Reading this, I just realized that all this time I was using the wrong computing analogy to understand the brain: neurons are not the transistors of the brain, dendrites are!
More importantly, neurons are nothing like transistors. Each neuron contains significant computing power. For example, in the retina of a frog there is a neuron that accepts as inputs dendrites from an area of light-sensitive cells (rods and cones) and only fires when a small, round, moving object is in the field of view. These neurons are colloquially called "bug perceivers." (2)
1 https://en.wikipedia.org/wiki/Neuron
2 "What the Frog's Eye Tells the Frog's Brain" (1959)
Seriously, molecular biologists devote decades of study and publish volumes figuring out how some small virus 'works'. A human neuron is orders of magnitude more complex than such a virus. To dismiss it as 'like an _____', is a (sometimes necessary) analogy of very little explanatory depth.
And yes, I'm aware neurons are complex. But that does not mean all of that complexity is relevant to the task of thinking; all living cells have machinery related to self-replication, self-maintenance and survival in a biological system.
Or to use another analogy - a smartphone is orders of magnitude more complicated than a flashlight. Yet in context of people illuminating their way during the night, a smartphone is just a flashlight.
Many synapses behave very much like memristors, though caveats apply.
I guess my point is that calling neurons analog or digital is not a very good analogy and should probably be avoided.
Perhaps a better model would be for each neuron to be modeled as a current source instead of a voltage source. The more parallel paths that current is funneled into, the smaller it gets, until it is below the threshold of detection. You can have your transistor model allowing/blocking that current, but never boosting it back up.
The voltage signal is propagated in axons sometimes over very long distances (a meter, e.g. from the foot to the spine). It is renewed (see "Nodes of Ranvier") along the path for as long as needed, at distances that depend on how, or if, the axon is myelinated. That's why the speed is so slow (ca. 120 m/s max., often much slower) - it's actual ion movement through lots and lots of opening and closing voltage triggered channels all along the axons, compared to a purely electrical signal like in a metal wire. This means while in a wire the electrical field is from start to end, leading to electrons exiting on the far end pretty much instantaneously, in an axon am electrical field is only local, and only strong enough to trigger anything for about 1-2 millimeters at best. Then there have to be a new set of channels triggered by the electrical field further up the axon that renew the signal by letting in ions.
Attenuation is more important in dendrites and neuron bodies, spatial distance of simultaneously incoming action potentials (via connected axons from other neurons) plays a role in determining whether a threshold is reached that would trigger firing an action potential from this neuron. Since a dendritic arbor can be quite extensive and have lots of branches location - where exactly along the branches does an incoming signal attach - matters a lot.
Computation happens at many levels - among dendrites, among axons (some neurons have multiple axons as well!), axonal branches, within the neuron, and also among various networks that arise (behavioral and structural networks). It's all a bit much, if you ask me.