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Path integration in complex number space.

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1 match 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 1 match
  1. [1] § paragraph 7 ↔ generate_figure2.py, lines 1–11 · score 0.71 · cross species, PI model, angular errors, Computational cost, Bracket, simulation

Paper

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The authors' code

Python · 170 lines · 7.2 KB · no license · 1 match

  1. #!/usr/bin/env python3
  2. """
  3. Figure 2 — PNAS Brief Report. Craddock, Miossec & Bouchekioua.
  4. Panel A: Placeholder for Müller & Wehner 1988 Fig. 1
  5. Panel B: PI Model simulation (seed=9798, 100 steps, raw waypoints)
  6. Panel C: Angular error histogram with verified cross-species bracket
  7. Panel D: Computational cost comparison
  8. """
  9. import numpy as np
  10. import matplotlib.pyplot as plt
  11. from matplotlib.lines import Line2D
  13. def complex_mult(a_re, a_im, b_re, b_im):
  14. return (a_re * b_re - a_im * b_im, a_re * b_im + a_im * b_re)
  16. def run_outward(num_steps, rng):
  17. traj = [{"x": 0.0, "y": 0.0, "I_re": 1.0, "I_im": 0.0}]
  18. z1_re, z1_im = 1.0, 1.0
  19. for _ in range(num_steps):
  20. prev = traj[-1]
  21. if rng.random() < 0.5: i_n_re, i_n_im = 0.0, 1.0
  22. else: i_n_re, i_n_im = 0.0, -1.0
  23. newI_re, newI_im = complex_mult(prev["I_re"], prev["I_im"], i_n_re, i_n_im)
  24. if i_n_im == 1.0:
  25. zd_re = 0.5 + rng.random() * 0.5; zd_im = rng.random() * 0.5
  26. else:
  27. zd_re = rng.random() * 0.5; zd_im = 0.5 + rng.random() * 0.5
  28. Iz1_re, Iz1_im = complex_mult(newI_re, newI_im, z1_re, z1_im)
  29. traj.append({"x": prev["x"] + Iz1_im + zd_im, "y": prev["y"] + Iz1_re + zd_re,
  30. "I_re": newI_re, "I_im": newI_im})
  31. return traj
  33. def run_return(traj, rng):
  34. last = traj[-1]
  35. X_n, Y_n = last["x"], last["y"]
  36. path = []
  37. for _ in range(10000):
  38. if X_n + Y_n < 1.0: break
  39. prevX, prevY = X_n, Y_n
  40. epsilon = -0.5 + rng.random()
  41. denom = prevX + prevY
  42. if abs(denom) > 0.001:
  43. X_n = prevX - (prevX / denom) + epsilon
  44. Y_n = prevY - (prevY / denom) + epsilon
  45. else: break
  46. path.append((X_n, Y_n))
  47. path.append((0.0, 0.0))
  48. return path
  50. def compute_ae(traj, ret):
  51. xs = np.array([p["x"] for p in traj])
  52. ys = np.array([p["y"] for p in traj])
  53. ideal = np.arctan2(-ys[-1], -xs[-1])
  54. if len(ret) > 1:
  55. dx, dy = ret[0][0] - xs[-1], ret[0][1] - ys[-1]
  56. ae = np.degrees(np.arctan2(dy, dx) - ideal)
  57. ae = ((ae + 180) % 360) - 180
  58. return abs(ae)
  59. return np.nan
  61. # ═══ Panel B: seed=9798, 100 steps, tort=3.10 ═══
  62. rng = np.random.default_rng(9798)
  63. traj = run_outward(100, rng)
  64. ret = run_return(traj, rng)
  65. xs_out = np.array([p["x"] for p in traj])
  66. ys_out = np.array([p["y"] for p in traj])
  67. ret_xs = [xs_out[-1]] + [p[0] for p in ret]
  68. ret_ys = [ys_out[-1]] + [p[1] for p in ret]
  69. pl = np.sum(np.sqrt(np.diff(xs_out)**2 + np.diff(ys_out)**2))
  70. d = np.sqrt(xs_out[-1]**2 + ys_out[-1]**2)
  71. print(f"Panel B: seed=9798, steps=100, tort={pl/d:.2f}, disp={d:.1f}")
  73. # ═══ Panel C: n=1000 ═══
  74. rng_c = np.random.default_rng(42)
  75. all_ae = []
  76. for _ in range(1000):
  77. t = run_outward(20, rng_c); r = run_return(t, rng_c)
  78. ae = compute_ae(t, r)
  79. if not np.isnan(ae): all_ae.append(ae)
  80. med_ae = np.median(all_ae); mean_ae = np.mean(all_ae)
  81. print(f"Panel C: n={len(all_ae)}, median={med_ae:.1f}, mean={mean_ae:.1f}")
  83. # ═══ FIGURE: 2x2 layout ═══
  84. fig = plt.figure(figsize=(7.5, 9))
  85. gs = fig.add_gridspec(2, 2, height_ratios=[1.2, 1], hspace=0.35, wspace=0.35)
  87. # ── Panel A: Placeholder ──
  88. ax_a = fig.add_subplot(gs[0, 0])
  89. ax_a.text(0.5, 0.5, 'Müller & Wehner\n(1988) Fig. 1\n\n[To be inserted\nwith permission]',
  90. ha='center', va='center', fontsize=9, fontstyle='italic',
  91. color='#666666', transform=ax_a.transAxes)
  92. ax_a.set_xlim(0, 1); ax_a.set_ylim(0, 1)
