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The inherent capacity of neurons to learn order relations and support abstract reasoning.

Code ↔ Paper

4 matches between paragraphs of the paper and lines of its authors' code, computed by the harvester (lexical-v1). Click a colored paragraph or line to see its counterpart.

The 4 matches
  1. [1] § Methods › Theoretical analysis of the shape of 2D projections of the firing activity of soft rank selective neurons for items in an order, discussed in section “Explaining curved 2D projections of neu ↔ Figure S10 S11 S12.ipynb, lines 49–129 · score 0.64 · Gaussian tuning curves, rank selective neuron, fires, matrix
  2. [2] § Results › Explaining curved 2D projections of neural representations of linear orders in the human brain ↔ Figure S10 S11 S12.ipynb, lines 49–129 · score 0.61 · Gaussian tuning, rank selective neurons, S10, S12, firing, encode
  3. [3] § Results › Rapid reconfiguration of the internal model when separately learned orders are combined on the basis of new information ↔ Figure S7.ipynb, lines 639–725 · score 0.56 · boundary pair, train short, train long, lowest, seeds, phase
  4. [4] § Methods › Details to combining separately learned orders into one order, discussed in section “Rapid reconfiguration of the internal model when separately learned orders are combined on the basis of n ↔ Figure2+S2+S3.ipynb, lines 1–129 · score 0.53 · 0.2–1 %, neurons fire, sparsity, dimension

