A spike train, the output of a neuron, can reasonably be represented as a binary signal (1=spike, 0=no spike) at 1 kHz. In fact, I'd say this is nearly standard.

The neuron's membrane potential, a combination of its current inputs, intrinsic properties, and recent history, changes on faster timescales. Whatever processing the brain does probably does not use this information itself, but one probably wants access to it anyway for experimental and practical reasons.

The signal that an extracellular electrode picks up isn't that 1 kHz spike train. It's many of them mixed together and you'll need finer features to tell them apart. In other words, you can't just invoke Nyquist, sample at 2 kHz, and call it a day.