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Intrinsic Cause-Effect Power: The Tradeoff Between Differentiation and Specification.

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The authors' code

Jupyter notebook · 174 lines · 6.3 KB · no license

  1. # %% [markdown]
  2. # # fig-growth.ipynb
  4. # %%
  5. %load_ext autoreload
  6. %autoreload 2
  8. from pathlib import Path
  9. import pickle
  11. import matplotlib.pyplot as plt
  12. from joblib import Parallel, delayed
  13. import numpy as np
  14. import pyphi
  15. import pandas as pd
  17. from systems import fig6d
  19. # %%
  20. FIGURE_DIR = Path('figures')
  21. DATA_DIR = Path('data')
  23. C, E = pyphi.Direction.CAUSE, pyphi.Direction.EFFECT
  25. # %% [markdown]
  26. # ### Figure 6D Network
  28. # %%
  29. args = np.linspace(0.001, 8, num=32, endpoint=True)
  30. args.round(2)
  32. # %%
  33. def worker(k):
  34. network, subsystem = fig6d(k=k)
  35. return pyphi.new_big_phi.sia(subsystem)
  37. # %%
  38. print(f'{pyphi.config.REPERTOIRE_DISTANCE=}')
  39. print(f'{pyphi.config.REPERTOIRE_DISTANCE_SPECIFICATION=}')
  40. print(f'{pyphi.config.REPERTOIRE_DISTANCE_DIFFERENTIATION=}')
  42. # %%
  43. COMPUTE_NEW_DATA = False
  45. # %%
  46. if COMPUTE_NEW_DATA:
  47. results = Parallel(n_jobs=min(32, len(args)), verbose=100)(
  48. delayed(worker)(k) for k in args
  49. )
  51. # %%
  52. if COMPUTE_NEW_DATA:
  53. with open(DATA_DIR / 'fig6d_results.pkl', 'wb') as f:
  54. pickle.dump(results, f)
  56. # %%
  57. with open(DATA_DIR / 'fig6d_results.pkl', 'rb') as f:
  58. results = pickle.load(f)
  60. # %%
  61. def make_dataframe(results):
  62. return pd.DataFrame({
  63. 'k': args,
  64. 'phi': [result.phi for result in results],
  65. 'phi_spec': [result.phi.specification for result in results],
  66. 'phi_diff': [result.phi.differentiation for result in results],
  67. 'phi_cause': [result.cause.phi for result in results],
  68. 'phi_effect': [result.effect.phi for result in results],
  69. 'phi_cause_spec': [result.cause.phi.specification for result in results],
  70. 'phi_cause_diff': [result.cause.phi.differentiation for result in results],
  71. 'phi_effect_spec': [result.effect.phi.specification for result in results],
  72. 'phi_effect_diff': [result.effect.phi.differentiation for result in results],
  73. 'intrinsic_differentiation_cause': [result.intrinsic_differentiation[C] for result in results],
  74. 'intrinsic_differentiation_effect': [result.intrinsic_differentiation[E] for result in results],
  75. 'intrinsic_information_cause': [result.system_state.cause.intrinsic_information for result in results],
  76. 'intrinsic_information_effect': [result.system_state.effect.intrinsic_information for result in results],
  77. })
  79. # %%
  80. df = make_dataframe(results)
  81. df.to_csv(DATA_DIR / 'fig6d_results.csv', index=False)
  83. # %%
  84. df = pd.read_csv(DATA_DIR / 'fig6d_results.csv')
  86. # %%
  87. def make_plot(data):
  88. import matplotlib.pyplot as plt
  90. fig, axes = plt.subplots(2, 3, figsize=(18, 8), sharex=True)
  92. # intrinsic_differentiation_{cause,effect}
  93. axes[0, 0].plot(data['k'], data['intrinsic_differentiation_cause'], marker='o', linestyle='-', label='Intrinsic Diff Cause', alpha=0.8)
  94. axes[0, 0].plot(data['k'], data['intrinsic_differentiation_effect'], marker='s', linestyle='--', label='Intrinsic Diff Effect', alpha=0.8)
  95. axes[0, 0].set_title('intrinsic_differentiation_{cause,effect}')
  96. axes[0, 0].set_ylabel('Value')
  97. axes[0, 0].legend()
  98. axes[0, 0].grid(True)
  100. # intrinsic_information_{cause,effect}
  101. axes[0, 1].plot(data['k'], data['intrinsic_information_cause'], marker='^', linestyle='-', label='Intrinsic Info Cause', alpha=0.8)
  102. axes[0, 1].plot(data['k'], data['intrinsic_information_effect'], marker='v', linestyle='--', label='Intrinsic Info Effect', alpha=0.8)
  103. axes[0, 1].set_title('intrinsic_information_{cause,effect}')
  104. axes[0, 1].legend()
  105. axes[0, 1].grid(True)
  107. # phi, phi_spec, phi_diff
  108. axes[0, 2].plot(data['k'], data['phi'], marker='o', color='tab:blue', linestyle='-', label='phi', alpha=0.8)
  109. axes[0, 2].plot(data['k'], data['phi_spec'], marker='s', color='tab:orange', linestyle='--', label='phi_spec', alpha=0.8)
  110. axes[0, 2].plot(data['k'], data['phi_diff'], marker='^', color='tab:green', linestyle='-.', label='phi_diff', alpha=0.8)
  111. axes[0, 2].set_title('phi, phi_spec, phi_diff')
  112. axes[0, 2].set_ylabel('phi')
  113. axes[0, 2].legend()
  114. axes[0, 2].grid(True)
  116. # phi_{cause,effect}
  117. axes[1, 0].plot(data['k'], data['phi_cause'], marker='o', linestyle='-', label='phi_cause', alpha=0.8)
  118. axes[1, 0].plot(data['k'], data['phi_effect'], marker='s', linestyle='--', label='phi_effect', alpha=0.8)
  119. axes[1, 0].set_title('phi_{cause,effect}')
  120. axes[1, 0].set_xlabel('k')
  121. axes[1, 0].set_ylabel('phi')
  122. axes[1, 0].legend()
  123. axes[1, 0].grid(True)
  125. # phi_{cause,effect}_{spec,diff}
  126. axes[1, 1].plot(data['k'], data['phi_cause_spec'], marker='o', linestyle='-', label='cause_spec', alpha=0.8)
  127. axes[1, 1].plot(data['k'], data['phi_cause_diff'], marker='s', linestyle='--', label='cause_diff', alpha=0.8)
  128. axes[1, 1].plot(data['k'], data['phi_effect_spec'], marker='^', linestyle='-.', label='effect_spec', alpha=0.8)
  129. axes[1, 1].plot(data['k'], data['phi_effect_diff'], marker='v', linestyle=':', label='effect_diff', alpha=0.8)
  130. axes[1, 1].set_title('phi_{cause,effect}_{spec,diff}')
  131. axes[1, 1].set_xlabel('k')
  132. axes[1, 1].legend()
  133. axes[1, 1].grid(True)
  135. axes[1, 2].axis('off')
  137. fig.suptitle('Quantities vs inverse temperature k')
  138. plt.tight_layout(rect=[0, 0, 1, 0.97])
  139. return fig
  142. # %%
  143. fig = make_plot(df)
  145. # %%
  146. def make_plot(data):
  147. import matplotlib.pyplot as plt
  149. fig, axes = plt.subplots(1, 2, figsize=(9, 4), sharex=True, sharey=True)
  151. # First subplot: intrinsic specification and intrinsic differentiation
  152. axes[0].plot(data['k'], data['phi_diff'], marker='o', color='tab:orange', linestyle='-', label='intrinsic differentiation', alpha=1.0)
  153. axes[0].plot(data['k'], data['phi_spec'], marker='o', color='tab:blue', linestyle='-', label='intrinsic specification', alpha=1.0)
  154. axes[0].set_xlabel(r'$K$ (inverse temperature)', fontsize=16)
  155. axes[0].set_ylabel('ibits', fontsize=16)
  156. # axes[0].legend(fontsize=14, labelcolor='#555555')
  157. axes[0].grid(True)
  159. # Second subplot: phi only
  160. axes[1].plot(data['k'], data['phi'], marker='o', color='black', linestyle='-', label=r'$\varphi_s$', alpha=1.0)
  161. axes[1].set_xlabel(r'$K$ (inverse temperature)', fontsize=16)
  162. # axes[1].legend(fontsize=14)
  163. axes[1].grid(True)
  165. fig.legend(fontsize=14, labelcolor='#555555')
  167. for ax in axes:
  168. ax.tick_params(axis='both', labelsize=16)
  170. plt.tight_layout()
  171. return fig, axes
  173. fig, ax = make_plot(df)
  174. fig.savefig(FIGURE_DIR / "fig-growth__specialized-majority-values.svg")

