Firing patterns and bifurcation analysis of neurons under electromagnetic induction

Qixiang Wen; Shenquan Liu; Bo Lu; Qixiang Wen; Shenquan Liu; Bo Lu

doi:10.3934/era.2021034

Electronic Research Archive

2021, Volume 29, Issue 5: 3205-3226. doi: 10.3934/era.2021034

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Firing patterns and bifurcation analysis of neurons under electromagnetic induction

Qixiang Wen ^{1
,},
Shenquan Liu ^{1
,
,},
Bo Lu ^{2
,}

1.
School of Mathematics, South China University of Technology, Guangzhou 510640, China
2.
School of Science and Mathematics, Henan Institute of Science and Technology, Xinxiang 453003, China

Received: 01 November 2020 Revised: 01 March 2021 Published: 15 April 2021
Primary: 37N25, 34C23; Secondary: 37N30, 92C20

Based on the three-dimensional endocrine neuron model, a four-dimensional endocrine neuron model was constructed by introducing the magnetic flux variable and induced current according to the law of electromagnetic induction. Firstly, the codimension-one bifurcation and Interspike Intervals (ISIs) analysis were applied to study the bifurcation structure with respect to external stimuli and parameter $ k_0 $, and two dynamical behaviors were found: period-adding and period-doubling bifurcation leading to chaos. Besides, Hopf bifurcation was specially discussed corresponding to the transformation of the state. Secondly, the different firing patterns such as regular bursting, subthreshold oscillations, fast spiking, mixed-mode oscillations (MMOs) etc. can be observed by changing the external stimuli and the induced current. The neuron model presented more firing activities under strong coupling strength. Finally, the codimension-two bifurcation analysis shown more details of bifurcation. At the same time, the Bogdanov-Takens bifurcation point was also analyzed and three bifurcation curves were derived.
- Electromagnetic induction,
- bursting,
- fast-slow dynamics,
- bifurcation
Citation: Qixiang Wen, Shenquan Liu, Bo Lu. Firing patterns and bifurcation analysis of neurons under electromagnetic induction[J]. Electronic Research Archive, 2021, 29(5): 3205-3226. doi: 10.3934/era.2021034

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Abstract

Based on the three-dimensional endocrine neuron model, a four-dimensional endocrine neuron model was constructed by introducing the magnetic flux variable and induced current according to the law of electromagnetic induction. Firstly, the codimension-one bifurcation and Interspike Intervals (ISIs) analysis were applied to study the bifurcation structure with respect to external stimuli and parameter $ k_0 $, and two dynamical behaviors were found: period-adding and period-doubling bifurcation leading to chaos. Besides, Hopf bifurcation was specially discussed corresponding to the transformation of the state. Secondly, the different firing patterns such as regular bursting, subthreshold oscillations, fast spiking, mixed-mode oscillations (MMOs) etc. can be observed by changing the external stimuli and the induced current. The neuron model presented more firing activities under strong coupling strength. Finally, the codimension-two bifurcation analysis shown more details of bifurcation. At the same time, the Bogdanov-Takens bifurcation point was also analyzed and three bifurcation curves were derived.

References

[1]	Optical magnetic detection of single-neuron action potentials using quantum defects in diamond. Proc. Natl. Acad. Sci. U.S.A. (2016) 113: 14133-14138.
[2]	Multi-timescale systems and fast-slow analysis. Math. Biol. (2017) 287: 105-121.
[3]	Analysis of the Takens-Bogdanov bifurcation on m-parameterized vector fields. Internat. J. Bifur. Chaos Appl. Sci. Engrg. (2010) 20: 995-1005.
[4]	Dynamics of neurons in the pre-B\begin{document}$ \ddot{o} $\end{document}tzinger complex under magnetic flow effect. Nonlinear Dynam (2018) 94: 1961-1971.
[5]	Two-parameter bifurcation analysis of firing activities in the Chay neuronal model. Neurocomputing (2008) 72: 341-351.
[6]	H. Gu, Experimental observation of transition from chaotic bursting to chaotic spiking in a neural pacemaker, Chaos, 23 (2013), 023126. doi: 10.1063/1.4810932
[7]	F. Han, Z. Wang, Y. Du, X. Sun and B. Zhang, Robust synchronization of bursting Hodgkin-Huxley neuronal systems coupled by delayed chemical synapses, Int. J. Non. Linear. Mech, 70 (2015), 105-111. doi: 10.1016/j. ijnonlinmec. 2014.10.010
[8]	Neural excitability, spiking and bursting. Internat. J. Bifur. Chaos Appl. Sci. Engrg. (2000) 10: 1171-1266.
[9]	M. S. Kafraj, F. Parastesh and S. Jafari, Firing patterns of an improved Izhikevich neuron model under the effect of electromagnetic induction and noise, Chaos, Solitons Fractals, 137 (2020), 109782, 11 pp. doi: 10.1016/j. chaos. 2020.109782
[10]	Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^nd$ edition, Springer-Verlag, New York, 2004.
[11]	J. Li, Y. Wu, M. Du and W. Liu, Dynamic behavior in firing rhythm transitions of neurons under electromagnetic radiation, Acta Phys. Sin, 64 (2015), 030503. doi: 10.7498/aps. 64.030503
[12]	Bistability of cerebellar Purkinje cells modulated by sensory stimulation. Nature Neurosci. (2005) 8: 202-211.
[13]	B. Lu, S. Liu, X. Liu, X. Jiang and X. Jang, Bifurcation and spike adding transition in Chay-Keizer model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650090, 13 pp. doi: 10.1142/S0218127416500905
[14]	Multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Neurocomputing (2016) 205: 375-381.
[15]	Bifurcation analysis and diverse firing activities of a modified excitable neuron model. Cognitive Neurodynamics (2019) 13: 393-407.
[16]	Pattern formation in diffusive excitable systems under magnetic flow effects. Phys. Lett. A (2017) 381: 2264-2271.
[17]	Glucose modulates \begin{document}${[Ca^{2+}]}_i$\end{document} oscillations in pancreatic islets via ionic and glycolytic mechanisms. Biophys. J (2006) 91: 2082-2096.
[18]	J. Rinzel, Bursting oscillations in an excitable membrane model, Ordinary Partial Differ. Equations, Springer Berlin Heidelberg, Berlin, Heidelberg 1151 (1985), 304-316. doi: 10.1007/BFb0074739
[19]	Wild oscillations in a nonlinear neuron model with resets: (I) bursting, spike-adding and chaos. Discrete. Contin. Dyn. Syst. Ser. B (2017) 22: 3967-4002.
[20]	Real-time detection of stimulus response in cultured neurons by high-intensity intermediate-frequency magnetic field exposure. Integr. Biol (2018) 10: 442-449.
[21]	A. Saito, K. Wada, Y. Suzuki and S. Nakasono, The response of the neuronal activity in the somatosensory cortex after high-intensity intermediate-frequency magnetic field exposure to the spinal cord in rats under anesthesia and waking states, Brain Res., 1747 (2020), 147063. doi: 10.1016/j. brainres. 2020.147063
[22]	From plateau to pseudo-plateau bursting: Making the transition. Bull. Math. Biol. (2011) 73: 1292-1311.
[23]	Full system bifurcation analysis of endocrine bursting models. J. Theor. Biol. (2010) 264: 1133-1146.
[24]	Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle. Phys. D (1993) 62: 263-274.
[25]	Mixed-mode oscillations and bifurcation analysis in a pituitary model. Nonlinear Dynam. (2018) 94: 807-826.

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