Paper

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The authors' code

Jupyter notebook · 329 lines · 10 KB · MIT · 2 matches

  1. # %%
  2. import numpy as np
  3. import pandas as pd
  4. from sklearn.decomposition import PCA
  5. import matplotlib.pyplot as plt
  8. def generate_data_adjusted(num_objects, num_dimensions, num_encoded_per_dim):
  9. data = np.zeros((num_objects, num_dimensions))
  10. for dim in range(num_dimensions):
  11. for rank in range(num_encoded_per_dim):
  12. obj = dim + 1 - rank
  13. if 1 <= obj <= num_objects:
  14. data[obj - 1, dim] = obj - 3.5
  16. return data
  18. # Example usage
  19. num_objects = 6
  21. num_encoded_per_dim = 6 # Adjust the number of objects to encode per dimension
  22. num_dimensions = num_objects+num_encoded_per_dim-1
  23. # Generate the data
  24. data = generate_data_adjusted(num_objects, num_dimensions, num_encoded_per_dim)
  25. # data = data[:,1:-1]
  27. # Perform PCA to reduce dimensions from 10 to 2
  28. pca = PCA(n_components=2)
  29. data_pca = pca.fit_transform(data)
  31. # Plot the PCA results in 2D
  32. plt.figure(figsize=(4,4))
  33. plot_data = np.zeros((2,len(data_pca)))
  34. for i, (x, y) in enumerate(data_pca):
  35. plt.scatter(x, y, s=300, color='white', edgecolor='tab:blue', linewidths=2,zorder=1) # Large white circles
  36. plt.text(x, y, f'{i + 1}', ha='center', va='center', fontsize=12, fontweight='bold', color='tab:blue', zorder=2) # Bold object index
  37. plot_data[0,i]=x
  38. plot_data[1,i]=y
  39. plt.plot(plot_data[0],plot_data[1],zorder=0,color='tab:blue', lw=2)
  40. # Set titles and labels with consistent font size
  41. # plt.title("2D Visualization of 12 Objects Using PCA", fontsize=12)
  42. plt.xlabel("PC 1", fontsize=12)
  43. plt.ylabel("PC 2", fontsize=12)
  44. plt.axis('equal')
  45. # plt.title()
  46. plt.show()
  49. # %%
  50. import numpy as np
  51. from sklearn.decomposition import PCA
  52. import matplotlib.pyplot as plt
  55. def generate_data_gaussian(num_objects, num_dimensions, sigma=1.0, amplitude=1.0, center_mode="shifted"):
  56. """
  57. Build an (num_objects x num_dimensions) response matrix where each "dimension"
  58. corresponds to a rank-selective neuron with a Gaussian tuning curve over object rank.
  60. - num_objects: number of items/ranks (1..num_objects)
  61. - num_dimensions: number of neurons/dimensions
  62. - sigma: fixed std dev for all Gaussians (same variance)
  63. - amplitude: peak firing rate scaling
  64. - center_mode:
  65. * "shifted": centers sweep over a slightly wider range than [1, num_objects]
  66. (useful if you previously had extra dims for edge effects)
  67. * "within": centers evenly spaced strictly within [1, num_objects]
  68. """
  69. ranks = np.arange(1, num_objects + 1)[:, None] # shape (N, 1)
  71. if center_mode == "within":
  72. centers = np.linspace(1, num_objects, num_dimensions)
  73. elif center_mode == "shifted":
  74. # Spread centers across the wider range that arises when using extra dimensions
  75. # similar spirit to your previous construction with num_dimensions = N + K - 1.
  76. extra = (num_dimensions - num_objects) / 2.0
  77. centers = np.linspace(1 - extra, num_objects + extra, num_dimensions)
  78. else:
  79. raise ValueError("center_mode must be 'shifted' or 'within'")
  81. centers = centers[None, :] # shape (1, M)
  83. # Gaussian tuning: exp(-(r - mu)^2 / (2*sigma^2))
  84. data = amplitude * np.exp(-0.5 * ((ranks - centers) / sigma) ** 2)
  86. # Optional: mean-center each neuron's activity across objects (often helps PCA look like manifolds)
  87. # data = data - data.mean(axis=0, keepdims=True)
  89. return data
  92. # ----------------------------
  93. # Example usage (your settings)
  94. # ----------------------------
  95. num_objects = 6
  96. num_encoded_per_dim = 6
  97. num_dimensions = num_objects + num_encoded_per_dim - 1
  99. # Generate Gaussian-tuned data
  100. sigma = 1.0 # fixed variance across neurons (variance = sigma^2)
  101. amplitude = 1.0
  102. data = generate_data_gaussian(
  103. num_objects=num_objects,
  104. num_dimensions=num_dimensions,
  105. sigma=sigma,
  106. amplitude=amplitude,
  107. center_mode="shifted" # try "within" if you don't want extra edge centers
  108. )
  110. # PCA to 2D
  111. pca = PCA(n_components=2)
  112. data_pca = pca.fit_transform(data)
  114. # Plot
  115. plt.figure(figsize=(4, 4))
  116. plot_data = np.zeros((2, len(data_pca)))
  118. for i, (x, y) in enumerate(data_pca):
  119. plt.scatter(x, y, s=300, color='white', edgecolor='tab:blue', linewidths=2, zorder=1)
  120. plt.text(x, y, f'{i + 1}', ha='center', va='center',
  121. fontsize=12, fontweight='bold', color='tab:blue', zorder=2)
  122. plot_data[0, i] = x
  123. plot_data[1, i] = y
  125. plt.plot(plot_data[0], plot_data[1], zorder=0, color='tab:blue', lw=2)
  126. plt.xlabel("PC 1", fontsize=12)
  127. plt.ylabel("PC 2", fontsize=12)
  128. plt.axis('equal')
  129. plt.show()
  132. # %%
  133. import numpy as np
  134. import matplotlib.pyplot as plt
  136. def generate_data_gaussian(num_objects, num_dimensions, sigma=1.0, amplitude=1.0, center_mode="shifted"):
  137. ranks = np.arange(1, num_objects + 1)[:, None] # (N,1)
  139. if center_mode == "within":
  140. centers = np.linspace(1, num_objects, num_dimensions)
  141. elif center_mode == "shifted":
  142. extra = (num_dimensions - num_objects) / 2.0
  143. centers = np.linspace(1 - extra, num_objects + extra, num_dimensions)
  144. else:
  145. raise ValueError("center_mode must be 'shifted' or 'within'")
  147. centers = centers[None, :] # (1,M)
