Computational modeling of neurodynamical systems often deploys neural networks and symbolic dynamics. A particular way for combining these approaches within a framework called vector symbolic architectures leads to neural automata. An interesting research direction we have pursued under this framework has been to consider mapping symbolic dynamics onto neurodynamics, represented as neural automata. This representation theory, enables us to ask questions, such as, how does the brain implement Turing computations. Specifically, in this representation theory, neural automata result from the assignment of symbols and symbol strings to numbers, known as G\"odel encoding. Under this assignment symbolic computation becomes represented by trajectories of state vectors in a real phase space, that allows for statistical correlation analyses with real-world measurements and experimental data. However, these assignments are usually completely arbitrary. Hence, it makes sense to address the problem question of, which aspects of the dynamics observed under such a representation is intrinsic to the dynamics and which are not. In this study, we develop a formally rigorous mathematical framework for the investigation of symmetries and invariants of neural automata under different encodings. As a central concept we define patterns of equality for such systems. We consider different macroscopic observables, such as the mean activation level of the neural network, and ask for their invariance properties. Our main result shows that only step functions that are defined over those patterns of equality are invariant under recodings, while the mean activation is not. Our work could be of substantial importance for related regression studies of real-world measurements with neurosymbolic processors for avoiding confounding results that are dependant on a particular encoding and not intrinsic to the dynamics.

翻译：神经动力学系统的计算建模通常采用神经网络和符号动力学。在一种称为向量符号架构的框架下，将这两种方法结合起来的特定方式产生了神经自动机。我们在此框架下探索的一个有趣研究方向是考虑将符号动力学映射到神经动力学上，并以神经自动机的形式表示。这种表示理论使我们能够提出诸如“大脑如何实现图灵计算”之类的问题。具体而言，在该表示理论中，神经自动机源于将符号和符号串分配给数字（即哥德尔编码）的过程。通过这种分配，符号计算由实相空间中状态向量的轨迹表示，从而能够与实际测量和实验数据进行统计相关性分析。然而，这些分配通常完全是任意的。因此，有必要解决以下问题：在这种表示下观察到的动力学中，哪些方面是动力学本身固有的，哪些并非如此。在本研究中，我们开发了一个形式严密的数学框架，用于研究不同编码下神经自动机的对称性和不变量。作为核心概念，我们定义了此类系统的相等性模式。我们考虑了不同的宏观可观测量，例如神经网络的平均激活水平，并探讨其不变性性质。我们的主要结果表明，只有基于这些相等性模式定义的阶跃函数在重新编码下保持不变，而平均激活则不然。我们的工作对于使用神经符号处理器对实际测量进行相关回归研究具有重要价值，有助于避免因依赖于特定编码而非动力学本身而导致的混杂结果。