^{*}

ID and AN conceived and designed the experiments. ID performed the experiments and analyzed the data. AN contributed reagents/materials/analysis tools. ID and AN wrote the paper.

The authors have declared that no competing interests exist.

The utilization of symbols such as words and numbers as mental tools endows humans with unrivalled cognitive flexibility. In the number domain, a fundamental first step for the acquisition of numerical symbols is the semantic association of signs with cardinalities. We explored the primitives of such a semantic mapping process by recording single-cell activity in the monkey prefrontal and parietal cortices, brain structures critically involved in numerical cognition. Monkeys were trained to associate visual shapes with varying numbers of items in a matching task. After this long-term learning process, we found that the responses of many prefrontal neurons to the visual shapes reflected the associated numerical value in a behaviorally relevant way. In contrast, such association neurons were rarely found in the parietal lobe. These findings suggest a cardinal role of the prefrontal cortex in establishing semantic associations between signs and abstract categories, a cognitive precursor that may ultimately give rise to symbolic thinking in linguistic humans.

Humans and animals share an evolutionarily old quantity representation system that allows the estimation of set size or number of events [

In contrast, humans familiar with number symbols are able to grasp exact cardinalities and to execute even the most abstract calculations. Humans learn to use number symbols as mental tools during childhood. Prior to the utilization of signs as numerical symbols [

Several studies in humans point to the prefrontal cortex (PFC) and the intraparietal sulcus (IPS) as key structures for both non-symbolic [

To that aim, we trained monkeys to assign visual shapes to numerical categories and recorded from single cells in both candidate regions. We report that many neurons in the PFC encoded the learned numerical value of a visual shape. In contrast, such association neurons were rarely found in the parietal lobe. Overall, the results suggest that the PFC is the prime source for the linking of signs to numerical categories in monkeys and may serve as a neuronal precursor for number symbol encoding.

We trained two rhesus monkeys in a delayed match-to-sample protocol to discriminate small numerosities (one to four) in multiple-dot patterns (

(A) Dot protocol designed as delayed match-to-sample task. The monkeys were required to release a lever if sample and test displays contained the same number of items, or to keep holding it otherwise (probability = 0.5). The dots' position and size were varied between sample and test display and changed in each trial.

(B) Shape protocol designed as delayed association task. Task conditions were identical to those of the dot protocol, but during the sample period the numerical information was cued by an Arabic numeral. The numerals' size and position changed in each trial.

(C and D) Standard (first row) and control (second to fourth row) stimuli. (C) In the dot protocol, we controlled for non-numerical cues (dot circumference, area, configuration, and density). (D) Four different fonts (“Arial” in the standard protocol; “Times New Roman,” “Souvenir BT,” and “Lithograph Light” in the controls) were presented in the shape protocol.

We ensured that non-numerical parameters in the dot protocol could not be used by the monkeys to solve the task by varying and controlling low-level visual features. For each session, 100 different images per numerosity were generated with pseudo-randomly varied visual properties. Sample and test stimuli were never identical. All four quantities were presented in each session with one standard and one control condition. Different control conditions were applied day by day. Controls in the dot protocol included dot displays with constant circumference, linear configuration, and constant density across all presented quantities (see

Both monkeys learned reliably to associate numerical values with the visual shape of numerals. Average performance in the dot protocol (

(A and B) Dot protocol.

(C and D) Shape protocol.

The curves show how often the monkeys judged the first test and sample numerosity to be equal. The numerical value in the first test display is shown on the

We recorded 692 randomly selected neurons from the lateral PFC of the monkeys while they performed the tasks. Intermingled presentation of both protocols during each session allowed us to investigate individual neurons' responses to both dot and shape protocols. Many neurons were selective to numerical category and discharged strongest to specific (direct or associated) numerical values, irrespective of the protocol. Neuron 1, in

(A–E) Neuron 1 showed highest firing rates for numerical value two in the dot (A) and shape (B) protocols in the early sample phase. Top panels in (A) and (B) show dot raster histograms (each dot represents an action potential); bottom panels are the corresponding color-coded spike density histograms (averaged and smoothed with a 100-ms Gaussian kernel for illustrative purposes only). The first 500 ms indicates the fixation period. Black vertical lines mark sample onset (500 ms) and offset (1,300 ms). (C) Tuning functions in the sample period for the dot and shape protocols, calculated from the raw firing rates in a 400-ms latency shifted window. (D) Time course of original CCs (red) and chance CCs (SPs, blue). (E) Time course of discriminability between CCs and SP quantified as the AUROC. The horizontal bar above the

(F–J) Neuron 2 exhibiting four as preferred numerical value in the sample and delay period. Same layout as in (A–E); tuning functions were derived from the second ANOVA window of the sample period. (I and J) Neuron 2 associated numerical values in both protocols throughout the entire sample and delay period.

