{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T23:02:50Z","timestamp":1772838170687,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T00:00:00Z","timestamp":1653350400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>A numerical parameter estimation method, based on input-output integro-differential polynomials in a bounded-error framework is investigated in this paper. More precisely, the measurement noise and parameters belong to connected sets (in the proposed work, intervals). First, this method, based on the Rosenfeld\u2013Groebner elimination algorithm, is presented. The latter provides differential equations containing derivatives, sometimes of high order. In order to improve the numerical results, a pretreatment of the differential relations is done and consists in integration. The new relations contain, essentially, integrals depending only on the outputs. In comparison with the initial relations, they are less sensitive to measurement noise. Finally, the impact of the size of the measurement noise domain on the estimated intervals is studied.<\/jats:p>","DOI":"10.3390\/a15060179","type":"journal-article","created":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T22:04:06Z","timestamp":1653429846000},"page":"179","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Bounded-Error Parameter Estimation Using Integro-Differential Equations for Hindmarsh\u2013Rose Model"],"prefix":"10.3390","volume":"15","author":[{"given":"Carine","family":"Jauberthie","sequence":"first","affiliation":[{"name":"LAAS-CNRS, Universit\u00e9 de Toulouse, CNRS, UPS, 7 Avenue du Colonel Roche, 31400 Toulouse, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7596-0941","authenticated-orcid":false,"given":"Nathalie","family":"Verdi\u00e8re","sequence":"additional","affiliation":[{"name":"LMAH, University of Normandie, UNIHAVRE, FR-CNRS-3335, ISCN, 76600 Le Havre, France"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,24]]},"reference":[{"key":"ref_1","unstructured":"Boulier, F., Lazard, D., Ollivier, F., and Petitot, M. (1997). Computing Representation for Radicals of Finitely Generated Differential Ideals, Universit\u00e9 Lille I. Technical Report."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1109\/TAC.1965.1098172","article-title":"More about process identification","volume":"10","author":"Loeb","year":"1965","journal-title":"Automatica"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sira-Ramirez, H., Rodriguez, C.G., Romero, J.C., and Ju\u00e1rez, A.L. (2014). Algebraic Identification and Estimation Methods, Wiley. Feedback Control Systems.","DOI":"10.1002\/9781118730591"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Verdi\u00e8re, N., Jauberthie, C., and Trav\u00e9-Massuy\u00e8s, L. (2018, January 12\u201315). Improvements in bounded error parameter estimation using distribution theory. Proceedings of the European Control Conference 2018, Limassol, Cyprus.","DOI":"10.23919\/ECC.2018.8550607"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Verdi\u00e8re, N., and Jauberthie, C. (2020, January 25). Parameter Estimation Procedure Based on Input-Output Integro-Differential Polynomials. Application to the Hindmarsh-Rose Model. Proceedings of the European Control Conference 2020, Saint Petersbourg, Russia.","DOI":"10.23919\/ECC51009.2020.9143670"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Jaulin, L., Kieffer, M., Didrit, O., and Walter, E. (2001). Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics, Springer. [1st ed.]. An Emerging Paradigm.","DOI":"10.1007\/978-1-4471-0249-6"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Boulier, F., Korporal, A., Lemaire, F., Perruquetti, W., Poteaux, A., and Ushirobira, R. (2014, January 8\u201312). An Algorithm for Converting Nonlinear Differential Equations to Integral Equations with an Application to Parameter Estimation from Noisy Data. Proceedings of the Computer Algebra in Scientific Computing 2014, Warsaw, Poland.","DOI":"10.1007\/978-3-319-10515-4_3"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1038\/296162a0","article-title":"A model of the nerve impulse using two first-order differential equations","volume":"296","author":"Hindmarsh","year":"1982","journal-title":"Nature"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1098\/rspb.1984.0024","article-title":"A model of neuronal bursting using three coupled first order differential equations","volume":"221","author":"Hindmarsh","year":"1984","journal-title":"Proc. R. Soc. Lond. Biol. