December  2019, 8(4): 883-902. doi: 10.3934/eect.2019043

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

Received  January 2019 Published  June 2019

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.

Citation: Cecilia Cavaterra, Denis Enăchescu, Gabriela Marinoschi. Sliding mode control of the Hodgkin–Huxley mathematical model. Evolution Equations & Control Theory, 2019, 8 (4) : 883-902. doi: 10.3934/eect.2019043
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.  Google Scholar

[2]

V. BarbuP. ColliG. GilardiG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

F. R. ChavaretteJ. M. BalthazarM. 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.  Google Scholar

[6]

Y. CheJ. WangB. DengX. 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.  Google Scholar

[7]

Y. CheB. LiuH. LiM. LuJ. 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.  Google Scholar

[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.  Google Scholar

[9]

P. ColliG. GilardiG. 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.  Google Scholar

[10] J. Cronin, Mathematical Aspects of Hodgkin-Huxley Neural Theory, Cambridge Univ. Press, Cambridge, 1987.  doi: 10.1017/CBO9780511983955.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

W. E. FitzgibbonM. 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.  Google Scholar

[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.   Google Scholar

[21]

J. L. Lions, Quelques Méthodes de R ésolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

V. BarbuP. ColliG. GilardiG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

F. R. ChavaretteJ. M. BalthazarM. 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.  Google Scholar

[6]

Y. CheJ. WangB. DengX. 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.  Google Scholar

[7]

Y. CheB. LiuH. LiM. LuJ. 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.  Google Scholar

[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.  Google Scholar

[9]

P. ColliG. GilardiG. 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.  Google Scholar

[10] J. Cronin, Mathematical Aspects of Hodgkin-Huxley Neural Theory, Cambridge Univ. Press, Cambridge, 1987.  doi: 10.1017/CBO9780511983955.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

W. E. FitzgibbonM. 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.  Google Scholar

[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.   Google Scholar

[21]

J. L. Lions, Quelques Méthodes de R ésolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Graphics $ v(t, 0) $ (left), $ v(t, x) $ (center), $ n $, $ m $, $ h $ (right) for $ v_0 = 4.82 $, $ v^* = 0 $, $ \rho = 0 $
Figure 2.  Graphics $ v(t, 0) $ (left), $ v(t, x) $ (center), $ n $, $ m $, $ h $ (right) for $ v_0 = 4.82 $, $ v^* = 0 $, $ \rho = 20 $
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 $
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 $
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 $
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 $
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