标题： 具有由规则动力学驱动的收缩递归网络中的平滑嵌入：神经表示的综合方法 显示英文标题

标题： Smooth embeddings in contracting recurrent networks driven by regular dynamics: A synthesis for neural representation

摘要： 具有时间序列预测能力的循环神经网络常常会发展出保留生成输入的动力系统定性结构的潜在轨迹。 最近的实证工作记录了训练后的循环模型中拓扑保持的潜在组织，而最近在储备计算中的理论结果建立了同步映射为嵌入的条件。 在这里，我们将这些线索综合成一个统一的描述，说明在广泛且生物学相关的输入类别——不变圆周和环面上的规则动力学——中，收缩循环网络何时会产生平滑且拓扑保持的内部表示。 我们的贡献是一个综合框架，该框架整合了（i）收缩储备的广义同步和嵌入保证，（ii）确保在轻微约束下同步映射可微的规则机制，以及（iii）一个基础系统视角，在该视角中，生成输入流的不变流形被视为驱动系统。 在这个规则设置中，通常在混沌吸引子分析中被视为限制的条件变得温和，并且可以被标准的收缩架构轻松满足。 该框架阐明了循环电路中的表征内容本质上是历史性的：网络状态编码的是输入历史的有限窗口，而不是瞬时刺激。 通过在共同假设下整合不同的实证和理论结果，这种综合得出了关于预测训练的循环电路何时（或不应）形成平滑潜在嵌入的具体、可测试的期望，以及所需状态维度如何随着驱动动力学的内在维度变化。 显示英文摘要

摘要： Recurrent neural networks trained for time-series prediction often develop latent trajectories that preserve qualitative structure of the dynamical systems generating their inputs. Recent empirical work has documented topology-preserving latent organization in trained recurrent models, and recent theoretical results in reservoir computing establish conditions under which the synchronization map is an embedding. Here we synthesize these threads into a unified account of when contracting recurrent networks yield smooth, topology-preserving internal representations for a broad and biologically relevant class of inputs: regular dynamics on invariant circles and tori. Our contribution is an integrated framework that assembles (i) generalized synchronization and embedding guarantees for contracting reservoirs, (ii) regularity mechanisms ensuring differentiability of the synchronization map under mild constraints, and (iii) a base-system viewpoint in which the invariant manifold generating the input stream is treated as the driving system. In this regular setting, the conditions commonly viewed as restrictive in chaotic-attractor analyses become mild and readily satisfied by standard contractive architectures. The framework clarifies how representational content in recurrent circuits is inherently historical: the network state encodes finite windows of input history rather than instantaneous stimuli. By consolidating disparate empirical and theoretical results under common assumptions, the synthesis yields concrete, testable expectations about when prediction-trained recurrent circuits should (or should not) form smooth latent embeddings and how required state dimension scales with the intrinsic dimension of the driving dynamics.