fig-growth.ipynb at commit 483c23a, no license · at the source

Overview

Authors: William G. P. Mayner1, William Marshall2, Giulio Tononi1
ORCID iDs: William G. P. Mayner
  1. Department of Psychiatry, University of Wisconsin–Madison, Madison, WI 53719, USA
  2. Department of Mathematics & Statistics, Brock University, St. Catharines, ON L2S 3A1, Canada
Institutions: University of Wisconsin–Madison (United States); Brock University (Canada)
Journal: Entropy (Basel, Switzerland), volume 28, issue 4, article 410
Dates: received 28 December 2025; accepted 2 April 2026; published online 4 April 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.3390/e28040410 · PMID 42072535 · PMCID PMC13114617 · OpenAlex W4416373148
Open access: gold, a free copy (OpenAlex)
Preprint: arxiv:2510.03881
Status: code verified
Categories: cognitive (subfield)
Keywords: consciousness, integrated information, intrinsic, differentiation, intrinsic specification, complexity
Topic: Embodied and Extended Cognition (Cognitive Neuroscience, Neuroscience), according to OpenAlex
Funding: Department of Obstetrics and Gynecology, University of Wisconsin-Madison; Natural Sciences and Engineering Research Council of Canada (RGPIN-2019-05418)
Citations: not cited yet (Europe PMC); 21 references in the paper

