Experimental evidence indicates that intrinsic temporal dynamics operating across multiple time scales are closely associated with the emergence of periodic spatial activity of increasing complexity. However, how information encoded in grid-like firing patterns for path integration is processed across these intrinsic time scales remains unclear. To address this question, we introduce adaptive time scales through a leak term in recurrent neural networks (RNNs), forming leaky RNNs discretized from the continuous attractors of firing rate models. Our results demonstrate that leaky RNNs substantially enhance the emergence of well-defined and highly regular hexagonal firing patterns. Compared with vanilla RNNs lacking a leak term, the trained leaky RNNs produce more accurate position estimates while generating reliable grid-cell-like representations. Furthermore, under identical noise conditions, leaky RNNs consistently exhibit more stable dynamics and better-defined grid structures. The learned dynamics also give rise to stable torus attractors with a clear central hole, supporting robust and regular grid-like activity. Overall, the dynamic leak acts as a low-pass filtering mechanism that protects recurrent neural circuitry from noise, stabilizes network dynamics, and improves path-integration accuracy in recurrent neural networks.

翻译：实验证据表明，跨多时间尺度运行的固有时间动力学与逐渐复杂的周期性空间活动的涌现密切相关。然而，在路径积分过程中，以网格样放电模式编码的信息如何在这些固有时间尺度上被处理仍不清楚。为解决这一问题，我们通过递归神经网络中的泄漏项引入自适应时间尺度，形成从连续吸引子放电率模型离散化的泄漏递归神经网络。结果表明，泄漏递归神经网络显著增强了清晰规则六边形放电模式的涌现。与缺乏泄漏项的普通递归神经网络相比，训练后的泄漏递归神经网络在生成可靠网格细胞样表征的同时，产生了更精确的位置估计。此外，在相同噪声条件下，泄漏递归神经网络始终表现出更稳定的动力学和更清晰的网格结构。学习到的动力学还产生了具有清晰中心孔的稳定环面吸引子，支持稳健规则的网格样活动。总体而言，动态泄漏作为一种低通滤波机制，保护递归神经回路免受噪声干扰、稳定网络动力学，并提高了递归神经网络中的路径积分精度。