  148. data = amplitude * np.exp(-0.5 * ((ranks - centers) / sigma) ** 2)
  149. return data, centers.flatten()
  151. # Settings (match your previous example)
  152. num_objects = 6
  153. num_encoded_per_dim = 6
  154. num_dimensions = num_objects + num_encoded_per_dim - 1
  155. sigma = 1.0
  156. amplitude = 1.0
  157. center_mode = "shifted"
  159. # For a continuous-looking plot, sample ranks densely
  160. ranks_dense = np.linspace(1, num_objects, 500)[:, None] # (R,1)
  162. # Create centers
  163. if center_mode == "within":
  164. centers = np.linspace(1, num_objects, num_dimensions)
  165. else:
  166. extra = (num_dimensions - num_objects) / 2.0
  167. centers = np.linspace(1 - extra, num_objects + extra, num_dimensions)
  169. # Compute dense tuning curves
  170. tuning_dense = amplitude * np.exp(-0.5 * ((ranks_dense - centers[None, :]) / sigma) ** 2)
  172. # Plot all tuning curves
  173. plt.figure(figsize=(6, 4))
  174. for j in range(num_dimensions):
  175. plt.plot(ranks_dense[:, 0], tuning_dense[:, j])
  177. plt.xlabel("Rank")
  178. plt.ylabel("Response")
  179. plt.title(f"Gaussian tuning curves (sigma={sigma}, M={num_dimensions}, mode={center_mode})")
  180. plt.xticks(range(1, num_objects + 1))
  181. plt.ylim(-0.05, 1.05)
  182. plt.show()
  185. # %%
  186. import numpy as np
  187. import matplotlib.pyplot as plt
  188. from sklearn.decomposition import PCA
  191. def make_centers_fixed_neurons(num_objects, num_neurons, spacing_scale=1.0):
  192. """
  193. spacing_scale > 1 : sparser
  194. spacing_scale < 1 : denser
  195. """
  196. base = np.linspace(1, num_objects, num_neurons)
  197. center = base.mean()
  198. return center + spacing_scale * (base - center)
  200. def gaussian_tuning(ranks, centers, sigma=1.0, amplitude=1.0):
  201. """
  202. ranks: (R,)
  203. centers: (M,)
  204. returns: (R, M)
  205. """
  206. ranks = ranks[:, None]
  207. centers = centers[None, :]
  208. return amplitude * np.exp(-0.5 * ((ranks - centers) / sigma) ** 2)
  211. def add_noise(tuning, noise_scale=0.0, rng=None):
  212. """
  213. Add i.i.d. Gaussian noise to tuning curves.
  214. """
  215. if noise_scale <= 0:
  216. return tuning
  217. if rng is None:
  218. rng = np.random.default_rng(1)
  219. noisy = tuning + rng.normal(0.0, noise_scale, size=tuning.shape)
  220. return np.clip(noisy, 0.0, None) # firing rates ≥ 0
  223. def plot_tuning_and_pca(
  224. num_objects=6,
  225. num_neurons=11,
  226. sigma=1.0,
  227. amplitude=1.0,
  228. noise_scale=0.0,
  229. center_mode="shifted",
  230. dense_points=600,
  231. seed=0,
  232. connect_points=True,
  233. mean_center_neurons=False,
  234. ):
  235. rng = np.random.default_rng(seed)
  237. # neuron preferred ranks
  238. centers = make_centers_fixed_neurons(num_objects, num_neurons, spacing_scale=1.0)
  240. # ---------- Left: continuous tuning curves ----------
  241. ranks_dense = np.linspace(1, num_objects, dense_points)
  242. tuning_dense = gaussian_tuning(ranks_dense, centers, sigma, amplitude)
  243. tuning_dense = add_noise(tuning_dense, noise_scale, rng)
  245. # ---------- Right: PCA on object representations ----------
  246. ranks_discrete = np.arange(1, num_objects + 1)
  247. data = gaussian_tuning(ranks_discrete, centers, sigma, amplitude)
  248. data = add_noise(data, noise_scale, rng)
  250. if mean_center_neurons:
  251. data = data - data.mean(axis=0, keepdims=True)
  253. pca = PCA(n_components=2)
  254. data_pca = pca.fit_transform(data)
  256. # ---------- Plot ----------
  257. fig, axes = plt.subplots(1, 2, figsize=(6, 2.8))
  259. # Left panel
  260. ax = axes[0]
  261. for j in range(num_neurons):
  262. ax.plot(ranks_dense, tuning_dense[:, j], lw=1)
  263. ax.set_title(f"Tuning curves (σ={sigma_norm}, n={noise_scale})")
  264. ax.set_xlabel("Rank")
  265. ax.set_ylabel("Response")
  266. ax.set_xticks(range(1, num_objects + 1))
  267. ax.set_ylim(-0.05 * amplitude, 1.3 * amplitude)
  269. # Right panel
  270. ax = axes[1]
  271. xs, ys = data_pca[:, 0], data_pca[:, 1]
  272. for i in range(num_objects):
  273. ax.scatter(xs[i], ys[i], s=300, facecolor="white",
  274. edgecolor="tab:blue", linewidth=2, zorder=2)
  275. ax.text(xs[i], ys[i], f"{i+1}",
  276. ha="center", va="center",
  277. fontsize=12, fontweight="bold",
  278. color="tab:blue", zorder=3)
  280. if connect_points:
  281. ax.plot(xs, ys, color="tab:blue", lw=2, zorder=1)
  283. ax.set_title("PCA of object representations")
  284. ax.set_xlabel("PC 1")
  285. ax.set_ylabel("PC 2")
  286. ax.axis("equal")
  287. # Get current limits
  288. x_min, x_max = ax.get_xlim()
  289. y_min, y_max = ax.get_ylim()
  291. # Enlarge by a fraction (e.g. 10%)
  292. pad = 0.1
  293. ax.set_xlim(x_min - pad * (x_max - x_min),
  294. x_max + pad * (x_max - x_min))
  295. ax.set_ylim(y_min - pad * (y_max - y_min),
  296. y_max + pad * (y_max - y_min))
  299. plt.tight_layout()
  300. plt.show()
  303. # --------------------
  304. # Example run
  305. # --------------------
  306. def sigma_from_norm(num_objects, sigma_norm, eps=1e-6):
  307. rank_range = max(num_objects - 1, 1) # N-1 (avoid 0 when N=1)
  308. return max(sigma_norm * rank_range, eps)
  310. num_objects= 6
  311. sigma_norm = 0.3 # in (0,1], fraction of full rank range
  312. sigma = sigma_from_norm(num_objects, sigma_norm)
  314. plot_tuning_and_pca(
  315. num_objects=num_objects,
  316. num_neurons=6, # <-- explicit, meaningful parameter
  317. sigma=sigma,
  318. noise_scale=0.0,
  319. center_mode="shifted",
  320. mean_center_neurons=False,
  321. seed=0
  322. )
  325. # %%
  326. 1/6*0.33
  328. # %%

