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Mean field derivation and validity verification of neural networks coupled by Izhikevich neurons

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Abstract

A significant aim of neuroscience and nonlinear dynamics is to understand how brain function emerges from the collective dynamics of neuronal networks. However, theoretical analysis and macroscopic understanding of neural networks are difficult because of their high dimension and computational complexity. Therefore, the concept of the mean field is proposed. Nevertheless, existing mean field models still need to improve. For example, the mean field model of neural networks composed of one-dimensional quadratic integral-and-fire (IF) neurons lacks adaptive variables and cannot simulate the necessary dynamic behavior of burst firing. The mean field of the neural network composed of the two-dimensional IF neurons lacks the expression related to membrane potential, and membrane potential is the critical variable of neural dynamics. In order to solve the above problems, this paper proposes a more accurate and comprehensive mean field model (a three-dimensional nonsmooth differential equation composed of mean membrane potential, mean adaptive variable, and mean firing rate) by using the conservation law of the number of neurons, which can not only analyze burst firing but also have expressions related to membrane potential. Moreover, the agreement degree between the original model and the derived mean field model, and the validity of this mean field model are verified by comparing the two models through numerical simulation.

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Availability of data and materials

The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request.

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Funding

Funding was received from the National Natural Science Foundation of China (No.12262002, No.11872154), Natural Science Foundation of Guangxi Province (2021GXNSFAA196076, 2021GXNSFFA196006), and The Guangxi Science and Technology base and Talent Project (Grant No.AD22080047).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Center for Applied Mathematics of Guangxi, Guangxi University, Nanning, 530004, China

    Jieqiong Xu, Junjie Wang, Qixiang Xu & Jimin Qiu

  2. Scientific Research Center of Engineering Mechanics, Guangxi University, Nanning, 530004, China

    Jieqiong Xu

  3. Beijing Institute of Control and Electronic Technology, Beijing, 102308, China

    Jie Fang

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Contributions

Jieqiong Xu conceived, designed, and performed the study, analyzed the data, reviewed drafts of the paper, and approved the final draft. Junjie Wang analyzed the data and authored or reviewed drafts of the paper. Qixiang Xu, Jie Fang, and Jimin Qiu analyzed the data and reviewed drafts of the paper.

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Correspondence to Junjie Wang.

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Xu, J., Wang, J., Xu, Q. et al. Mean field derivation and validity verification of neural networks coupled by Izhikevich neurons. Nonlinear Dyn 111, 22567–22593 (2023). https://doi.org/10.1007/s11071-023-09009-1

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