The NAND (Not AND) gate is all you need to build a universal computer. In other words, any computation that can be done by your desktop computer, can be accomplished by some combination of NAND gates. If you believe the brain is computable (i.e. can be simulated by a computer) then in principle, this is all you need to construct a brain. There are multiple ways to build a NAND gate out of neuro-wetware. A simple example takes just two neurons. A single neuron can act as an AND gate by having a spiking threshold high enough such that two simultaneous synaptic events are required for it to fire. This neuron then inhibits the second neuron that is always active except when the first neuron receives two simultaneous inputs and fires. A network of these NAND circuits can do any computation a brain can do. In this sense, we already have all the elementary components necessary to construct a brain. What we do not know is how to put these circuits together. We do not know how to do this by hand nor with a learning rule so that a network of neurons could wire itself. However, it could be that the currently known neural plasticity mechanisms like spike-timing dependent plasticity are sufficient to create a functioning brain. Such a brain may be very different from our brains but it would be a brain nonetheless.
The fact that there are an infinite number of ways to creating a NAND gate out of neuro-wetware implies that there are an infinite number of ways of creating a brain. You could take two neural networks with the same set of neurons and learning rules, expose them to the same set of stimuli and end up with completely different brains. They could have the same capabilities but be wired differently. The brain could be highly sensitive to initial conditions and noise so any minor perturbation would lead to an exponential divergence in outcomes. There might be some regularities (like scaling laws) in the connections that could be deduced but the exact connections would be different. If this were true then the connections would be everything and nothing. They would be so intricately correlated that only if taken together would they make sense. Knowing some of the connections would be useless. The real brain is probably not this extreme since we can sustain severe injuries to the brain and still function. However, the total number of hard-wired conserved connections cannot exceed the number of bits in the genome. The other connections (which is almost all of them) are either learned or are random. We do not know which is which.
To clarify my position on the Hopfield Hypothesis, I think we may already know enough to create a brain but we do not know enough to understand our brain. This distinction is crucial. What my lab has been interested in lately is to understand and discover new treatments for cognitive disorders like Autism (e.g. see here). This implies that we need to know how perturbations at the cellular and molecular levels affect the behavioural level. This is an obviously daunting task. Our hypothesis is that the bridge between these two extremes is the canonical cortical circuit consisting of recurrent excitation and lateral inhibition. We and others have shown that such a simple circuit can explain the neural firing dynamics in diverse tasks such as working memory and binocular rivalry (e.g. see here). The hope is that we can connect the genetic and molecular perturbations to the circuit dynamics and then connect the circuit dynamics to behavior. In this sense, we can circumvent the really hard problem of how the canonical circuits are connected to each other. This may not lead to a complete understanding of the brain or the ability to treat all disorders but it may give insights into how genes and medication act on cognitive function.