  93. for spine in ax_a.spines.values(): spine.set_linestyle('--'); spine.set_color('#aaa')
  94. ax_a.set_xticks([]); ax_a.set_yticks([])
  95. ax_a.set_title('(A) C. fortis', fontsize=9, fontweight='bold',
  96. loc='left', fontstyle='italic', pad=6)
  98. # ── Panel B: Raw trajectory ──
  99. ax_b = fig.add_subplot(gs[0, 1])
  100. ax_b.plot(xs_out, ys_out, color='black', linewidth=0.4, solid_capstyle='round', zorder=2)
  101. ax_b.plot(ret_xs, ret_ys, color='black', linewidth=0.6,
  102. linestyle=(0, (2, 2)), zorder=3)
  103. ax_b.plot(0, 0, 'ks', markersize=4, zorder=5)
  104. ax_b.annotate('N', (0, 0), textcoords="offset points", xytext=(-12, -8),
  105. fontsize=11, fontweight='bold', fontstyle='italic')
  106. ax_b.plot(xs_out[-1], ys_out[-1], 'ko', markersize=4, zorder=5)
  107. ax_b.annotate('F', (xs_out[-1], ys_out[-1]), textcoords="offset points",
  108. xytext=(6, 4), fontsize=11, fontweight='bold', fontstyle='italic')
  109. ax_b.set_aspect('equal'); ax_b.axis('off')
  110. ax_b.set_title('(B) PI Model simulation', fontsize=9,
  111. fontweight='bold', loc='left', pad=6)
  112. leg_b = [Line2D([0],[0], color='k', lw=0.4, label='Outbound'),
  113. Line2D([0],[0], color='k', lw=0.6, ls='--', label='Homeward')]
  114. ax_b.legend(handles=leg_b, loc='upper left', fontsize=6.5, frameon=False)
  116. # ── Panel C: Angular error histogram ──
  117. ax_c = fig.add_subplot(gs[1, 0])
  119. ax_c.hist(all_ae, bins=40, color='#888888', edgecolor='white',
  120. linewidth=0.3, alpha=0.85, zorder=2)
  121. ax_c.axvline(med_ae, color='black', linewidth=1.2, linestyle='-', zorder=4)
  122. ax_c.axvline(mean_ae, color='black', linewidth=1.2, linestyle='--', zorder=4)
  124. # Single verified cross-species bracket
  125. y_pos = 310 * 0.68
  126. ax_c.plot([10, 25], [y_pos, y_pos], 'k-', linewidth=1.2, zorder=5)
  127. ax_c.plot([10, 10], [y_pos - 7, y_pos + 7], 'k-', linewidth=0.9, zorder=5)
  128. ax_c.plot([25, 25], [y_pos - 7, y_pos + 7], 'k-', linewidth=0.9, zorder=5)
  129. ax_c.text(26.5, y_pos, 'Cross-species PI error\n(Etienne & Jeffery 2004, Fig. 3)',
  130. fontsize=6, fontstyle='italic', va='center', ha='left')
  132. leg_c = [
  133. plt.Rectangle((0,0), 1, 1, fc='#888888', alpha=0.85, label='Model (n = 1,000)'),
  134. Line2D([0],[0], color='black', lw=1.2, ls='-', label=f'Median ({med_ae:.1f}\u00b0)'),
  135. Line2D([0],[0], color='black', lw=1.2, ls='--', label=f'Mean ({mean_ae:.1f}\u00b0)'),
  136. ]
  137. ax_c.legend(handles=leg_c, loc='upper right', fontsize=5.5, frameon=True,
  138. framealpha=0.95, edgecolor='#ccc')
  140. ax_c.set_xlabel('Absolute angular error (\u00b0)', fontsize=8)
  141. ax_c.set_ylabel('Count', fontsize=8)
  142. ax_c.set_xlim(0, 75)
  143. ax_c.set_ylim(0, 320)
  144. ax_c.tick_params(labelsize=7)
  145. ax_c.spines['top'].set_visible(False); ax_c.spines['right'].set_visible(False)
  146. ax_c.set_title('(C) Homing angular error', fontsize=9,
  147. fontweight='bold', loc='left', pad=6)
  149. # ── Panel D: Computational cost ──
  150. ax_d = fig.add_subplot(gs[1, 1])
  151. models = ['Complex\nnumber\n(this study)', 'Cartesian\nvector', 'Polar\ncoord.']
  152. arith_ops = [10, 5, 5]
  153. trig_cost = [0, 24, 48]
  154. x = np.arange(len(models))
  155. width = 0.30
  156. ax_d.bar(x - width/2, arith_ops, width, color='black', label='Arithmetic', zorder=2)
  157. ax_d.bar(x + width/2, trig_cost, width, color='#aaaaaa', label='Trigonometric', zorder=2)
  158. ax_d.set_xticks(x)
  159. ax_d.set_xticklabels(models, fontsize=6.5)
  160. ax_d.set_ylabel('Operations per step', fontsize=8)
  161. ax_d.tick_params(labelsize=7)
  162. ax_d.spines['top'].set_visible(False); ax_d.spines['right'].set_visible(False)
  163. ax_d.legend(fontsize=6, loc='upper left', framealpha=0.9, edgecolor='#ccc')
  164. ax_d.set_ylim(0, 55)
  165. ax_d.set_title('(D) Computational cost', fontsize=9,
  166. fontweight='bold', loc='left', pad=6)
  168. plt.savefig('figure2_pnas.png', dpi=600, bbox_inches='tight', facecolor='white')
  169. plt.savefig('figure2_pnas.tiff', dpi=600, bbox_inches='tight', facecolor='white')
  170. print("\nSaved: figure2_pnas.png / .tiff")

