The brain is neither analog nor digital. The brain encodes information sent between cells as pulses. The timing of these pulses is where the information is encoded.
As such it is transmitting a fourier transform of the signal it is transmitting between the cells (but with different basis functions, not cosines). When a pulse is received, this will result in a sudden jump of the action potential, and then slowly die down. If a second pulse comes while the first is still in action it gets multiplied by the current value of (this particular) input potential. Rapid pulsing (3-4 ms between pulses) means something along the lines of "multiply membrane potential by 10 for every pulse", whereas a pulse width of ~350ms means "keep the membrane potential constant".
Any one of these pulses may push a neuron "over the edge" and have it transmit a pulse of it's own (or a series of pulses as pulsing the axon lowers the membrane potential by a factor. This may or may not cause it to go below the trigger level, and so sometimes the axon needs to pulse multiple times to get the membrane potential back below the trigger).
This is a beautiful way to encode simplifications of analog signals. It's log scale in the sense that 2 rapid pulses encode that the function value should be an order of magnitude larger than 2 slow pulses. It is extremely sparse (pulsing speeds rarely exceed 1hz, and only for short times) so it is quite energy efficient. Your brain quite literally does nothing at all unless there's actual information to process. I find it a bit misleading that wikipedia says that the frequency of pulses encodes the info : it's true, but makes one think this frequency is constant, when in fact it is changing rapidly. Does the word frequency really make sense when any 2 measurements of pulse frequency will be different ?
Normally one would call pulses digital, because they are 1 or 0. Values in between are interpreted as being either 1 or 0, definitely not analog. But the time spacing between the pulses is what matters, and this is measured to astonishing accuracy by a chemical reaction. A measurement like that, one would normally call analog. So it doesn't really match either signal type ... It is very different from any analog or digital signal I know about.
One might say your neurons contain an encoded, but uncompressed mp3 of the signal they're transferring between neuron membranes : it's transmitting in the frequency domain, but without the compression that reduces the zeroes, so it's sending very long series of zeroes, interspersed with the very occasional one. This works well because transmitting a zero is done by doing nothing at all.