Figure S10 S11 S12.ipynb at commit 6eef732, under MIT · at the source

Overview

Authors: Yukun Yang1, Wolfgang Maass1
ORCID iDs: Yukun Yang, Wolfgang Maass
  1. Institute of Machine Learning and Neural Computation, Graz University of Technology, Graz, Austria
Institutions: Graz University of Technology (Austria)
Journal: Nature communications, volume 17, issue 1, article 9436
Dates: received 22 July 2025; accepted 20 July 2026; published online 5 August 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.1038/s41467-026-76102-5 · PMID 42686686 · PMCID PMC13538631 · OpenAlex W7172506211
Open access: gold, a free copy (OpenAlex)
Status: code verified
Categories: computational modeling (no new data) (modality), human (organism), cognitive (subfield)
Methods: Smoothing, state filtering, decompositions, Single-unit activity, calcium imaging
Keywords: Network models, Learning algorithms, Electrical and electronic engineering, Decision
MeSH: Brain*, Learning*, Models, Neurological*, Neurons*, Animals, Cognition, Decision Making, Humans, Neuronal Plasticity (* major topic)
Topic: Neural and Behavioral Psychology Studies (Cognitive Neuroscience, Neuroscience), according to OpenAlex
Funding: Austrian Science Fund FWF (10.55776/COE12); National Science Foundation (NSF) (EFRI BRAID project 2318152)
Citations: not cited yet (Europe PMC); 54 references in the paper

Abstract

Brains extract relations between objects and concepts and integrate them into cognitive maps for decision-making. But it remains unclear how they achieve that. Here we present a rigorous theory showing that single neurons can already learn to extract ranks of items in a linear order with a simple local rule for synaptic plasticity. The resulting model explains human brain data on the emergence of cognitive maps from linear orders, accounts for the terminal item effect in transitive inference, and enables rapid reconfiguration of internal representations when new evidence appears. We also present a theoretical explanation for the surprising fact that 2D projections of neural representations of linear orders in the brain are curved rather than linear. Since the model requires only local synaptic plasticity in shallow networks, it is suited for relational learning and fast inference on low-energy edge devices. We demonstrate this on the neuromorphic chip Loihi 2.

Reproduced under the paper's license (CC BY), from the paper cited above.

