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# Sliding mode control of the Hodgkin–Huxley mathematical model

• * Corresponding author: Gabriela Marinoschi
• 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.

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

 Citation: • • 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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