There is very little that we understand about the larger picture of how learning happens in the brain. We have some understanding of how learning happens on very small scales, I'm talking plasticity at a single synapse. But even restricting ourselves to a single synapse, there is much we don't know. At the least, it's clear that synapses and dendrites have impressive computational capacity but making detailed measurements is currently beyond the reach of our experimental apparati. We can measure signals in dendrites and synapses, but not at a high enough spatiotemporal resolution to answer the big questions.

And we're starting to bump against fundamental limits of these apparati. Most modern neurobiology uses genetically encoded fluorescent sensors read out by rather expensive 2-photon microscopes. The sensors aren't as crisp as one wishes - there is a huge subfield dedicated just to deconvolving these fluorescent sensor readings into what the neurons are actually doing. And there's only so much further the 'scopes can be pushed.

The point being: it's really quite difficult to overstate just how overwhelmingly complex the brain is and how far we are from understanding even little really specific bits of it, let alone the whole thing.

That being said, the redwood center for theoretical neuroscience does some excellent work bridging the cutting edge of theory neuro and machine learning - towards the larger picture of how the brain works. You might be surprised at how 'rudimentary' the questions we're trying to solve in that domain are. Most work focuses on the visual system - far easier to study something when you have a good idea of what it's supposed to do (as opposed to, say, cortex).

I am not aware of anything resembling a grand theory that makes experimentally verifiable predictions. I am pretty sure I would have heard of such a thing if it existed.