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Dynamics of neurons in the pre-Bötzinger complex under magnetic flow effect

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Abstract

Breathing is a complex rhythmic movement. The dynamics of neuron firing activity has important implications in understanding the causes of pathological respiratory rhythm. Studies on electrophysiology have shown that the internal and external bioelectricity of nervous system may play important roles on firing activities of neuron. In this paper, the magnetic flow is added as a new variable to the Butera model to investigate the effect of electromagnetic induction on neuronal activities. The effect of magnetic flow on membrane potential is described by imposing additive memristive current on the membrane variable. The memristive current depends on the variation of magnetic flow. Direct and alternating currents of external stimulus are also added to the membrane potential. Dynamics of this modified model is discussed to consider the influence of magnetic flux on the membrane potential under different conditions of direct and alternating currents. The results show that the magnetic flux makes the pre-BötC neuron oscillate under a lower value of the direct current. Regular bursting and the mixed modes bursting types can be observed by changing the external condition and stimulus. Further studies on the combination effects of the parameter \({k_1}\) and the direct and alternating currents are performed by two-parameter bifurcation analysis.

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Acknowledgements

This research is supported by National Natural Science Foundation of China under Grant No. 11472009, Construction Plan for Innovative Research Team of North China University of Technology under Grant No. XN018010, and Scientific Research for Undergraduate of North China University of Technology.

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Authors and Affiliations

  1. School of Science, North China University of Technology, Beijing, 100144, China

    Lixia Duan, Qinyu Cao & Zijian Wang

  2. Department of Mathematics, University of Texas at Arlington, Arlington, TX, 76019, USA

    Jianzhong Su

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  1. Lixia Duan
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  2. Qinyu Cao
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Correspondence to Lixia Duan.

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Appendix

Appendix

For \(x\in \{mP,m,h,n\}\), the function of \(x_{\infty }(v)\) takes the form \(x_{\infty }(v)=\{1+\exp (v-\theta _{x}/\sigma _{x})\}^{-1}\). For \(x\in \{h,n\}\), the function of \(\tau _{x}(v)\) takes the form \(\tau _{x}(v)=\tau _{x}/\cosh [(v-\theta _{x})/2\sigma _{x}]\). The parameter values are shown in Table 1.

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Duan, L., Cao, Q., Wang, Z. et al. Dynamics of neurons in the pre-Bötzinger complex under magnetic flow effect. Nonlinear Dyn 94, 1961–1971 (2018). https://doi.org/10.1007/s11071-018-4468-7

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