Hmm... after more searching, I found this page, which says:

The faster the processor runs, the larger the energy required to maintain the bit in the predefined 1 or 0 state. You can spend a lot of time arguing about a sensible value but something like the following is not too unreasonable: The Landauer switching limit at finite (GHz) clock speed:

Energy to switch 1 bit > 100 k_B T ln(2)

So biological neurons still don't seem to be near the physical limit since they fire at only around 100 hz and according to my previous link dissipates millions to billions times more than k_B T ln(2).