Our human-specific symbolic number skills that underpin science and technology spring from nonsymbolic set size representations. Despite the significance of numerical competence, its single-neuron mechanisms in the human brain are unknown. We therefore recorded from single neurons in the medial temporal lobe of neurosurgical patients that performed a calculation task. We found that distinct groups of neurons represented either nonsymbolic or symbolic number, but not both number formats simultaneously. Numerical information could be decoded robustly from the population of neurons tuned to nonsymbolic number and with lower accuracy also from the population of neurons selective to number symbols. The tuning characteristics of selective neurons may explain why set size is represented only approximately in behavior, whereas number symbols allow exact assessments of numerical values. Our results suggest number neurons as neuronal basis of human number representations that ultimately give rise to number theory and mathematics.

As a neuronal correlate of numerosity representations, electrophysiological recordings from the association cortex of monkeys showed neurons that are tuned to a specific preferred numerosity of visual and auditory items. Such number neurons have also been postulated by neural network models (). In humans, number neurons have been suggested based on blood flow changes in functional imaging studies (), as well as the combined synaptic mass signals from hundreds of neurons measured with electrocorticography (ECoG) (). Despite the progress that has been made using functional imaging and ECoG recordings, the mechanism of how single neurons, the anatomical and functional units of the brain, encode nonsymbolic or symbolic numerical information in humans remains unknown. We addressed this question and recorded from single neurons in the MTL of neurosurgical patients that performed a calculation task and were implanted with intracranial electrodes ().

Single neuron activity in human hippocampus and amygdala during recognition of faces and objects.

Studies in humans () and nonhuman primates () indicated parts of the parietal and prefrontal cortices as a core number system that processes nonsymbolic and symbolic numerical magnitude. However, the wider cortical number network also incorporates areas of the medial temporal lobe (MTL) (), such as the hippocampus, parahippocampal cortex, entorhinal cortex, and amygdala. The MTL comprises highly associative brain areas that are directly and reciprocally connected with the frontal number network (), and human MTL neurons are known for their selectivity to abstract categories (). Functional imaging studies in humans showed that the hippocampal system—among many other functions outside of the number domain—is also involved in learning to count and arithmetic skill acquisition, specifically during childhood (). Hippocampal-frontal circuit reorganization plays an important role in children’s shift from effortful counting to efficient memory-based solving of mathematical problems ().

Effects of problem size and arithmetic operation on brain activation during calculation in children with varying levels of arithmetical fluency.

Dual pathways connecting the dorsolateral prefrontal cortex with the hippocampal formation and parahippocampal cortex in the rhesus monkey.

Numbers are fundamental to science and technology. Despite counting and arithmetic requiring years of training, the origins of our symbolic number capabilities are deeply rooted in our ancestry (). Human adults without formal education (), pre-linguistic human infants (), and nonhuman animals () can approximately estimate numerosity, the number of items in a set. These intuitive nonsymbolic capabilities are harnessed and qualitatively transformed by children when they begin to learn symbolic counting and mathematics in school (). This intimate relationship between set size estimation and precise counting suggests that symbolic arithmetic abilities build on nonsymbolic numerical capacities.

Non-symbolic arithmetic abilities and mathematics achievement in the first year of formal schooling.

Finally, we explored whether MTL neurons also encoded the calculation rules (addition and subtraction) in an abstract manner, independent from the rule notation (word or calculation symbol as rule cues). Cells selective to nonsymbolic numerical rules have been found in monkey cortex (). We determined calculation rule-selective units by applying a sliding-window 4-factor ANOVA (with the factors mathematical rule [addition and subtraction], rule cue [word and symbol], numerical value of operand 1 [1–5], and format [symbolic and nonsymbolic]; α = 0.01) during the calculation rule phase and the rule delay phase. Figure S9 displays two rule-selective neurons. The neuron in Figure S9 A showed a selective increase whenever an addition was required (reddish discharges), whereas the neuron in Figure S9 B selectively enhanced discharges whenever a subtraction was cued (blueish colors). These rule-selective response increases were abstract and independent from the notation of the rule cue (word or symbol). In total, we found only a small proportion of 2% of abstract calculation rule cells, but this fraction was significantly larger than expected by chance ( Figure S10 ). In addition, a significant fraction of 3% of the cells encoded the rule cue (calculation word or symbol) during the calculation rule phase ( Figure S10 ).

