Abstract
We consider distributed parameter identification problems for the FitzHugh–Nagumo model of electrocardiology. The model describes the evolution of electrical potentials in heart tissues. The mathematical problem is to reconstruct physical parameters of the system through partial knowledge of its solutions on the boundary of the domain. We present a parallel algorithm of Newton–Krylov type that combines Newton’s method for numerical optimization with Krylov subspace solvers for the resulting Karush–Kuhn–Tucker system. We show by numerical simulations that parameter reconstruction can be performed from measurements taken on the boundary of the domain only. We discuss the effects of various model parameters on the quality of reconstructions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Akcelik, V. (2002). Multiscale Newton–Krylov Methods for inverse acoustic wave propagation. Ph.D. thesis, Carnegie Mellon University, Pittsburgh, Pennsylvania.
Akcelik, V., Biros, G., Ghattas, O., Hill, J., Keyes, D., & van Bloemen Waanders, B. (2006). Parallel algorithms for PDE-constrained optimization. In: M. Heroux, P. Raghaven, & H. Simon (Eds.), Frontiers of parallel computing. SIAM.
Argentina, M., Coullet, P., & Krinsky, V. (2000). Head-on collisions of waves in an excitable FitzHugh–Nagumo system: a transition from wave annihilation to classical wave behavior. Journal of Theoretical Biology, 205, 47–52.
Balay, S., Buschelman, K., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C. & et al. (2007). PETSc Homepage. http://www.mcs.anl.gov/petsc.
Banks, H. T., & Kunisch, K. (1989). Estimation techniques for distributed parameter systems. Boston, MA: Birkhäuser.
Benzi, M., Haber, E., & Hanson, L. (2006). Multilevel algorithms for large-scale interior point methods in bound constrained optimization. Technical Report TR-2006-002-A, Department of Mathematics and Computer Science, Emory University. 16p.
Bernus, O., Verschelde, H., & Panfilov, A. V. (2002). Modified ionic models of cardiac tissue for efficient large scale computations. Physics in Medicine and Biology, 47, 1947–1959.
Biegler, L., Ghattas, O., Heinkenschloss, M., & Bloemen-Waanders, B. v. (Eds.) (2003). Large-scale PDE-constrained optimization. In Lecture Notes in Computational Science and Engineering. Berlin: Springer-Verlag.
Biros, G., & Ghattas, O. (2005a). Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part I: The Krylov-Schur solver. SIAM Journal on Scientific Computing, 27, 687–713.
Biros, G., & Ghattas, O. (2005b). Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part II: The Lagrange–Newton solver and its application to optimal control of steady viscous flows. SIAM Journal on Scientific Computing, 27, 714–739.
Brooks, D. H., Ahmad, G. F., MacLeod, R. S., & Maratos, G. M. (1999). Inverse electrocardiography by simultaneous imposition of multiple constraints. IEEE Transactions on Biomedical Engineering, 46, 3–18.
Bub, G., Shrier, A., & Glass, L. (2002). Spiral wave generation in heterogeneous excitable media. Physical Review Letters, 88, 058101.
Chen, X., & Oshita, Y. (2006). Periodicity and uniqueness of global minimizers of an energy functional containing a long-range interaction. SIAM Journal on Mathematical Analysis, 37, 1299–1332.
Cheng, L. K., Bodley, J. M., & Pullan, A. J. (2003). Comparison of potential- and activation-based formulations for the inverse problem of electrocardiology. IEEE Transactions on Biomedical Engineering, 50, 11–22.
Colli-Franzone P., & Pavarino, L. F. (2004). A parallel solver for reaction-diffusion systems in computational electrocardiology. Mathematical Models and Methods in Applied Sciences, 14, 883–911.
Courtemanche, M., Skaggs, W., & Winfree, A. T. (1990). Stable 3-dimensional action-potential circulation in the FitzHugh–Nagumo model. Physica D, 41, 173–182.
