By "ether" I mean to no post-synaptic cell in particular.
If this is the case, how common is it for an axon terminal to not be part of a synapse?
Related question: do axon terminals form only as part of a synapse? Or are they formed ahead of time, and then connect to form synapses as needed?
By "ether" I mean to no post-synaptic cell in particular.
Yes; this is an important mechanism, as it allows the volume conduction of neuroactive chemicals. For example, many brainstem nucleus, such as the locus coeruleus, are small in size, yet affect much or even most of the cortex by this mechanism. Neurotransmitters are released and diffuse in the fluid-filled extracellular space (what you call "ether"), affecting many cells at once without making specific synaptic contact with any of them.
This mode of conduction is slow and spatially nonspecific compared to synaptic transmission, but it underlies much of neuromodulation, such as the setting of cortical states via norepinephrine, dopamine, serotonin etc.
If this is the case, how common is it for an axon terminal to not be part of a synapse?
This depends on what brain systems you're talking about. Intracortical connections and most direct afferent and efferent projections are mostly synaptic (since they require speed), but many other systems are greatly, or even mostly, volume conducting. Again, the brainstem nuclei are a good example, but gut neurons are another.
Related question: do axon terminals form only as part of a synapse? Or are they formed ahead of time, and then connect to form synapses as needed?
Some synapse quickly, others never synapse.
By definition, axon terminals are part of a synapse:
The somewhat enlarged, often club-shaped endings by which axons make synaptic contacts with other nerve cells or with effector cells. Also called end-feet, neuropodia, terminal boutons.
Structural remodeling of synapses and formation of new synaptic connections has been observed since 1999.1 Neurotrophins are proteins that activate neurons to produce more and new synaptic connections, allowing survival of neurons, regeneration, and neuroplasticity.2
The very complex pathways involved in neuroplasticity would suggest that neurons don't put out random axons or dendrites, but do so purposefully to increase synaptic density and neuronal health.
A random growth of dendrites or axons would be both wasteful and deleterious to a highly integrated nervous system, and therefore is as unlikely to be normal as non-directionally-oriented and non-innervated skeletal muscle cell formation.