Next, we analyzed selectivity to number in the delay 2 phase, again separately for nonsymbolic and symbolic number format. In the delay 2 phase, all the information necessary to solve the calculation is available to the subjects. The delay 2 phase may therefore be regarded as the calculation result phase. For statistical analysis, we applied a sliding-window 6-factor ANOVA (with the same factors as above, plus main factor numerical value corresponding to the result of the calculation [0–9]; α = 0.01). Neither for the nonsymbolic nor for the symbolic format was the proportion of neurons selective to the calculation result higher than expected by chance ( Figure S8 ).

The responses of a single neuron throughout the whole trial are shown in Figure S7 . This neuron was significantly tuned to numerosity 5 of operand 1 during the operand 1 phase and of operand 2 during the operand 2 phase ( Figure S7 , upper histograms). This neuron also showed strong responses to the numerical values of the operand 2 during the symbolic format ( Figure S7 , lower histograms); however, it was also selective to the numeral protocol and thus not counted as an exclusively numeral-selective cell. Overall, the highest proportion of neurons selective to the nonsymbolic numerical value of operand 2 in the MTL was found in the parahippocampal cortex (20%), followed by the hippocampus (6%; Figure S8 ).

After analysis of the responses to the operand 1, we also examined selectivity to the numerical value of operand 2 separately for nonsymbolic and symbolic number format. For the nonsymbolic format, 7.7% (45/585) of the tested neurons showed activity that varied exclusively with the number of operand 2 items during operand 2 presentation, irrespective of the dot array layout (5-factor sliding-window ANOVA with the factors numerical value of operand 1 [1–5], numerical value of operand 2 [0–5], protocol [standard and control], “mathematical rule” [addition and subtraction], and “rule cue” [word and symbol]; α = 0.01). Twenty-two of the units selective to nonsymbolic operand 1 (n = 92) were also tuned to nonsymbolic operand 2; of those, 9 cells had the same preferred number. Given that 20% of the selective units are expected to share the preferred number by chance (5 number values), this proportion of 9 cells was significantly higher (p < 0.05 in binomial test). The finding that cells that responded both to operand 1 and operand 2 tended to show the same preferred numerosity was also confirmed by a correlation analysis (Pearson’s r = 0.64; p = 0.0013; Figure S6 ). For the symbolic format, only a chance proportion of 1.5% (9/585) responded exclusively to the numerical value of the operand 2 during the presentation of the operand 2.

In addition, we analyzed the coding capacity and dynamics of the population of number-selective neurons by performing a multi-dimensional state-space analysis (see Supplemental Information ) for nonsymbolic and symbolic numbers separately. At each point in time, the activity of n recorded neurons is defined by a point in n-dimensional space, with each dimension representing the activity of a single neuron. This results in trajectories that are traversed for different neuronal states, i.e., the five different numerical values in the nonsymbolic ( Figure 6 A, left) and symbolic format ( Figure 6 A, right). These trajectories reflect the instantaneous firing rates of the respective neuronal population as they evolve over time. To evaluate the temporal evolution of population numerical tuning in each format, we measured Euclidian distances between trial trajectories in the whole population space corresponding to the activity to the five numerical values. In the nonsymbolic format, the trajectory distances systematically increased with numerical distance (p < 0.001; permutation test for all trajectories; see Supplemental Information ), starting shortly after onset of operand 1 until the end of the memory delay 1. The distances between the population trajectories confirm the findings based on single selective neurons: the closer two numerosities were in the numerical continuum, the more similar were the corresponding patterns of population activity and vice versa ( Figure 6 B, left). This argues for a numerical distance effect in the population data. In the symbolic format, the trajectory distances were much less pronounced but likewise tended to increase with numerical distance (p < 0.001 for 1 versus 4 in a permutation test), reflecting the remnants of a distance effect ( Figure 6 B, right). A comparison of the trajectory distances also suggests that MTL neurons responded longer lasting to the nonsymbolic format and throughout the working memory period (i.e., delay 1). In contrast, the responses to the symbolic format were more confined to the sample phase of operand 1. Again, this analysis yielded similar results when performed for the whole population of single units ( Figure S5 ).

