# American Institute of Mathematical Sciences

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
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##### References:
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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