But what sort of dynamical system it is is another question. Probably a human brain is not ONE sort of dynamical system but many different sorts cooperating in all sorts of subtle and complex ways, some of them virtual machines, some neural machines, some chemical machines, and some combinations of several types. What sort of mathematics is relevant will depend on details of the system. E.g. if the main interactions between components are via forces, voltages, fields, etc. But do not expect that kind of interaction to be detected by opening a system up and doing visual inspection or physical measurements.
Structured messages are more likely to be implemented at an abstract level in a virtual machine.
The space of types of systems is VAST, and anyone who claims to have understood more than a tiny corner of it is almost certainly deceiving himself or herself. Likewise anyone who claims to have identified a corner into which brains and minds fit. I regard much of the debate as vacuous religious warfare. One issue of substance raised by Doug Watt and others in recent psyche-b correspondence is whether the architecture of human and other animal brains is best understood in terms of very large numbers of dynamical sytems operating in parallel on different spatial and temporal scales, with fairly close coupling between the diverse small scale processes and the more central, high level major controlling processes. My guess is that that will be needed to explain some aspects of human mentality but not all.