(B) Intertrajectory distances, averaged across pairs of trajectories with the same numerical distance. Dashed lines represent the average distances for trajectories obtained for label-shuffled data.

(A) Average state-space trajectories, reduced to the three principal dimensions for visualization, for the sub-populations of numerosity-selective (left) and numeral-selective (right) units. Each trajectory depicts the temporal evolution in the time window 0–1,850 ms. Circles indicate boundaries between experimental periods (Cl.R., calculation rule; Del.1, delay 1; Fix., fixation; Op.1, operand 1).

Next, we trained and tested the classifier on the firing rates of each neuron obtained by averaging across the time window that had turned out significant in the cross-training classification. The resulting confusion matrices show robust accuracy (65.6% ± 2.5%) for the five numerosities in the nonsymbolic format represented by the diagonal ( Figure 5 C, left). The probability of misclassification of trials increased the closer two classes were in the numerical space (“distance effect”; Figure 5 D, left). Also for number symbols, the numerical values could be classified significantly above chance level but with lower accuracy (38.8% ± 2.9%; Figure 5 C, right). Misclassifications hardly varied as a function of numerical distance for number symbols ( Figure 5 D, right). At absolute numerical distances 2, 3, and 4, classification probabilities obtained for the classifiers (n = 32) trained on nonsymbolic and symbolic number neurons were almost identical and significantly higher for symbolic than for nonsymbolic number (p < 0.01; t test). This indicates a sharper transition from the preferred to all nonpreferred numbers and thus greater selectivity in neurons tuned to symbolic number. When applied to the entire set of single units regardless of numerosity selectivity (585 units), this analysis yielded qualitatively similar results ( Figure S4 ).

We therefore explored how the two populations of numerosity-selective and numeral-selective neurons encode numerical values. To evaluate the neuronal populations’ information carried about number, we first trained a multi-class support vector machine (SVM) classifier to discriminate numerical values based on the spiking activity of selective MTL neurons (see Supplemental Information ). After training, the classifier was tested with novel data from the same neuronal population to explore how well it could predict number categories based on the information extracted from trials used for classifier training. Initially, we performed a temporal cross-training classification to assess the classifier’s accuracy in identifying the correct numerical values when tested on the activity from a given time period after being trained on other time periods of the trials. With a chance performance of 20% (for five classes), the classifier accuracy was significantly higher for both nonsymbolic and symbolic number throughout the operand 1 and delay 1 phases, albeit with better performance during the nonsymbolic-format trials ( Figures 5 A and 5B ).

(D) Classification probability as a function of numerical distance. The dashed line represents chance level; shaded areas indicate SEM; asterisks represent significant differences between adjacent numerical distances ∗∗∗ p < 0.001).

(C) Confusion matrix derived when training an SVM on firing rates, averaged across the significant time windows in the temporal cross-training classification (B). Values on the main diagonal represent correct classification.

(B) Accuracy for training and testing on identical time periods (main diagonal of matrices in A). The dashed line represents chance level (20% for five classes). Black bars above the data indicate significance (p < 0.01) when testing against performance for SVMs trained on shuffled data in a permutation test. Shaded areas indicate SEM.

