Learning-induced reorganization of number neurons and emergence of numerical representations in a biologically inspired neural network

Abstract

Number sense, the ability to decipher quantity, forms the foundation for mathematical cognition. How number sense emerges with learning is, however, not known. Here we use a biologically-inspired neural architecture comprising cortical layers V1, V2, V3, and intraparietal sulcus (IPS) to investigate how neural representations change with numerosity training. Learning dramatically reorganized neuronal tuning properties at both the single unit and population levels, resulting in the emergence of sharply-tuned representations of numerosity in the IPS layer. Ablation analysis revealed that spontaneous number neurons observed prior to learning were not critical to formation of number representations post-learning. Crucially, multidimensional scaling of population responses revealed the emergence of absolute and relative magnitude representations of quantity, including mid-point anchoring. These learnt representations may underlie changes from logarithmic to cyclic and linear mental number lines that are characteristic of number sense development in humans. Our findings elucidate mechanisms by which learning builds novel representations supporting number sense.

Fig. 1. Architecture of number deep neural network (nDNN) adapted from the biologically-inspired CORnet-S.

nDNN consists of four layers that model hierarchy and recurrent circuit dynamics in areas V1, V2, V3, and IPS of the dorsal visual processing stream. The architecture of nDNN is adapted from CORnet-S, a biologically inspired network architecture for visual object categorization. The nDNN is trained to map non-symbolic representation of numbers to their symbolic representation. The nDNN includes feedforward and recurrent (shown by looped arrows within a layer in the figure) connections.

Fig. 2. Overview of the key goals and analysis steps.

Goal 1 investigated how neurons reorganize with learning and determine whether numerosity training preserves the integrity of spontaneous number neurons (SPONs). Goal 2 examined how learning changes the properties of individual neurons along the information processing hierarchy from V1, V2, and V3 to IPS. We examined several neuronal properties, including neuronal tuning, stability, selectivity, and distance effects at a single unit level, how they change with numerosity training, and how they correspond to neuronal recordings in primate IPS. Goal 3 investigated how distributed population-level representations change with learning. Goal 4 examined latent structure of distributed population-level neural representations, the coding properties that emerge from these representations, and how these coding properties relate to number sense. Goal 5 identified neuronal tuning and distributed population-level representational features that predict network accuracy. In each goal we examined neural reorganization in each nDNN layer along the information processing hierarchy from V1, V2, V3 to IPS. We also performed control analyses using alternate training methods to check the robustness of the findings.

Fig. 3. Reorganization of nDNN with numerosity training.

(A–E, L: numerosity neurons identified based on being selectively sensitive to numerosity, but not condition; F–J, M: numerosity neurons identified based on being sensitive to numerosity, regardless of whether they are also sensitive to stimulus condition). A, F Testing accuracy of the pre-trained network and across 50 epochs of numerosity training: full network (blue); network with persistent spontaneous (P-SPON) neurons in IPS layer ablated (pink) does not suffer performance degradation; network with all neurons in IPS layer except P-SPON ablated (green) suffers significant degradation in performance. B, G Number sensitive neurons as a proportion of total number of neurons in each layer, pre- and post-training. C, H Reorganization of SPONs with numerosity training: P-SPONs (dark blue), drop number sensitivity (red), or switch numerosity (light blue). D, I Reorganization of non-SPONs with numerosity training: Proportion of non-SPONs that remain non-sensitive to any number (dark blue), and that change to being number-sensitive (red). E, J Proportion of number sensitive post training that are newly trained or switched numerosities (dark blue) versus those that are P-SPONs and retain their SPON numerosity (red). K Reorganization between numerosity neurons that are exclusively sensitive to numerosity (selective), those that are sensitive to both numerosity and stimulus condition (non-selective), and those that are not sensitive to numerosity. (a, b) show the distribution of these neurons in each layer pre and post training. (c) Shows how the selective numerosity neurons reorganize post-training, with switch indicating a switch to a different preferred numerosity. (d) Shows a similar reorganization plot for neurons that are non-selective numerosity neurons pre-training. (e) Shows a similar reorganization plot for neurons that were not numerosity sensitive pre-training. L, M Proportion of number-sensitive neurons in each layer and training epoch that are (a) P-SPONs, (b) New+Switch, and (c) neither (transient). The contribution of New+Switch increases as we move across higher layers and increases with training epoch. Source data are provided as a Source Data file.

Fig. 4. Changes in key neuronal properties of selective number sensitive neurons with numerosity training.

