Spatiotemporal flows of neural activity, such as traveling waves, have been observed throughout the brain since the earliest recordings; yet there is still little consensus on their functional role. Recent experiments and models have linked traveling waves to visual and physical motion, but these observations have been difficult to reconcile with standard accounts of topographically organized selectivity and feedforward receptive fields. Here, we introduce a theoretical framework that formalizes and generalizes the connection between 'motion' and flowing neural dynamics in the language of equivariant neural network theory. We consider 'motion' not only in physical or visual spaces, but also in more abstract representational spaces, and we argue that recurrent traveling-wave-like dynamics are not just useful but necessary for accurate and stable processing of any signal undergoing such motion. Formally, we show that for any non-trivial recurrent neural network to process a sequence undergoing a flow transformation (such as visual motion) in a structured equivariant manner, its hidden state dynamics must actively realize a homomorphic representation of the same flow through recurrent connectivity. In this ''spatiotemporal perspective on dynamical computation'', traveling waves and related flows are best understood as faithful dynamic representations of stimulus flows; and consequently the natural inclination of biological systems towards such dynamics may be viewed as an innate inductive bias towards efficiency and generalization in the spatiotemporally-structured dynamical world they inhabit.

翻译：自最早的记录以来，神经活动的时空流（如行波）在整个大脑中已被观测到；然而，关于其功能作用仍鲜有共识。最近的实验和模型已将行波与视觉和物理运动联系起来，但这些观察结果难以与地形组织选择性和前馈感受野的标准解释相协调。本文引入一个理论框架，该框架利用等变神经网络理论的语言，形式化并推广了“运动”与流动神经动力学之间的联系。我们不仅考虑物理或视觉空间中的“运动”，还考虑更抽象的表征空间中的“运动”，并认为循环的行波状动力学不仅有用，而且对于准确稳定地处理任何经历此类运动的信号是必要的。形式上，我们证明，对于任何非平凡的循环神经网络要以结构化的等变方式处理经历流变换（如视觉运动）的序列，其隐藏状态动力学必须通过循环连接主动实现同一流的同态表征。在这种“动态计算的时空视角”中，行波及相关的流最好被理解为刺激流的忠实动态表征；因此，生物系统对此类动力学的自然倾向可被视为对其所栖息的时空结构化动态世界中效率与泛化的一种先天归纳偏置。