{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T23:52:53Z","timestamp":1784245973475,"version":"3.55.0"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,22]],"date-time":"2020-12-22T00:00:00Z","timestamp":1608595200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100011730","name":"Templeton World Charity Foundation","doi-asserted-by":"publisher","award":["TWCF0196"],"award-info":[{"award-number":["TWCF0196"]}],"id":[{"id":"10.13039\/501100011730","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Tiny Blue Dot Foundation","award":["UW 133AAG3451"],"award-info":[{"award-number":["UW 133AAG3451"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Integrated information theory (IIT) provides a mathematical framework to characterize the cause-effect structure of a physical system and its amount of integrated information (\u03a6). An accompanying Python software package (\u201cPyPhi\u201d) was recently introduced to implement this framework for the causal analysis of discrete dynamical systems of binary elements. Here, we present an update to PyPhi that extends its applicability to systems constituted of discrete, but multi-valued elements. This allows us to analyze and compare general causal properties of random networks made up of binary, ternary, quaternary, and mixed nodes. Moreover, we apply the developed tools for causal analysis to a simple non-binary regulatory network model (p53-Mdm2) and discuss commonly used binarization methods in light of their capacity to preserve the causal structure of the original system with multi-valued elements.<\/jats:p>","DOI":"10.3390\/e23010006","type":"journal-article","created":{"date-parts":[[2020,12,22]],"date-time":"2020-12-22T12:42:28Z","timestamp":1608640948000},"page":"6","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Computing Integrated Information (\u03a6) in Discrete Dynamical Systems with Multi-Valued Elements"],"prefix":"10.3390","volume":"23","author":[{"given":"Juan D.","family":"Gomez","sequence":"first","affiliation":[{"name":"Department of Psychiatry, Wisconsin Institute for Sleep and Consciousness, University of Wisconsin-Madison, Madison, WI 53719, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9737-6260","authenticated-orcid":false,"given":"William G. P.","family":"Mayner","sequence":"additional","affiliation":[{"name":"Department of Psychiatry, Wisconsin Institute for Sleep and Consciousness, University of Wisconsin-Madison, Madison, WI 53719, USA"},{"name":"Neuroscience Training Program, University of Wisconsin-Madison, Madison, WI 53719, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maggie","family":"Beheler-Amass","sequence":"additional","affiliation":[{"name":"Department of Psychiatry, Wisconsin Institute for Sleep and Consciousness, University of Wisconsin-Madison, Madison, WI 53719, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Giulio","family":"Tononi","sequence":"additional","affiliation":[{"name":"Department of Psychiatry, Wisconsin Institute for Sleep and Consciousness, University of Wisconsin-Madison, Madison, WI 53719, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1000-9917","authenticated-orcid":false,"given":"Larissa","family":"Albantakis","sequence":"additional","affiliation":[{"name":"Department of Psychiatry, Wisconsin Institute for Sleep and Consciousness, University of Wisconsin-Madison, Madison, WI 53719, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,22]]},"reference":[{"key":"ref_1","unstructured":"Thomas, R., and D\u2019Ari, R. 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