A Normalized tuning curves for the pre-trained network: The plots show the mean normalized activation values (by input stimuli (1 to 9, on the x-axis), grouped by neurons of each preferred numerosity (PN), for layers (a) V1, (b) V2, (c) V3, and (d) IPS. B Normalized tuning curves for the post-training network for layers (a) V1, (b) V2, (c) V3, and (d) IPS. C Tuning precision: Acuity of the tuning curves as measured by the weighted average precision of the best fitting Gaussian tuning curves for each numerosity. Training leads to improving precision across layers, but primarily in IPS. Each dot represents the tuning precision for a preferred numerosity from 1 to 9. D Stability: the rank correlation (Kendall’s tau) or preservation of relative rank order of numerosities across conditions. Values >0 indicate better than chance level agreement, and values close to 1 indicate almost perfect rank order preservation across conditions. Improvements in stability increase as we move from V1 to IPS. E Selectivity: the proportion of comparisons where neuronal responses are higher for the PN. Higher selectivity indicates higher consistency of preference for the PN. Selectivity improves post training only in the IPS layer. F Numerical distance effect (NDE) is calculated as the average slope of selectivity versus input distance for each neuron, and averaged over number sensitive neurons. Improvements in NDE are highest in the IPS layer. The circles represent median values, the thick bars show the IQR (50% CI) and the thin bars show the 95% CI. D–F The circles represent median values, the thick bars show the IQR (50% CI) and the thin bars show the 95% CI. G NDE in the IPS layer: (a) Average selectivity and (b) Activation difference are both shown as a function of the numerical distance between pairwise input stimuli. The distance effects increase sharply from pre-trained (blue) to post-training (red). Source data are provided as a Source Data file.

Fig. 5. Correspondence between selective numerosity tuning curves in layer IPS of nDNN model and IPS subdivision of parietal cortex in monkeys.

A Normalized tuning curve for numerosity in monkey IPS adapted from Viswanathan & Nieder. All rights reserved. © PNAS 2013. Error bars indicate SEM. B Normalized tuning curves in our nDNN model (Fig. 4A, B) averaged across all numerosities showing tuning curves up to a distance of 4 units from the preferred numerosity, compared to tuning curves in monkeys. Numerosity tuning curves in layer IPS of the nDNN were similar to the those reported in the IPS of monkeys. Layer IPS units showed high-levels of similarity only after training. V1, V2, and V3 units did not show similarity with neuronal recordings either prior to after training. The normalized tuning curve (black) for numerosity in monkey IPS adapted from Viswanathan and Nieder. All rights reserved. © PNAS 2013. Error bars indicate SEM. Source data are provided as a Source Data file.

Fig. 6. Neural tuning in nDNN as a function of log-ratio numerosity.

A (A) Behavioral tuning for number discrimination and (B) inferred neural tuning in the IPS, as a function of the log-ratio of input numerosities, for adults and children. Tuning curves demonstrate a U-shaped curve with slightly higher tapering (steeper rise away from $\log$-ratio 1) for adults compared to children. Adapted from Kersey & Cantlon. B Neuronal tuning based on selectivity as a function of $\log$-ratio of input numerosities being compared shows a similar sharp U-curve in the layer (d) IPS of our nDNN model, but not in the earlier layers (a) V1, (b) V2, and (c) V3 (blue: pre-trained; red: post-training). C Neuronal tuning based on difference in activation as a function of $\log$-ratio of input numerosities being compared shows a similar sharp U-shaped curve in layer (d) IPS of our nDNN model but not in the earlier layers (a) V1, (b) V2, and (c) V3 (blue: pre-trained; red: post-training). Source data are provided as a Source Data file.

Fig. 7. Neural Representation Similarity of distributed population-level response and relation to Numerical Distance Effect.

A, B Neural representation similarity (NRS) calculated based on pairwise similarity between the mean activation across neurons in each layer (a) V1, (b) V2, (c) V3, and (d) IPS, for each value of the input stimuli. This shows us how well differentiated each input stimuli are, compared to other input values, in terms of their neuronal representations, pre-training (A) and post-training (B). The influence of training in the whole layer level RSA can be seen strongly in IPS, to a progressively smaller extent in V3 and negligible in V2 and V1. C, D The NRS is condensed to map the dissimilarity (1- average NRS) averaged as a function of each unique value of difference between inputs, that is, directly measure the numerical distance effect between representations of numerosities at a distributed level. A robust representation should show a sharply increasing linear trend in the average similarity with increasing input difference. The condensed RSA as a function of input difference is calculated for (a) the whole layer, (b) P-SPONs based on selective numerosity neurons, (c) New+Switch based on selective numerosity neurons, (d) P-SPONs based on all numerosity neurons, and (e) New+Switch based on all numerosity neurons. Pre-training (C), these linear trends have a very small slope. This slope increases with numerosity training (D), with significantly larger increases as we move from lower to higher layers, especially in IPS. E, F The NRS is condensed to map the dissimilarity (1- average NRS) averaged as a function of each unique value of $\log$-ratio between inputs, that is, directly measure the ratio effect between representations of numerosities at a distributed level. This is shown for (a) the whole layer, (b) P-SPONs based on selective numerosity neurons, (c) New+Switch based on selective numerosity neurons, (d) P-SPONs based on all numerosity neurons, and (e) New+Switch based on all numerosity neurons. A robust representation should show a sharply increasing linear trend in the average similarity with increasing $\log$-ratio. Source data are provided as a Source Data file.