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1113\/jphysiol.1952.sp004764","article-title":"A quantitative description of membrane current and its application to conduction and excitation in nerve","volume":"117","author":"Hodgkin","year":"1952","journal-title":"J. Physiol."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Izhikevich, E.M. (2007). Dynamical Systems in Neuroscience, MIT Press.","DOI":"10.7551\/mitpress\/2526.001.0001"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Tokuda, I., Parlitz, U., Illing, L., Kennel, M., and Abarbanel, H. (2003, January 26\u201329). Parameter estimation for neuron models. Proceedings of the AIP Conference, San Diego, CA, USA.","DOI":"10.1063\/1.1612220"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Corson, N., Lanza, V., and Verdi\u00e8re, N. (2016). Hopf bifurcations in a chain of coupled Hindmarsh-Rose system. Acta Biotheor., 65.","DOI":"10.1007\/s10441-016-9288-x"},{"key":"ref_14","first-page":"375","article-title":"State and parameter estimation using unconstrained optimization","volume":"84.","author":"Parlitz","year":"2011","journal-title":"Phys. R. E"},{"key":"ref_15","unstructured":"Steur, E. (2006). Parameter Estimation in Hindmarsh-Rose Neurons. [Ph.D. Thesis, Technische Universiteit Eindhoren]."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Denis-Vidal, L., Joly-Blanchard, G., Noiret, C., and Petitot, M. (2001, January 4\u20136). An algorithm to test identifiability of non-linear systems. Proceedings of the 5th IFAC NOLCOS, Saint Petersburg, Russia.","DOI":"10.1016\/S1474-6670(17)35173-X"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.arcontrol.2013.04.002","article-title":"Fault detection and identification relying on set-membership identifiability","volume":"37","author":"Jauberthie","year":"2013","journal-title":"Annu. Rev. Control."},{"key":"ref_18","unstructured":"Fliess, M., Mboup, M., Mounier, H., and Sira-Ramirez, H. (2022, April 07). Questioning Some Paradigms of Signal Processing via Concret Examples. Available online: https:\/\/hal.inria.fr\/inria-00001059\/file\/signalg.pdf."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1016\/0005-1098(93)90106-4","article-title":"Set inversion via interval analysis for nonlinear bounded-error estimation","volume":"29","author":"Jaulin","year":"1993","journal-title":"Automatica"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"446","DOI":"10.3182\/20120711-3-BE-2027.00374","article-title":"Interval Methods for Control-Oriented Modeling of the Thermal Behavior of High-Temperature Fuel Cell Stacks","volume":"45","author":"Rauh","year":"2012","journal-title":"IFAC Proc. Vol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/j.ifacol.2018.03.058","article-title":"An Interval Approach for Parameter Identification and Observer Design of Spatially Distributed Heating Systems","volume":"51","author":"Rauh","year":"2018","journal-title":"IFAC-PapersOnLine"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"AbdelAty, A.M., Fouda, M.E., and Eltawil, A. (2022). Parameter Estimation of Two Spiking Neuron Models With Meta-Heuristic Optimization Algorithms. Front. Neuroinform., 16.","DOI":"10.3389\/fninf.2022.771730"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"10","DOI":"10.3389\/fninf.2015.00010","article-title":"Parameter estimation of neuron models using in-vitro and in-vivo electrophysiological data","volume":"9","author":"Lynch","year":"2015","journal-title":"Front. Neuroinform."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1515\/amcs-2016-0057","article-title":"Set-membership identifiability of nonlinear models and related parameter estimation properties","volume":"26","author":"Jauberthie","year":"2016","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1016\/S0377-0427(02)00659-3","article-title":"Sensitivity analysis for dynamic systems with time-lags","volume":"151","author":"Rihan","year":"2003","journal-title":"J. Comput. Appl. Math."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/6\/179\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:18:01Z","timestamp":1760138281000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/6\/179"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,24]]},"references-count":25,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["a15060179"],"URL":"https:\/\/doi.org\/10.3390\/a15060179","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,24]]}}}