{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:57:40Z","timestamp":1760245060031,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,11,24]],"date-time":"2020-11-24T00:00:00Z","timestamp":1606176000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012190","name":"Ministry of Science and Higher Education of the Russian Federation","doi-asserted-by":"publisher","award":["075-15-2020-808","0729-2020-0061"],"award-info":[{"award-number":["075-15-2020-808","0729-2020-0061"]}],"id":[{"id":"10.13039\/501100012190","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002261","name":"RFBR","doi-asserted-by":"publisher","award":["20-32-70081"],"award-info":[{"award-number":["20-32-70081"]}],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007155","name":"MRC","doi-asserted-by":"publisher","award":["MR\/R02524X\/1"],"award-info":[{"award-number":["MR\/R02524X\/1"]}],"id":[{"id":"10.13039\/501100007155","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Integrated information has been recently suggested as a possible measure to identify a necessary condition for a system to display conscious features. Recently, we have shown that astrocytes contribute to the generation of integrated information through the complex behavior of neuron\u2013astrocyte networks. Still, it remained unclear which underlying mechanisms governing the complex behavior of a neuron\u2013astrocyte network are essential to generating positive integrated information. This study presents an analytic consideration of this question based on exact and asymptotic expressions for integrated information in terms of exactly known probability distributions for a reduced mathematical model (discrete-time, discrete-state stochastic model) reflecting the main features of the \u201cspiking\u2013bursting\u201d dynamics of a neuron\u2013astrocyte network. The analysis was performed in terms of the empirical \u201cwhole minus sum\u201d version of integrated information in comparison to the \u201cdecoder based\u201d version. The \u201cwhole minus sum\u201d information may change sign, and an interpretation of this transition in terms of \u201cnet synergy\u201d is available in the literature. This motivated our particular interest in the sign of the \u201cwhole minus sum\u201d information in our analytical considerations. The behaviors of the \u201cwhole minus sum\u201d and \u201cdecoder based\u201d information measures are found to bear a lot of similarity\u2014they have mutual asymptotic convergence as time-uncorrelated activity increases, and the sign transition of the \u201cwhole minus sum\u201d information is associated with a rapid growth in the \u201cdecoder based\u201d information. The study aims at creating a theoretical framework for using the spiking\u2013bursting model as an analytically tractable reference point for applying integrated information concepts to systems exhibiting similar bursting behavior. The model can also be of interest as a new discrete-state test bench for different formulations of integrated information.<\/jats:p>","DOI":"10.3390\/e22121334","type":"journal-article","created":{"date-parts":[[2020,11,24]],"date-time":"2020-11-24T09:06:28Z","timestamp":1606208788000},"page":"1334","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Integrated Information in the Spiking\u2013Bursting Stochastic Model"],"prefix":"10.3390","volume":"22","author":[{"given":"Oleg","family":"Kanakov","sequence":"first","affiliation":[{"name":"Faculty of Radiophysics, Lobachevsky State University of Nizhny Novgorod, 603950 Nizhny Novgorod, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7687-3065","authenticated-orcid":false,"given":"Susanna","family":"Gordleeva","sequence":"additional","affiliation":[{"name":"Institute of Biology and Biomedicine, Lobachevsky State University of Nizhny Novgorod, 603950 Nizhny Novgorod, Russia"},{"name":"Center for Technologies in Robotics and Mechatronics Components, Innopolis University, 420500 Innopolis, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7540-1130","authenticated-orcid":false,"given":"Alexey","family":"Zaikin","sequence":"additional","affiliation":[{"name":"Institute of Information Technology, Mathematics and Mechanics, Lobachevsky State University of Nizhny Novgorod, 603950 Nizhny Novgorod, Russia"},{"name":"Institute for Women\u2019s Health and Department of Mathematics, University College London, London WC1E 6BT, UK"},{"name":"Centre for Analysis of Complex Systems, Sechenov University, 119991 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Tononi, G. 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