{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T15:01:45Z","timestamp":1788879705526,"version":"build-2803163510"},"reference-count":61,"publisher":"Public Library of Science (PLoS)","issue":"4","license":[{"start":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T00:00:00Z","timestamp":1777420800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Western Michigan University College of Arts and Sciences","award":["Interdisciplinary Research Initiative Award"],"award-info":[{"award-number":["Interdisciplinary Research Initiative Award"]}]},{"DOI":"10.13039\/100005746","name":"Michigan Space Grant Consortium","doi-asserted-by":"publisher","award":["Fellowships"],"award-info":[{"award-number":["Fellowships"]}],"id":[{"id":"10.13039\/100005746","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100005746","name":"Michigan Space Grant Consortium","doi-asserted-by":"publisher","award":["Fellowships"],"award-info":[{"award-number":["Fellowships"]}],"id":[{"id":"10.13039\/100005746","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007159","name":"Western Michigan University","doi-asserted-by":"publisher","award":["Graduate Student Research Grant"],"award-info":[{"award-number":["Graduate Student Research Grant"]}],"id":[{"id":"10.13039\/100007159","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Western Michigan University College of Arts and Sciences","award":["Interdisciplinary Research Initiative Award"],"award-info":[{"award-number":["Interdisciplinary Research Initiative Award"]}]},{"name":"Western Michigan University College of Arts and Sciences","award":["Interdisciplinary Research Initiative Award"],"award-info":[{"award-number":["Interdisciplinary Research Initiative Award"]}]},{"name":"Western Michigan University College of Arts and Sciences","award":["Interdisciplinary Research Initiative Award"],"award-info":[{"award-number":["Interdisciplinary Research Initiative Award"]}]},{"name":"National Defense Science and Engineering Graduate Fellowship Program"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"<jats:p>\n                    We describe an application of optimal control theory to\n                    <jats:italic>in vivo<\/jats:italic>\n                    intracellular stimulation of pressure-sensitive mechanosensory (P-cell) neurons of the leech\n                    <jats:italic>Hirudo verbana<\/jats:italic>\n                    . The control objective seeks optimal stimuli that balance the minimization of stimulation current energy and the error of tracking an action potential evoked by a high-energy rectangular pulse. Tracking a known neuron response mitigates controllability and numerical solution issues and avoids the need to constrain stimulation currents. The reduced second-order neuron conductance model used in optimization was not fit to the target P-cell, but parameters were instead selected based on an assumed saddle-node on invariant circle bifurcation. Optimal stimuli that provided a range of tracking performance and energy minimization were computed prior to experimental work. Remarkably, simulated and biological neurons show the same tracking performance decrease at higher levels of stimulus current energy reduction. Numerical analysis of neuron model responses to optimal current perturbations revealed a phase space separatrix between regions with and without action potential trajectories, demonstrating high sensitivity to optimal current shape in a high-energy reduction case and verifying local optimality. This proof-of-concept study demonstrates that a control strategy based on reproducing action potential shapes can compute reduced-energy current stimulation waveforms that are effective in biological neurons. This effectiveness may extend to other biological neurons since the optimization method was applied to the same reduced-order model for other bifurcations and to a six-dimensional neuron. This method may be useful in research and future clinical applications, particularly as technological advances expand intracellular stimulation to applications previously limited to less effective extracellular methods. The high sensitivity of the neuron response to the optimal current waveform shapes could be useful in drug discovery, neuron disease diagnosis, toxin identification, and provide insights into neuron dynamics.