    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:
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##### References:
  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  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.  Google Scholar  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  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  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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  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.  Google Scholar  J. Cronin, Mathematical Aspects of Hodgkin-Huxley Neural Theory, Cambridge Univ. Press, Cambridge, 1987.  doi: 10.1017/CBO9780511983955.  Google Scholar  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  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  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  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  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  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  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  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  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.  Google Scholar  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  J. L. Lions, Quelques Méthodes de R ésolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969. Google Scholar  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  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  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 Graphics $v(t, 0)$ (left), $v(t, x)$ (center), $n$, $m$, $h$ (right) for $v_0 = 4.82$, $v^* = 0$, $\rho = 0$ Graphics $v(t, 0)$ (left), $v(t, x)$ (center), $n$, $m$, $h$ (right) for $v_0 = 4.82$, $v^* = 0$, $\rho = 20$ Graphics $v(t, 0)$ (left), $v(t, x)$ (center), $n$, $m$, $h$ (right) for $v_0 = 4.82$, $v^* = 0.5\sin(4/\pi *t)+0.6$, $\rho = 20$ Graphics $v(t, 0)$ (left), $v(t, x)$ (center), $n$, $m$, $h$ (right) for $v_0 = 4.82$, $v^* = 0$, $\rho = 0$, $g_K = 3.8229$ Graphics $v(t, 0)$ (left), $v(t, x)$ (center), $n$, $m$, $h$ (right) for $v_0 = 4.82$, $v^* = 0$, $\rho = 20$, $g_K = 3.8229$ 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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