Qualitative parameter estimation for a class of relaxation oscillators

Lyapunov method; neuroscience; nonlinear system; parameter estimation; relaxation oscillator; singular perturbation; Global asymptotic stability; Qualitative estimation; Qualitative parameters; Redundant parameters; Relaxation oscillation; Sigmoidal nonlinearity; Singular perturbations; Control and Systems Engineering

Abstract :

[en] Motivated by neuroscience applications, we introduce the concept of qualitative estimation as an adaptation of classical parameter estimation to nonlinear systems characterized by i) the presence of possibly many redundant parameters, ii) a small number of possible qualitatively different behaviors, iii) the presence of sharply different characteristic timescales and, consequently, iv) the generic impossibility of quantitatively modeling and fitting experimental data. As a first application, we illustrate these ideas on a class of nonlinear systems with a single unknown sigmoidal nonlinearity and two sharply separated timescales. This class of systems is shown to exhibit either global asymptotic stability or relaxation oscillations depending on a single ruling parameter and independently of the exact shape of the nonlinearity. We design and analyze a qualitative estimator that estimates the distance of the ruling parameter from the unknown critical value at which the transition between the two behaviors happens without using any quantitative fitting of the measured data.

Disciplines :

Engineering, computing & technology: Multidisciplinary, general & others

Author, co-author :

Tang, Ying; Université de Lorraine, CRAN, UMR 7039, France ; CNRS, CRAN, UMR 7039, France

Franci, Alessio ; Université de Liège - ULiège > Département d'électricité, électronique et informatique (Institut Montefiore) > Brain-Inspired Computing ; Department of Mathematics, Universidad National Autónoma de México, Mexico

Postoyan, Romain; Université de Lorraine, CRAN, UMR 7039, France ; CNRS, CRAN, UMR 7039, France

Language :

English

Title :

Qualitative parameter estimation for a class of relaxation oscillators

Publication date :

July 2017

Event name :

IFAC World Congress

Event date :

2017

Audience :

International

Journal title :

IFAC-PapersOnLine

ISSN :

2405-8971

eISSN :

2405-8963

Publisher :

Elsevier B.V.

Volume :

Issue :

Pages :

2929 - 2934

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://api.elsevier.com/content/article/PII:S2405896317310376?httpAccept=text/xml

Funding text :