\n                  <\/jats:p>","DOI":"10.1371\/journal.pone.0345922","type":"journal-article","created":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T17:50:47Z","timestamp":1777485047000},"page":"e0345922","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":1,"title":["In vivo efficacy of current stimuli computed by optimal tracking of neuron membrane voltage"],"prefix":"10.1371","volume":"21","author":[{"given":"Alexandra C.","family":"Ferguson","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9938-1047","authenticated-orcid":true,"given":"Damon 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Effect of the cellular composition of neural tissue on its spatio-temporal filtering properties","volume":"11","author":"B Tahayori","year":"2014","journal-title":"J Neural Eng"},{"key":"pone.0345922.ref019","volume-title":"Spikes: exploring the neural code","author":"F Rieke","year":"1999","edition":"1"},{"key":"pone.0345922.ref020","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-22492-2","volume-title":"The geometry of biological time","author":"AT Winfree","year":"1980"},{"key":"pone.0345922.ref021","volume-title":"Optimal control","author":"FL Lewis","year":"1995","edition":"2"},{"key":"pone.0345922.ref022","volume-title":"Modern control systems","author":"RC Dorf","year":"2021","edition":"14"},{"key":"pone.0345922.ref023","volume-title":"Nonlinear control","author":"HK Khalil","year":"2014"},{"key":"pone.0345922.ref024","doi-asserted-by":"crossref","DOI":"10.1201\/9780429398490","volume-title":"Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering","author":"SH Strogatz","year":"2024","edition":"3"},{"issue":"4","key":"pone.0345922.ref025","doi-asserted-by":"crossref","first-page":"5","DOI":"10.18500\/0869-6632-2018-26-4-5-58","article-title":"Nonlinear dynamical models of neurons: review","volume":"26","author":"AS Dmitrichev","year":"2018","journal-title":"Izvestiya VUZ Appl Nonlinear Dynam"},{"key":"pone.0345922.ref026","first-page":"4847","article-title":"Computation of reduced energy input current stimuli for neuron phase models","volume":"2014","author":"J Anyalebechi","year":"2014","journal-title":"Annu Int Conf IEEE Eng Med Biol Soc"},{"issue":"3","key":"pone.0345922.ref027","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/s00422-011-0427-9","article-title":"Exploring optimal current stimuli that provide membrane voltage tracking in a neuron model","volume":"104","author":"M Ellinger","year":"2011","journal-title":"Biol Cybern"},{"issue":"4","key":"pone.0345922.ref028","article-title":"Current stimuli that provide membrane voltage tracking in a six dimensional neuron model","volume":"135","author":"ME Koelling","year":"2013","journal-title":"J Dynamic Syst Measurem Control"},{"key":"pone.0345922.ref029","unstructured":"Ellinger M. Exploration of stimulus current energy reduction and bifurcation dynamics in conductance-based neuron models using optimal control theory. Western Michigan University; 2015. https:\/\/scholarworks.wmich.edu\/dissertations\/583"},{"key":"pone.0345922.ref030","article-title":"Generating neuron membrane action potentials using optimal input current stimuli in the medicinal leech","author":"DA Miller","year":"2015"},{"key":"pone.0345922.ref031","article-title":"Using experimentally informed models to find optimal neural stimuli in the medicinal leech.","volume-title":"2016 neuroscience meeting planner (Program No 561.25)","author":"AC Ferguson","year":"2016"},{"key":"pone.0345922.ref032","article-title":"Optimization and experimental application of current stimuli to leech pressure-sensitive mechanosensory cells","author":"AF Ferguson","year":"2017"},{"key":"pone.0345922.ref033","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/j.cnsns.2016.03.017","article-title":"Minimum energy control for a two-compartment neuron to extracellular electric fields","volume":"40","author":"G-S Yi","year":"2016","journal-title":"Commun Nonlinear Sci Num Simul"},{"issue":"12","key":"pone.0345922.ref034","doi-asserted-by":"crossref","first-page":"1747","DOI":"10.1007\/s10483-020-2689-9","article-title":"A review of computational modeling and deep brain stimulation: applications to Parkinson\u2019s disease","volume":"41","author":"Y Yu","year":"2020","journal-title":"Appl Math Mech"},{"issue":"5","key":"pone.0345922.ref035","doi-asserted-by":"crossref","first-page":"055004","DOI":"10.1088\/1741-2552\/aad416","article-title":"Non-rectangular waveforms are more charge-efficient than rectangular one in eliciting network-mediated responses of