We examine the dynamical properties of a single-layer convolutional recurrent network with a smooth sigmoidal activation function, for small values of the inputs and when the convolution kernel is unitary, so all eigenvalues lie exactly at the unit circle. Such networks have a variety of hallmark properties: the outputs depend on the inputs via compressive nonlinearities such as cubic roots, and both the timescales of relaxation and the length-scales of signal propagation depend sensitively on the inputs as power laws, both diverging as the input to 0. The basic dynamical mechanism is that inputs to the network generate ongoing activity, which in turn controls how additional inputs or signals propagate spatially or attenuate in time. We present analytical solutions for the steady states when the network is forced with a single oscillation and when a background value creates a steady state of ongoing activity, and derive the relationships shaping the value of the temporal decay and spatial propagation length as a function of this background value.

在输入较小且卷积核是幺正矩阵的条件下，我们考察了具有平滑sigmoid激活函数的单层卷积递归网络的动力学特性，其输出通过诸如立方根等压缩非线性函数与输入相关，且弛豫时间和信号传播的长度尺度都以幂律形式对输入敏感，当输入趋近于0时两者均发散。基本动力学机制是网络中的输入产生持续的活动，并进而控制附加输入或信号在空间上的传播抑或时间上的衰减。我们提供了当单个振荡强迫网络或背景值引发持续活动时的稳态的解析解，并导出了时间衰减和空间传播长度与背景值的关系。

卷积循环神经网络在其临界点附近的动力学