Cox, S. J. (2006). An adjoint method for channel localization. Mathematical Medicine and Biology, 23, 139–152.
Cox, S. J., & Griffith, B. E. (2001). Recovering quasi-active properties of dendritic neurons from dual potential recordings. Journal of Computational Neuroscience, 11, 95–110.
Cox S. J., & Ji, L. (2003) Discerning ionic currents and their kinetics from input impedance data. Bulletin of Mathematical Biology, 63, 909–932.
Cox, S. J., & Wagner, A. (2004). Lateral overdetermination of the FitzHugh–Nagumo system. Inverse Problems, 20, 1639–1647.
Dauby, P. C., Desaive, T., & Croisier, H. (2006). Standing waves in the FitzHugh–Nagumo model of cardiac electrical activity. Physical Review E, 73, 021908.
Davidenko, J. M., Pertsov, A. V., Salomonsz, R., Baxter, W., & Jalefe, J. (1992). Stationary and drifting spiral waves of excitation in isolated cardiac-muscle. Nature, 355, 349–351.
Dennis, Jr J. E., & Schnabel, R. B. (1996). Numerical methods for unconstrained optimization and nonlinear equations. In Classics in Applied Mathematics. Philadelphia: SIAM.
Eisenstat S. C., & Walker, H. F. (1994). Globally convergent inexact Newton methods, 4, 393–422.
Elmer, C. E., & Van Vleck, E. S. (2005). Spatially discrete FitzHugh–Nagumo equations. SIAM Journal on Applied Mathematics, 65, 1153–1174.
Engl, H. W., Hanke, M., & Neubauer, A. (1996). Regularization of inverse problems. Dordrecht: Kluwer.
FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1, 445–466.
Franzone, P. C., Pavarino, L. F., & Taccardi, B. (2005). Simulating patterns of excitation, repolarization and action potential duration with cardiac bidomain and monodomain models. Mathematical Biosciences, 197, 35–66.
Gao, W., & Wang, J. (2004). Existence of wavefronts and impulses to FitzHugh–Nagumo equations. Nonlinear Analysis, 57, 667–676.
Haber, E., Ascher, U., & Oldenburg, D. (2000). On optimization techniques for solving nonlinear inverse problems. Inverse Problems, 16, 1263–1280.
Hansen, P. C., & O’Leary, D. P. (1993). The use of L-curve in the regularization of discrete ill-posed problems. SIAM Journal on Scientific Computing, 14, 1487–1503.
Hoffman, D. A., Magee, J. C., Colbert, C. M., & Johnston, D. (1997). K + Channel regulation of signal propagation in dendrites of hippocampal pyramidal neurons. Nature, 387, 869–875.
Isakov, V. (1998). Inverse problems for partial differential equations. New York: Springer-Verlag.
Knoll, D. A., & Keyes, D. E. (2004). Jacobian-free Newton–Krylov methods: A survey of approaches and applications. Journal of Comparative Physiology, 193, 357–397.
Krupa, M., Sandstede, B., & Szmolyan, P. (1997). Fast and slow waves in the FitzHugh–Nagumo equation. Journal of Differential Equations, 133, 49–97.
MacLeod, R. S., & Brooks, D. H. (1998). Recent progress in inverse problems in electrocardiology. IEEE Engineering in Medicine and Biology Magazine, 17, 73–83.
Moreau-Villéger, V., Delingette, H., Sermesant, M., Ashikaga, H., Faris, O., & McVeigh, E., et al. (2006). Building maps of local apparent conductivity of the epicardium with a 2D electrophysiological model of the heart. IEEE Transactions on Biomedical Engineering 53(8), 1457–1466.
Murillo, M., & Cai, X.-C. (2004). A fully implicit parallel algorithm for simulating the non-linear electrical activity of the heart. Numerical Linear Algebra with Applications, 11, 261–277.
Murray, J. D. (1993). Mathematical biology. Berlin: Springer
Nagumo, J., Arimoto, S., & Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. Proceedings of the Institute of Radio Engineers, 50, 2061–2070.