Repositories

Its files are read in the Code ↔ Paper reader above, with 4 matches between paragraphs and lines of code.

superrrpotato/Relationship-learning

License: MIT
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Commit: 6eef732c9ea219209c78ed3f4dadd176afe7cda3, 16 June 2026
Languages: Jupyter (12), Python (1)
Size: 18 files, 13 scripts
Software Heritage: not archived
Found in: “Code availability”
Holds: README, license file, environment (requirements.txt), 12 notebooks
Not found: CITATION.cff, tests, continuous integration, documentation
Tools: NumPy (13 files), Matplotlib (12 files), scikit-learn (11 files), PyTorch (8 files), SciPy (4 files), pandas (2 files)
Availability: 1 check, the latest on 27 September 2026: the link answers
  • 27 September 2026: the link answers
15 files
At the source: github.com/superrrpotato/Relationship-learning

Zenodo 20729243

License: MIT
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Size: 1 file
Software Heritage: not checked
Found in: the references
Not found: README, license file, CITATION.cff, environment file, tests, continuous integration, documentation
Tools: NumPy (13 files), Matplotlib (12 files), scikit-learn (11 files), PyTorch (8 files), SciPy (4 files), pandas (2 files)
Availability: 1 check, the latest on 27 September 2026: the link answers (HTTP 200)
  • 27 September 2026: the link answers (HTTP 200)
15 files
At the source:

Code availability

Example code implementing the proposed learning rule, together with the code used to generate all figures, is publicly available on GitHub at (https://github.com/superrrpotato/Relationship-learning) and archived on Zenodo53.

Reproduced under the paper's license (CC BY), from the paper cited above.

Tracing map

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  • 2 repositories of the authors' code, each at its verified commit, with its license and how the link was found in the paper;
  • 26 scripts, each with its path and the digest of its content;
  • 4 matches between paragraphs of the paper and lines of the code (method lexical-v1);
  • neither the text of the paper nor the code itself.

Its JSON (tracing-map.json) is deposited on Zenodo with its DOI once the map is validated.

Data

No dataset and no data link were found in the paper.

Data Availability Statement

All data used in the experiments were synthetically generated. The code used to generate these data is publicly available on GitHub at (https://github.com/superrrpotato/Relationship-learning) and archived on Zenodo53.

Example code implementing the proposed learning rule, together with the code used to generate all figures, is publicly available on GitHub at (https://github.com/superrrpotato/Relationship-learning) and archived on Zenodo53.

Reproduced under the paper's license (CC BY), from the paper cited above.

Versions

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Version 1, 27 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 2 authors, 4 keywords, 9 MeSH terms, 2 funders, 42 references.

Cite

This paper

Yang, Y., & Maass, W. (2026). The inherent capacity of neurons to learn order relations and support abstract reasoning. Nature communications, 17(1), 9436. https://doi.org/10.1038/s41467-026-76102-5

BibTeX

@article{yang2026inherent,
author = {Yang, Yukun and Maass, Wolfgang},
title = {{The inherent capacity of neurons to learn order relations and support abstract reasoning}},
journal = {Nature communications},
year = {2026},
month = aug,
volume = {17},
number = {1},
pages = {9436},
publisher = {Nature Publishing Group},
issn = {2041-1723},
doi = {10.1038/s41467-026-76102-5},
url = {https://doi.org/10.1038/s41467-026-76102-5},
pmid = {42686686},
pmcid = {PMC13538631}
}

RIS

TY - JOUR
AU - Yang, Yukun
AU - Maass, Wolfgang
TI - The inherent capacity of neurons to learn order relations and support abstract reasoning
T2 - Nature communications
J2 - Nat Commun
PY - 2026
DA - 2026/08/05
VL - 17
IS - 1
SP - 9436
SN - 2041-1723
PB - Nature Publishing Group
DO - 10.1038/s41467-026-76102-5
UR - https://doi.org/10.1038/s41467-026-76102-5
LA - en
ER -

CSL-JSON

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"id": "10.1038/s41467-026-76102-5",
"type": "article-journal",
"title": "The inherent capacity of neurons to learn order relations and support abstract reasoning",
"container-title": "Nature communications",
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"family": "Yang",
"given": "Yukun"
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],
"container-title-short": "Nat Commun",
"volume": "17",
"issue": "1",
"page": "9436",
"DOI": "10.1038/s41467-026-76102-5",
"PMID": "42686686",
"PMCID": "PMC13538631",
"ISSN": "2041-1723",
"publisher": "Nature Publishing Group",
"URL": "https://doi.org/10.1038/s41467-026-76102-5",
"language": "en",
"issued": {
"date-parts": [
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2026,
8,
5
]
]
}
}

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