This paper is concerned with the modeling and analysis of two of the most commonly used recurrent neural network models (i.e., Hopfield neural network and firing-rate neural network) with dynamic recurrent connections undergoing Hebbian learning rules. To capture the synaptic sparsity of neural circuits we propose a low dimensional formulation. We then characterize certain key dynamical properties. First, we give biologically-inspired forward invariance results. Then, we give sufficient conditions for the non-Euclidean contractivity of the models. Our contraction analysis leads to stability and robustness of time-varying trajectories — for networks with both excitatory and inhibitory synapses governed by both Hebbian and anti-Hebbian rules. For each model, we propose a contractivity test based upon biologically meaningful quantities, e.g., neural and synaptic decay rate, maximum in-degree, and the maximum synaptic strength. Then, we show that the models satisfy Dale’s Principle. Finally, we illustrate the effectiveness of our results via a numerical example.

中文翻译：

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使用赫布学习的神经突触网络的建模和收缩性

本文关注的是两种最常用的循环神经网络模型（即 Hopfield 神经网络和放电率神经网络）的建模和分析，这些模型具有遵循 Hebbian 学习规则的动态循环连接。为了捕捉神经回路的突触稀疏性，我们提出了一种低维公式。然后我们表征某些关键的动力学特性。首先，我们给出了受生物学启发的前向不变性结果。然后，我们给出了模型的非欧几里得收缩性的充分条件。我们的收缩分析导致了时变轨迹的稳定性和鲁棒性——对于具有受赫布规则和反赫布规则控制的兴奋性和抑制性突触的网络。对于每个模型，我们提出基于生物学有意义的量的收缩性测试，例如神经和突触衰减率、最大入度和最大突触强度。然后，我们证明模型满足戴尔原理。最后，我们通过数值示例说明了结果的有效性。