How much intrinsic noise is there in a neuron? This question would deserve a longer post, but here I will just make a few remarks. In vitro, when the membrane potential is recorded in current-clamp, little noise is seen. There could be hidden noise in the spike generating process (i.e., in the sodium channels), but when a time-varying current in injected somatically into a cortical neuron, the spike trains are also highly reproducible (Mainen & Sejnowski, 1995). This means that the main source of intrinsic noise in vivo is synaptic unreliability.
Transmission at a given synapse is unreliable, in general. That is, there is a high probability of transmission failure, in which there is a presynaptic spike but no postsynaptic potential. However, an axon generally contacts a postsynaptic neuron at multiple release sites, which we may consider independent. If there are N sites with a transmission probability p, then the variance of the noise represents a fraction x=(1-p)/(pN) of the variance of the signal (expected PSP size). We can pick some numbers from Branco & Staras (2009). There seems to be quite different numbers depending on studies, but it gives an order of magnitude. For cat and rat L2/3 pyramidal cells, we have for example N=4 and p=0.5 (ref. 148). This gives x=0.25. Another reference (ref. 149) gives x=0.07 for the same cells.
These numbers are not that big. But it is possible that transmission probability is lower in vivo. So we have to recognize that synaptic noise might be substantial. However, even if it is true, it is an argument in favor of the stochasticity of neural computation, not in favor of rate-based computation. In addition, I would like to add that synaptic unreliability has little impact on theories based on synchrony and coincidence detection. Indeed, a volley of synchronous presynaptic spikes arriving at a postsynaptic neuron has an essentially deterministic effect, by law of large numbers. That is, synchronous input spikes are equivalent to multiple release sites. If there are m synchronous spikes, then the variance of the noise represents a fraction x=(1-p)/(pmN) of the variance of the signal (compound PSP). Taking the same numbers as above, if there are 10 synchronous spikes then we get x=0.025 (ref. 148) and x=0.007 (ref. 149), i.e., an essentially deterministic compound PSP. And we have shown that neurons are very sensitive to fast depolarizations in a background of noise (Rossant et al. 2011). The theory of synfire chains is also about the propagation of synchronous activity in a background of noise, i.e., taking into account synaptic unreliability.
In summary, the main source of intrinsic noise in neurons is synaptic noise. Experimental figures from the literature indicate that it is not extremely large but possibly substantial. However, as I noted in previous posts, the presence of large intrinsic noise does invalidate spiked-based theories but deterministic theories. In addition, synaptic noise has no impact on synchronous events, and therefore it is essentially irrelevant for synchrony-based theories.