This paper proposes a novel controller to control position, amplitude and frequency of periodic firing activity in Hindmarsh–Rose model based on Hopf bifurcation theory which is composed of linear control gain and nonlinear control gain. First, we select the activation of the fast ion channel as control parameter. Based on explicit criterion of Hopf bifurcation, a series of conditions are obtained to derive the linear gains of controller responsible for control of the location where the periodic firing activity occurs. Then, based on the control parameter, a series of conditions are obtained to derive the nonlinear gains of controller responsible for controlling the amplitude and frequency of periodic firing activity by using center manifold and normal form. Finally, the numerical experiments show that our controller can make the periodic firing activity occur at designed value and control the amplitude and frequency of periodic firing activity by adjusting nonlinear control gain of controller.

基于Hopf分岔理论，提出了一种新的控制器来控制Hindmarsh-Rose模型中周期性发射活动的位置、幅度和频率，该控制器由线性控制增益和非线性控制增益组成。首先，我们选择快离子通道的激活作为控制参数。基于Hopf分岔的明确标准，获得了一系列条件来推导负责控制周期性点火活动发生位置的控制器的线性增益。然后，在控制参数的基础上，利用中心流形和范式得到一系列条件，推导出负责控制周期性发射活动幅度和频率的控制器的非线性增益。最后，