Nii, S. (1997). Stability of traveling multiple-front (multiple-back) wave solutions of the FitzHugh–Nagumo equations. SIAM Journal on Mathematical Analysis, 28, 1094–1112.
Nocedal, J. & Wright, S. J. (1999). Numerical optimization. New York: Springer-Verlag.
Patel, S. G., & Roth, B. J. (2005). Approximate solution to the bidomain equations for electrocardiogram problems. Physical Review E, 72, 051931.
Pennacchio, M., Savare, G., & Franzone, P. C. (2006). Multiscale modeling for the bioelectric activity of the heart. SIAM Journal on Mathematical Analysis, 37, 1333–1370.
Pernarowski, M. (2001). Controllability of excitable systems. Bulletin of Mathematical Biology, 63, 167–184.
Petersson, J. H. (2005). On global existence for semilinear parabolic systems. Nonlinear Analysis, 60, 337–347.
Riccio, M. L., Koller, M. L., & Gilmour, P. F. (1999). Electrical restitution and spatiotemporal organization during ventricular fibrillation. Circulation Research, 84, 955– 963.
Roth, B. J. (2004). Art Winfree and the bidomain model of cardiac tissue. Journal of Theoretical Biology, 230, 445–449.
Saad, Y. (2003). Iterative methods for sparse linear systems, 2nd ed. Philadelphia: SIAM.
Scott, A. C. (1975). The electrophysics of a nerve fiber. Reviews of Modern Physics, 47, 487–533.
Shahidi, V., Savard, P., & Nadeau, R. (1994). Forward and inverse problem of electrocardiography: Modeling and recovery of epicardial potentials in humans. IEEE Transactions on Biomedical Engineering, 41, 249–256.
Sneyd, J., Dale, P. D., & Duffy, A. (1998). Traveling waves in buffered systems: applications to calcium waves. SIAM Journal on Applied Mathematics, 58, 1178–1192.
Suckley, R., & Biktashev, V. N. (2003). Comparison of asymptotics of heart and nerve excitability. Physical Review E, 68, 011902.
Tsai, J.-C., & Sneyd, J. (2005). Existence and stability of traveling waves in buffered systems. SIAM Journal on Applied Mathematics, 66, 237–265.
Vanier, M. C., & Bower, J.-M. (1999). A comparative survey of automated parameter-search methods for compartmental neural models. Journal of Computational Neuroscience, 7, 149–171.
Vogel, C. R. (1996). Non-convergence of the L-curve regularization parameter selection method. Inverse Problems, 12, 535–547.
Vogel, C. R. (2002). Computational methods for inverse problems. In Frontiers in applied mathematics. Philadelphia: SIAM.
Weiss, J. N., Garfinkel, A., Karagueuzian, H. S., Qu, Z., & Chen, P. S. (1999). Chaos and the transition to ventricular fibrillation—A new approach to antiarrhythmic drug evaluation. Circulation, 99, 2819–2826.
Willms, A. R., Baro, D. J., Harris-Warrick, R. M., & Guckenheimer, J. (1999). An improved parameter estimation method for Hodgkin–Huxley models. Journal of Computational Neuroscience, 6, 145–168.
Winfree, A. T. (1990). Stable particle-like solutions to the nonlinear-wave equations of 3-dimensional excitable media. SIAM Review, 32, 1–53.
Yamada, H., & Nozaki, K. (1990). Interaction of pulses in dissipative systems. FitzHugh–Nagumo equations. Progress of Theoretical Physics, 84, 801–809.
Author information
Authors and Affiliations
Corresponding author
Additional information
Action Editor: David Terman
Rights and permissions
About this article
Cite this article
He, Y., Keyes, D.E. Reconstructing parameters of the FitzHugh–Nagumo system from boundary potential measurements. J Comput Neurosci 23, 251–264 (2007). https://doi.org/10.1007/s10827-007-0035-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10827-007-0035-9