic infants can all compare and approximate numerical quantities[1]. Addition also seems to be a skill not only apparent in humans, but in monkeys as well.[2] In one study, Canton and Brannon gave monkeys and college students the same nonverbal arithmetic task. They were shown two sets of dots and had to choose from a second two sets which accurately reflected the sum of the first two. It should be noted that the college students did outperform the monkeys, but that the monkeys still computed sums at levels above chance and that their accuracy and response times followed the same patterns as the humans’.

But much of what we know about numerosity on the neuronal level also comes from studies on monkeys. By examining single-cell recordings from monkeys’ parietal and prefrontal cortices while they complete different numerosity tasks, Andreas Nieder has found neuronal specificity for different numerosities[3]. In other words, he discovered that many neurons have preferences – some fire most when the monkey is looking at two objects, others fire most when it’s looking at three, and so on.

In the first of the series of studies that will be discussed here, Nieder and Miller pointed out that numerosity representations, which seem high-level and sophisticated, are in fact similar to lower-level sensory (i.e., visual) representations[3]. Sensory processes exhibit two distinctive effects: size and distance effects. The size effect describes the increased difficulty for discriminating magnitudes as a set size increases. For example, quickly determining that a set contains 8 items is more difficult than determining that a set has 2 items. The distance effect specifies that discriminating between different magnitudes also becomes more difficult as the distance between the two numerosities decreases. For instance, differentiating between a set containing 8 items and another that has 7 is more difficult than differentiating between a set with 8 items and another with 2. In this experiment, the monkeys’ behavior on a numerosity task exhibited these same effects that emerge in psychophysical tasks.

The dominant paradigm in Nieder’s experiments is a delayed match-to-numerosity task. Monkeys were shown a display with a certain number of randomly placed dots (ranging from one to five), and after a short delay period, they were shown another display. In 50% of the trials, the number of dots in the test phase matched the number in the sample phase. In these cases, the monkeys were trained to release a lever they had been holding. In the other trials, the first display did not match the sample, so the monkeys had to continue to hold the lever down, releasing only when a new display, matching the original, was shown.

Recordings from neurons in the parietal and prefrontal cortices as monkeys completed this task showed that about one-third of the 352 neurons that the researchers examined showed significant tuning to one of the five tested numerosities. For example, a neuron might fire most when the monkey saw three dots. That same neuron would exhibit the next-highest firing rate for a quantity of four (size effect), and almost as much firing when shown two dots (distance effect). In other words, this neuron had a preference for about 3 items. As the quantities got larger, it became harder to tell the difference between two adjacent numerosities (i.e., 3 and 4), and easier as they became smaller (i.e., 2 and 3).

The graphs show firing rates for neurons with varying numerosity preferences. For example, in the second graph from the top, the neurons preferring 2 items fired far more when the monkey saw 2 dots, next most when he saw 3 dots, and next most for 1 dot. Reproduced from Miller & Nieder [3].

The graphs show firing rates for neurons with varying numerosity preferences. For example, in the second graph from the top, the neurons preferring 2 items fired far more when the monkey saw 2 dots, next most when he saw 3 dots, and next most for 1 dot.
Reproduced from Miller & Nieder [3].

In another study, Diester and Nieder found that the same low-level sensory psychophysical properties (size and distance effects) were evident in a numerosity task involving symbols instead of absolute quantities[4]. The researchers trained the monkeys to associate symbols (Arabic numerals) with the numerosities 1 through 4 and then had them complete the same delayed match-to-sample task as in the previous experiment. Only behavioral measures were collected in this experiment, but the monkeys’ error rates and reaction times indicated that as quantities got larger, the monkeys showed both size and distance effects. The evidence of the size and distance effects in this task indicates that the monkeys weren’t simply matching the visual appearance of one numeral to the next, but were instead using the numerals as symbols representing actual quantities and reasoning about those quantities.

In another extension, Nieder found that the behavioral and neuronal patterns shown in the earlier studies also held when monkeys processed numerosities in a different modality (i.e., auditory stimuli instead of visual)[5]. In this version, monkeys were trained to recognize numerosities either in the form of a sequence of dots or auditory pulses. When they completed the delayed match-to-sample task, they showed the distance and size effects regardless of the modality they were being tested in. In the prefrontal cortex (PFC), 18% of sampled neurons showed activity that varied significantly with the number of sounds during the auditory sample, and 29% showed such activity based on the number of dots in the visual protocol. He found that 60% of the auditory numerosity-selective neurons responded to both audio and visual numerosities. On average, these bimodal neurons preferred the same numerosity in both protocols. For example, a neuron tuned to 2 audio pulses was most likely also tuned to 2 dots presented sequentially. In the intraparietal sulcus (IPS), 10% of the sampled neurons were tuned to numerosity in the auditory protocol, and 11% were tuned to the number of visual dots. Of the auditory numerosity-selective neurons, 30% also had a numerosity preference in the visual protocol, but only neurons that preferred the numerosity one had the same sample-preferred numerosity in both modalities. In other words, neurons that preferred quantities greater than one had different numerosity preferences depending on the modality. Because Nieder found a significantly larger proportion of numerosity-selective neurons for both modalities in the PFC than the IPS, the results suggest that the PFC is especially important for abstract numerosity coding early in the learning process.

In subsequent studies, Nieder and his colleagues have found that the same behavioral and neuronal patterns exist even when the monkeys haven’t been explicitly trained to distinguish numerosities[6] and even when the numerosity task is embedded in tasks requiring different mental processes, like differentiating the color of various stimuli[7].

Numerosity is simultaneously a complex, high-level cognitive process and a less-impressive, low-level one. On the one hand, a number system forms a foundation upon which we scaffold almost every other sophisticated practice, from simple bartering to engineering massive bridges. On the other hand, many more basic mathematical abilities are not unique to humans, like approximating and adding quantities. Further, numerosity representations seem to share properties with lower-level perceptual properties, demonstrating that many of the complex number-related abilities that humans show are actually based in simpler cognitive processes. Maybe underlying our abilities to understand quantum mechanics (NB: I do not include myself in this camp, despite my use of the pronoun our), are abilities like being able to discern the difference between two approaching animals (maybe provoking a fight response) and ten (flight).


[1] Dehaene, S. (1992). Varieties of cognitive abilities. Cognition, 44,(1-2), 1-42.

[2] Cantlon, J. & Brannon, E. (2007). Basic math in monkeys and college students. PLoS Biol., 5(12): e328. doi:10.1371/journal.pbio.0050328

[3] Nieder, A. & Miller, E.K. (2003). Coding of cognitive magnitude: Compressed scaling of numerical information in the primate prefrontal cortex. Neuron, 37, 149-157.

[4] Diester, I. & Nieder, A. (2010). Numerical values leave a semantic imprint on associated signs in monkeys. Journal of Cognitive Science, 22, 174-183.

[5] Nieder, A. (2012). Supramodal numerosity selectivity of neurons in primate prefrontal and posterior parietal cortices. Proceedings of the National Academy of Sciences of the USA, 109, 11860-5.

[6] Viswanathan, P. & Nieder, A. (2013). Neuronal correlates of a visual ‘sense of number’ in primate parietal and prefrontal cortices. Proceedings of the National Academy of Sciences of the USA, 110, 11187-11192.

[7] Moskaleva, M. & Nieder, A. (2014). Stable numerosity representations irrespective of magnitude context in macaque prefrontal cortex. European Journal of Neuroscience, 39, 866-874.