In short: yes, we can certainly reproduce artificial brains, but no, maybe it won’t be with our modern computers (Turing machines), because we just don’t know enough about the brain to know if we need new computers ( super-Turing or biologically engineered brains), or if current computers (with plenty of power / storage space) are enough to mimic the whole brain.
Long
Disclaimer: I work in the field of neuroscience research, and I am interested in both the neurobiological side and the computational (artificial intelligence).
Most answers suggest, as a true OP postulate, that simulating neurons is enough to preserve the state of the entire brain and, thus, simulate the whole brain.
This is not true.
The brain is more than just neurons.
Firstly, there is a connection, synapses that are of paramount importance, perhaps even more than neurons.
Secondly, there are glial cells, such as astrocytes and oligodendrocytes, which also have their own connectivity and communication system.
Thirdly, neurons are heterogeneous, which means that there is not only one template model of a neuron that we could scale to the required number to simulate the brain, we also need to identify several types of neurons and place them in the right places. In addition, types can be continuous, so you can actually have neurons that are halfway between three different types ...
Fourth, we know little about the rules for processing and managing brain information. Of course, we found that the cerebellum works pretty much like an artificial neural network using stochastic gradient descent, and that the dopaminergic system works like TD learning, but then we have no idea about the rest of the brain, even memory is inaccessible (although we assume that this is something close to the Hopfield network, but there is no exact model yet).
Fifth, there are many other examples from current research in the field of neurobiology and computational neuroscience that show the complexity of the dynamics of brain objects and networks, which this list can go on and on.
So, in the end, your question cannot be answered, because we just don’t know enough about the brain to find out if our modern computers (Turing machines) are enough to reproduce the complexity of biological brains to bring about a full range of cognitive functions.
However, the field of biology is getting closer and closer to the field of computer science, as you can see with biologically engineered viruses and cells that are programmed in the same way as you design a computer program, and genetic treatments that basically process a living system based on its template is a "class" (genome). Therefore, I dare say that when we know quite well about the architecture and dynamics of the brain, reproduction in silicone will not be a problem: if our current computers cannot reproduce the brain due to theoretical limitations, we will develop new computers. And if only biological systems can reproduce the brain, we can program the artificial biological brain (we can already 3D-print functional bubbles, skin and veins and hearts, etc.).
Therefore, I would dare to say (even if it may be inconsistent, here is my own statement) that yes, artificial brains will undoubtedly be possible someday , but will it be like a Turing machine a computer, a Turing super-simulator or a biologically designed the brain remains to be seen depending on our progress in knowledge of brain mechanisms.