ON type retinal ganglion cells","volume":"15","author":"J-I Lee","year":"2018","journal-title":"J Neural Eng"},{"issue":"4","key":"pone.0345922.ref036","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1115\/1.2338654","article-title":"Optimal inputs for phase models of spiking neurons","volume":"1","author":"J Moehlis","year":"2006","journal-title":"J Comp Nonlin Dyn"},{"issue":"2","key":"pone.0345922.ref037","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/s10827-014-0499-3","article-title":"Locally optimal extracellular stimulation for chaotic desynchronization of neural populations","volume":"37","author":"D Wilson","year":"2014","journal-title":"J Comput Neurosci"},{"issue":"4","key":"pone.0345922.ref038","doi-asserted-by":"crossref","first-page":"1413","DOI":"10.3934\/dcds.2010.28.1413","article-title":"Charge-balanced spike timing control for phase models of spiking neurons","volume":"28","author":"P Danzl","year":"2010","journal-title":"DCDS"},{"issue":"1","key":"pone.0345922.ref039","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s00285-020-01501-1","article-title":"Optimal open-loop desynchronization of neural oscillator populations","volume":"81","author":"D Wilson","year":"2020","journal-title":"J Math 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excitation of nerve fibers","volume":"52","author":"S Jezernik","year":"2005","journal-title":"IEEE Trans Biomed Eng"},{"issue":"3","key":"pone.0345922.ref044","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pone.0090480","article-title":"Energy-optimal electrical-stimulation pulses shaped by the Least-Action Principle","volume":"9","author":"NI Krouchev","year":"2014","journal-title":"PLoS One"},{"issue":"4","key":"pone.0345922.ref045","doi-asserted-by":"crossref","first-page":"046009","DOI":"10.1088\/1741-2560\/7\/4\/046009","article-title":"Energy-efficient waveform shapes for neural stimulation revealed with a genetic algorithm","volume":"7","author":"A Wongsarnpigoon","year":"2010","journal-title":"J Neural Eng"},{"issue":"1","key":"pone.0345922.ref046","doi-asserted-by":"crossref","first-page":"13582","DOI":"10.1038\/s41598-017-13671-y","article-title":"Energy-efficient waveform for electrical stimulation of the cochlear nerve","volume":"7","author":"M Yip","year":"2017","journal-title":"Sci Rep"},{"issue":"1","key":"pone.0345922.ref047","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/s12264-023-01095-w","article-title":"Neuronomodulation of excitable neurons","volume":"40","author":"Y Chen","year":"2024","journal-title":"Neurosci Bull"},{"issue":"9","key":"pone.0345922.ref048","doi-asserted-by":"crossref","first-page":"1804","DOI":"10.1016\/j.bios.2005.09.008","article-title":"Toxin detection based on action potential shape analysis using a realistic mathematical model of differentiated NG108-15 cells","volume":"21","author":"DK Mohan","year":"2006","journal-title":"Biosens Bioelectron"},{"issue":"9","key":"pone.0345922.ref049","doi-asserted-by":"crossref","first-page":"1195","DOI":"10.1016\/j.conengprac.2004.10.008","article-title":"Feedback control of Hodgkin\u2013Huxley nerve cell dynamics","volume":"13","author":"F Fr\u00f6hlich","year":"2005","journal-title":"Control Eng 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filter control of a model of spatiotemporal cortical dynamics","volume":"5","author":"SJ Schiff","year":"2008","journal-title":"J Neural Eng"},{"issue":"7","key":"pone.0345922.ref054","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pcbi.1008053","article-title":"Estimation of neuron parameters from imperfect observations","volume":"16","author":"JD Taylor","year":"2020","journal-title":"PLoS Comput Biol"},{"issue":"5","key":"pone.0345922.ref055","doi-asserted-by":"crossref","first-page":"168","DOI":"10.3390\/a15050168","article-title":"Approaches to parameter estimation from model neurons and biological neurons","volume":"15","author":"A Nogaret","year":"2022","journal-title":"Algorithms"},{"issue":"2","key":"pone.0345922.ref056","doi-asserted-by":"crossref","DOI":"10.1103\/PRXLife.2.023007","article-title":"Inferring the dynamics of ionic currents from recursive piecewise data assimilation of approximate neuron models","volume":"2","author":"SA Wells","year":"2024","journal-title":"PRX Life"},{"key":"pone.0345922.ref057","unstructured":"Miller D, Ferguson A, Ellinger M, Koelling M, Jellies J, Linn C. 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