Sliding mode control of the Hodgkin–Huxley mathematical model

Cecilia Cavaterra; Denis Enăchescu; Gabriela Marinoschi

doi:10.3934/eect.2019043

Article Contents

2019, Volume 8, Issue 4: 883-902. Doi: 10.3934/eect.2019043

This issue Previous Article Existence and extinction in finite time for Stratonovich gradient noise porous media equations Next Article

Sliding mode control of the Hodgkin–Huxley mathematical model

1.
Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy
2.
"Gheorghe Mihoc-Caius Iacob" Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Calea 13 Septembrie 13, Bucharest, Romania
3.
Research Group of the Project PN-Ⅲ-P4-ID-PCE-2016-0372, Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania
4.
Istituto di Matematica Applicata e Tecnologie Informatiche "E. Magenes", CNR, Via Ferrata 1, 27100 Pavia, Italy

^* Corresponding author: Gabriela Marinoschi
^* Corresponding author: Gabriela Marinoschi

Received: December 31, 2018

Early access: June 2019

Published: June 2019

Abstract Full Text(HTML) Figure(6) Related Papers Cited by

Abstract

In this paper we deal with a feedback control design for the action potential of a neuronal membrane in relation with the non-linear dynamics of the Hodgkin-Huxley mathematical model. More exactly, by using an external current as a control expressed by a relay graph in the equation of the potential, we aim at forcing it to reach a certain manifold in finite time and to slide on it after that. From the mathematical point of view we solve a system involving a parabolic differential inclusion and three nonlinear differential equations via an approximating technique and a fixed point result. The existence of the sliding mode and the determination of the time at which the potential reaches the prescribed manifold are proved by a maximum principle argument. Numerical simulations are presented.

Keywords:

Mathematics Subject Classification: Primary: 35K55, 35K57, 35Q92; Secondary: 93B52, 92C30.

Citation:

Full Text(HTML)

Figure 1. Graphics $ v(t, 0) $ (left), $ v(t, x) $ (center), $ n $, $ m $, $ h $ (right) for $ v_0 = 4.82 $, $ v^* = 0 $, $ \rho = 0 $

Download: Full-size image PowerPoint slide

Figure 2. Graphics $ v(t, 0) $ (left), $ v(t, x) $ (center), $ n $, $ m $, $ h $ (right) for $ v_0 = 4.82 $, $ v^* = 0 $, $ \rho = 20 $

Download: Full-size image PowerPoint slide

Figure 3. Graphics $ v(t, 0) $ (left), $ v(t, x) $ (center), $ n $, $ m $, $ h $ (right) for $ v_0 = 4.82 $, $ v^* = 0.5\sin(4/\pi *t)+0.6 $, $ \rho = 20 $

Download: Full-size image PowerPoint slide

Figure 4. Graphics $ v(t, 0) $ (left), $ v(t, x) $ (center), $ n $, $ m $, $ h $ (right) for $ v_0 = 4.82 $, $ v^* = 0 $, $ \rho = 0 $, $ g_K = 3.8229 $

Download: Full-size image PowerPoint slide

Figure 5. Graphics $ v(t, 0) $ (left), $ v(t, x) $ (center), $ n $, $ m $, $ h $ (right) for $ v_0 = 4.82 $, $ v^* = 0 $, $ \rho = 20 $, $ g_K = 3.8229 $

Download: Full-size image PowerPoint slide

Figure 6. Graphics $ v(t, 0) $ (left), $ v(t, x) $ (center), $ n $, $ m $, $ h $ at $ x = 0.5 $ (right) for $ v_0 = 0.5sin(4\pi x)+0.6 $, $ v^* = 0 $, $ \delta = 50 $, $ g_K = 36 $, $ \rho = 50 $