Fig. 8. Multidimensional scaling of population-level responses reveals latent two-dimensional representations of absolute and relative magnitude.

A, B Multidimensional scaling (MDS) reveals a low-dimensional representation of each input stimuli (1–9) in a two-dimensional space. The two-dimensional representations of each input at each epoch were obtained from multiple groups of neurons, using (a) the whole layer, (b) P-SPONs based on selective numerosity neurons, (c) New+Switch based on selective numerosity neurons, (d) P-SPONs based on all numerosity neurons, and (e) New+Switch based on all numerosity neurons, for A pre-trained and B post-training networks. The two-dimensional representations are color coded by layer (V1: light blue, V2: green, V3: red, IPS: dark blue). The MDS reveals the emergence of a clear two-dimensional arch structure in V3 and especially in IPS, post numerosity training. This structure is not present for the pre-trained network. This representation also shows that whilst whole layer representations depend on SPONs pre-training (compare Aa and Ad), they shift to newly trained and switched neurons post training (compare Ba and Be), as well as the fact that they depend not just on the selective numerosity neurons (b, c) but on all numerosity neurons including those sensitive to both numerosity and condition (d, e). C–G The two dimensions obtained from the MDS using (C) all neurons, (D) P-SPONs based on selective numerosity neurons, (E) New+Switch based on selective numerosity neurons, (F) P-SPONs based on all numerosity neurons, and (G) New+Switch based on all numerosity neurons, are evaluated for covariance with key input stimulus properties. These include the stimulus magnitude (perceptual property), and distance of the input from the mid-point of the stimuli space (cognitive property). Across these sets the pre-training covariances are low, but post-training covariance between (a) dimension 1 and magnitude, and between (d) dimension 2 and distance from mid-point of the stimuli space, increase post-training in the IPS layer, especially in (A) and (G). The covariance between (b) dimension 2 and magnitude, and (c) dimension 1 and distance from mid-point remain low post-training. Source data are provided as a Source Data file.

Fig. 9. Multidimensional scaling of population-level responses reveals magnitude and midpoint anchoring in layer IPS of the nDNN.

A The plots show each MDS dimension as a function of the input stimuli magnitude, pre (blue) and post (red) training. The first dimension encodes a unimodal monotonic representation of the input stimuli magnitude. B The plots show each MDS dimension as a function of the distance from mid-point of the stimuli space, pre (blue) and post (red) training. The second-dimension codes for the distance of the input from the mid-point, with the response profile showing that representation increases faster as the distance from mid-point increases. For both (a) and (b), the dimensions are shown for (a, f) the whole layer, (b, g) P-SPONs based on selective numerosity neurons, (c, h) New+Switch based on selective numerosity neurons, (d, i) P-SPONs based on all numerosity neurons, and (e, j) New+Switch based on all numerosity neurons, with a–e showing dimension 1, and f–j showing dimension 2. C For each of the two MDS representational dimensions in layer IPS, the distance between each pair of input values is calculated, and the average distance of each input from all other values is converted into a relative similarity measure. High relative similarity of an input value implies higher propensity to confuse the input with other input values, and thus influence the output variability and errors. Training reduces the similarity between inputs, but also changes the shape of the similarity curves. It reduces the average similarity of the end-points (dimension 1), and reduces the similarity of the mid-point of the stimuli space (dimension 2). D The MDS dimensions for each numerosity can be translated to measure the “distance” between consecutive numerosities and create a latent “number line” in each MDS dimension. This is normalized and shown in the plots, for the pre-trained and post-training MDS representations. The pre-trained number lines show a logarithmic shape. The post-training number lines have a near-linear profile in the first dimension, and cyclic profile with a mid-point anchor (reference point) in the second dimension. Source data are provided as a Source Data file.

Fig. 10. Control Analyses.

The key distributed properties including A NRS representation, B NRS based distance effect, C NRS based $\log$-ratio effect, D MDS representations, E covariance between MDS dimension 1 and magnitude, and F covariance between MDS dimension 2 and mid-point anchor. These properties are all replicated for the three control analyses conducted, and in A–F the subplots show (a) Main: primary analysis based on accuracy reaching 99%+ levels; (b) Epoch 1: control for limited single epoch training where accuracy increases to 77%, similar to human accuracy levels; (c) RMS prop; control for change in training method from Adam optimizer to RMS propagation; and (d) SGD: control for change in training method from Adam optimizer to stochastic gradient descent. Source data are provided as a Source Data file.