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Does this mean that the brain happened to evolve with one area using dopamine while another uses seretonin, but with no real practical difference? Ie. could the brain have evolved another way (say, switching the two) and still function the same? Is there any fundamental reason why one neurotransmitter would be used in place of another, or was it just the result of evolution in those particular parts of the brain?
I agree with lucid_dream in that evolution of neurotransmission might well have gone some other way. Perhaps, also it is the way you think about it that causes you confusion. To the best of our understanding, neurotransmitters and their receptors are what they are and where they are in a particular organism because, in the course of history, such arrangement allowed to enhance the fitness of the organism. In other words, natural selection (evolution) is the only "fundamental" reason that things are the way we find them. Of course, you must remember that an organism is much more than neurotransmitters and receptors and it is the arrangement of all the components that determines whether an organism will survive or become extinct. Regarding the practical differences you are right in a certain sense, the brain did "happen" to evolve one way but not the other. However, it happened so because it was useful under the
circumstances. If something else happened to be useful first or happened to be more useful, we might live in a very different world, or not live period, or notice hardly any change at all.
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Or do neurons of a certain transmitter seek out other neurons with similar transmitters during the growth and development of the brain? Do neurons favor one type of transmitter over another? This could lead to regions dominated by certain transmitters. How does one neuron "decide" while the brain is growing that a certain receptor should open K+ channels while another neuron decides the exact same receptor should open Na+ channels? Is it completely random, but doesn't affect the brain because the inherent plasticity makes up for it? Or is there some order that defines which neurons do what?
These are all very interesting and rather complicated questions. From what we know, neuronal growth during brain development is guided by different chemical factors. Certain molecules in the extracellular environment encourage neurons to grow or migrate in a particular direction whereas others discourage them from doing so. So to answer your question, neurons do not seek each other out based on the identity of their neurotransmitters.
To answer your second question, most neurons do indeed synthesize one type of neurotransmitter over others. This, however, does not necessarily lead to regions dominated by certain neurotransmiters because the neurons that receive neurotransmitter signals (so-called post-synaptic neurons) usually receive connections from many (pre-synaptic) neurons releasing different neurotransmitters. The post-synaptic neurons correspondingly have different neurotransmiter receptors capable of detecting different neurotransmitter molecules.
I'm not familiar with the book you are quoting but I can see how it can be confusing. To begin with, the receptors you are thinking about are themselves a channel through the neuronal membrane. They are called ligand-gated ion channels and they do not so much open a channel as form it. Let me elaborate a little: these receptor-channels are proteins made of several (3-5) subunits each of which crosses the neuronal membrane so that some of the protein is inside and some of it outside the cell. In an inactive state, no ions can pass through the receptor, but when a neurotransmiter molecule (usually two) chances along and binds to the binding site on the extracellular side of the receptor the subunits shift from their positions changing the structure of the protein such that a pore is created in the middle of the protein allowing the passage of ions into and out of the cell. It is the size of the pore, the amino-acid structure of the receptor protein and some additional factors that determine which ion or ions will be able to pass through the pore.
So if you want to allow the passage of Na+ ions only in one case and of K+ ions only in the other case you would need two different proteins. Ligand-gated ion channels, however, differentiate mostly between positively and negatively charged ions. So a ligand-gated channel permeable to Na+ will likely be permeable to other monovalent cations (like K+) but not anions (like Cl-). Inhibition and excitation is a bit of a different matter that in addition to permeability has to do with the direction of ion movement. This is determined by the ionic concentrations inside and outside the cell as well as the resting membrane potential. Perhaps, if you are really interested in the subject matter you should consider some general neuroscience textbooks like: "From Neuron to Brain" by Nicholls or "Principles of Neural Science" by Schwartz, Jessell, and Kandel.
I hope this helps.