generate_figure2.py at commit a561537, no license · at the source

Overview

Authors: Paul Craddock1, Yannick Miossec1, Youcef Bouchekioua2
ORCID iDs: Paul Craddock, Youcef Bouchekioua
  1. Department of Psychology, University of Lille, Villeneuve d’Ascq 59653, France
  2. Division of Engineering in Medicine, Department of Medicine, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA 02115
Institutions: Université de Lille (France); Brigham and Women's Hospital (United States); Harvard University (United States)
Journal: Proceedings of the National Academy of Sciences of the United States of America, volume 123, issue 22, article e2603690123
Dates: received 31 January 2026; accepted 29 April 2026; published online 29 May 2026; in print 2 June 2026
Type: Brief report · Language: English
License: CC BY-NC-ND
Identifiers: DOI 10.1073/pnas.2603690123 · PMID 42213748 · PMCID PMC13229285 · OpenAlex W7162805726
Open access: hybrid, a free copy (OpenAlex)
Status: code verified
Categories: other (organism)
Keywords: path integration, complex numbers, spatial navigation, head direction cells
MeSH: Spatial Navigation*, Animals, Ants, Brain Stem, Hippocampus, Models, Neurological, Neurons (* major topic)
Topic: Memory and Neural Mechanisms (Cognitive Neuroscience, Neuroscience), according to OpenAlex
Citations: not cited yet (Europe PMC); 15 references in the paper

Abstract

The abstract is not reproduced here: the paper's license (CC BY-NC-ND) does not allow it. Read it in the paper, at the publisher or on Europe PMC.