Qualitative parameter estimation for a class Qualitative parameter estimation★for a class Qualitativoef praelraaxmaetitoenr eosstciimllatoiorns ★★for a class of relaxation oscillators★ of relaxation oscillators YingTang∗,∗∗AlessioFranci∗∗∗RomainPostoyan∗,∗∗ Ying Tang ∗,∗∗ Alessio Franci ∗∗∗ Romain Postoyan ∗,∗∗ Ying Tang∗,∗∗ Alessio Franci∗∗∗ Romain Postoyan∗,∗∗ ∗∗∗ Université de Lorraine, CRAN, UMR 7039, France ∗Univer∗∗sitéde Lorraine, CRAN, UMR 7039, France ∗ Univer∗∗∗∗s∗itCC´eNRS,NdeRSL,orCCrRAN,RaiAneN,,CUMURMARRN,7039,70U3M9,RFF7rraa0nn3cc9ee, France {ying.ta∗∗ngCC,NRS,NRromain.postoyanoSm,aCCiRAN,Rn.ApNos,tUMUoMyaRRn}7039,7@univ-lorraine.fr0u3n9i,vFF-rrlaaonnrccreeaine.fr ∗∗∗ {ying.tangC,NRroSm,aCiRn.ApNos,tUoMyaRn}7@0u3n9i,vF-rlaonrcreaine.fr ∗∗∗∗∗∗ Department of Mathematics, Universidad National Autónoma de Department of Mathematics, Universidad National Autónoma de ∗∗∗ México, Mexico afranci@ciencias.unam.mx México, Mexico afranci@ciencias.unam.mx México, Mexico afranci@ciencias.unam.mx Abstract: Motivated by neˇroscience applications, fle introdˇce the concept of qualitative estimation as an adaptation of classical parameter estimation to nonlinear systems characterized estimation as an adaptation of classical parameter estimation to nonlinear systems characterized by i) the presence of possibly many redˇndant parameters, ii) a small nˇmber of possible qˇal-ityatii)vethlye dpirfefesreenncteboefhpaovsisoirbsl,yiimi)atnhye rperdeˇsnendcaentofpashraamrpeltyerdsi,ffieir)eantsmchaalrlancˇtmerbisetricotfimpoessciablleesqaˇnadl-, itatively different behaviors, iii) the presence of sharply different characteristic timescales and, ctoantsiveeqlˇyendtiflfye,rievn)ttbheehagvenioerrsi,ciiimi)ptohsesipbrielisteyncoef oqfˇashnatirtpaltyivdeilfyfemreondteclhinagraacntderifsittticintgimexespcearliemseanntda,l dcoantase∆qAˇsenatfliyr,stiva)ptphliecagteinoner,ifcleimilplˇosstsriabtielittyheosfeqidˇeaanstiotnataivcellayssmoofdneolninligneaanrdsyfisttteinmgsefxliptehriamseingtalel data∆ As a first application, fle illˇstrate these ideas on a class of nonlinear systems flith a single ˇntkan∆oAflsnasfiigrmstoaidpapllincaotniloinne,afrleityillˇanstdrattfelothshesaerpidlyeasseponaraatceldastsimofenscoanlleins∆eaTrhsiyssctleamsss oflfitshysatesminsglies ˇnknofln sigmoidal nonlinearity and tflo sharply separated timescales∆ This class of systems is shnokfnlnofltno seixghmiboiidt aelitnhoenrlignleoabraitlyaasnymd ptftlotischastrapblyilisteypaorratreedlaxtiamtieosncaolescs∆illTahtiiosncsladsespoefnsdyinstgemons ias sihnogfllenrtˇolinexghpibairtameiethteer agnlodbainldaespyemnpdteonttilcysotfabthileityexoarctreslhaaxpaetioonf tohsecilnlaotniolinnseadreitpye∆nWdiengdeosnigna single rˇling parameter and independently of the exact shape of the nonlinearity∆ We design ainndgleanraˇllyinzeg apaqruaamlietatetirveanedstiimndaetpoerntdheanttleystoimf tahteesetxhaectdsishtaapneceofofththeenroˇnlliinngeapraitrya∆mWeteerdefrsoigmn tahnedˇannkanlyofzlenacrqiutiaclaitlavtaivlˇeeeasttimflhaticohr tthheatreasntsimitiaotnesbtehtfeledeinsttahnecetfloof bthehearvˇiloirnsghpaaprpaemnestfelritfhrooˇmt the ˇnknofln critical valˇe at flhich the transition betfleen the tflo behaviors happens flithoˇt ˇsthseinginˇgnaaknnyyofqˇqlˇnaacnnrttiiitttiaacttaiilvvveeafittinfliˇtteinaggt foolffhthetihche tmmheeeaatssrˇreˇarnesdditdadioantatab∆∆etfleen the tflo behaviors happens flithoˇt ˇsing any qˇantitative fitting of the measˇred data∆ ˇsing any qˇantitative fitting of the measˇred data∆ ˇ©si2n0g17a, nIFyAqCˇ a(Innttietrantaivtieonfaitl tFinedgeroaftitohneomf Aeautsoˇmreadticd Catoan∆trol) Hosting by Elsevier Ltd. All rights reserved. Keywords: parameter estimation, relaxation oscillator, singˇlar pertˇrbation, neˇroscience, Keywords: parameter estimation, relaxation oscillator, singˇlar pertˇrbation, neˇroscience, nonoenlinenylwinoeraadrrs:ssysypsatetreamm,,eLLteyyraapˇpesˇtnnimoovvatmmioeenthot,hroeddlaxation oscillator, singˇlar pertˇrbation, neˇroscience, nonlinear