Advanced Search
Article Contents
Article Contents

Design of the monodomain model by artificial neural networks

  • *Corresponding author: Sébastien Court

    *Corresponding author: Sébastien Court 
Abstract Full Text(HTML) Figure(4) / Table(1) Related Papers Cited by
  • We propose an optimal control approach in order to identify the nonlinearity in the monodomain model, from given data. This data-driven approach gives an answer to the problem of selecting the model when studying phenomena related to cardiac electrophysiology. Instead of determining coefficients of a prescribed model (like the FitzHugh-Nagumo model for instance) from empirical observations, we design the model itself, in the form of an artificial neural network. The relevance of this approach relies on the approximation capacities of neural networks. We formulate this inverse problem as an optimal control problem, and provide mathematical analysis and derivation of optimality conditions. One of the difficulties comes from the lack of smoothness of activation functions which are classically used for training neural networks. Numerical simulations demonstrate the feasibility of the strategy proposed in this work.

    Mathematics Subject Classification: Primary: 49K20, 35A01, 41A46, 68T07; Secondary: 35D30, 35K40, 35K45, 35K58, 35B30, 35M99, 41A99, 49J45, 49N15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  With $ (v_0,w_0) = (v_{{\rm{ FH}}},w_{{\rm{ FH}}})(0) = (2,0) $: The solution $ v_{{\rm{ FH}}} $ of the FitzHugh-Nagumo model (47) in blue, and in red the solution $ v $ of (46) with the neural network as $ \Phi = (\phi_v,\phi_w) $. (For interpretation of the references to color in this figure legend, the reader is invited to refer to the digital version of this article.)

    Figure 2.  With $ (v_0,w_0) = (v_{{\rm{ AP}}},w_{{\rm{ AP}}})(0) = (0.75,0.75) $: The solution $ (v_{{\rm{ AP}}},w_{{\rm{ AP}}}) $ of the Aliev-Panfilov model (49) in dashed blue and green respectively, in red and magenta respectively, the solution $ (v,w) $ of (46) with the trained neural networks as $ \Phi = (\phi_v,\phi_w) $. Left: with Architecture 1. Right: with Architecture 2

    Figure 3.  At time $ t = 9.00 $, the computed solution $ (v_{{\rm{ AP}}},w_{{\rm{ AP}}}) $ of the Aliev-Panfilov PDE model corresponding to (51) represented in dashed blue and green respectively, compared with the computed solution $ (v,w) $ of (50) with the neural network as $ \Phi = (\phi_v,\phi_w) $, represented in red and magenta, respectively. Here the neural network was trained with Architecture 1

    Figure 4.  Same as Figure 3, but with the neural network trained with Architecture 2 as $ \Phi = (\phi_v,\phi_w) $