(A) Classification accuracy for decoding numerosity information when training a multi-class support vector machine (SVM) on instantaneous firing rates at a given time point and testing on another one for numerosity-selective (left) and numeral-selective (right) neurons.

So far, our data suggest two main findings at the level of individual neurons. First, the representation of nonsymbolic number was abundant and comparable to the core number network in nonhuman primates (), whereas the representation of symbolic numbers was sparse in the MTL. Second, neurons responsive to nonsymbolic or symbolic number formats are largely segregated in the MTL; abstract neurons that encode the same numerical value in both nonsymbolic and symbolic formats were rarely found.

Representation of the quantity of visual items in the primate prefrontal cortex.

Overall, the numeral-selective neurons’ preference covered the entire range of numbers 1–5 ( Figure 4 A, right), and their normalized activity for each preferred numeral formed overlapping tuning functions ( Figure 4 B, right). The decline of activity from the preferred to the nonpreferred numerals was brisk and categorical, with only a mild progressive decrease with numerical distance, hardly showing a neuronal numerical distance effect ( Figure 4 C, right). At absolute numerical distance 1, the normalized firing rates obtained for symbolic number (n = 16) were significantly lower compared to nonsymbolic number (n = 92; p < 0.05; t test), indicating higher selectivity for (or sharper tuning to) symbolic number. When comparing the neuronal latencies to reach number-selectivity, neurons tuned to nonsymbolic (990 ms) and symbolic number (880 ms) did not differ significantly (p = 0.23; Mann-Whitney U test).

When participants calculated with Arabic numerals (symbolic format), a smaller but significant proportion of the recorded neurons (3%; p < 0.001 in binomial test; p= 0.01) responded selectively to numerals during operand 1 presentation and the subsequent working memory delay 1 (2-factor sliding-window ANOVA, with factors numerical value × protocol; α = 0.01; Figure S1 , right). The highest fraction of such numeral-selective neurons in the MTL was again found in the parahippocampal cortex (6%), followed by the amygdala (4%; Figure 3 , lower columns). Six numeral-selective neurons (1% of all neurons) were also tuned to nonsymbolic number, which was more than expected by chance (p < 0.05 in binomial test; p= 0.16 × 0.03 = 0.005, or 0.5%). Of these, four neurons had identical preferred numerical values for nonsymbolic and symbolic number. This correlation did not reach significance ( Figure S3 A), possibly due to the small sample size. Next, we investigated whether the preferred numbers of units tuned to nonsymbolic numerosity might be correlated with their (non-significant) tuning to symbolic numerals ( Figure S3 B) and vice versa ( Figure S3 C). Neither correlation reached significance, indicating that numerosity and abstract numerals are encoded by two largely distinct neuronal populations. Two neurons tuned to the same value in both nonsymbolic and symbolic formats are depicted in Figures 2 C and 2D. The neuron in Figure 2 C as well as the neuron in Figure 2 D showed maximum responses to quantity 5 in both the nonsymbolic and symbolic format. In contrast, the two neurons shown in Figures 2 A and 2B were only significantly tuned to dot numerosities, but not to numerals. Again, a cross-validation analysis confirmed the reliability of the preferred numeral determination (average correlation coefficient r = 0.57; p < 0.05). The proportion of neurons selective to symbolic number for each subject is shown in Table S1 . As for nonsymbolic number, the firing rates of numeral-selective neurons were significantly higher compared to the non-selective neurons (p < 0.01; Mann-Whitney U test; Figure S2 , right).