system, Lyapˇnov method nonlinear system, Lyapˇnov method 1∆ INTRODUCTION 1∆ INTRODUCTION 1∆ INTRODUCTION Online parameter estimation of dynamical models of neˇ-Online parameter estimation of dynamical models of neˇ-rOonnlainl eacptairvaitmyetmerigehsttilmeadtiotno onfedflynpaemrsipcaecl tmivoeds eilns onfenˇerˇo-rsocineanlceasc∆tIinvietypilmepigsyhtfolreiandsttaoncnee,fhlavpienrgspaecccteisvsesinirneanl-etˇimroe-srocineanlceasc∆tIinvietypilmepigsyhtfolreiandsttaoncnee,fhlavpienrgspaecccteisvsesinirneanl-etˇimroe-to the gains governing the excitation/inhibition balance to the gains governing the excitation/inhibition balance flithin popˇlations of neˇrons might provide important in-ffolirtmhiantipoonpaˇblaotˇitonthseofonne-gˇorionngsemleicgthrtopphroyvsiidoleoigmicpaol ratcatnivtiitny-, formation aboˇt the on-going electrophysiological activity, afonrdmtahtˇiosnhaeblpoˇdtetvheeloopnin-ggoninegflesletrcatrtoegpiheysstioolodgeitceacltaocrtievviteyn, predict seizˇres∆ This task is challenging becaˇse of the predict seizˇres∆ This task is challenging becaˇse of the redˇndancy and the large variability of biophysical param-redˇndancy and the large variability of biophysical param-etxehrisbiatcinrogss ipmoiplaˇrlaaticotnivsitoyf npeaˇttreornnss∆anDdisnpeaˇrarotenaclocmirbciˇniats-tions of biophysical parameters are indeed knofln to lead tions of biophysical parameters are indeed knofln to lead ttoiontsheofsabmioephaycstiicvailtypapraatmtertnersatarteheindceeleldˇlakrnolfelvneltoGloeladd-tmoantheet saalm∆ (e20a0c1t)iv∆ iTtyhepsaatmteerndeagtenthereatceedllˇplaaramleevterlizGatoilodn-mtoantheet saalm∆ (e20a0c1t)iv∆ iTtyhepsaatmteerndeagtenthereatceedllˇplaaramleevterlizGatoilodn-pmraonpeerttyalp∆ r(o2p00a1g)a∆teTshaetstahmeendeeˇgreonnearlacteirdcˇpiatrlaemveeltrMizaartdioenr p(2r0o1p1e)r∆tyInpraodpdaigtaiotnes, abtiotphheysniecˇalropnaarlacmirectˇeirts leslvoefllMy avradreyr, (p2r0o1p1e)r∆tyInpraodpdaigtaiotnes, abtiotphheysniecˇalropnaarlacmirectˇeirts leslvoeflllMy avradreyr, (in2d01ˇ1c)in∆ gInshadrpdittiroann,sibtiioonpshybseitcfalleepnarqaˇmaleitteartsivselloyfldlyiffevraernyt, i(n2d0ˇ11ci)n∆ gInshadrpdittiroann,sibtiioonpshybseitcfalleepnarqaˇmaleitteartsivselloyfldlyiffevraernyt, iancdtiˇvciitnygmshoadreps (tsrapniksiintigonosr beˇtrfslteieng,qˇhaealilttahtyivoelryedpiiflfeprteinct, aincdtiˇvciitnygmshoadreps (tsrpaniksiintigonosr bbeˇtrfslteienng,qˇhaealilttahtyivoelryedpiiflfeeprteinct, etc∆) at the crossing of critical parameter sets∆ In this etc∆) at the crossing of critical parameter sets∆ In this ceotcn∆t)exatt, qthˇeanctritoastsiivnegmoofdcerlintigcaalndpafritatminegteorf esxetpse∆riImnenthtaisl context, qˇantitative modeling and fitting of experimental dcoantategxetn, eqrˇicaanltliytactoinvestmitˇotdeelainngilal-npdosfeitdtipnrgobofleemxpaenrdimaennetafl estimation approach is therefore needed∆ estimation approach is therefore needed∆ Online parameter estimation for dynamical systems has Online parameter estimation for dynamical systems has been flidely stˇdied, in particˇlar for linear systems, see been flidely stˇdied, in particˇlar for linear systems, see ★been flidely stˇdied, in particˇlar for linear systems, see ★ This work was supported by the ANR under the ffrant SEPICOT ★★ This work was supported by the ANR under the ffrant SEPICOT This work was supported by the ANR under the ffrant SEPICOT (ANR 12 JS03 004 01), by the French “Réffion Grand Est” throuffh a fellowship ffrant 2016-2017, and by DGAPA-UNAM under the ffrant PeAlloPwIIsThipIAff1r0a5n8t1260.16-2017, and by DGAPA-UNAM under the ffrant PfeAlloPwIIshiTpIAff105816.rant 2016-2017, and by DGAPA-UNAM under the ffrant PAPIIT IA105816. 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 2984 CPoepery rriegvhietw © u 2n0d1e7r rIFesApConsibility of International Federation of Automa2t9ic8 4Control. Copyright © 2017 IFAC 2984 C10.1016/j.ifacol.2017.08.651opyright © 2017 IFAC 2984

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