That's a really profound question that deserves serious research, IMO. Communication is easy enough to define using Shannon's theory of information. I would argue that computation is subset of communication: signals go in one end of a 'processor' & come out the other (so the processor acts as a 'channel' in communication theory), but the data comes out modified in some non-trivial way, which distinguishes computation/processing of information from mere replication of the signals at the output.

A trivial modification would be something like (X,Y) -> (Y,X), just re-ordering cables. A non-trivial modification would be something like (X,Y) -> (X XOR Y, X AND Y). Both of these 'channels' have 2 bits of capacity, but intuitively only the second one is actually doing any information processing/computing.

I haven't found a satisfactory way to formalize my intuition about what is 'non-trivial' and to quantify how much 'processing' is being done by a general system, in such a way that one could meaningfully compare a brain and a CPU, for example. I think it's a great research question.

I think Noam Chomsky even believes that human's language ability was selected for to aid computation not communication. Therefore in humans: communication IS computation.

Asking the real questions.

Axons are connected to the neuron and exchange a lot of chemicals between them. For example, the mitochondria in axons go back and fort to the neuron IIRC. It's not very googlable but neurons have a memory of how likely they are to trigger an action potential, based on past stimuluses/timings. No one knows where this memory is encoded. And is there one action potential per axon? If so then it would be more logical for this memory (and computation which leads to mutation to the memory) to be stored in the axon (if).