    Table 1.  Description of the two types of architectures

    Architecture act. function nb of layers width of hidden layers
    1 GCU 7 2
    2 Tanh 5 8
     | Show Table
    DownLoad: CSV
  • [1] R. R. Aliev and A. V. Panfilov, A simple two-variable model of cardiac excitation, Chaos, Solitons & Fractals, 7 (1996), 293-301.  doi: 10.1016/0960-0779(95)00089-5.
    [2] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. Ⅰ, vol. 89 of Monographs in Mathematics, Birkhäuser Boston Inc., Boston MA, 1995, Abstract linear theory. doi: 10.1007/978-3-0348-9221-6.
    [3] H. Amann and P. Quittner, Semilinear parabolic equations involving measures and low regularity data, Trans. Amer. Math. Soc., 356 (2004), 1045-1119.  doi: 10.1090/S0002-9947-03-03440-8.
    [4] H. Amann and P. Quittner, Optimal control problems governed by semilinear parabolic equations with low regularity data, Adv. Differential Equations, 11 (2006), 1-33. 
    [5] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.
    [6] J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Functional Analysis, 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5.
    [7] T. Breiten and K. Kunisch, Riccati-based feedback control of the monodomain equations with the FitzHugh-Nagumo model, SIAM J. Control Optim., 52 (2014), 4057-4081.  doi: 10.1137/140964552.
    [8] T. Breiten and K. Kunisch, Compensator design for the monodomain equations with the FitzHugh-Nagumo model, ESAIM Control Optim. Calc. Var., 23 (2017), 241-262.  doi: 10.1051/cocv/2015047.
    [9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
    [10] S. L. Brunton, M. Budišić, E. Kaiser and J. N. Kutz, Modern koopman theory for dynamical systems, arXiv, arXiv: 2102.12086.
    [11] S. L. BruntonJ. L. Proctor and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proceedings of the National Academy of Sciences, 113 (2016), 3932-3937.  doi: 10.1073/pnas.1517384113.
    [12] S. L. Brunton, J. L. Proctor and J. N. Kutz, Sparse identification of nonlinear dynamics with control (sindyc)*, IFAC-PapersOnLine, 49 (2016), 710-715, 10th IFAC Symposium on Nonlinear Control Systems NOLCOS 2016. doi: 10.1016/j.ifacol.2016.10.249.
    [13] Y. CheL.-H. GengC. HanS. Cui and J. Wang, Parameter estimation of the fitzhugh-nagumo model using noisy measurements for membrane potential, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 023139.  doi: 10.1063/1.4729458.
    [14] C. ChristofC. MeyerS. Walther and C. Clason, Optimal control of a non-smooth semilinear elliptic equation, Math. Control Relat. Fields, 8 (2018), 247-276.  doi: 10.3934/mcrf.2018011.
    [15] G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems, 2 (1989), 303-314.  doi: 10.1007/BF02551274.
    [16] G. Dal Maso, An Introduction to $\Gamma$-Convergence, vol. 8 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston Inc., Boston MA, 1993. doi: 10.1007/978-1-4612-0327-8.
    [17] R. DenkM. Hieber and J. Prüss, $\mathscr R$-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type, Mem. Amer. Math. Soc., 166 (2003), viii+114.  doi: 10.1090/memo/0788.
    [18] K. DisserA. F. M. ter Elst and J. Rehberg, On maximal parabolic regularity for non-autonomous parabolic operators, J. Differential Equations, 262 (2017), 2039-2072.  doi: 10.1016/j.jde.2016.10.033.
    [19] R. O. Doruk and L. Abosharb, Estimating the parameters of fitzhugh-nagumo neurons from neural spiking data, Brain Sciences, 9 (2019). doi: 10.3390/brainsci9120364.
    [20] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466, URL http://www.sciencedirect.com/science/article/pii/S0006349561869026. doi: 10.1016/S0006-3495(61)86902-6.
    [21] H. FrankowskaE. M. Marchini and M. Mazzola, Necessary optimality conditions for infinite dimensional state constrained control problems, J. Differential Equations, 264 (2018), 7294-7327.  doi: 10.1016/j.jde.2018.02.012.
    [22] S. FrescaA. ManzoniL. Dedè and A. Quarteroni, Deep learning-based reduced order models in cardiac electrophysiology, PLOS ONE, 15 (2020), 1-32.  doi: 10.1371/journal.pone.0239416.
    [23] M. O. Gani and T. Ogawa, Stability of periodic traveling waves in the Aliev-Panfilov reaction-diffusion system, Commun. Nonlinear Sci. Numer. Simul., 33 (2016), 30-42.  doi: 10.1016/j.cnsns.2015.09.002.
    [24] S. GöktepeJ. Wong and E. Kuhl, Atrial and ventricular fibrillation: Computational simulation of spiral waves in cardiac tissue, Archive of Applied Mechanics, 80 (2010), 569-580.  doi: 10.1007/s00419-009-0384-0.
    [25] I. GoodfellowY. Bengio and  A. CourvilleDeep Learning, Adaptive Computation and Machine Learning, MIT Press, Cambridge, MA, 2016. 