Average tuning curves were calculated by averaging the normalized activity for all numerosity-selective neurons that preferred a given numerosity. Neural activity formed overlapping tuning functions with progressively reduced activity as distance from the preferred quantity increased ( Figure 4 B, left). To compare the decay of activity from the preferred quantity across all neurons tuned to preferred numerosities 1–5, we plotted the normalized firing rates as a function of absolute numerical distance from the preferred numerosity. For example, the normalized firing rate to numerosity 2 and 4 of a cell tuned to numerosity 3 (3 therefore is absolute numerical distance 0) were plotted at absolute numerical distance 1. The pooled function for all selective neurons compared to a function from random tuning curves is shown in Figure 4 C, left. On average, activity dropped off progressively with numerical distance across all preferred numerosities, an effect that is not observed for random tuning curves. This finding reflects a neuronal correlate of the well-known “numerical distance effect,” the behavioral observation that discrimination progressively enhances as numerical distance between two quantities increases (). A cross-validation analysis (see Supplemental Information ) yielded high reproducibility of preferred numerosity for the population of numerosity-selective units (average correlation coefficient r = 0.83; p < 0.0001), indicating that the preferred numerosity of the neurons was reliable and robust.

When the participants calculated with numerosities (nonsymbolic format), a substantial proportion of the tested neurons (16%; p ≪ 0.001 in binomial test; p= 0.01; see also Supplemental Information for verification with shuffled data) showed activity that varied exclusively with the number of items during operand 1 presentation and the working memory delay 1 that followed, irrespective of the dot array layout (2-factor sliding-window ANOVA, with factors “numerical value” × “protocol”; α = 0.01; Figure S1 , left). Four of such numerosity-selective neurons are shown in Figure 2 , left column. Each cell is tuned to numerosity; it shows peak activity for one of the numerosities, its preferred numerosity, and a systematic decrease of activity the more the number of items deviates from the preferred value. The highest fraction of such numerosity-selective neurons in the MTL was found in the parahippocampal cortex (29%), followed by the hippocampus (18%; Figure 3 , upper columns). The selective neurons’ preference covered the entire tested range of numerosities, albeit with most neurons preferring numerosity “five” ( Figure 4 A, left). The proportion of neurons selective to nonsymbolic number for each subject is shown in Table S1 . Firing rates were generally low in the MTL, but the firing rates of numerosity-selective neurons were significantly higher compared to the non-selective neurons (p < 0.0001; Mann-Whitney U test; Figure S2 , left).

(C) Averaged normalized activity across all preferred numerosities (left) and numerals (right) as a function of absolute numerical distance (black line). Asterisks above the graph represent significant differences between responses to adjacent numerical distances; asterisks below the dashed line indicate significant differences between recorded and random tuning curves ( ∗ p < 0.05 and ∗∗∗ p < 0.001). Error bars denote SEM.

(B) Average tuning curves of neurons tuned to the five numerosities (left) and numerals (right).

(A) Frequency distribution of the preferred number of neurons tuned to numerosity (left) and numerals (right).

Proportions of single units with significant main effects for “numerical value” (NUM: 1–5) or “protocol” (PROT: standard and control) and interactions (NUM × PROT) in a 2-factor ANOVA evaluated at α = 0.01, separately for each format and MTL region (AMY, amygdala; EC, entorhinal cortex; HIPP, hippocampus; PHC, parahippocampal cortex). All analyses refer to exclusively number-selective (NUM-ONLY) units, i.e., neurons with an effect for numerical value but no concurrent effects for protocol or interaction. Numbers of significant neurons were subjected to a Bonferroni-corrected (n = 4) binomial test; asterisks indicate significance ( ∗ p < 0.05, ∗∗ p < 0.01, and ∗∗∗ p < 0.001).

(C and D) Hippocampal neuron #1 (C) and neuron #2 (D) responding to both nonsymbolic and symbolic number 5.

(A and B) Two parahippocampal neurons only responsive to nonsymbolic number with preferred numerosity 1 (A) and 3 (B).

Responses of four example neurons to both nonsymbolic numerosities (left column) and symbolic numerals (right column). The left panels depict a density plot of the recorded action potentials (color darkness indicates number of overlapping wave forms according to color scale at the bottom). Panels show single-cell response rasters for many repetitions of the format (each dot represents an action potential) and averaged instantaneous firing rates below. The first 500 ms represent the fixation period. Colors correspond to the five different operand 1 values. Gray shaded areas represent significant number discrimination periods according to the sliding-window ANOVA (color-coded p values above each panel). Insets show the number tuning functions.

