Anti-control of periodic firing in HR model in the aspects of position, amplitude and frequency
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Anti-control of periodic firing in HR model in the aspects of position, amplitude and frequency Tao Dong1
•
Huiyun Zhu1
Received: 10 September 2019 / Revised: 8 August 2020 / Accepted: 16 August 2020 Ó Springer Nature B.V. 2020
Abstract 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. Keywords HR model Periodic firing Anti-control Hopf bifurcation
Introduction The Hindmarsh and Rose (HR) model (1984), proposed by J. L. Hindmarsh and R. M. Rose in 1984, is a very famous neuron model because it establishes mathematical model describing many electrical activities of the biological neurons such as bursting, periodic firing activity, spiking firing activity and so on (Hindmarsh and Rose 1984; Wouapi et al. 2020; Wu et al. 2016; Han et al. 2019; Lange and Hasler 2008; Mondal et al. 2019). There are various studies about the HR neuron model have been conducted (Hayati et al. 2016; Wang et al. 2020; Yu et al. 2020; Lakshmanan et al. 2017; Lu et al. 2019; Rajagopal et al. 2019; Ge et al. 2019; Jhou et al. 2012; Li and He 2019; Bao et al. 2020; Khanday et al. 2018). The HR model can be described as follows:
& Tao Dong [email protected] 1
Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, College of Electronics and Information Engineering, Southwest University, Chongqing 400715, People’s Republic of China
8 _ ¼ yðtÞ axðtÞ3 þ bxðtÞ2 fzðtÞ þ I; > xðtÞ > < _ ¼ c dxðtÞ2 yðtÞ; yðtÞ > > : zðtÞ _ ¼ rðsðxðtÞ xc Þ zðtÞÞ;
ð1Þ
where xðtÞ is the voltage across the cell’s membrane, yðtÞ and zðtÞ describe the exchange of ions through the neuron membrane by means of fast and slow ion channels respectively.r is related to the concentration of calcium ions, representing the efficiency of the slow channels to exchange ions.I is the current that enters the neuron. b and d denote activation and inactivation of the fast ion channel. s and xc describe activation and inactivation of the slow ion channel.a; c; f are constants. Following (Dong and Xia 2019), we choose a ¼ 1 ðmV
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