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. 2023 Jun 29;14:3843. doi: 10.1038/s41467-023-39548-5

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© The Author(s) 2023

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 9 — 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.