We recorded from 585 single neurons in the medial temporal lobes (153 amygdala, 126 parahippocampal cortex, 107 entorhinal cortex, and 199 hippocampus) of nine human subjects performing the calculation tasks. In order to explore pure number representations, and to avoid confounds with cognitive factors later in the task, we focus on the presentation of the first operand (operand 1) and the subsequent working memory phase (delay 1); the remaining task phases are considered toward the end of the results. Random presentation of either the nonsymbolic or symbolic format from trial to trial allowed us to investigate an individual neuron’s responses to each of the formats individually, but also to both formats, in an unbiased way.

Participants performed simple sequential addition and subtraction tasks using a computer display ( Figure 1 A). Task involvement ensured that numbers shown as operands were consciously processed. Numerical values of the operands ranged from 1 to 5. In half of the shuffled trials, the numerical values were presented nonsymbolically as the number of randomly placed dots in an array (numerosity). In the other half, Arabic numerals were shown as symbolic number representations. Both nonsymbolic and symbolic numbers were shown in standard and control displays in order to control for low-level visual features ( Figure 1 B; see Supplemental Information ). Arithmetic symbols or words were applied for addition and subtraction instructions ( Figure 1 C). Average performance of all participants was close to ceiling for all tested quantities and calculations (performance range 90.3%–99.8%).

(C) Example stimuli for the different calculation rules indicated by arithmetic symbols (“+” and “−”) and written words (“und” [add] and “weniger” [subtract]), respectively.

(B) Example operand 1 stimuli for the nonsymbolic and symbolic format for standard and control protocols.

(A) Experimental design of the calculation task. After visual fixation on the screen, the first number (operand 1) was followed by a brief delay, after which the addition or subtraction rule was presented, followed in turn by a delay and then the second number (operand 2). After another brief delay, subjects were required to indicate the calculated result (ranging from 0 to 9) on a number panel.

Discussion

Diester and Nieder, 2010 Diester I.

Nieder A. Numerical values leave a semantic imprint on associated signs in monkeys. Livingstone et al., 2014 Livingstone M.S.

Pettine W.W.

Srihasam K.

Moore B.

Morocz I.A.

Lee D. Symbol addition by monkeys provides evidence for normalized quantity coding. Nieder, 2009 Nieder A. Prefrontal cortex and the evolution of symbolic reference. Using single-cell recordings in subjects performing a calculation task, we have shown that single neurons in the MTL of humans are tuned to numerical values in nonsymbolic dot displays and symbolic numerals. The data about nonsymbolic number coding from humans can now be compared to those of nonhuman primates. In addition, our MTL recordings show how the capacity to represent symbolic number is represented in this part of our brain. This capacity to link number to visual signs has precursors in nonhuman primates (), but ultimately the symbolic number system is uniquely human ().

Nieder, 2012 Nieder A. Supramodal numerosity selectivity of neurons in primate prefrontal and posterior parietal cortices. Diester and Nieder, 2007 Diester I.

Nieder A. Semantic associations between signs and numerical categories in the prefrontal cortex. We have discovered two largely segregated populations of tuned number neurons in the human MTL that process either nonsymbolic or symbolic numerical quantity. The representation of nonsymbolic and symbolic number information by two distinct populations of tuned number neurons may either be inherited from the core number system or a special feature of the human MTL. Neurons in the prefrontal cortex of monkeys have been shown to respond abstractly by integrating visual and auditory numerosity (). Of course, number neurons in nonhuman primates operate strictly within the nonsymbolic format, but in monkeys trained to associate visual shapes with varying numbers of items, the responses of prefrontal neurons to the visual shapes reflected the associated numerical value in a behaviorally relevant way ().