Background It is a crucial task of brain science researches to explore functional connective maps of Biological Neural Networks (BNN). The maps help to deeply study the dominant relationship between the structures of the BNNs and their network functions. Results In this study, the ideas of linear Granger causality modeling and causality identification are extended to those of nonlinear Granger causality modeling and network structure identification. We employed Radial Basis Functions to fit the nonlinear multivariate dynamical responses of BNNs with neuronal pulse firing. By introducing the contributions from presynaptic neurons and detecting whether the predictions for postsynaptic neurons’ pulse firing signals are improved or not, we can reveal the information flows distribution of BNNs. Thus, the functional connections from presynaptic neurons can be identified from the obtained network information flows. To verify the effectiveness of the proposed method, the Nonlinear Granger Causality Identification Method (NGCIM) is applied to the network structure discovery processes of Spiking Neural Networks (SNN). SNN is a simulation model based on an Integrate-and-Fire mechanism. By network simulations, the multi-channel neuronal pulse sequence data of the SNNs can be used to reversely identify the synaptic connections and strengths of the SNNs. Conclusions The identification results show: for 2–6 nodes small-scale neural networks, 20 nodes medium-scale neural networks, and 100 nodes large-scale neural networks, the identification accuracy of NGCIM with the Gaussian kernel function was 100%, 99.64%, 98.64%, 98.37%, 98.31%, 84.87% and 80.56%, respectively. The identification accuracies were significantly higher than those of a traditional Linear Granger Causality Identification Method with the same network sizes. Thus, with an accumulation of the data obtained by the existing measurement methods, such as Electroencephalography, functional Magnetic Resonance Imaging, and Multi-Electrode Array, the NGCIM can be a promising network modeling method to infer the functional connective maps of BNNs.

中文翻译：

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非线性格兰杰因果关系法识别脉冲神经元网络结构

背景探索生物神经网络（BNN）的功能连接图是脑科学研究的一项重要任务。这些地图有助于深入研究 BNN 的结构与其网络功能之间的主导关系。结果本研究将线性格兰杰因果关系建模和因果关系识别的思想扩展到非线性格兰杰因果关系建模和网络结构识别的思想。我们采用径向基函数来拟合具有神经元脉冲发射的 BNN 的非线性多元动态响应。通过引入突触前神经元的贡献，并检测突触后神经元脉冲发射信号的预测是否有所改善，我们可以揭示 BNN 的信息流分布。因此，突触前神经元的功能连接可以从获得的网络信息流中识别出来。为了验证所提出方法的有效性，将非线性格兰杰因果关系识别方法（NGCIM）应用于尖峰神经网络（SNN）的网络结构发现过程。SNN是一种基于Integrate-and-Fire机制的仿真模型。通过网络模拟，可以利用 SNN 的多通道神经元脉冲序列数据反向识别 SNN 的突触连接和强度。结论 识别结果表明：对于2~6节点小型神经网络、20节点中型神经网络、100节点大型神经网络，NGCIM高斯核函数的识别准确率分别为100%、99.64% , 98.64%, 98.37%, 98.31%, 84.87% 和 80.56%, 分别。识别精度明显高于相同网络规模的传统线性格兰杰因果识别方法。因此，通过积累现有测量方法（如脑电图、功能磁共振成像和多电极阵列）获得的数据，NGCIM 可以成为一种很有前景的网络建模方法，用于推断 BNN 的功能连接图。