What analog function exactly are we talking about here? I'm genuinely curious. There's certainly a fading art for lots of analog computing techniques, like taking the arctan or log of an input signal. But how is that useful for neural computing?

I worked for a while as the software/FPGA guy on a high-power RF amplifier for MRI machines. I worked with some very talented RF engineers. There were obviously some parts of the chain that were analog (like the preamps and high power amp itself). But lots of it was being replaced with digital (DSP techniques) in the next generation and the RF engineers loved it. Filters no longer needed a 15 minute tuning session by a technician with a network analyzer. A Hilbert transform is far easier and better in digital than analog. Design changes could be experimented with in a day instead of PCB spins that took weeks.