Repositories

Its files are read in the Code ↔ Paper reader above, with 1 match between paragraphs and lines of code.

YuYuB/Path_Integration_Simulation_Dr.Craddock_2026

License: none: the authors keep all their rights
State: the link answers, verified on 28 September 2026
Evidence: files inventoried
Commit: a56153753ad9b662b8f786e410d3f63d98f37499, 25 March 2026
Languages: Python (2)
Size: 3 files, 2 scripts
Software Heritage: not archived
Found in: “Data, Materials, and Software Availability”
Not found: README, license file, CITATION.cff, environment file, tests, continuous integration, documentation
Tools: Matplotlib (2 files), NumPy (2 files)
Availability: 1 check, the latest on 28 September 2026: the link answers
  • 28 September 2026: the link answers
2 files
At the source: github.com/YuYuB/Path_Integration_Simulation_Dr.Craddock_2026

Zenodo 19262464

License: CC-BY-4.0
State: the link answers, verified on 28 September 2026
Evidence: files inventoried
Size: 1 file
Software Heritage: not checked
Found in: “Data, Materials, and Software Availability”
Not found: README, license file, CITATION.cff, environment file, tests, continuous integration, documentation
Tools: Matplotlib (2 files), NumPy (2 files)
Availability: 1 check, the latest on 28 September 2026: the link answers (HTTP 200)
  • 28 September 2026: the link answers (HTTP 200)
2 files
At the source: doi.org/10.5281/zenodo.19262464

The paper's code and data availability statement is in the Data section.

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  • 4 scripts, each with its path and the digest of its content;
  • 1 match between paragraphs of the paper and lines of the code (method lexical-v1);
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Data

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Code and data availability statement

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Read it in the paper: doi.org/10.1073/pnas.2603690123.

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Version 1, 28 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 3 authors, 4 keywords, 7 MeSH terms, 12 references.

Cite

This paper

Craddock, P., Miossec, Y., & Bouchekioua, Y. (2026). Path integration in complex number space. Proceedings of the National Academy of Sciences of the United States of America, 123(22), e2603690123. https://doi.org/10.1073/pnas.2603690123

BibTeX

@article{craddock2026path,
author = {Craddock, Paul and Miossec, Yannick and Bouchekioua, Youcef},
title = {{Path integration in complex number space}},
journal = {Proceedings of the National Academy of Sciences of the United States of America},
year = {2026},
month = may,
volume = {123},
number = {22},
pages = {e2603690123},
publisher = {National Academy of Sciences},
issn = {0027-8424},
doi = {10.1073/pnas.2603690123},
url = {https://doi.org/10.1073/pnas.2603690123},
pmid = {42213748},
pmcid = {PMC13229285}
}

RIS

TY - JOUR
AU - Craddock, Paul
AU - Miossec, Yannick
AU - Bouchekioua, Youcef
TI - Path integration in complex number space
T2 - Proceedings of the National Academy of Sciences of the United States of America
J2 - Proc Natl Acad Sci U S A
PY - 2026
DA - 2026/05/29
VL - 123
IS - 22
SP - e2603690123
SN - 0027-8424
PB - National Academy of Sciences
DO - 10.1073/pnas.2603690123
UR - https://doi.org/10.1073/pnas.2603690123
LA - en
ER -

CSL-JSON

{
"id": "10.1073/pnas.2603690123",
"type": "article-journal",
"title": "Path integration in complex number space",
"container-title": "Proceedings of the National Academy of Sciences of the United States of America",
"author": [
{
"family": "Craddock",
"given": "Paul"
},
{
"family": "Miossec",
"given": "Yannick"
},
{
"family": "Bouchekioua",
"given": "Youcef"
}
],
"container-title-short": "Proc Natl Acad Sci U S A",
"volume": "123",
"issue": "22",
"page": "e2603690123",
"DOI": "10.1073/pnas.2603690123",
"PMID": "42213748",
"PMCID": "PMC13229285",
"ISSN": "0027-8424",
"publisher": "National Academy of Sciences",
"URL": "https://doi.org/10.1073/pnas.2603690123",
"language": "en",
"issued": {
"date-parts": [
[
2026,
5,
29
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]
}
}

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