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However, this equivalence between the deterministic notions of phase is broken in the presence of noise. Different notions of phase reduction for a stochastic oscillator can be defined either in terms of mean\u2013return-time sections or as the argument of the slowest decaying complex eigenfunction of the Kolmogorov backwards operator. Although both notions of phase enjoy a solid theoretical foundation, their relationship remains unexplored. Here, we quantitatively compare both notions of stochastic phase. We derive an expression relating both notions of phase and use it to discuss differences (and similarities) between both definitions of stochastic phase for (i) a spiral sink motivated by stochastic models for electroencephalograms, (ii) noisy limit-cycle systems-neuroscience models, and (iii) a stochastic heteroclinic oscillator inspired by a simple motor-control system.<\/jats:p>","DOI":"10.1007\/s00422-022-00929-6","type":"journal-article","created":{"date-parts":[[2022,3,23]],"date-time":"2022-03-23T13:13:04Z","timestamp":1648041184000},"page":"219-234","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Quantitative comparison of the mean\u2013return-time phase and the stochastic asymptotic phase for noisy oscillators"],"prefix":"10.1007","volume":"116","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3258-6275","authenticated-orcid":false,"given":"Alberto","family":"P\u00e9rez-Cervera","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5617-127X","authenticated-orcid":false,"given":"Benjamin","family":"Lindner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7533-6770","authenticated-orcid":false,"given":"Peter J.","family":"Thomas","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,3,23]]},"reference":[{"issue":"5","key":"929_CR1","doi-asserted-by":"publisher","first-page":"763","DOI":"10.1038\/nn.3081","volume":"15","author":"T Akam","year":"2012","unstructured":"Akam T, Oren I, Mantoan L, Ferenczi E, Kullmann DM (2012) Oscillatory dynamics in the hippocampus support dentate gyrus\u2013CA3 coupling. 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