{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T03:04:28Z","timestamp":1784862268422,"version":"3.55.0"},"update-to":[{"DOI":"10.1371\/journal.pcbi.1013629","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T00:00:00Z","timestamp":1762992000000}}],"reference-count":92,"publisher":"Public Library of Science (PLoS)","issue":"11","license":[{"start":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T00:00:00Z","timestamp":1762300800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100018260","name":"Tiny Blue Dot Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100018260","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010269","name":"Wellcome Trust","doi-asserted-by":"publisher","award":["210920\/Z\/18\/Z"],"award-info":[{"award-number":["210920\/Z\/18\/Z"]}],"id":[{"id":"10.13039\/100010269","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000275","name":"Leverhulme Trust","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000275","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.ploscompbiol.org"],"crossmark-restriction":false},"short-container-title":["PLoS Comput Biol"],"abstract":"<jats:p>A key feature of information theory is its universality, as it can be applied to study a broad variety of complex systems. However, many information-theoretic measures can vary significantly even across systems with similar properties, making normalisation techniques essential for allowing meaningful comparisons across datasets. Inspired by the framework of Partial Information Decomposition (PID), here we introduce Null Models for Information Theory (NuMIT), a null model-based non-linear normalisation procedure which improves upon standard entropy-based normalisation approaches and overcomes their limitations. We provide practical implementations of the technique for systems with different statistics, and showcase the method on synthetic models and on human neuroimaging data. Our results demonstrate that NuMIT provides a robust and reliable tool to characterise complex systems of interest, allowing cross-dataset comparisons and providing a meaningful significance test for PID analyses.<\/jats:p>","DOI":"10.1371\/journal.pcbi.1013629","type":"journal-article","created":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T18:43:36Z","timestamp":1762368216000},"page":"e1013629","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":4,"title":["Null models for comparing information decomposition across complex systems"],"prefix":"10.1371","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2247-8710","authenticated-orcid":true,"given":"Alberto","family":"Liardi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fernando E.","family":"Rosas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Robin L.","family":"Carhart-Harris","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"George","family":"Blackburne","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel","family":"Bor","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pedro A. 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Nonnegative decomposition of multivariate information. arXiv preprint 2010. https:\/\/arxiv.org\/abs\/1004.2515"},{"issue":"1","key":"pcbi.1013629.ref007","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/cogs.12142","article-title":"Information processing and dynamics in minimally cognitive agents","volume":"39","author":"RD Beer","year":"2015","journal-title":"Cogn Sci."},{"key":"pcbi.1013629.ref008","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.bandc.2015.09.004","article-title":"Partial information decomposition as a unified approach to the specification of neural goal functions","volume":"112","author":"M Wibral","year":"2017","journal-title":"Brain Cogn."},{"issue":"7","key":"pcbi.1013629.ref009","doi-asserted-by":"crossref","first-page":"318","DOI":"10.3390\/e19070318","article-title":"Measuring multivariate redundant information with pointwise common change in surprisal","volume":"19","author":"R Ince","year":"2017","journal-title":"Entropy."},{"issue":"9","key":"pcbi.1013629.ref010","doi-asserted-by":"crossref","first-page":"474","DOI":"10.3390\/e19090474","article-title":"The partial information decomposition of generative neural network models","volume":"19","author":"T Tax","year":"2017","journal-title":"Entropy."},{"key":"pcbi.1013629.ref011","doi-asserted-by":"crossref","unstructured":"Proca AM, Rosas FE, Luppi AI, Bor D, Crosby M, Mediano PA. Synergistic information supports modality integration and flexible learning in neural networks solving multiple tasks. arXiv preprint 2022. https:\/\/arxiv.org\/abs\/2210.02996","DOI":"10.32470\/CCN.2023.1113-0"},{"issue":"3","key":"pcbi.1013629.ref012","article-title":"Gene regulatory network inference from single-cell data using multivariate information measures","volume":"5","author":"TE Chan","year":"2017","journal-title":"Cell Syst."},{"issue":"1","key":"pcbi.1013629.ref013","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1186\/s12859-018-2217-z","article-title":"Evaluating methods of inferring gene regulatory networks highlights their lack of performance for single cell gene expression data","volume":"19","author":"S Chen","year":"2018","journal-title":"BMC Bioinformatics."},{"key":"pcbi.1013629.ref014","doi-asserted-by":"crossref","unstructured":"Finn C, Lizier JT. 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