Is Artificial General Intelligence possible with our current machine learning models?
The short answer is, we don't know! This is an open question in AI research.

We know how neurons transmit signals, and can simulate that in a straightforward way: that's how layered perceptron models work. If we had a way to measure all the connections in a human brain, and enough computing power (which would be a LOT of computing power), we could simulate it. Or we could just make a neural network with as many neurons and connections as a human brain has. There's no theoretical missing link, we're just orders and orders of magnitude away in scale.

But it's an unanswered question whether the human mind exists entirely within neurons or not. It's a philosophical question more than a computer science one, and comes down to monism versus dualism.

Those who believe there's no theoretical barrier to AGI (just a problem of scale) tend to call their belief the Strong Church-Turing Thesis, which claims that a universal Turing machine could simulate a human mind, given unbounded resources and time. Or, equivalently, that a human mind is a Turing machine, or that there's nothing a human mind can do that a computer cannot do given time and resources for it.