ABSTRACT
The purpose of this article is to present some recent applications of computer algebra to answer structural and numerical questions in applied sciences. A first example concerns identifiability which is a pre-condition for safely running parameter estimation algorithms and obtaining reliable results. Identifiability addresses the question whether it is possible to uniquely estimate the model parameters for a given choice of measurement data and experimental input. As discussed in this paper, symbolic computation offers an efficient way to do this identifiability study and to extract more information on the parameter properties. A second example addressed hereafter is the diagnosability in nonlinear dynamical systems. The diagnosability is a prior study before considering diagnosis. The diagnosis of a system is defined as the detection and the isolation of faults (or localization and identification) acting on the system. The diagnosability study determines whether faults can be discriminated by the mathematical model from observations. These last years, the diagnosability and diagnosis have been enhanced by exploitting new analytical redundancy relations obtained from differential algebra algorithms and by the exploitation of their properties through computer algebra techniques.
- Stefania Audoly, Giuseppina Bellu, Leontina D'Angió, Maria Saccomani, and Claudio Cobelli. Global identifiability of nonlinear models of biological systems. IEEE Transactions on Bio-medical Engineering, 48:55--65, 2001.Google ScholarCross Ref
- Giuseppina Bellu, Maria Saccomani, Stefania Audoly, and Leontina D'Angiò. DAISY: A new software tool to test global identifiability of biological and physiological systems. Computer Methods and Programs in Biomedicine, 28 (1):52--61, 2007.Google ScholarCross Ref
- Upinder S. Bhalla and James M. Bower. Exploring parameter space in detailed single neuron models: simulations of the mitral and granule cells of the olfactory bulb. Journal of Neurophysiology, 69(6):1948--1965, 1993.Google ScholarCross Ref
- François Boulier. Study and implementation of some algorithms in differential algebra. PhD thesis, Université des Sciences et Technologie de Lille - Lille I, June 1994.Google Scholar
- François Boulier, Daniel Lazard, François Ollivier, and Michel Petitot. Computing representation for radicals of finitely generated differential ideals. Technical report, Université Lille I, LIFL, 59655, Villeneuve d'Ascq, 1997.Google Scholar
- Christopher W. Brown. QEPCAD B: A program for computing with semialgebraic sets using CADs. SIGSAM BULLETIN, 37:97--108, 2003.Google ScholarDigital Library
- Michael J. Chapman, Keith R. Godfrey, Michael J. Chappell, and Neil D. Evans. Structural identifiability of nonlinear systems using linear/nonlinear splitting. International Journal of Control, 76:209--216, 2003.Google ScholarCross Ref
- Michael J. Chappell and Keith R. Godfrey. Structural identifiability of the parameters of a nonlinear batch reactor model. Mathematical Biosciences, 108:245--251, 1992.Google ScholarCross Ref
- Changbo Chen, James H. Davenport, John P. May, Marc Moreno Maza, Bican Xia, and Rong Xiao. Triangular decomposition of semi-algebraic systems. Journal of Symbolic Computation, 49:3--26, 2013.Google ScholarDigital Library
- David Cox, John Little, and Donal O'Shea. Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. 2nd ed. 03 2015.Google Scholar
- Dávid Csercsik, Katalin M. Hangos, and Gábor Szederkényi. Identifiability analysis and parameter estimation of a single Hodgkin--Huxley type voltage dependent ion channel under voltage step measurement conditions. Neurocomputing, 77(1):178--188, 2012.Google ScholarDigital Library
- Lilianne Denis-Vidal and Ghislaine Joly-Blanchard. Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems. Automatica, 40:287--292, 2004.Google ScholarDigital Library
- Lilianne Denis-Vidal, Ghislaine Joly-Blanchard, and Céline Noiret. Some effective approaches to check identifiability of uncontrolled nonlinear systems. Mathematics and Computers in Simulation, 57:35--44, 2001.Google ScholarDigital Library
- Lilianne Denis-Vidal, Ghislaine Joly-Blanchard, Céline Noiret, and Michel Petitot. An algorithm to test identifiability of non-linear systems. In Proceedings of 5th IFAC Symposium on Nonlinear Control Systems, volume 7, pages 174--178, St Petersburg, Russia, 2001.Google ScholarCross Ref
- Sette Diop and Michel Fliess. Nonlinear observabiliy, identifiability and persistant trajectories. In Proceedings of the 30th IEEE Conference on Decision and Control, pages 714--719, Brighton, UK, 1991.Google Scholar
- Jean-Charles Faugère. A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, ISSAC '02, page 75--83, New York, NY, USA, 2002. Association for Computing Machinery.Google ScholarDigital Library
- Michel Fliess and Torkel Glad. An algebraic approach to linear and nonlinear control. In Essays on control: perspectives in the theory and it application, volume 7, pages 223--267, Cambridge, MA, Birkhauser, 1993.Google ScholarCross Ref
- James L. Hindmarsh and R.M. Rose. A model of the nerve impulse using two first-order differential equations. Nature, 296(5853):162--164, 1982.Google ScholarCross Ref