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.
[2]	V. Barbu, P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a nonlinear phase-field system, SIAM J. Control Optim., 55 (2017), 2108-2133. doi: 10.1137/15M102424X.
[3]	E. N. Best, Null space in the Hodgkin-Huxley equations, A critical test, Biophys. J., 27 (1979), 87-104. doi: 10.1016/S0006-3495(79)85204-2.
[4]	C. Cavaterra and M. Grasselli, Robust exponential attractors for singularly perturbed Hodgkin-Huxley equations, J. Differential Equations, 246 (2009), 4670-4701. doi: 10.1016/j.jde.2008.12.025.
[5]	F. R. Chavarette, J. M. Balthazar, M. Rafikov and H. A. Hermini, On non-linear dynamics and an optimal control synthesis of the action potential of membranes (ideal and non-ideal cases) of the Hodgkin-Huxley (HH) mathematical model, Chaos, Solitons and Fractals, 39 (2009), 1651-1666. doi: 10.1016/j.chaos.2007.06.016.
[6]	Y. Che, J. Wang, B. Deng, X. Wei and C. Han, Bifurcations in the hodgkin–huxley model exposed to DC electric fields, Neurocomputing, 81 (2012), 41-48. doi: 10.1016/j.neucom.2011.11.019.
[7]	Y. Che, B. Liu, H. Li, M. Lu, J. Wang and X. Wei, Robust stabilization control of bifurcations in Hodgkin-Huxley model with aid of unscented Kalman filter, Chaos, Solitons and Fractals, 101 (2017), 92-99. doi: 10.1016/j.chaos.2017.04.045.
[8]	P. Colli, M. Colturato, Global existence for a singular phase field system related to a sliding mode control problem, Nonlinear Anal. Real World Appl. 41 (2018), 128-151. doi: 10.1016/j.nonrwa.2017.10.011.
[9]	P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for phase field system related to tumor growth, Appl. Math.Optimiz., 79 (2019), 647-670. doi: 10.1007/s00245-017-9451-z.
[10]	J. Cronin, Mathematical Aspects of Hodgkin-Huxley Neural Theory, Cambridge Univ. Press, Cambridge, 1987. doi: 10.1017/CBO9780511983955.
[11]	J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comp. Appl. Math., 6 (1980), 19-26. doi: 10.1016/0771-050X(80)90013-3.
[12]	R. Ozgur Doruk, Feedback controlled electrical nerve stimulation: A computer simulation, Computer Methods and Programs in Biomedicine, 99 (2010), 98-112. doi: 10.1016/j.cmpb.2010.01.006.
[13]	R. Ozgur Doruk, Control of repetitive firing in Hodgkin-Huxley nerve fibers using electric fields, Chaos, Solitons & Fractals, 52 (2013), 66-72. doi: 10.1016/j.chaos.2013.04.003.
[14]	J. W. Evans, Nerve axon equations Ⅰ: Linear approximations, Indiana Univ. Math. J., 21 (1972), 877-885. doi: 10.1512/iumj.1972.21.21071.
[15]	J. W. Evans, Nerve axon equations Ⅱ: stability at rest, Indiana Univ. Math. J., 22 (1972), 75-90. doi: 10.1512/iumj.1973.22.22009.
[16]	J. W. Evans, Nerve axon equations Ⅲ: stability of the nerve impulse, Indiana Univ. Math. J., 22 (1972), 577-593. doi: 10.1512/iumj.1973.22.22048.
[17]	J. W. Evans and N. A. Shenk, Solutions to axon equations, Biophys. J., 10 (1970), 1090-1101. doi: 10.1016/S0006-3495(70)86355-X.
[18]	W. E. Fitzgibbon, M.E. Parrott, Y.You, Global dynamics of singularly perturbed Hodgkin-Huxley equations, In: Semigroups of Linear and Nonlinear Operations and Applications (eds. G. Ruiz Goldstein, J. Goldstein), Springer Science+ Business Media B.V., Dordrecht, (1993), 159–176.
[19]	W. E. Fitzgibbon, M. E. Parrott and Y. You, Finite dimensionality and uppper semicontinuity of the global attractor of singularly perturbed Hodgkin Huxley systems, J. Diff. Equations, 129 (1996), 193-237. doi: 10.1006/jdeq.1996.0116.
[20]	A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544.
[21]	J. L. Lions, Quelques Méthodes de R ésolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.
[22]	M. Mascagni, An initial–boundary value problem of physiological significance for equations of nerve conduction, Comm. Pure Appl. Math., 42 (1989), 213-227. doi: 10.1002/cpa.3160420206.
[23]	L. F. Shampine and M. W. Reichelt, The MATLAB ODE Suite, SIAM Journal on Scientific Computing, 18 (1997), 1-22. doi: 10.1137/S1064827594276424.
[24]	R. D. Skeel and M. Berzins, A method for the spatial discretization of parabolic equations in one space variable, SIAM Journal on Scientific and Statistical Computing, 11 (1990), 1-32. doi: 10.1137/0911001.