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In this work, we analyze the responses resulting from modulation of a localized input in each of three classes of model neural networks that have been recognized in the literature for their capacity to produce robust three-phase rhythms: coupled fast-slow oscillators, near-heteroclinic oscillators, and threshold-linear networks. Triphasic rhythms, in which each phase consists of a prolonged activation of a corresponding subgroup of neurons followed by a fast transition to another phase, represent a fundamental activity pattern observed across a range of central pattern generators underlying behaviors critical to survival, including respiration, locomotion, and feeding. To perform our analysis, we extend the recently developed local timing response curve (lTRC), which allows us to characterize the timing effects due to perturbations, and we complement our lTRC approach with model-specific dynamical systems analysis. Interestingly, we observe disparate effects of similar perturbations across distinct model classes. Thus, this work provides an analytical framework for studying control of oscillations in nonlinear dynamical systems and may help guide model selection in future efforts to study systems exhibiting triphasic rhythmic activity.<\/jats:p>","DOI":"10.1162\/neco_a_01586","type":"journal-article","created":{"date-parts":[[2023,4,10]],"date-time":"2023-04-10T21:03:00Z","timestamp":1681160580000},"page":"1028-1085","update-policy":"https:\/\/doi.org\/10.1162\/mitpressjournals.corrections.policy","source":"Crossref","is-referenced-by-count":4,"title":["Sensitivity to Control Signals in Triphasic Rhythmic Neural Systems: A Comparative Mechanistic Analysis via Infinitesimal Local Timing Response Curves"],"prefix":"10.1162","volume":"35","author":[{"given":"Zhuojun","family":"Yu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, OH 44106, U.S.A. zhuojun.yu@case.edu"}]},{"given":"Jonathan E.","family":"Rubin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. jonrubin@pitt.edu"}]},{"given":"Peter J.","family":"Thomas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Applied Mathematics, and Statistics, Department of Biology, and Department of Electrical, Control and Systems Engineering, Case Western Reserve University, Cleveland, OH 44106, U.S.A. pjthomas@case.edu"}]}],"member":"281","published-online":{"date-parts":[[2023,5,12]]},"reference":[{"issue":"4","key":"2023050921091554700_B1","doi-asserted-by":"publisher","first-page":"1195","DOI":"10.1142\/S0218127404009806","article-title":"Heteroclinic contours in neural ensembles and the winnerless competition principle","volume":"14","author":"Afraimovich","year":"2004","journal-title":"International Journal of Bifurcation and Chaos"},{"issue":"1","key":"2023050921091554700_B2","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1063\/1.1539951","article-title":"Noisy heteroclinic networks","volume":"13","author":"Armbruster","year":"2003","journal-title":"Chaos: An Interdisciplinary Journal of Nonlinear Science"},{"issue":"1","key":"2023050921091554700_B3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00440-010-0264-0","article-title":"Noisy heteroclinic networks","volume":"150","author":"Bakhtin","year":"2011","journal-title":"Probability Theory and Related Fields"},{"issue":"2","key":"2023050921091554700_B4","doi-asserted-by":"publisher","DOI":"10.1007\/s00285-022-01859-4","article-title":"Heteroclinic cycling and extinction in May-Leonard models with demographic stochasticity","volume":"86","author":"Barendregt","year":"2023","journal-title":"Journal of Mathematical Biology"},{"key":"2023050921091554700_B5","doi-asserted-by":"crossref","DOI":"10.1017\/9780511734809","volume-title":"Control theory for physicists","author":"Bechhoefer","year":"2021"},{"key":"2023050921091554700_B6","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1016\/j.mbs.2016.07.003","article-title":"Multi-timescale systems and fast-slow analysis","volume":"287","author":"Bertram","year":"2017","journal-title":"Mathematical Biosciences"},{"key":"2023050921091554700_B7","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611973884","volume-title":"Finite dimensional linear systems","author":"Brockett","year":"2015"},{"issue":"4","key":"2023050921091554700_B8","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1162\/089976604322860668","article-title":"On the phase reduction and response dynamics of neural oscillator populations","volume":"16","author":"Brown","year":"2004","journal-title":"Neural Computation"},{"issue":"1","key":"2023050921091554700_B9","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.brainresrev.2007.06.028","article-title":"Organizing network action for locomotion: Insights from studying insect walking","volume":"57","author":"B\u00fcschges","year":"2008","journal-title":"Brain Research Reviews"},{"issue":"1","key":"2023050921091554700_B10","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1152\/jn.1999.82.1.382","article-title":"Models of respiratory rhythm generation in the pre-Botzinger complex. 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