(K–O) Neuron 3 exhibited two as preferred numerical value in the delay period. Same layout as in (A–E); tuning functions were derived from the second ANOVA window of the delay period.

For a quantitative analysis of the neurons' selectivity to numerical values, we first calculated a two-way analysis of variance (ANOVA) (with factors numerical value [i.e., 1, 2, 3, 4] × stimulus condition [i.e., standard versus control],

Neurons that were ANOVA-selective in both protocols (especially those with the identical preferred numerical value in both protocols) constitute a potential neural substrate for long-term numerical associations. In addition to mere selectivity in the dot and shape protocols, however, neurons should have similar tuning functions for the (direct and associated) numerical values in both protocols. To test this hypothesis, and to investigate the time course of association, we performed a sliding cross-correlation analysis between each neuron's tuning functions in the shape and dot protocols for all 210 ANOVA-selective cells and derived the cross-correlation coefficients (CCs; see

(A) Diagram showing the temporal evolution of significant (as determined by a sliding ROC analysis; see

(B) Distribution of response-latency-corrected time points at which neurons started to associate numerical values.

(C) Population tuning curves. Normalized discharges and averaged tuning curves of all association neurons to the dot (gray) and shape (black) protocols. Data are plotted as a function of numerical distance from the preferred numerical value.

Interestingly, the tuning functions of association neurons showed a distance effect [

Is the association of numerical values by single PFC neurons really relevant for the monkeys' behavior? If association neurons constitute a neuronal correlate for the monkeys' ability to link signs with numerosities, the tuning correlations for both protocols should be weakened whenever the monkeys failed to associate visual shapes with their corresponding numerosities in error trials. To address this issue, we calculated the CCs of association neurons between correct trials in the dot protocol and error trials in the shape protocol. Because of the monkeys' low overall error rates, error trials were only available for a subset of numerical values (e.g., 2, 3, and 4) for many neurons. Only neurons recorded during errors to two or more numerical values were included into the error trial analysis. This criterion was fulfilled by 153 out of the 157 association neurons. As shown in

(A) Temporal profile of CCs during correct trials (upper panel) and error trials (bottom panel) for the same association neurons (running average rectangular filter, window size five data points). Neurons are sorted by time of maximal correlation.

(B) Time course of mean CCs across cells for correct and error trials (running average rectangular filter, window size five data points; shaded areas ± standard error of the mean).

During PFC recordings, we simultaneously recorded from 437 neurons in the fundus of the IPS (see

(A and B) Venn diagrams summarizing the results from the two-factor ANOVA and cross-correlation analysis in the PFC (A) and IPS (B). Data from the sample and delay period are combined. Numbers correspond to the numbers of neurons selective for each class.

(C) Lateral view of a monkey brain. Circles represent schematic locations of recording sites in the frontal and parietal lobe. AS, arcuate sulcus; CS, central sulcus; PS, principal sulcus; STS, superior temporal sulcus; LS, lateral sulcus.

(D) Frequency of association neurons. Proportions correspond to the added numbers of neuron classes (i.e., association neurons, ANOVA-selective neurons for both protocols, and ANOVA-selective neurons for shape or dot protocol; ***,

In contrast to the abundance of significantly tuned IPS neurons for the shape and dot protocols, only very few IPS neurons were selectively tuned to both protocols (

Correlation time course and correlation strength (as measured by the ROC values) were fundamentally different between PFC and IPS neurons (

The association strength (measured as the AUROC) of 157 association PFC (A) and eight association IPS (B) neurons. AUROC values were sorted independently for each time bin. Higher AUROC values indicate stronger association, i.e., more similar tuning functions to the dot and shape protocols. Black lines correspond (from right to left) to sample onset and offset. Time is aligned to the midpoint of the sliding windows.

We trained monkeys to associate quantitative categories with inherent meaning (i.e., numerosities) with a priori meaningless visual shapes. After this long-term learning process was completed, a large proportion of PFC neurons (23%) encoded plain numerical values, irrespective of whether they had been presented as a specific number of dots or as a visual shape. The activity of association neurons predicted the monkeys' judgment performance; if the monkeys failed to match the correct number of dots to the learned shapes, discharge patterns were drastically de-correlated. The population of these PFC cells represented the numerical association throughout the entire trial, providing crucial information to bridge the association over time. In contrast, only 2% of all recorded IPS neurons associated signs with numerosities. These findings suggest the PFC as the prime source in the mapping process of visual shapes to cardinalities.

