I think this to mean that bits of matter somehow represent ideas, in the same way that codes represent objects in, say, computer systems. It seems natural, even obvious.
The problem is that even very simple mental operations can generate enormously divergent patterns of neural activity. Very simple stimulus and response patterns in mice are subject to what is called 'representational drift' - the same stimulus evokes responses in very different regions in the brain over time. Same thing happens with humans, albeit even more complicated. I read that long neurological studies attempted to trace characteristic patterns of activity in human brains when learning simple tasks, like memorising a new word, but that the activities were so divergent that researchers could detect no consistent pattern despite years of studies (see Why Us?, James le Fanu.)
Furthermore, consider the way in which a divergence of symbolic forms can be used to convey the same idea. A number can be represented by a variety of symbols, but they all specify the same idea. So the meaning of the idea is in some sense separable from its physical form. The mind, of course, can recognise such equivalences and translate one form to another - but again, can that be understood as a physical process? I think rather that it's a pretty cogent argument for dualism.
Right, when memes are said to live in the host, it isn't in a particular set of synapses we're talking about, it's a set of processes that give rise to a corresponding, similar-enough, set of mental phenomenon. Memes are abstractions that live as part of the emergent system of conciousness in their hosts. However, I think memes can still be understood as physical processes. The evidence for this is that people with damaged brains stop being able to understand ideas. If there is a powerful idea in a society, one that dominates their conciousness, and that society and its texts are destroyed, the meme vanishes until some lost text is found and translated by archeologists. It doesn't hang out in the ether, or if it does, no empirical evidence for it can be produced. So maybe it is the case that the idea lives on in an eternal realm, but the eternal realm is not necissary to explain ideas.
That you can't pinpoint the physical location of an idea, and that the activity that makes up the idea changes from moment to moment isn't at all incompatible with the findings of neuroscience, it's what we should expect. If ideas corresponded to hardwired structures then we'd have a finite memory capacity and would loose very specific bits of information with age and neuronal death, which isn't what we see. I would also disagree with the code analogy. I think brains as computers analogies generally do more harm than good in explaining things. When you write code, the meaning of your operations doesn't shift over time. Individual strings remain constant. That's not how brains work. The pattern of neuronal activity associated with something as simple as a smell varies over time, eventually corresponding to entirely different sets of neuronal activity. Since the subjective mental phenomena don't appear to change over time, this appears to suggest that the process, not the medium in which it occurs, creates the mental phenomena.
It's like how ecosystems exist but aren't located in a singular location as well. The movement of ideas works the same way that a terraforming operation could recreate an ecosystem in any physical space using none of the same material.
So I don't think you need non-material ideas to make sense of ideas. You just need a model where there are myriad ways to represent the same idea. The other problem for eternal ideas is that, if they are not material, how do they interact with our material brains? It seems like you'd need some version of Decartes pineal gland in place for that.
Which is not to address that the "material world" is itself a subjective abstraction made up of ideas, and that, in every sense, our understanding is the product of ideas. I find valid arguments for forms of idealism or dualism in that direction, just not in Plato's original direction of pointing to seemingly eternal ideas.