Misconception #2: “Neural responses are variable in vivo, therefore neural codes can only be based on rates”. Again, this is a category error. Neural variability (assuming this means randomness) is about determinism vs. stochasticity, not about rate vs. timing. There can be stochastic or deterministic spike-based theories.

I will expand on this point, because it is central to many argumentations in favor of rate-based theories. There are two ways to understand the term "variable" and I will first discard the meaning based on temporal variability. Interspike intervals (ISIs) are highly variable in the cortex (Softky and Koch, 1993), and their distribution is close to an exponential (or Gamma) function, as for Poisson processes (possibly with a refractory period). This could be interpreted as the sign that spike trains are realizations of random point processes. This argument is very weak, because the exponential distribution is also the distribution with maximum entropy for a given average rate, which means that maximizing the information content in the timing of spikes of a single train also implies an exponential distribution of ISIs. Temporal variability cannot distinguish between rate-based and spike-based theories.

Therefore the only reasonable variability-based argument in support of the rate-based view is the variability of spike trains across trials. In the cortex (but not so much in some early sensory areas such as the retina and some parts of the auditory brainstem), both the timing and number of spikes produced by a neuron in response to a given stimulus varies from one trial to another. This means that the response of a neuron to a stimulus cannot be described by a deterministic function. In other words, the stimulus-output relationship of neurons is stochastic. This is the only fact that this observation tells us (note that we may also argue that stochasticity only reflects uncertainty on hidden variables). That this stochasticity is entirely captured by an intrinsic time-varying rate signal is pure speculation at this stage. Therefore, the argument of spike train variability is about stochastic vs. deterministic theories, not about rate-based vs. spike-based theories. It only discards deterministic spike-based theories based on absolute spike timing. However, the prevailing spike-based theories are based on relative timing across different neurons (for example synchrony or rank order), not on absolute timing.

In fact, the argument can be returned against rate-based theories. It is often written or implied that rate-based theories take into account biological variability, whereas spike-based theories do not. But actually, quite the opposite is true. Rate-based theories are fundamentally deterministic, and a deterministic description is obtained at the cost of averaging noisy responses over many neurons, or over a long integration time. On the other hand, spike-based theories take into account individual spikes, and therefore do not rely on averaging. In other words, it is not that rate-based descriptions account for more observed variability, it is just that they acknowledge that neural responses are noisy, but they do not account for any variability at all. Accounting for more variability would require stochastic spike-based accounts. This confusion may stem from the fact that spike-based theories are often described in deterministic terms. But as stressed above, rate-based theories are also described in deterministic terms.

Throwing dice can be described by deterministic laws of mechanics. The fact that the outcomes are variable does not invalidate the laws of mechanics. It simply means that noise (or chaos) is involved in the process. Therefore criticizing spike-based theories for not being stochastic is not a fair point, and stochasticity of neural responses cannot be a criterion to distinguish between rate-based and spike-based theories.