Previous studies showed that neurons in the PFC encode learned associations between two purely sensory stimuli without intrinsic meaning (e.g., the association of a certain color with a specific sound, or pairs of pictures) [

The described association neurons and their response characteristics suggest such cells as neuronal correlates of semantic association. We observed that many neurons associated visual shapes with numerical values transiently, and not until the end of the delay period (

While quantity representations are spontaneously developed [

Even though numerosity-selective neurons in IPS are relatively abundant and encode numerical information earlier than PFC neurons [

Support for this assumption comes from recent functional magnetic resonance imaging studies with human children. In contrast to adults, preschoolers lacking proficiency with number symbols show elevated PFC activity when dealing with symbolic cardinalities [

During cultural evolution, humans invented number symbols as mental tools. Number symbols endow our species with an exact understanding of cardinality and the ability to execute the most complicated calculations. Given that the first ancient number symbols have been dated back to only a couple of thousand years ago [

We trained two monkeys to match either a set of dots with another set of dots (delayed match-to-sample task, or dot protocol; see

The stimuli for the dot protocol were randomly arranged black dots displayed on a gray background (diameter 6° of visual angle). For each session, 100 different images per numerosity were generated with pseudo-randomly varied visual features: the diameter of the dots ranged from 0.5 to 0.9° of visual angle, and their positions were restricted only by the border of the gray background circle and the fact that they were not allowed to overlap each other. Sample and test stimuli were never identical. All four quantities were presented in each session with one standard and one control condition. Controls in the dot protocol included dot displays with constant circumference (the summed circumference of the dots was constant, such that dot size decreased as dot number increased, as opposed to in the standard condition), linear configuration (i.e., all dots were linearly arranged), and constant density (i.e., constant mean distance between dots) across all presented quantities (see

Recordings were made from one left and one right hemisphere of the ventral convexity of the lateral PFC and in the fundus of the IPS of two rhesus monkeys (

To determine the neuronal response latencies, averaged spike density histograms were derived with a 1-ms resolution, smoothed by a sliding window with a kernel bin width of 10 ms for all sample stimuli. A 200-ms time window before stimulus onset was used as baseline. If five consecutive time bins after stimulus onset reached a value higher than the maximum of the baseline period, response latency was defined by the first of these time bins. A default latency of 100 ms was used if no value could be calculated.

Putative association neurons were preselected based on a two-factor ANOVA. To account for different temporal response phases, spike rates were tested in four adjacent, nonoverlapping time windows. The first window (400 ms) started at the beginning of the sample period and was shifted by the neurons' response latencies. The second window (400 ms) followed right after the first one and covered the rest of the sample period. The subsequent two windows (450 ms each) covered the first and second part of the delay period. Selectivity for numerical values was calculated based on these discharge rates separately for the dot and shape protocols using a two-way ANOVA with main factors “numerical value” (one to four) and “stimulus condition” (standard and control). Cells were considered to be numerosity-selective only if they showed a significant main effect to “numerosity” in one of the four analysis windows, but no significant “stimulus condition” or interaction effect.

To derive averaged numerosity-filter functions, the tuning functions of individual neurons were normalized by dividing all spike rates of the tuning functions by the maximum activity, thus setting the activity at the preferred numerical value to 100%. Pooling the resulting normalized tuning curves across the entire population of association cells resulted in averaged numerosity-filter functions (see

The correlation analysis aimed to extract tuning similarities of individual neurons to numerical values in the shape and dot protocols.

Out of the set of all trials (

The tuning functions _{shape} and _{dot} were composed of the spike rates of a given neuron obtained in the shape and dot protocols, respectively. Spike rates were obtained by averaging across the raw spike trains for 100 ms (see shaded windows in

The CCs provided a measure to quantify the similarity between tuning to the shape and dot protocols. The rationale behind this was the following. A neuron that was ANOVA-selective in both protocols constituted a potential neuronal association substrate between shapes and numerical values. In addition to the mere selectivity in the dot and shape protocols, however, neurons should have similar tuning functions for the (direct and associated) numerical values in both protocols. Neurons showing different tunings to the numerical values in the two protocols cannot be regarded as association neurons and should be excluded. The normalized cross-correlation is an appropriate method for filtering for these criteria. The cross-correlation takes a neuron's entire tuning functions _{shape}(_{dot}(_{shape} and _{dot} are subtracted from each spike rate, and has the advantage of normalization, which allows comparison across all cells. The normalized CC was calculated as follows:

The SP is supposed to represent the chance correlation level, irrespective of numerical values. For its calculation we abolished the relationship between neural activity and numerical value by randomly assigning each neural response a numerical value (

To determine whether a given cell in a given time bin responded more similarly to shape and dot stimuli than expected by chance, we performed a ROC analysis [

It needs to be emphasized that significant correlations are not caused by similar overall response modulations in the dot and shape protocols without being related to numerical value.

