The article seems to use "turing machine" or "emulator" in a more figurative sense, not to mean that the spider's neural network literally simulates simple rules that emulate a brain. The analogy to timeslicing makes more sense: the state of a computation can take less neural-network space than the ongoing computation, so switch between them and remember the variable values.

Also, the broader speculation in this article reminds me of https://www.xkcd.com/505/