Abstract

Integrated information theory (IIT) starts from the existence of consciousness and characterizes its essential properties: every experience is intrinsic, specific, unitary, definite, and structured. IIT then formulates existence and its essential properties operationally in terms of cause–effect power of a substrate of units. Here, we address IIT’s operational requirements for existence by considering that, to have cause–effect power, to have it intrinsically, and to have it specifically, substrate units in their actual state must both (i) ensure the intrinsic availability of a repertoire of cause–effect states, and (ii) increase the probability of a specific cause–effect state. We showed previously that requirement (ii) can be assessed by the intrinsic difference of a state’s probability from maximal differentiation. Here, we show that requirement (i) can be assessed by the intrinsic difference from maximal specification. These points and their consequences for integrated information are illustrated using simple systems of micro units. When applied to macro units and systems of macro units such as neural systems, a tradeoff between differentiation and specification is a necessary condition for intrinsic existence—and therefore, according to IIT, for consciousness.

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

Repository

Its files are read in the Code ↔ Paper reader above.

wmayner/intrinsic-information

License: none: the authors keep all their rights
State: the link answers, verified on 28 September 2026
Evidence: files inventoried
Commit: 483c23a3c9c0b2075ead17cd803bf7ec6ad45b6a, 27 December 2025
Languages: Jupyter (4), Python (3)
Size: 31 files, 7 scripts
Software Heritage: not archived
Found in: “Data Availability Statement”
Holds: README, environment (environment.yml), 4 notebooks
Not found: license file, CITATION.cff, tests, continuous integration, documentation
Tools: NumPy (7 files), Matplotlib (4 files), pandas (4 files), SciPy (2 files), seaborn (2 files)
Availability: 1 check, the latest on 28 September 2026: the link answers
  • 28 September 2026: the link answers
8 files
At the source: github.com/wmayner/intrinsic-information

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

Tracing map

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  • 1 repository of the authors' code, each at its verified commit, with its license and how the link was found in the paper;
  • 7 scripts, each with its path and the digest of its content;
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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

Code and data used in this work are available at https://github.com/wmayner/intrinsic-information (27 December 2025).

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

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Version 1, 28 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 3 authors, 6 keywords, 2 funders, 19 references.

Cite

This paper

Mayner, W. G. P., Marshall, W., & Tononi, G. (2026). Intrinsic Cause-Effect Power: The Tradeoff Between Differentiation and Specification. Entropy (Basel, Switzerland), 28(4), 410. https://doi.org/10.3390/e28040410

BibTeX

@article{mayner2026intrinsic,
author = {Mayner, William G. P. and Marshall, William and Tononi, Giulio},
title = {{Intrinsic Cause-Effect Power: The Tradeoff Between Differentiation and Specification}},
journal = {Entropy (Basel, Switzerland)},
year = {2026},
month = apr,
volume = {28},
number = {4},
pages = {410},
publisher = {Multidisciplinary Digital Publishing Institute (MDPI)},
issn = {1099-4300},
doi = {10.3390/e28040410},
url = {https://doi.org/10.3390/e28040410},
pmid = {42072535},
pmcid = {PMC13114617}
}

RIS

TY - JOUR
AU - Mayner, William G. P.
AU - Marshall, William
AU - Tononi, Giulio
TI - Intrinsic Cause-Effect Power: The Tradeoff Between Differentiation and Specification
T2 - Entropy (Basel, Switzerland)
J2 - Entropy (Basel)
PY - 2026
DA - 2026/04/04
VL - 28
IS - 4
SP - 410
SN - 1099-4300
PB - Multidisciplinary Digital Publishing Institute (MDPI)
DO - 10.3390/e28040410
UR - https://doi.org/10.3390/e28040410
LA - en
ER -

CSL-JSON

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"author": [
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"family": "Mayner",
"given": "William G. P."
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"page": "410",
"DOI": "10.3390/e28040410",
"PMID": "42072535",
"PMCID": "PMC13114617",
"ISSN": "1099-4300",
"publisher": "Multidisciplinary Digital Publishing Institute (MDPI)",
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