    [26] K. Groeger, A $W^{1, p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), 679-687.  doi: 10.1007/BF01442860.
    [27] K. Groeger, $W^{1, p}$-estimates of solutions to evolution equations corresponding to nonsmooth second order elliptic differential operators, Nonlinear Anal., 18 (1992), 569-577.  doi: 10.1016/0362-546X(92)90211-V.
    [28] K. Groeger and J. Rehberg, Resolvent estimates in $W^{-1, p}$ for second order elliptic differential operators in case of mixed boundary conditions, Math. Ann., 285 (1989), 105-113.  doi: 10.1007/BF01442675.
    [29] P. Grohs, D. Perekrestenko, D. Elbrächter and H. Bölcskei, Deep neural network approximation theory, arXiv, arXiv: 1901.02220.
    [30] R. Haller-Dintelmann and J. Rehberg, Maximal parabolic regularity for divergence operators including mixed boundary conditions, J. Differential Equations, 247 (2009), 1354-1396.  doi: 10.1016/j.jde.2009.06.001.
    [31] M. HieberN. KajiwaraK. Kress and P. Tolksdorf, The periodic version of the Da Prato-Grisvard theorem and applications to the bidomain equations with FitzHugh-Nagumo transport, Ann. Mat. Pura Appl. (4), 199 (2020), 2435-2457.  doi: 10.1007/s10231-020-00975-6.
    [32] M. Hieber and J. Prüss, On the bidomain problem with FitzHugh-Nagumo transport, Arch. Math. (Basel), 111 (2018), 313-327.  doi: 10.1007/s00013-018-1188-7.
    [33] M. Hieber and J. Prüss, Bounded $H^\infty$-calculus for a class of nonlocal operators: The bidomain operator in the $L_q$-setting, Math. Ann., 378 (2020), 1095-1127.  doi: 10.1007/s00208-019-01916-2.
    [34] K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks, 4 (1991), 251-257.  doi: 10.1016/0893-6080(91)90009-T.
    [35] K. KahemanJ. N. Kutz and S. L. Brunton, Sindy-pi: A robust algorithm for parallel implicit sparse identification of nonlinear dynamics, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 476 (2020), 20200279.  doi: 10.1098/rspa.2020.0279.
    [36] K. Kunisch and M. Wagner, Optimal control of the bidomain system (Ⅰ): The monodomain approximation with the Rogers-McCulloch model, Nonlinear Anal. Real World Appl., 13 (2012), 1525-1550.  doi: 10.1016/j.nonrwa.2011.11.003.
    [37] K. Kunisch and M. Wagner, Optimal control of the bidomain system (Ⅱ): Uniqueness and regularity theorems for weak solutions, Ann. Mat. Pura Appl. (4), 192 (2013), 951-986.  doi: 10.1007/s10231-012-0254-1.
    [38] Y. LeCunY. Bengio and G. Hinton, Deep learning, Nature, 521 (2015), 436-444.  doi: 10.1038/nature14539.
    [39] M. LeshnoV. Y. LinA. Pinkus and S. Schocken, Multilayer feedforward networks with a nonpolynomial activation function can approximate any function", Neural Networks, 6 (1993), 861-867.  doi: 10.1016/S0893-6080(05)80131-5.
    [40] D. A. Messenger and D. M. Bortz, Weak sindy for partial differential equations, Journal of Computational Physics, 443 (2021), 110525.  doi: 10.1016/j.jcp.2021.110525.
    [41] C. Meyer and L. M. Susu, Optimal control of nonsmooth, semilinear parabolic equations, SIAM J. Control Optim., 55 (2017), 2206-2234.  doi: 10.1137/15M1040426.
    [42] M. P. Nash and A. V. Panfilov, Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias, Progress in Biophysics and Molecular Biology, 85 (2004), 501-522, Modelling Cellular and Tissue Function. doi: 10.1016/j.pbiomolbio.2004.01.016.
    [43] S. Pagani and A. Manzoni, Enabling forward uncertainty quantification and sensitivity analysis in cardiac electrophysiology by reduced order modeling and machine learning, International Journal for Numerical Methods in Biomedical Engineering, 37 (2021), e3450, URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cnm.3450.
    [44] J. PrüssG. Simonett and M. Wilke, Critical spaces for quasilinear parabolic evolution equations and applications, J. Differential Equations, 264 (2018), 2028-2074.  doi: 10.1016/j.jde.2017.10.010.
    [45] M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim., 7 (1997), 26-33.  doi: 10.1137/S1052623494266365.
    [46] Y. Renard and Konstantinos Poulios, GetFEM: Automated FE modeling of multiphysics problems based on a generic weak form language, 2020, URL https://hal.archives-ouvertes.fr/hal-02532422/file/gwfl_paper_prepub.pdf.
    [47] H. Triebel, Theory of Function Spaces, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2010, Reprint of 1983 edition [MR0730762], Also published in 1983 by Birkhäuser Verlag [MR0781540].
    [48] F. Tröltzsch, Optimal Control of Partial Differential Equations, vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010, Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels. doi: 10.1090/gsm/112.
  • 加载中




Article Metrics

HTML views(1831) PDF downloads(59) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint