Very back-of-envelop calculation:

The quantum scale is characterized by an action comparable to the Planck constant, hbar=6.6x10^-16^ eV s. A quick wikipedia search tells us that the typical electric impulse needs at least 25 mV (from -70 mV in the resting state to -55 mV in the threshold), and lasts around 1 ms, giving an action of 2.5x10^-5^ eV s, which is 300 billion times larger that the quantum noise.