- James L. Hindmarsh and R.M. Rose. A model of neuronal bursting using three coupled first order differential equations. Proceedings of the Royal Society of London B: Biological Sciences, 221(1222):87--102, 1984.Google ScholarCross Ref
- Alan L. Hodgkin and Andrew F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4):500--544, 1952.Google ScholarCross Ref
- Hoon Hong, Alexey Ovchinnikov, Gleb Pogudin, and Chee Yap. Global identifiability of differential models. Communications on Pure and Applied Mathematics, 73(9):1831--1879, 2020.Google ScholarCross Ref
- Eugene M. Izhikevich. Which model to use for cortical spiking neurons? IEEE transactions on neural networks, 15(5):1063--1070, 2004.Google Scholar
- Eugene M. Izhikevich. Dynamical Systems in Neuroscience. MIT press, 2007.Google Scholar
- David Janzén, Mats Jirstrand, Michael Chappell, and Neil Evans. Three novel approaches to structural identifiability analysis in mixed-effects models. Computer Methods and Programs in Biomedicine, 171:141--152, 2019.Google ScholarDigital Library
- Jack Lee, Bruce H. Smaill, and Nicolas Smith. Hodgkin--Huxley type ion channel characterization: an improved method of voltage clamp experiment parameter estimation. Journal of Theoretical Biology, 242(1):123--134, 2006.Google ScholarCross Ref
- Lennart Ljung and Torkel Glad. On global identifiability for arbitrary model parametrizations. Automatica, 30:265--276, 1994.Google ScholarDigital Library
- Nicolette Meshkat, Chris Anderson, and Joseph J. DiStefano III. Finding identifiable parameter combinations in nonlinear ODE models and the rational reparameterization of their input-output equations. Mathematical Biosciences, 233(1):19--31, 2011.Google ScholarCross Ref
- Nicolette Meshkat, Marisa Eisenberg, and Joseph J. DiStefano III. An algorithm for finding globally identifiable parameter combinations of nonlinear ODE models using Groebner Bases. Mathematical Biosciences, 222(2):61--72, 2009.Google ScholarCross Ref
- Alexey Ovchinnikov, Anand Pillay, Gleb Pogudin, and Thomas Scanlon. Computing all identifiable functions of parameters for ODE models. Systems & Control Letters, 157:105030, 2021.Google ScholarCross Ref
- Hannu Pohjanpalo. System identifiability based on the power series expansion of the solution. Mathematical Biosciences, 41:21--33, 1978.Google ScholarCross Ref
- Maria Saccomani, Stefania Audoly, Giuseppina Bellu, and Leontina D'Angiò. Examples of testing global identifiability of biological and biomedical models with the DAISY software. Computers in Biology and Medicine, 40 (4):402--407, 2010.Google ScholarDigital Library
- Maria Saccomani, Stefania Audoly, and Leontina D'Angiò. Parameter identifiability of nonlinear systems: the role of initial conditions. Automatica, 39:619--632, 2004.Google ScholarDigital Library
- Mohab Safey El Din. RAGLib (Real Algebraic Geometry Library). Available at https://www-polsys.lip6.fr/ safey/RAGLib, 2007.Google Scholar
- Mohab Safey El Din. Testing sign conditions on a multivariate polynomial and applications. Mathematics in Computer Science, 1:177--207, 12 2007.Google ScholarCross Ref
- Andreas Schaefer, Moritz Helmstaedter, Bert Sakmann, and Alon Korngreen. Correction of conductance measurements in non-space-clamped structures: 1. Voltage-gated K+ channels. Biophysical Journal, 84(6):3508--3528, 2003.Google ScholarCross Ref
- Hans Stigter and Jaap Molenaar. A fast algorithm to assess local structural identifiability. Automatica, 58:118--124, 2015.Google ScholarDigital Library
- Michael Vanier and James Bower. A comparative survey of automated parametersearch methods for compartmental neural models. Journal of Computational Neuroscience, 7(2):149--171, 1999.Google ScholarCross Ref
- Nathalie Verdière and Sébastien Orange. Diagnosability in the case of multifaults in nonlinear models. Journal of Process Control, 69, 2018.Google Scholar
- Nathalie Verdière, Lilianne Denis-Vidal, Ghislaine, and Dominique Domurado. Identifiability and estimation of pharmacokinetic parameters of ligands of macrophage mannose receptor. International Journal of Applied Maths and Computer Science, 15, 2005.Google Scholar
- Nathalie Verdière and Sébastien Orange. A systematic approach for doing an a priori identifiability study of dynamical nonlinear models. Mathematical Biosciences, 308:105--113, 2019.Google ScholarCross Ref
- Eric Walter and Yves Lecourtier. Global approaches to identifiability testing for linear and nonlinear state space models. Mathematics and Computers in Simulation, 24:472--482, 1982.Google ScholarCross Ref
- Allan Willms, Deborah Baro, Ronald Harris-Warrick, and John Guckenheimer. An improved parameter estimation method for Hodgkin-Huxley models. Journal of Computational Neuroscience, 6(2):145--168, 1999.Google ScholarCross Ref
- Bican Xia. DISCOVERER: a tool for solving semi-algebraic systems. ACM Communications in Computer Algebra, 41(3):102--103, 09 2007.Google ScholarDigital Library
Index Terms
- Applications of Computer Algebra to Parameter Analysis of Dynamical Systems
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