  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.
