Continuous-time recurrent neural networks are widely used as models of neural dynamics and also have applications in machine learning. But their dynamics are not yet well understood, especially when they are driven by external stimuli. In this article, we study the response of stable and unstable networks to different harmonically oscillating stimuli by varying a parameter ρ, the ratio between the timescale of the network and the stimulus, and use the dimensionality of the network’s attractor as an estimate of the complexity of this response. Additionally, we propose a novel technique for exploring the stationary points and locally linear dynamics of these networks in order to understand the origin of input-dependent dynamical transitions. Attractors in both stable and unstable networks show a peak in dimensionality for intermediate values of ρ, with the latter consistently showing a higher dimensionality than the former, which exhibit a resonance-like phenomenon. We explain changes in the dimensionality of a network’s dynamics in terms of changes in the underlying structure of its vector field by analysing stationary points. Furthermore, we uncover the coexistence of underlying attractors with various geometric forms in unstable networks. As ρ is increased, our visualisation technique shows the network passing through a series of phase transitions with its trajectory taking on a sequence of qualitatively distinct figure-of-eight, cylinder, and spiral shapes. These findings bring us one step closer to a comprehensive theory of this important class of neural networks by revealing the subtle structure of their dynamics under different conditions.

中文翻译：

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激发速率神经元的谐波驱动随机网络中动态跃迁的研究。

连续时间递归神经网络被广泛用作神经动力学模型，并且在机器学习中也有应用。但是它们的动力学尚未得到很好的理解，特别是当它们是由外部刺激驱动时。在本文中，我们通过改变参数ρ来研究稳定和不稳定网络对不同谐波振荡刺激的响应。，网络时间尺度与刺激之间的比率，并使用网络吸引子的维数作为此响应复杂性的估算。此外，我们提出了一种探索这些网络的固定点和局部线性动力学的新颖技术，以了解与输入有关的动态过渡的起源。稳定和不稳定网络中的吸引子在ρ的中间值时都显示出一个维数峰值，后者始终显示出比前者更高的尺寸，前者表现出类似共振的现象。通过分析静态点，我们根据网络矢量场的基础结构的变化来解释网络动力学的维数变化。此外，我们发现不稳定网络中具有各种几何形式的潜在吸引子并存。随着ρ的增加，我们的可视化技术显示网络经过一系列相变，其轨迹呈现出一系列定性不同的8字形，圆柱和螺旋形。这些发现通过揭示在不同条件下动力学的微妙结构，使我们更接近这一重要神经网络类的综合理论。