We calculated the CCs, the SP, and the area under the ROC curve (AUROC) in sliding windows (100-ms duration, shifted by 25 ms; see shaded area in

We evaluated the link between neuronal responses and behavior by analyzing the influence of erroneous judgments on the neuronal association. To that aim, we calculated CCs between the neuronal tuning functions based on error trials in the shape protocol and neuronal tuning functions obtained from correct trials in the dot protocol. Since the monkeys made very few errors, we often did not collect error trials for all tested numerical values. In these cases we restricted the analysis to the numerical values for which we obtained neuronal data during error trials (at least two numerical values). We compared these error-related CCs with CCs based on correct trials (again restricted to the same numerical values).

Was the proportion of neurons tuned to the same numerical value in both the dot and shape protocols higher than expected by chance? Since some neurons were tuned to numerosity in the dot protocol while others were encoding numerical information in the shape protocol, neurons encoding both formats may simply emerge by chance. We therefore compared the actual frequency of neurons with identical preferred numerical values in both protocols to chance occurrence based on probability calculations. To that aim, we considered the following three events: a cell is shape-selective, a cell is dot-selective, and a cell is selective for shapes and dots, formally written as
_{pred} was compared to the observed probability calculated as the percentage of cells with the same preferred quantity in both protocols in the pool of cells that were ANOVA-selective in both the dot and shape protocols. We calculated binomial tests with _{pred} as test proportion. The observed fractions in the PFC differed significantly from the test proportions during sample and delay period (_{pred} = 0.30, and _{pred} = 0.31, respectively). The fraction of neurons in the IPS with the same preferred numerical value in both protocols was very small but differed significantly from the predicted frequency during the sample and delay period (_{pred} = 0.32, and _{pred} = 0.25, respectively). The results are depicted as fractions of the entire sample of recorded neurons (both selective and unselective) in

This analysis represents a parallel argumentation line to the cross-correlation analysis. It shows on a stochastic basis that associations of visual signs and numerical values is not a coincidence.

(A–D) Performance of monkey 1 for standard (A) and control (B) trials in the dot protocol and standard (C) and control (D) trials in the shape protocol. Same layout as in

(E–H) Performance of monkey 2. Layout as in (A–D).

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(A and B) Dot raster histograms showing discharges to all trials of the dot (A) and shape (B) protocols. The numerical value is color coded.

(C) Subsets of eight randomly drawn trials per numerical value and protocol (1,000 repetitions with replacement). The numbers correspond to the numerical value shown during the sample period of the respective trial. From these trials, spike rates were calculated over 100-ms windows (indicated by the shaded time windows in [A] and [B]).

(D and E) The spike rates were arranged in a numerically ascending order for the dot (D) and shape (E) protocols, thereby forming tuning curves. With these resulting tuning curves for the dot and shape protocols, the CC was calculated.

(F–H) For the SP, we randomly shuffled discharges to the numerical values (F) and calculated the tuning functions with this shuffled dataset (G and H). With these resulting dummy tuning curves, the SP was calculated. The CCs and the SP were compared by ROC analysis.

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(A–H) Application flow of correlation analysis. Layout as in

(I–K) Comparison of results obtained by the two different shuffling methods for the three example neurons shown in

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(A and B) Dot raster and spike density histograms (100-ms smoothing Gaussian kernel) for the dot (A) and shape (B) protocols. Neuron showing similar response modulations in the dot and shape protocols, but not as a function of numerical value.

(C) The CC (red line) has values close to zero. The SP (blue line) resembles the CC.

(D) The area under the curve obtained by the ROC analysis fluctuates around 0.5. No significant correlation is detected. The black dashed lines depict the significance criterion (mean ± three standard deviations during fixation period); the gray dashed line represents the chance level.

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(A and B) Distributions of preferred numerical values one to four in PFC neurons during sample (A) and delay (B) period. Gray and black bars correspond to the dot and shape protocols, respectively.

(C and D) Distributions of preferred numerical values in IPS neurons during sample (C) and delay (D) period.

(E) Frequency of neurons with identical preferred numerical values in both protocols. The predicted frequencies were compared with the observed data shown in (A–D) (***,

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(A and B) Normalized responses averaged for neurons preferring the same sample quantity for the dot (A) and shape (B) protocols. Error bars represent the standard error of the mean.

(C) Distribution of rate differences between preferred and least preferred numerical value in the dot (gray) and shape (black) protocols for all associative neurons.

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We thank S. Jacob, O. Tudusciuc, and D. Vallentin for critical reading of the manuscript.

analysis of variance

area under the receiver operating characteristic curve

cross-correlation coefficient

intraparietal sulcus

prefrontal cortex

receiver operating characteristic

shuffle predictor